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Star Wars Episode VII – The Force Awakens
14th January 2016 ReviewsRichard Hewison
This review is full of spoilers, so please don't read any further if you've not seen The Force Awakens yet! Also, I've only seen the movie once so I might have blinked and missed the odd clue or reference to things I mention in the review.
Of course, it goes without saying that I'm a Star Wars fan. I was ten and a half years old when I got to see the original 'Star Wars' in a UK cinema. As it was for millions of other people, it was unlike anything else I'd ever seen before. It was a game changer in every way possible. From that moment I was hooked, and the next five or six years were a blur of waiting for, watching or reading about Star Wars any way that I could. The Empire Strikes Back was – and still is – the best movie sequel ever, and whilst Return of the Jedi was a bit of a let down, in hindsight it was okay compared to the prequels that emerged years later.
Anyway, fast-forward almost 38 years. The cinema scheduling deities had deemed that the release of Episode VII in the UK was to fall on our oldest son's 9th birthday. As a result, he had been planning his 2015 birthday party the moment he knew. He was naturally extremely excited to be seeing a Star Wars movie in a cinema for the first time, on the day of release and on his actual birthday. Could it get any better?
So, five kids and two adults went into the multiplex, clutching 3D glasses and munch bags after the children had pigged-out on pizza (from the Hutt) immediately after school. On the way back in the car, we asked all the children for their ratings and they all – to a boy – gave it 20/10, possibly fuelled by an overdose of pizza, sweets, popcorn and Caprisun. Whilst it is tempting to write my review using the voice of my inner ten-year old self, I am sadly way past the stage where Star Wars is a huge part of my life. So, for an almost 50-something Star Wars fan with a family, a mortgage and everything else that goes with that, what did I think?
The good news is that The Force Awakens looks and sounds like a Star Wars movie. The complete over-use of CGI in the prequels (not to mention the plots and the majority of the acting being well below par) meant that those films felt entirely different to the Star Wars movie universe I had grown up with, and as such I never liked them much. Thankfully, the makers of Episode VII have sensibly gone back to using CGI sparingly and subtly, improving practical sets and effects with CGI rather than replacing them entirely. As a result, the Star Wars universe looks and feels real (and used) once again.
The new main cast of characters are all good, which is a relief. You get to really like Rey and Finn as the story unfolds, although you also begin to realise that the trailers definitely lead people to certain conclusions about both characters that turn out to be wrong. I'm not sure whether to feel angry at being tricked by the trailers or wishing that they simply hadn't shown certain scenes – or revealed certain lines – in the marketing campaign at all. On the other hand, I'm glad that the trailers didn't give away too much so there were still plenty of surprises.
Out of the other new characters, Poe Dameron and Captain Phasma get far less screen time than I was expecting. The new main villain – Kylo Ren – is just different enough not to be a direct Darth Vader clone (pun definitely intended!), with his out-of-control hissy-fit during the movie being in stark contrast to the calm way that Vader dealt with set-backs in the original trilogy. My theory about the new evil Emperor type character – Snoke – is that his gigantic hologram is overcompensating and he'll turn out to be smaller than Yoda. You read it here first.
The new droid on the block – BB8 – is great. He's technically impressive and for a ball with a small movable domed head, he can convey a surprising range of emotion. Once or twice I did find myself thinking about other film robots inspired by Star Wars though. In particular – and it's a coincidence that they were also made by Disney – Old B.O.B. and V.I.N.Cent from The Black Hole sprung to mind. Odd how the brain works sometimes!
A number of people have said it so I will to. The Force Awakens does echo a number of story elements from the original Star Wars movie. It has a desert planet, a droid with something the bad guys are after. It has a local who dreams of being elsewhere and a weapon that can destroy planets. The main baddie is a guy dressed in black, there is a battle to destroy the weapon and there is a light sabre duel (or two). Oh, and a lead character dies.
I'll admit to having mixed feelings about the return of Han Solo and General Leia. Maybe I've seen too many interviews with Harrison Ford and Carrier Fisher down the years, but when I saw them on screen and heard them speak all I saw were the actors, not the characters. Once or twice I also felt that Harrison Ford was behaving more like Indiana Jones than Han Solo, although that's more a criticism of the script than the actor.
Chewbacca was fine, but it might have been an idea to add a bit of streaked grey hair here and there. I know Wookies are meant to live for hundreds of years, but a subtle nod to the passing of time might have made Harrison Ford's grey locks a little less startling in comparison! It was also good to see the Millennium Falcon flying again, complete with a new radar dish to replace the old one that got knocked off during the battle of Endor in Return of the Jedi. Me, a geek?
It's a fairly major plot strand throughout the movie, and yet the search for Luke Skywalker doesn't dominate the film. The film's McGuffin is that BB8 holds an incomplete piece of a star map that shows where Luke is hiding. It's quite a flimsy plot device, as is having R2-D2 wake up in his brief cameo and produce the rest of the missing information that ultimately solves the puzzle. Talking of cameos, C-3PO didn't really need to make an appearance either, but I guess it was nice to confirm that old goldenrod is still in one piece.
The reveal that Darth Vader is Ren's Grandfather was unexpected (at least to me) but also a bit disappointing in the way it was handled. It certainly doesn't have the same impact as the scene in The Empire Strikes Back when Vader reveals who he is to a defeated and desperate Luke Skywalker. Soliloquy's are a very unnatural way of handling a revelation of that magnitude, and once it had been revealed I was half-expecting Ren to pick the melted helmet up and say 'Alas Poor Vader, I never knew my Grandfather at all'. Maybe not.
The demise of a major classic Star Wars character doesn't occur very often, but I felt strangely detached from the scene when it did. I know other people felt it quite moving, but for me it happened and was discarded in a fairly casual way. I can see how they wanted to show Kylo Ren (aka Ben Solo) tempted by the good side before deciding – via his actions – to choose the dark path. The scene harks back to choices made in the previous trilogy. Killing his father was something Luke Skywalker couldn't do, and so it becomes an important rite of passage for Kylo Ren to do what his Uncle could not. However, I'd argue that Han's descent into the carbon freezing chamber in The Empire Strikes Back was far more dramatic and memorable, but that's probably just me.
The new weapon – Starkiller base – is where my loyalty to Star Wars struggles the most. I immediately saw countless problems with the whole premise of the base. Even though you never actually saw either Death Star jump to light speed so they could traverse the galaxy, you never questioned their ability to do so or why they needed to. Unless I blinked and missed it, there was no mention of the Starkiller Base host planet being able to move. If it can't move, how can the weapon destroy anything unless its target is (a) very close by, and (b) happens to be lined up with the base so it can be directly hit. If the target isn't in the same solar system, then the actual projectile fired from the base is likely to take a few years minimum to reach it's intended target unless it's being hurled faster-than-light speed. Not to mention the fact that if the planet can't move, then sucking the energy out of the local star to power the weapon is a pretty daft thing to do!
One of the reasons I felt disappointed with Return of the Jedi back in the early 1980s was that they had a second, bigger Death Star. That felt like George Lucas had run out of ideas and as a result, it was a much more typical Hollywood sequel than The Empire Strikes Back had been. The Force Awakens does the same thing, but just makes the threat bigger again. Harrison Ford even mentions this during the movie, as if poking fun at it would somehow make up for it's lack of originality.
My other disappointment was that there was no real clue as to how big a threat the First Order are. Although I wouldn't want the movie to spend much screen time explaining the whole political situation (the original Star Wars disposed of such talk in a couple of conversations then moved on), knowing if the galaxy was now run by the New Republic and how big the First Order really were would have clarified a few things for me. As it was, it felt like the New Republic and the Resistance were two totally separate entities and there was no sense of scale to the conflicts. I had no idea how important any of the battles in the movie were. If there was a line or two that explained all that, then I must have missed it in the cinema.
Overall though the film was entertaining, well acted, technically superb and felt like a proper Star Wars movie. Certain parts of the plot were disappointing and I can't help feeling that the film was really made just to introduce the new characters and then set things up for Episode VIII, which has already begun filming. Not that that's a problem, but hopefully the next film can be a little more original.
Welcome back (proper) Star Wars. We've missed you!
About Richard Hewison
I'm a creative writer at heart, trying to break free from the shackles of others and express myself via my writing.	View all posts by Richard Hewison →
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1 thought on "Star Wars Episode VII – The Force Awakens"
Humayun Mirza says:
29th June 2016 at 10:50 am
Terrific review. An articulate and detailed analysis of the latest Star Wars film that is a welcome return to the original. I tried hard to be the 11 year old boy who was amazed when the first film was released 39 years ago. Like you, I watched it with my family and was particularly keen on passing on the torch to my 7 year old son. I am his father.
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« Howard Webb, gone from the pitch but not gone in terms of influence
The origins of the Arsenal badge and the club's motto »
MISSION IMPOSSIBLE? The power of the referee.
The beautiful game is one of the easiest competitions to manipulate because of the absolute power the referee and his assistants enjoy.
It is also one of the hardest games to police because of the difficulty in interpreting an official's decision, except when it is obviously wrong according to the Laws.
So we might ask: is it mission impossible to prevent some officials from negatively influencing the game? For example (and in these examples "he" meaning the ref, refers to "he or she"
1) He does NOT have to explain his decisions to anyone including the players, their captain, the managers or even his assessor. Nor is he obliged to talk to anyone other than his assessor and the governing body for officials.
2) He is the ONLY official timekeeper, not even his assistant can overrule his decisions as to adding time or when to stop time.
3) Once play has been restarted he CANNOT ever return to change his decision nor can he be overruled after the fact UNLESS he has committed a direct error with regards to the LAWS, regardless of all the video evidence in the world. That is why they don't show replays on the big screen.
4) Most players and many managers have a very limited knowledge and understanding of the LAWS and therefore cannot argue intelligently with the official about a call. Therefore he can call what he likes and easily outwit the other stakeholders by using his ¨superior ¨ knowledge of the LAWS.
5) He is the final arbitrator on points of fact and that is what prevents the FIFA, EUFA and FA dinosaurs from implementing video referees….they fear the consequences of taking this status away. Many officials don't want to be scrutinized by video recording and assessing because they fear for their privileged status being reviewed.
6) He does not need to judge INTENT anymore before rendering a decision but rather must consider the end result of a player's actions regardless of whether it was intentional or not. That does NOT mean that he must always punish an action but it does mean that he doesn't have to justify his decision to anyone, unless he makes a serious technical mistake as I mentioned previously.
7) He knows that he and his assistants will make far fewer mistakes than the 22 on-field players will during a game so he feels secure in the knowledge that he is less visible than them….this is of course false security. However he knows that the higher authorities will ALWAYS give him the benefit of the doubt.
8) He has the authority to allow play to continue, even when a foul is committed, if it offers the offended team an advantage to create a goal-scoring opportunity. He can also stop play regardless of an advantage arising because he feels that the offence must be punished immediately. Ultimately, one of the hardest calls to make is to recognize and give advantage and many officials take the safer route of making a call.
9) He can caution a player based on an extensive list of behaviours, all of which are open to misinterpretation or misunderstanding. He can send a player off if another carding offence occurs or a serious first time foul happens, and rarely will his decision be questioned or even reviewed.
10) He can award ¨mystery¨, very minor or ¨invisible¨ fouls in order to warn or control a player who is obnoxious, to calm down and stay out of his face. I have seen this done countless times and it serves as a painful reminder to the player(s) involved that he is the ¨boss¨. It is a sweet form of revenge as we say.
There are so many more powers the officials enjoy but they also always face extreme risks when trying to officiate 22 excited players, whose idiosyncrasies and personalities can quickly distort a game. So if it is mission impossible to prevent the skewing of games by officials, what are we to do?
Here are a few suggestions:
1) Video technology including goal-line and video refereeing reviews is a great start.
2) The simplification of some of the laws would help make his job easier.
3) Total transparency by the officials and particularly their governing organizations would be welcome.
4) An immediate expulsion for players repeatedly harassing and haranguing the officials.
5) A new governing authority other than the PGMOL. the new body being composed of neutral Football experts like retired referees, university sports instructors, coaching instructors etc.
6) A more transparent and in-depth assessment process like Walter's referee reviews and seasonal analysis done by a team of referees.
7) A clear and well documented process for promotion of officials from lower leagues that is publicly available and reputable.
I am sure our erudite readers can suggest even more options that will illuminate our beautiful game.
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August 9th, 2014 | Category: Arsenal stories | 19 comments - (Comments are closed)
19 comments to MISSION IMPOSSIBLE? The power of the referee.
Here are some other suggestions.
1. Establish a refereeing 'academy' for all sports where officials can be trained to an better-than-acceptable standard. Sharing refereeing 'techniques' across sports would be a significant step forward. Just watch how officials handle a field hockey match and you'll see what I mean. And why not open it up to foreign applicants who'd pay through the nose for the privilege thereby raising standards globally. Officials could take refresher or remedial courses to maintain standards. It can even be handed over to the private sector if you were capitalistically minded.
2. The creation of refereeing teams. Using the same personnel throughout the season thereby eliminating some of the vagaries of miscommunication. There'd be an established team leader, though promotion can still take place where warranted.
3. Post-match refereeing press conference and Q&A. Though this would have to be 'some' days after a match as it first has to be reviewed on VT. Only by being challenged can the public gain confidence in officials.
4. A kind-of refereeing 'ombudsman' that would oversee the whole shebang … academy, self-discipline, etc answerable only the the Sports Minister and the appropriate Parliamentary committee.
5. Two on-pitch referees instead of one increasing the proximity of officials to significant incidents.
6. On-pitch team-consulting of officials for important decisions pretty much as they have in the MLB.
7. And/or an off-pitch video official in constant communication. This is guaranteed to happen one day, so why not now?
8. Clear and published guidelines to officials prior to each season from the PL or FA presented during a press conference that includes a Q&A. Make sure everyone's on the same page.
9. Sacking, shooting, culling of all the corrupt, old men in football. Bring in women and other 'minorities'. None of points 1 to 8 will ever be instigated while these 'fat cat' dinosaurs are in charge.
colario
Up until I read this I thought it was shakespeare who wrote "A midsummer's Night Dream'.
I was wrong. Ii was Walter from Belgium. 🙂
Every individual decision can be debated and, usually, justified – albeit often with very contorted logic. If not it can be excused as an innocent mistake.
The only way that a case can be build against any referee (or club) is through statistical evidence painstakingly compiled over a period of time. This is what Untold has done with the referee reviews, what I did with the penalty analysis and also – occasionally – what other people have done. E.g. the analysis of "Fergie time" a couple of years back where it was demonstrated that Man Utd got more injury time when they were losing matches.
Statistical evidence is very powerful but, sadly, not well understood by many people. For example, most in the media do not have a statistical or even technical background.
Or, to put it another way: Once is unfortunate, twice is a coincidence but three times….
bjtgooner
The refs have demonstrated repeated incompetence and bias – in the EPL, in Europe and at the World Cup. They are not fit for purpose, bring on the technology.
oldgroover
Some good ideas, a few of which are (under the current system) impractical. e.g All of the PGMOL referees are promoted from the Championship (who recruit from Div 1 & 2), where most are part timers (learning their trade literally on the hoof), and have a full time job. If they graduated from academies they would be looking to be full time and not have to slog through the lower divisions. So yes, full timers properly trained to elite standard at academies is the ideal, but in reality I can't see it happening without complete restructuring of the refereeing system right through the divisions.
The post match interviews was tried for a while and for some reason dumped. I'd like to see it return.
The ombudsman could infringe FIFA rules as it might involve government interference in FA business.
The other points are worth a try though, particularly number 9
EA's FIFA series proves that if shock sensors were imbedded in players socks and kit we could have an automated referee call fouls, pushes, excessive contact, offsides etc without error, with a clearly defined parameter and without the human element/bias.
'Hi Webby. It's Mikey here. Howard you to day?
'Is that Rilly you Mikey?
'Sure thing Webby. Why do you ask?'
'No offence boss but you can't be too sure of what hear or for that matter what you see these days'.
'Strange you should say that Webby, becausse that is what I am calling you about.
'Oh yes boss'. Said with the sound of doubt in his voice.
'Well Webby you know we refs always get the decision right first time and the cameras and their obnoxious replays always tell a lie and those idiots who call themselves 'supporters' always believe the camera replays.'
'Yes boss.'
'Well I have been thinking, its time we stood up for ourselves and you Webby are the man to do it.'
'What I want you do to do is to retire!' (Webby drops his phone so doesn't hear Mikey's next bit first time.
Not being able to see Webby drop his phone Really Micky continues.
'Don't worry you get your £50 000 silence hand shake and we give you a great job your goonner like this one.
You watch the game on TV. then you go on TV and explain our — no I mean the decisions that no one understands. You will be our lead in 'referee openess'. You are answer to all this nonsense about technology'.
We pay you a great salary just for watching TV. (no more exhausting yourself running up and down the pitch surroundered by a bunch of mindless idiots) and explain why we are right and technology is wrong.
In addition to all that your name will be in the media almost everyday, you will be one of the most famous people in the country.'
What say you Webby?'
'Thanks boss you really are a true boss.'
'Well Webby I can see the headlines now.
'Howard's really Mike's man'. I'll email you the details. Bye for now.'
About a minute later Reaily Micky calls his friend Webby again.
'Hi Webby. I for got to tell you. You can throw your whistle away as you wont be needing it in your new job. Nope. There will be no whistle blowing in your new job. See yer'.
Personally, I think the game is fine as it is. There are academies and a very structured review processes, we don't need any more. Instigating all these fandango technological strategies will take the passion and zip out of the game. The role of the referee is also to man manage and he does this by applying common sense, whether it's understood by the players or those watching. Ultimately, he wants to keep 22 players on the field to maintain a contestable game. Pundits often scrutinise referees without detail knowledge of the laws themselves. Apart from goal line technology, leave the game alone and let's stop making the game sterile. Stop pandering to the player's egos and leave the refs alone. Yes they make mistakes but how many refs command £150k plus a week to make as many blunders as the players do a game? Players who turn to refereeing often say to me just how difficult it is on the other side and can see why we make the decisions we do. I suggest all Pundits and those pointing the finger try a few games themselves
Micheal Ram
I think the most important aspect of electing a referee is character and profile background. Knowing and experiencing the law is not enough, the ability to apply and enforce the law according to moral and safety requirement is also important. A strong official must be humble and conscious to be respected by everyone. His actions should be louder than any criticism, scandal or act of self-interest. David Webb is a man of law (police), so is not easily intimidated by power and corruption. How about moral stand and efficiency? I suggest referees of all sports to be randomly elected form the police force and army. A seasonal and experience man of honour do the job unlike any other. Trust me.
One of the biggest reasons for the officiating status quo in football are the clubs , managers and fans themselfs who bemoan every wrong or controversial decision going against them but bend over backwards to justify the same decision going their way.
Players are coached from a very young age to cheat . Managers are incouraged to manipulate and influence referees , and partisan fans are only too happy to excuse their idols from any guilt for diving , time wasting, shirt pulling and holding as long as it benefits the team.
If the clubs, players,fans and the media that cover football , stood in a united fashion for changes being implemented into the refereeing aspects of the game , the governing bodies would have no choice but to comply.
The quality of refereeing in football is low but it matches closely the levels of integrity of most coaches, players and fans. We only get what we deserve.
I think that in truth people are going a bit overboard with some of the suggestions for referee performance improvement.
Devices in socks? Quasi religious intervention (this was somebody on UA the other day), Have I had a humour bypass today? And I don't get the continuing comparisons with field hockey, a completely dissimilar sport, but I'm sure the angry chap who keeps calling me disingenuous will clue me in.
There's only so much extra technological assistance that can be applied before a whole new IT infrastructure is installed throughout the Premier & Championship Leagues, and lets not take the power from the man in the middle by adding extra men for him to be confused by, but make sure he (or she) has reliable assistance if required.
The regional imbalance (to prevent bias) must also be addressed as should the issue of unnecessary PGMOL secrecy.
That's right Tom, but nobody ever thinks to bring that up
H Bo
@wam it's a cool idea but to be honest I'm with @zac and @oldgroover it's all too much, however I do think there should be so bit more explanation than a few hand signals now I'm nit completely sure as only watch footie really but there is a sport stateside I think American football where the referees decision are called around the stadium. Generally if they make the calls too perfect what will we debate when the season starts and it's part of what makes the game so interesting and emotional.
It's a rather dangerous thing to assume that all players, managers and fans are cheats, certainly there is a large element of that, but I don't think it is universal. Further, we like to think that Arsenal as a team and as a club (in terms of cheating) are one of the cleanest around.
The point about introducing technology is that there should then be a level playing field; at present Arsenal appear to be punished when they commit fouls – ok that is correct – but the teams we play against are often unpunished for similar or more major offenses – and as for penalties …..
@H Bo
'Generally if they make the calls too perfect what will we debate when the season starts'
How about we debate the quality of the football and the performance of players which is what we go to the match for. Why should I have to come away from a game knowing that my team was robbed because of a terrible ref decision that everyone can see was wrong but according to you placate myself by saying saying 'that's OK, it gives us something interesting to talk about'. Arsenal have been on the receiving end of more dodgy calls than any other team over the last few years which has probably cost us a couple of trophies and you tell us it makes for interesting debate. Give me a break please.
omgarsenal
Excellent debate and here are a few more points to consider:
1)A study done by the German FA indicated that officials got 82% of their calls correct, the first time and 88% when they communicated with their assistants and reconsidered their initial decisions. These were German referees but I imagine this is true for other nations as well.
2)FIFA's technical committee reported that on average, a referee has 1.5 seconds to see the incident, another 2 seconds to consider and judge it and another 2 seconds to make the call. That is 5.5 seconds to perform under pressure while running and trying to keep eye contact with the play, the ball (not always the same thing) and his/her assistant.
3)The top paid referee in any professional league in the world is on average paid 30 times less than what the lowest paid professional player in the premier division he is officiating gets on average. So for example, Webb would get about 1000 quid per game, while Sanogo would get about 28k per game/week.
4)Referees do not get any substitutions unless they are injured and unable to continue. They run an average of 10km per match and in some cases even more. While they usually do not touch the ball or have any physical contact, they have to run backwards, sideways and forwards as well as turning abruptly and all the while looking at the entire half/quarter they are in.
5)Most referees worldwide don't have official assistants so they also have to keep up with the offside line and become excellent guessers and prayerful optimists.
6)FIFA considers that it takes between 5-7 years for a neophyte official to reach their best level as a referee.
7)Not all referees make equally good assistants and vice-versa. It is considered by all training groups that referees should have as much time as an assistant as a referee.
8)The art and science of instructing your assistants before the game and at half time is one that few referees are properly trained in and prepared for. This appears to be the case at all levels.
it would appear that somebody at FIFA is taking notice at last.
http://www.mirror.co.uk/sport/football/news/video-referees-football-move-closer-4029264#.U-cSQ1ZfAZx
Re the original post point 4:
Referees generally do not listen as it would be seen as a sign of weakness to change a decision after listening to a player, you could quote the rule book word for word and prove a referee 100% wrong and all you would likely get is a yellow card.
jayramfootball
The biggest problem is that a referee can apply the same set of laws in different ways and still be within the rules.
Football is game that can be manipulated easily without making big decisions. In fact the big decisions can often mask cheating. A referee can ensure that he equalised the big decisions for each club, though make them more crucial for one club (i.e. a penalty to a favoured team that is 1-0 down evened up by awarding a penalty to the other team when they are 4-1 down). Looking at the big decisions it would be seen as one penalty decision each but it does not explain the flow of game.
Even then, it's the smaller decisions that can often have a bigger impact. Penalising one team's defenders for aggressively handling an opposing striker whilst letting the other team away with it is one of the biggest ways to cheat in football but it is rarely mentioned. The ability for a team to make the ball stick with the front man and have players play off him is one of the most important areas of the modern game. A referee can make this nearly impossible for one team whilst ensuring multiple dangerous free kicks for the other. This form of cheating happens all the time.
Bottom line – if refs want to cheat, or are invented to do so, then there is nothing that can be done about it… at least not until the laws and methods of refereeing are changed.

redpajama_set_name: RedPajamaCommonCrawl


text:
\section{Introduction}

An optical vortex is a light field exhibiting a pure screw phase
dislocation along the propagation axis; i.e., an azimuthal phase
dependence $\exp ( i \ell \varphi)$. The integer $\ell$ plays the role
of a topological charge: the phase changes its value by $\ell$ cycles
of $2 \pi$ in any closed circuit about the axis, while the amplitude
is zero there~\cite{Torres:2011vn}.

One of the most intriguing properties of vortices is that they carry
orbital angular momentum (OAM). This was first realized by Allen and
coworkers~\cite{Allen:1992zr} for the important instance of
Laguerre-Gauss (LG) laser modes. Furthermore, they demonstrated that
these modes carry an OAM of $\ell\hbar$ per photon along the
propagation direction.

A useful feature of optical OAM is that it can be easily
manipulated and transferred; this has opened new horizons in various
fields, ranging from mechanical
micro-manipulation~\cite{Padgett:2010tg} to imaging
sciences~\cite{Maurer:2011oj,Uribe-Patarroyo:2013dw}, as well as
potential astronomical~\cite{Elias:2008hc,Tamburini:2011qt} and
communication applications~\cite{Wang:2012bs}. Beyond optical
wavelengths, OAM now plays a major role in
electron~\cite{Uchida:2010uq,Verbeeck:2010qd,
  McMorran:2011cl,karimi2012spin}, x-ray~\cite{Peele:2002wd,
  Nugent:2009vs,Hemsing:2013li} and radio frequency
engineering~\cite{Thide:2007hb,Mohammadi:2010km,Tamburini:2012vp}.

The core observation that individual photons also carry OAM brings the
most exciting possibilities for employing this variable in the quantum
regime, and a number of uses has already been
demonstrated~\cite{Mair:2001fv,Molina:2004dz,Oemrawsingh:2004fu,
Marrucci:2011we,Molina:2007qa,Fickler:2012bs}

Despite this intense activity, very little attention has been paid
thus far to the radial index $p$ of the LG modes. Usually, it is
stated that for $p > 0$, the modes are multiringed with $p + 1$
radial nodes. Beyond this short mention, no physical meaning is
attached to this quantity. Two recent papers, however, have presented
challenging and interesting insights into this
issue~\cite{Karimi:2012yt,Plick2013:sf}. Our purpose here is to
present a simple comprehensive analysis of this variable.

The two aforementioned papers considered optical modes, governed by
the paraxial wave equation. These modes are ultimately suitably
rescaled wave functions of the stationary states of a two-dimensional
quantum oscillator, under the Schr\"{o}dinger
equation~\cite{Nienhuis:2004}. Since this latter system can properly
model other interesting vortices arising in different media (as in
plasmas~\cite{Mikhailovskii:1987ys},
superfluids~\cite{Salomaa:1987ly}, and Bose-Einstein
condensates~\cite{Salomaa:1987ly}), the oscillator will serve as our
thread, bearing in mind that the results can be immediately translated
to the optical case.

\section{Stationary states of a two-dimensional oscillator}

\subsection{Cartesian coordinates}

To be as self-contained as possible, we briefly review the example of
an isotropic two-dimensional quantum harmonic oscillator of mass $m$
and natural frequency $\omega$, with coordinates in two orthogonal
axes, say $x$ and $y$~\cite{Dodonov:1989lq,Cohen:2006bh}.  The Hamiltonian can be
compactly written as $\op{H} = \hbar \omega ( \op{n} +
\op{\openone})$, where the total number operator $\op{n}$ is
\begin{equation}
  \label{eq:Ndef}
  \op{n} = \op{n}_{x} + \op{n}_{y} =
  \op{a}_{x}^{\dagger} \op{a}_{x} + \op{a}_{y}^{\dagger} \op{a}_{y}  \, ,
\end{equation}
and the annihilation and creation operators fulfil the canonical
commutation relations $[ \op{a}_{j}, \op{a}_{k}^{\dagger} ]
=\delta_{jk} \op{\openone}$, with $j, k\in \{x,y\}$. Since the spectrum
of $\op{n}_{j}$ is composed of all non-negative integers $n_{j}$, the
energies are given by $E_{n_{x}, n_{y}} = \hbar \omega
(n_{x} + n_{y} + 1)$ and these eigenvalues are $(n_{x} + n_{y} + 1)$-fold
degenerate.

Elements of the Fock basis are the common eigenvectors of $\op{n}_{x}$
and $\op{n}_{y}$:
\begin{equation}
  \label{eq:4}
  | n_{x},  n_{y} \rangle = \frac{1}{\sqrt{n_{x} ! n_{y} !}}
  ( \op{a}_{x}^{\dagger} )^{n_{x}} (\op{a}_{y}^{\dagger}) ^{n_{y}}
  |0, 0 \rangle    \, ,
\end{equation}
where $|0, 0 \rangle$ is the ground state.  The stationary states of
the oscillator are the product of Hermite-Gauss modes, as the
oscillations in each axes are kinematically independent:
\begin{eqnarray}
  \label{eq:5}
  \Psi_{n_{x}  n_{y}} (x, y) & =  & \sqrt{\frac{\alpha^{2}}{\pi \; 2^{n_{x} + n_{y}} n_{x}! n_{y}!}}
  H_{n_{x}} (\alpha x) H_{n_{y}} ( \alpha y) \nonumber \\
  & \times & \exp [-  \alpha^{2} (x^2+y^2)/2 ] \, ,
\end{eqnarray}
with $\alpha = \sqrt{m\omega/\hbar}$. To recover the equivalent beam
solutions, one needs to take $\alpha = \sqrt{2}$, since the paraxial
wave equation (in adimensional coordinates) coincides with the
Schr\"{o}dinger equation for the oscillator when $m=2\hbar$ and
$\omega=1$.

For our purposes, the solution at $t=0$ is enough. The wave function at
any other time can be obtained in a simple way by using the explicit
form of the propagator.  For beams, where the role of time is played
by the coordinate $z$ along the symmetry axis, this propagation brings
about additional interesting points, such as the Gouy phase.

\subsection{Cylindrical coordinates}

The axes $x$ and $y$ do not enjoy a privileged role in the
problem. Since the energy is invariant under rotations in the $xy$
plane, we could as well have chosen any other rotated reference
frame. To take a better advantage of this symmetry, we consider the
$z$-component of the angular momentum, $ \op{L}_{z} = \hbar
\op{\ell}$, with $\op{\ell} = i (\op{a}_{y}^\dagger\op{a}_{x}
-\op{a}_{x}^\dagger \op{a}_{y} )$, and use the rotated bosonic
operators~\cite{Sanchez-Soto:2013gf}
\begin{equation}
  \label{eq:7}
  \op{a}_{\pm}= \frac{1}{\sqrt{2}}  ( \op{a}_{x} \mp i  \op{a}_{y}) \, ,
  \qquad
  \op{a}_{\pm}^{\dagger} = \frac{1}{\sqrt{2}}
  ( \op{a}_{x}^\dagger \pm i  \op{a}_{y}^\dagger) \, ,
\end{equation}
where $[\op{a}_{j},\op{a}_{k}^{\dagger}] =\delta_{jk} \op{\openone}$,
with $j, k \in \{+,-\}$. We can then check that
\begin{equation}
  \label{eq:8}
  \op{n} =  \op{n}_{+} + \op{n}_{-}  \, ,
  \qquad
  \op{\ell} = \op{n}_{+} - \op{n}_{-} \, ,
\end{equation}
whose interpretation is direct: the system can be envisioned now as
consisting of ``quanta'' with positive (counterclockwise rotation
around $z$) and negative (clockwise rotation around $z$) orbital
angular momentum.

The Fock basis $\{ |n_{+}, n_{-} \rangle \}$ of the common
eigenvectors of $\op{n}_{+}$ and $\op{n}_{-}$ can be constructed much
in the same way as in Eq.~(\ref{eq:4}).  However, it will prove useful
to consider instead the continuous set
\begin{equation}
  \label{eq:11}
  | \eta  \rangle =
  \exp \left (
    -\frac{1}{2} |\eta |^2 +
    \eta  \op{a}_{+}^{\dagger}  -\eta ^{\ast} \op{a}_{-}^{\dagger} +
    \op{a}_{+}^{\dagger} \op{a}_{-}^{\dagger}
  \right ) |0,0\rangle \, ,
\end{equation}
parametrized by the complex number $\eta = r \exp (- i \varphi )$. The
states $|\eta \rangle$ constitute an orthonormal basis, whose
properties have been reviewed in depth in Ref.~\cite{Hong-yi:1994rw}.
In the representation they generate [$\psi (\eta ) = \langle \eta |
\psi \rangle$], the action of the basic operators is
\begin{eqnarray}
  \label{eq:12}
  \op{a}_{+} \psi (\eta  ) & = & \left ( \frac{\eta}{2} +
    \frac{\partial}{\partial \eta^{\ast}} \right ) \psi (\eta ) \, ,
  \nonumber \\
& & \\
  \op{a}_{-} \psi (\eta  ) & = & -  \left ( \frac{\eta^{\ast}}{2} +
    \frac{\partial}{\partial \eta} \right ) \psi(\eta ) \, , \nonumber
\end{eqnarray}
while for the adjoints we have
\begin{equation}
  \label{eq:adj}
  \op{a}_{+}^{\dagger} = \op{a}_{-} - \eta^{\ast} \, ,
  \qquad
  \op{a}_{-}^{\dagger} = \op{a}_{+} - \eta \, .
\end{equation}
Since the exponential acting on the vacuum in Eq.~\eqref{eq:11} is not
unitary, the creation and destruction operators are not conjugates one
of the other under the usual boson conjugation.

Given the above, $\op{n}$ and $\op{\ell}$ act in this space as
\begin{eqnarray}
  \label{eq:NLdif}
  \op{n} & \mapsto &  \frac{r^2}{2} -
    \frac{1}{2} \left (\frac{\partial^{2}}{\partial r^{2}} +
      \frac{1}{r}  \frac{\partial}{\partial r} +
      \frac{1}{r^2}\frac{\partial^{2}}{\partial \varphi^{2}} \right )
    - 1 \, , \nonumber \\
  & & \\
  \op{\ell} & \mapsto & - i \frac{\partial}{\partial \varphi} \, .
\nonumber
\end{eqnarray}

As $[\op{\ell}, \op{n} ] = 0$, the basis $\{ |n_{+},n_{-} \rangle \}$ can
be reinterpreted as common eigenvectors of $\op{n}$ and
$\op{\ell}$, with eigenvalues $n= n_{+} +  n_{-}$ and $\ell = n_{+} -
n_{-}$, respectively. The stationary states in this basis can be
readily obtained using Eqs.~(\ref{eq:NLdif}); the final result is
\begin{equation}
  \label{eq:14}
  \Psi_{n \ell} (r, \varphi ) =
  A_{n \ell} (r) \, e^{i\ell \varphi} \, ,
\end{equation}
where the normalized amplitude is
\begin{equation}
  \label{eq:amp}
  A_{n \ell} (r) = \frac{\sqrt{2\alpha^{2} p!}}{\sqrt{(p + |\ell |) !}} \,
  e^{- \alpha^{2} r^2/2} (\alpha r)^{|\ell|}
  L^{|\ell |}_{p} (\alpha^{2} r^2) \, ,
\end{equation}
$L_{p}^{\ell} (x)$ are the generalized Laguerre polynomials and
we have written $p = (n - |\ell |)/2$. The probability distribution $|
\Psi_{n \ell} (r, \varphi)|^{2}$ shows $p$ dark concentric rings.

\section{Quantum optics with radial modes}

\subsection{The radial number operator}

Since the number of dark rings is $p= (n-|\ell|)/2$, the operator
\begin{equation}
  \label{eq:defP}
  \op{p}  = \frac{1}{2} (\op{n}- | \op{\ell} |) = \left \{
    \begin{array}{ll}
      \op{n}_{-}  \quad & \mathrm{for} \ \ell > 0  \, , \\
      & \\
      \op{n}_{+}  & \mathrm{for} \ \ell < 0 \, ,
    \end{array}
  \right .
\end{equation}
seems to be a sensible definition for the radial-number operator of
the Laguerre-Gauss modes.  According to Eq.~(\ref{eq:NLdif}), in
differential form it reads
\begin{equation}
  \label{eq:15}
  \op{p}  \mapsto  - \frac{1}{4}
  \left(\pdd{}{r}+\frac{1}{r}\pd{}{r}+\frac{1}{r^2}
    \pdd{}{\varphi}\right)+\frac{i}{2}\pd{}{\varphi}+\frac{1}{2}\frac{r^2}{2}-
  \frac{1}{2} \,.
\end{equation}
Incidentally, it coincides with the operator found in
Ref.~\cite{Plick2013:sf} by setting $r \rightarrow\alpha r$.

To simplify the following discussion, let us, for the time being,
relabel the stationary states $|n, \ell \rangle$ as $|p, \ell
\rangle$, where $p$ indicates the radial mode eigenvalue; i.e.,
\begin{equation}
  \label{eq:16}
  \op{p} | p, \ell \rangle = p | p, \ell \rangle \, .
\end{equation}
As heralded in the Introduction, we are interested in exploring the
Hilbert space associated with the radial number $p$, while keeping the
OAM $\ell$ fixed. At first sight, one might
look for the canonical conjugate variable to $\op{p}$. Since,
according to Eq.~(\ref{eq:defP}), $\op{p} = \op{n}_{-}$ or
$\op{n}_{+}$ (depending on the sign of $\ell$), such a variable would
be a phase $\op{\phi}_{-}$ or $\op{\phi}_{+}$. This means that if we
denote by $\op{e}= \exp (i\op{\phi}_{\pm})$ the exponential of such a
putative phase, the corresponding commutation relation will
read~\cite{Lynch:1995cq}
\begin{equation}
  \label{eq:PE}
  [ \op{e}, \op{p} ] = \op{e} \, .
\end{equation}
This immediately implies
\begin{equation}
  \label{eq:elad}
  \op{e} = \sum_{p=0}^{\infty} |p, \ell \rangle \langle p+1, \ell | \, ,
 \end{equation}
so that
\begin{equation}
  \label{eq:elad}
  \op{e} |p, \ell \rangle = |  p-1 , \ell \rangle \, ,
  \qquad
  \op{e}^{\dagger}  |p, \ell \rangle = |  p + 1 , \ell \rangle \, .
\end{equation}
Whereas the spectrum of $\op{\ell}$ is unbounded, including all the
integer numbers, the spectrum of $\op{p}$ is semibounded, as it
comprises only  non-negative integers. This indicates that the action of
$\op{e}$ as a ladder operator fails at $p=0$, and consequently it
cannot be unitary:
\begin{equation}
  \label{eq:enonu}
  \op{e} \op{e} ^{\dagger} = \op{\openone} \, ,
  \qquad
  \op{e} ^{\dagger} \op{e} =  \op{\openone} - \op{\mathcal{P}}_{0} \, ,
\end{equation}
where $\op{\mathcal{P}}_{0} = |0, \ell \rangle \langle 0, \ell |$ is
the projector on the ``vacuum.''

All these problems thus place this interpretation on shaky grounds.
For this reason, we prefer to follow an alternative route. To this
end, we observe that to increase ( decrease) the radial number by
one unit, with $\ell$ unchanged, we need to create (annihilate)
one positive quantum and one negative quantum, namely
\begin{eqnarray}
  \label{eq:17}
  \op{k}_{+} |p, \ell \rangle & =  &
  \op{a}_{-}^\dagger \op{a}_{+}^\dagger |p,\ell\rangle
  \propto |p+1,\ell \rangle  \, , \nonumber \\
  & & \\
  \op{k}_{-} |p,\ell\rangle & = & \op{a}_{-} \op{a}_{+} |p,\ell\rangle
  \propto |p-1,\ell\rangle \, . \nonumber
\end{eqnarray}
One can check that
\begin{equation}
  \label{eq:ccr}
  [\op{k}_{+} , \op{k}_{-} ]  =  - 2 \op{k}_{z} \, ,
  \quad
  [\op{k}_{z} ,\op{k}_{+} ]=\op{k}_{+} \, ,
  \quad
  [\op{k}_{z} ,\op{k}_{-} ] =- \op{k}_{-} \, ,
\end{equation}
with $\op{k}_{z}= (\op{n} + \op{\openone} )/2$. This means that if we
define $\op{k}_{\pm} = \op{k}_{x} \pm i \op{k}_{y}$, we have
\begin{equation}
  \label{eq:18}
  [\op{k}_{x} , \op{k}_{y} ]  =  - i \op{k}_{z} \, ,
  \quad
  [\op{k}_{y} ,\op{k}_{z} ] = i \op{k}_{x} \, ,
  \quad
  [\op{k}_{z} ,\op{k}_{x} ]= i \op{k}_{y} \, ,
\end{equation}
that is, they are the generators of the su(1, 1) algebra, as first
noticed in Ref.~\cite{Karimi:2012yt}.

\subsection{Radial coherent states}

To explore the issue in more detail, it is convenient to give some
basic background on some well-known irreducible representations (irreps)
of SU(1,1), which are excellently reviewed in
Ref.~\cite{Biedenharn:1965it} and whose role in quantum optics is
difficult to underestimate~\cite{Gerry:1985fk,Yurke:1986yg,
Buzek:1989wd,Vourdas:1992vl,Ban:1993pi,Brif:1996oj,Gerry:1997kn,
Agarwal:2001dw,Mias:2008cr,Marino:2012ct}.

The Casimir operator for this group is $\op{K}^{2} = \op{k}_{z}^{2} -
\op{k}_{x}^{2} - \op{k}_{y}^{2}$, which can be expressed as
$\op{K}^{2} = k (k-1) \op{\openone}$, where the Bargmann index $k$
labels the different irreps (this index plays the role of
spin for rotations). In our case, a simple calculation shows that $k =
(| \ell | + 1)/2$, so that $k = 1/2, 1, 3/2, \ldots$, which
corresponds to the so-called positive discrete series, for which $
\op{k}_{z}$ is diagonal and has a discrete spectrum. In the Fock basis
$\{ |n_{+}, n_{-} \rangle \}$, the basis states of the irrep $k$ are
$ \{ | k , k + p \rangle \}$, with $p=0, 1, \ldots$, and hence
\begin{eqnarray}
  \label{eq:diag}
  \op{k}_{z} |k, k+p \rangle  =
  (k+p)  |k, k+ p \rangle \, ,
\end{eqnarray}
while the ladder operators act as
\begin{eqnarray}
  \label{eq:ladd}
  \op{k}_{+} |k, k+p \rangle  & =  &
  \sqrt{(2k+p) (p+1)}  |k, k+ p +1 \rangle \, ,
  \nonumber \\
  & & \\
  \op{k}_{-} |k, k+p \rangle & =  &
  \sqrt{p (2k+p- 1 )}  |k, k+p - 1 \rangle \, .
  \nonumber
\end{eqnarray}
Note that we can make the identification $ |p,\ell\rangle
\leftrightarrow |k, k+ p \rangle$, provided $k = (| \ell | + 1)/2$.

Since $\op{k}_{-} |k,k \rangle = 0$, this state can be taken as the
vacuum. Indeed, $\op{D} (\xi) = \exp ( \xi \op{k}_{+} - \xi^{\ast}
\op{k}_{-} )$ are truly displacement operators, so according to
Perelomov prescription~\cite{Perelomov:1986kl}, the set
\begin{equation}
  \label{eq:21}
  | \zeta \rangle =  \op{D} (\xi) |k, k \rangle
\end{equation}
constitutes a family of \emph{bona fide} coherent states parametrized
by the pseudo-Euclidean unit vector $\mathbf{n} = (\sinh \omega \cos
\varphi, \sinh \omega \sin \varphi, \cosh \omega )$, with $\xi =
(\omega /2) \exp (i \varphi) $ and $\zeta = \tanh (\omega /2) \exp (-i
\varphi)$.

By expanding the exponential and employing the disentangling theorem, we
get the decomposition
\begin{equation}
  \label{eq:6}
  | \zeta \rangle = ( 1 - |\zeta |^{2})^{k} \sum_{p=0}^{\infty}
  \sqrt{\frac{\Gamma (2k + p)}{p! \Gamma (2k)}} \zeta^{p}
  |k, k + p \rangle \, ,
\end{equation}
and by projecting over the complete basis $| \eta \rangle$ we get the
corresponding wave function $\Psi_{\zeta} (r, \varphi)$ in the
transverse parameters. The expression can be simplified into an
exponential form using the identity
\begin{equation}
  \exp { \left(\frac{\gamma x}{\gamma -1}\right) } =
  (1-\gamma)^{1+|\ell|} \sum_{p=0}^{\infty} {\gamma}^{p}
  L_{p}^{|\ell|}(x) \, ,
\end{equation}
so the final result is
\begin{equation}
  \label{eq:rcs}
  \Psi_{\zeta} ( r , \varphi) =\sqrt{\frac{\alpha^{2}}{\pi|\ell|!}}
  \left [ \frac{1-|\zeta|^2}{(1-\zeta)^2}\right]^{\frac{|\ell|+1}{2}} \!\!
  e^{\frac{\zeta+1}{\zeta -1} \,\alpha^{2} r^2/2}  (\alpha r)^{|\ell|}
  e^{i \ell\varphi} \, .
\end{equation}
We see that the Perelomov coherent states are polynomial-Gauss modes
at $t=0$; a subfamily of Hypergeometric-Gauss modes upon
evolution, as discussed in Ref.~\cite{Karimi:2007fj}. They are also
eigenstates of the OAM, and shape invariant in the time evolution.

\begin{figure}
 \includegraphics[width=0.80\columnwidth]{Figure1}
  \caption{(Color online). Probability distribution $W_{p}$ for a
    coherent state  $| \zeta \rangle $ with $\ell =1$ as as a function
  of $p$ and the ring average number  $\bar{p}$.}
  \label{fig:dist}
\end{figure}

The average number of sharp rings in the state $ | \zeta \rangle$ is
\begin{equation}
  \label{eq:9}
  \bar{p} = \frac{| \zeta |^{2}}{|\zeta|^{2} - 1} (| \ell | + 1) \, ,
\end{equation}
and the statistical distribution of rings $W_{p} =| \langle p,\ell |
\zeta \rangle |^{2} $ is
\begin{eqnarray}
  \label{eq:13}
  W_{p}  =  \frac{(| \ell | +1)^{|\ell| + 1} (p + |\ell|)!}{p! | \ell |!}
\frac{ \bar{p}^{p}}{(\bar{p} +|\ell| +1)^{p + |\ell| +1}} \, .
\end{eqnarray}
In Fig.~1 we have plotted this distribution for a coherent state with
$\ell =1$ and different values of $\bar{p}$. In Fig.~\ref{fig:Trans1},
we have plotted $| \Psi_{\zeta} (r, \varphi)|^{2}$ for a coherent
state with $\ell =1$ and $\langle \op{p} \rangle = 1$, and the
corresponding distribution for the Laguerre-Gauss eigenstate with
$p=1$. The striking differences can be appreciated at a glance.

It is worth mentioning that there is an alternative
definition of coherent states, due to Barut and
Girardello~\cite{Barut:1971yq}:
\begin{equation}
  \label{eq:BGdef}
  \op{k}_{-} | \zeta \rangle_{\mathrm{BG}} = \zeta | \zeta
  \rangle_{\mathrm{BG}}\, ,
\end{equation}
which appears as a reasonable generalization of the standard coherent
states as eigenstates of the annihilation operator. This equation can
be solved in the $|p, \ell \rangle $ basis, yielding
\begin{equation}
  \label{eq:BGst}
  | \zeta \rangle_{\mathrm{BG}} = \frac{| \zeta |^{\ell /2}}
 {\sqrt{I_{\ell} (2 |\zeta|)}}
\sum_{p=0}^{\infty}
\frac{\zeta^{p}} {\sqrt{p! (p + |\ell |)!}}
|p, \ell \rangle \, ,
\end{equation}
where $I_{\ell} (x)$ is the modified Bessel function. Projecting again in
the transverse coordinates we get, after some calculations,
\begin{equation}
  \label{eq:BGproj}
 \Psi_{\zeta,{\mathrm{BG}}} (r, \varphi) = \sqrt{\frac{\alpha^{2}}{\pi I_{| \ell |} ( 2 |\zeta |^{2})}}
e^{(\zeta^{2} - \alpha^{2} r^{2}/2)}
 J_{| \ell |} (  2 \zeta r) e^{i \ell   \varphi} \, .
\end{equation}
This set of coherent states are thus realized as Bessel-Gauss
modes. However, these solutions are not shape invariant, which runs
against the notion of coherence.

\begin{figure}
  \includegraphics[width=0.92\columnwidth]{Figure2}
  \caption{(Color online). Intensity profiles $| \Psi (r, \phi)|^{2}$
    for a radial coherent state $ | \zeta \rangle$ written as in
    \eqref{eq:rcs}, with $\langle p \rangle = 1$ and for an eigenstate
    $|p , \ell \rangle$ with $p=1$ and $\ell =1$. In the inset, we
    show the corresponding density plots, in the same order.}
  \label{fig:Trans1}
\end{figure}

\subsection{Radial intelligent and squeezed states}

The commutation relations (\ref{eq:ccr}) imply that these
operators cannot be measured simultaneously, which is reflected by the
uncertainty relation
\begin{equation}
  \label{eq:ur}
  \Delta \op{k}_{x} \, \Delta \op{k}_{y} \ge \textstyle{\frac{1}{2}}
| \langle \op{k}_{z} \rangle | \, ,
\end{equation}
where $\Delta \op{A} = [ \langle \op{A}^{2}
\rangle - \langle \op{A} \rangle^{2} ]^{1/2}$ stands for the
variance. According to the standard definition, squeezing
occurs whenever~\cite{Wodkiewicz:1985qf}
\begin{equation}
  \label{eq:10}
  (\Delta \op{k}_{x})^{2} \le \textstyle{\frac{1}{2}}
  |\langle  \op{k}_{z} \rangle |
  \qquad
  \mathrm{or}
  \quad
  (\Delta \op{k}_{y})^{2} \le \textstyle{\frac{1}{2}}
  |\langle  \op{k}_{z} \rangle |\, .
\end{equation}
Intelligent states are those for which (\ref{eq:ur}) holds as an
equality. The coherent states (\ref{eq:rcs}) and (\ref{eq:BGproj})
are intelligent but not squeezed.

Indeed, these intelligent states are solutions of the eigenvalue
problem~\cite{Jackiw:1968gd}
\begin{equation}
  \label{eq:defIS}
  (\op{k}_{x} - i \lambda \op{k}_{y} ) | \Psi_{\lambda } \rangle =
\Lambda |\Psi_{\lambda}  \rangle \, ,
\qquad
\lambda \in \mathbb{R} \, .
\end{equation}
Although they have been investigated from various
perspectives~\cite{Bergou:1991km,Gerry:1995kq,Puri:1996qm}, we follow
here the comprehensive approach of Ref.~\cite{Joanis:2010yq}, which
starts by noting that the coherent state $\exp ( i \tau \op{k}_{y} )
|k , k \rangle$ is intelligent provided $\lambda= \cosh \tau$ [with
eigenvalue $\Lambda = - (k + M) \sinh \tau$, and $M = 0, 1, \ldots$ an
integer number]. Then, the most general intelligent state can be
written as
\begin{equation}
  \label{eq:puf}
  | \Psi_{\ell, M} (\tau) \rangle = \exp (  i \tau \op{k}_{y} )
  | \kappa_{M}^{k}  (\tau ) \rangle \, ,
\end{equation}
where $\tau$ is the squeezing parameter and the seed state
$|\kappa_{M}^{k} \rangle$ can be expressed as
\begin{equation}
  \label{eq:2}
  | \kappa_{M}^{k} (\tau ) \rangle = \sum_{p=0}^{M}
c_{p}^{k} (\tau ) |k, k + p \rangle \, .
\end{equation}
The coefficients $c_{p}^{k}$ can be obtained as a recursion relation;
the final result reads
\begin{equation}
  \label{eq:1}
c_{p}^{k} =\binom{M}{p}
\displaystyle
\frac{\tanh^{p} \tau}{\binom{2 k+ p-1}{p}^{{1/2}}}
\, c_{0}^{k}\, ,
\end{equation}
and $c_{0}^{k}$ is fixed by the normalization of the state. The infinite family
(\ref{eq:puf}) of states parametrized by $k$ and $M$ is actually squeezed.

\begin{figure}
  \includegraphics[width=0.92\columnwidth]{Figure3}
  \caption{(Color online). Intensity profiles $| \Psi
    (r,\varphi)|^{2}$ for intelligent states with $M=10$ and $\tau =
    1/2$ for several values of $\ell$: $\ell = 1$, $\ell=10$, and
    $\ell=20$. The inset shows the corresponding density plots, in the
    same order.}
  \label{fig:variableell}
\end{figure}

Next, we need to project $\exp ( i \tau \op{k}_{y} )|\kappa_{M}^{k}
(\tau ) \rangle$ on the basis $| k , k + p \rangle $. To this end, we
recall that the action of $\exp ( i \tau \op{k}_{y} ) $ on a basis
state $|k, k + p\rangle$ is given in terms of SU(1,1) Wigner
$d$-functions~\cite{Ui:1970nh}
\begin{equation}
\exp ( i \tau \op{k}_{y} )|k, k + p\rangle
=\sum_{p^{\prime}=0}^\infty  d^k_{k+p,k+p^{\prime}} (- \tau ) \,
|k, k +p^{\prime}\rangle \, .
\end{equation}
In this way, we get,  expressed  in transverse coordinates
\begin{equation}
\Psi_{\ell M} (r, \varphi, \tau )= \sum_{p^{\prime}=0}^{\infty}
 \sum_{p=0}^{M}  d^k_{k+p^{\prime},k+p} (- \tau) \, c_{p}^{k} (\tau ) \,
 A_{p \ 2k-1 }(r) e^{i\ell\varphi} \, ,
\end{equation}
where $A_{p \ell}$ has been defined in Eq.~\eqref{eq:amp}.
The effect of increasing $\ell$, for fixed $p$ and $\tau$, is
illustrated by plotting the intensity profile $|
\Psi_{\ell,M}(r,\varphi, \tau) |^2$ as a function of $ r$ in
Fig.~\ref{fig:variableell}: this intensity tends
to a Gaussian-like shape. The effect of increasing $\tau$ for fixed
$\ell$ and $M$ is  illustrated in Fig.~\ref{fig:FFplot3D24}, and leads
to the appearance of rings as we increase $\tau$.

For large values of $\ell$ (more concretely, for  $k$ and  $p$ large,
but $p/k\ll 1$), one has at hand a compact asymptotic approximation
to the $d$ functions, namely~\cite{Rowe:2001uq}
\begin{equation}
  d^{k}_{k+p,k}(\tau) \simeq \frac{1}
  {[ (k+p)^2-k^2 ]^{1/4}} \,e^{-k(\tau-\tau_p)^2/2}\, ,
\end{equation}
with $\cosh \tau_{p}=(k+p)/k$ and whose Gaussian nature is evident.
\begin{figure}[t]
  \includegraphics[scale=0.6]{Figure4}
  \caption{(Color online). Intensity profiles $| \Psi
    (r,\varphi)|^{2}$ for intelligent states with $M=11$ and $\ell =3$
    for several values of $\tau$: $\tau = 16/5$, $\tau=13/10$, and
    $\tau =3/5$. The inset shows the corresponding density plots,
    in the same order.}
  \label{fig:FFplot3D24}
\end{figure}

\section{Concluding remarks}

In summary, we have provided a  handy toolbox to deal with the radial
index of Laguerre-Gauss modes and shown how it can be
used to construct a consistent quantum theory of this variable. We
stress that this is more than an academic curiosity, since recent
experiments in our laboratory~\cite{Karimi:2014by} have confirmed that
the radial degree of freedom of single  photons can be manipulated
individually in a quantum regime.  It is our hope that the results
presented here will inspire novel quantum protocols and algorithms
using  such a ``forgotten quantum number''.

\begin{acknowledgments}
  E. K. and R. W. B. acknowledge the support of the Canada Excellence
  Research Chairs (CERC) Program. The work of H. G. is supported by
  the Natural Sciences and Engineering Research Council (NSERC) of
  Canada. J. R. and Z. H. are grateful to the financial assistance of the
  Technology Agency of the Czech Republic (Grant TE01020229) and the
  Czech Ministry of Industry and Trade (Grant FR-TI1/364). G. L. is
  partially funded by EU FP7 (Grant Q-ESSENCE).  Finally, P. H. and
  L. L. S. S. acknowledge the support from the Spanish MINECO (Grant
  FIS2011-26786).
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


Looking for private in-home or in-studio Viola lessons? Our Happy Valley teachers are ready to get you started. Find your teacher today!
Correct play position, reading viola clef, scales, etudes, solo pieces, shifting, vibrato. Possibility to play in a student string orchestra!
Hello! I am offering beginning to advanced lessons for students of all ages (age 3 through adult). The methodology that I teach is directly from Oberlin and New England Conservatories. My teaching philosophy is to self motivate students through fun and a positive learning environment. I have over 10 years of teaching experience and have taught in a variety of settings across the US and Taiwan.
As a viola teacher, I focus on making sure all my students have a strong foundation. Playing in tune, keeping a steady tempo and playing with a solid tone are extremely important. As the student continues to work and perfect those, I begin to work on playing with feeling and expression. Putting your heart into the music is what sets one person's sound apart from others. I teach all styles of music and encourage my students to try new things.
I received my undergraduate degree in viola performance from the Eastman School of Music. In my 13 years teaching, I've taught over a hundred students from beginner to advanced players. Whatever you want to accomplish as a violist, I'm confident that I can guide you toward realizing your vision.
love writing music as well and have written music for Amtrak, Estée Lauder, Nabisco and other brands.
ong Kow Chinese School in Boston, MA. My daughter, Sarina, was one of her students. My daughter loves her so much and made good progress when Elisa was teaching her. Elisa made Sarina relax and comfortable so that she was enjoying the learning process; she emphasized on skill and poses, she tried hard to help the students to get to the best they could. She is reliable as you can talk with her about the timing change and make up lessions. She came to our house to say goodbye and hug Sarina before she left. She is a great teacher that you can count on and you will enjoy her teaching."
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Looming Demolition of Centre Artisanal?
The 1962 Centre Artisanal in Sèvres, designed by the office of Candilis Josic Woods, is under threat of demolition. The project is an early example of French structuralism and what Alison Smithson called 'mat-building'. The arts building with studio spaces is a precursor to the more famous projects for the city centre of Frankfurt am Main and the Berlin Free University, which was realized in the early 1970s.
Pierre Lagard, architect in Sèvres has started an action to prevent demolition. Alexis Josic is involved, too, and willing to advise on a possible restructuring of the project. After all, Candilis Josic Woods designed the building in such an elementary way to facilitate re-appropriation by future users.
To help save the building from demolition, please sign up with Pierre Lagard.
For more information check the website Désirs d'avenir.
CIAM 9 d'Aix-en-Provence, 1953. The book is the result of a colloquium organised in 2003 by Jean Lucien Bonillo, Claude Massu and Daniel Pinson.
'La Modernité critique' presents a broad reflection on the event of the ninth CIAM congress, including the issue of habitat, and the formation of Team 10. Of particular interest are two interviews with Blanche Lemco-Van Ginkel, who was involved with Team 10 in its early years, and Gérard Thurnauer, member of the Paris CIAM group, and co-founder of the firm Atelier Montrouge.
The book also contains various reflections on positions close to, or parallel of the CIAM discourse and the Team 10 group, such as Roland Simounet, John Habraken, Josep Lluis Sert, and Ernesto Rogers.
For more information visit the publisher's website: Editions Imbernon.
18 September, 18.30, Ken Frampton will deliver a lecture 'Structure, Identity and Existence in the Work of Team 10'.
21 September, 18.30, a short symposium will take place; invited participants include Tom Avermaete, Peter de Bretteville, Keith Krumwiede, Ana Miljacki and Alan Plattus.
For more information contact the Yale School of Architecture.
On 21 March 2006 the reconstructed sculpture pavilion of Aldo van Eyck was opened at the Kröller-Müller Museum in Otterlo. This provisional, yet poetic structure was originally designed for the 1966 sculpture exhibition at Park Sonsbeek, Arnhem. It was dismantled after the exhibition was closed, and gained an immediate cult status. Now, after years of preparations and fund raising the pavilion can be visited again.
For the occasion the museum set up a modest and elegant presentation of drawings for the pavilion, including hitherto unknown sketches from the Van Eyck archive. It could be visited until 11 June.
During the winter of 2005-2006 the Kröller-Müller Museum's Sculpture Garden has witnessed the reconstruction of the pavilion that Aldo van Eyck designed in 1965 for the 5th International Sculpture Exhibition, which was held at Park Sonsbeek in Arnhem from 27 May to 25 September 1966. This temporary pavilion was dismantled after the exhibition. Despite having existed for a mere few months, reactions to the pavilion were extremely positive, from both the general public and art and architecture enthusiasts alike. The pavilion has gone down as a highpoint in Dutch architectural history and is viewed as one of Van Eyck's masterpieces.
The Van Eyck pavilion is compact and closed, making it ideal for displaying small pieces of sculpture, which are richly represented in the museum's collection. Thanks to its intimacy and its contrast with the surrounding landscape it has an entirely different spatial effect and functionality to the pavilion that Gerrit Rietveld designed in 1955 for Park Sonsbeek and which has stood in the Kröller-Müller Museum's Sculpture Garden since 1965. The Rietveld pavilion's elementary and open character and its connection with the landscape makes it better suited to displaying large sculptures and projects. These considerations, combined with the design's great architectural value have been crucial in the decision to reconstruct this second pavilion in the garden. The reconstruction of the pavilion was discussed with Aldo van Eyck before his death. He made the necessary alterations to the design to unsure a longer lifespan than its original few months. These changes, mainly affecting the roof, have been implemented by the architect's widow Hannie van Eyck and Abel Blom from Buro Van Eyck.
For more information on the Kröller-Müller Museum visit their website.
Towards Open Form by Oskar Hansen is posthumously published by the Foksal Gallery Foundation in collaboration with the Warsaw Academy of Fine Arts Museum.
The English-Polish publication is edited by Jola Gola who also authored the monograph of Jerzy Soltan of 1995.
This book is the first full-scale pesentation of Oskar Hansen's Open Form and at the same time a monograph of his work. It is made up of two parts - a presentation of the different fields of Hansen's work, selected and arranged by the author himself, and a chronology showing events in his life and work on a temporal axis.
For the author, the book's visual design has become one of the ways of presenting his Open Form idea: the contents 'flow' before the reader's eyes, taking them from one topic to another. The non-textual introduction is an iteration from the Polish catalogue for the 1960 Milan Triennal. 'Bridging' the book's two parts are interviews with the artist: Joanna Mytkowska's in the first part, and fragments of Hans Ulrich Obrist's in the second.
The book can be ordered on-line from Revolver, Archiv für aktuelle Kunst.
On 16 September 2005 Jerzy Soltan died in Cambridge, Massachusetts.
Another Modern by Tom Avermaete focuses on the theory and practice of the French contributors to Team 10: Georges Candilis, Alexis Josic and Shadrach Woods (all disciples of Le Corbusier). Relying for the first time on the archives of both Georges Candilis and Shadrach Woods, it re-situates the work of the Candilis-Josic-Woods partnership (1952 - 1970) within the international framework of CIAM as well as within the French context of decolonization and modernization during les trente glorieuses (the 'thirty glorious years' between 1945 and 1975).
The author argues that the work of Candilis-Josic-Woods entails a new way of analysing, understanding and conceiving the built environment, an approach that is radically different to those previously propagated within CIAM. Avermaete reveals how this new approach to the built environment was elaborated in architectural and urban projects for three realms that changed drastically during the post-war period: residential culture, urban culture and mass leisure culture. This book addresses the projects and ideas of Candilis-Josic-Woods as a relentless quest for an inclusive design approach that acknowledges both the complexity of social life and its spatial translation.
The opening of this exhibition coincides with the launch of this website and with the book Team 10, 1953-1981, In Search of a Utopia of the Present, published by NAi Publishers. The exhibition, which has been organized in association with the Faculty of Architecture at the TU Delft, emphasizes the work of Team 10's core group: Alison and Peter Smithson from England, Aldo van Eyck and Jaap Bakema from the Netherlands, Giancarlo De Carlo from Italy and Georges Candilis and Shadrach Woods from France. This first major survey of Team 10's work draws upon material from collections around the world, including the Centre Pompidou, the Harvard University, the Università di Venezia, the ETH Zurich and the Fondation Le Corbusier, as well as private collections.
The exhibition "Team 10 - A Utopia of the Present" is on show at the NAi from September 25, 2005 until January 8, 2006. After this, the exhibition goes on tour to Milan, IFA Paris and Columbia University in New York.
See the NAi website for location and opening hours.
To See the World , an exhibition at Zach�ta National Gallery of Art, accompanied by a publication bearing the same title, presents the last project the artist completed before his death on 11th May 2005. Fragments of paintings will be arranged in front of the main entrance to the gallery displaying time-space relationships. Above them, a structure of roadblock spikes will appear to frame the sky and integrate the exterior with the interior, attracting viewers inside. The structure is a reference to the artist s first solo exhibition in the Salon Po Prostu at the Jewish Theatre in Warsaw (1957). In the following rooms will be presented paintings by Hansen that have not been on public display since that first exhibition. Another room will be filled with a different structure for organising space  a rounded shape primarily intended as a background for a visitor and (in keeping with the artist s original wish) for Hansen s students, who will present their works here during the exhibition. The exhibition will also feature some of the apparatuses constructed by the artist, i.e. the devices that students formerly used to complete exercises in his classes.
With its space-integrating structures and densely displayed paintings, the exhibition leads up to a large, empty space with a single armchair in the middle and two clocks ticking out of sync on two opposing walls. When making drafts for the exhibition the artist was already aware of his fast-approaching death, and the last room is intended to provoke reflection on the passage of time.
The exhibition and the publication have been prepared in cooperation with the Museum of the Academy of the Fine Arts in Warsaw that houses Oskar Hansen s collection.
NEW YORK, August 1, 2005) The archive of the New York architect, urban planner, teacher and theorist Shadrach Woods (1923 1973) has been donated to Columbia University s Avery Architectural and Fine Arts Library.
I am tremendously excited that this material has come to Columbia just as the critical reevaluation of Woods and his contemporaries is getting underway said Mark Wigley, Dean of the Graduate School of Architecture, Planning and Preservation.
Beginning his career in the office of Le Corbusier, Woods subsequently designed innovative housing in Morocco and Algeria with Georges Candilis, forming a partnership with the latter and Alexis Josic in 1954 in Paris. Their early projects inspired the formation of the Team 10 group and helped set the agenda for postwar modernist architects. As Candilis-Josic-Woods the firm won and executed several important commissions in France and elsewhere, most notably the new master town plan for Toulouse-le-Mirail (1961) and a plan for the Free University of Berlin (1963). Woods also lectured various universities including Cornell and Yale and from 1969 he was a professor at Harvard University s Graduate School of Design. His writings include Urbanism is Everybody s Business (with Joachim Pfeufer, 1968), What U Can Do (1970) and The Man in the Street: a Polemic on Urbanism (1975).
With his death a generation falters, stated the obituary for Woods in Architectural Design, written by fellow Team 10 member Peter Smithson. The archive, currently still being processed, is the gift of Val Woods. An oral history component and a program of research and exhibition are in development. Further relevant material is being actively sought from Shadrach Woods  former friends and colleagues.
A public event celebrating the arrival of the archive is scheduled for Monday, November 14, 2005 in Columbia s Avery Building. Gabriel Feld, Professor of Architecture at the Rhode Island School of Design, will lead a discussion of Shadrach Woods and the importance of his work. The event will also feature a viewing of the Woods Collection.
The Avery Architectural and Fine Arts Library is one of the most outstanding and comprehensive collections relating to architecture and the fine arts in the world. Avery collects a full range of primary and secondary sources for the advanced study of architecture, historic preservation, art history, decorative arts, city planning, real estate, and archeology. The Library contains more than 400,000 volumes, including 35,000 rare books, and receives approximately 1,000 periodicals. Avery s Drawings and Archives collection includes one million architectural drawings and records. The Library is home to the Avery Index to Architectural Periodicals, an operating program of the Getty Research Institute, which is the only comprehensive American guide to the current literature of architecture and design.
Columbia University Libraries is one of the top ten academic library systems in the nation, with 8.6 million volumes, over 65,650 serials, as well as extensive collections of electronic resources, manuscripts, rare books, microforms, and other nonprint formats. The collections and services are organized into 22 libraries, supporting specific academic or professional disciplines. Columbia Libraries employs more than 400 professional and support staff to assist faculty, students, and researchers in their academic endeavors.


Comments made during and shortly after the UNCLOS negotiations shed light on how participants intended the Convention to be interpreted. Ambassador Tommy T.B. Koh from Singapore, the President of UNCLOS III,32 recognized that the text of UNCLOS did not explicitly provide clarity on permissible military conduct in the EEZ, but he noted "it was the general understanding that the text we negotiated and agreed upon would permit such activities to be conducted."33 Furthermore, Ambassador Koh has spoken about certain coastal states trying to apply territorial sea sovereignty rights in the EEZ, which he noted "is not consistent with the intention of those of us who negotiated this text, and is not consistent with the correct interpretation of this part [Part V] of the Convention."34 The United States further points out that of the few statements made by China at the time of ratifying UNCLOS, only one related to military activities.35 The statement related only to activities in the territorial sea, saying nothing about China's objections to military activities in a coastal State's EEZ.
On how many issues does one witness this sort of unanimity across parties, agencies, and time? These people are true experts: not just on theory, but on how things play out in policy practice. There is a compelling reason for their unanimity: U.S. UNCLOS ratification is a great idea whose time has more than come.
Additionally, we need to reinforce the global institutions that the Law of the Sea was designed to create and support. This entails underwriting with our power and example peaceful dispute resolution based on international law and international institutions. Among these, the United States must ratify the UN Convention on the Law of the Sea (UNCLOS). As Peter Dutton testified before the House Foreign Affairs Committee in 2014, "American policy makers must realize that the contest for East Asia is one of both power and law. ... Acceding to [UNLCOS] and once again exercising direct leadership over the development of its rules and norms is the first and most critical step. ... My sense is that East Asian states, indeed many states around the world, are desperate for active American leadership over the norms and laws that govern legitimate international action."25 Once again, I agree fundamentally with my colleague.
The United States should ratify UNCLOS because doing so would further support the rules-and-norms- based international system that Washington is rightly trying to foster—in part as a means to ensure the following: (1) that neither force, nor even the threat of force, will be employed to resolve island and maritime claims disputes in a dynamic but increasingly-tense region; and (2) that such destabilizing approaches will not be encouraged anywhere else. Ratifying UNCLOS would also eliminate a perennial source of deflective criticism by China and understandable concern on the part of U.S. allies and partners. While the U.S. stance with regard to international maritime law is obviously far more sophisticated than this—including nuanced positions regarding the far-reaching applicability of customary international law— ratifying UNCLOS would nevertheless eliminate a perception that Washington is advocating "Do as I say, not as I do." The application of maritime law in practice is shaped over time, and China is already benefitting from U.S. vulnerability in this area—vulnerability caused by not joining 166 other nations in becoming a party to UNCLOS.
I can attest from personal experience to the extent to which China has cultivated a new generation of sharp, persistent maritime legal specialists who are active in the international arena. I believe that their concerted efforts can make a difference over time, a difference that would undermine the governance of the global maritime commons to our collective detriment.


text:
10 Unforgettable Treks and Walking Holidays in Europe
If you're looking for a trekking holiday that isn't in your backyard, you can find some of the best treks in Europe. Discover the beauty of the iconic Alps, walk the famous pilgrimage paths of Spain, or revel in the beauty of the Mediterranean by foot. Set aside some time on your getaway and indulge in these unforgettable walking holidays in Europe as told by local European bloggers and trekking experts.
Find "The Way" along this famous walking holiday in Europe
by Inma of A World to Travel
The Camino de Santiago or "The Way" in Spain is one of the most famous walking holidays in Europe. From the middle ages, pilgrims from all over Europe, moved by faith, have approached on foot, on horseback or by bicycle to Santiago de Compostela, in the stunning Galicia. At the end of the road, the remains of one of the apostles rest in the gorgeous cathedral in Santiago's old town. People have been trekking the Camino de Santiago for centuries.
Today, The Way people walk this historical route for a variety of reasons. From foodies to low-cost holiday seekers, sporty people, and many others; the Camino lacks the intense religious sense of yesteryear. Even so, its iconic nature makes it one of the best treks in Europe. Most people start their journey in Roncesvalles.
To obtain the pilgrimage, one needs to walk at least 100km. The stages vary in difficulty, and you can choose which one you take one based on your ability. There are low-cost (or even free) hostels for walkers to spend the night, making this a budget-friendly trek in Europe.
Image by Inma from A World of Travel
Soar among spires on the Haute Route in Switzerland
by Miguel of Travel Sauro
Considered one of the most scenic trails in the Alps (and in the world), the Haute Route Trek offers endless adventure and spanning vistas, making it one of the best treks in Europe. The original path, which can be completed in about 10-14 days, starts in the French town of Chamonix and ends in the Swiss town of Zermatt. That's right, you'll be crossing international borders on foot!
This circuit takes you along unforgettable landscapes such as rocky peaks, amazing glaciers, blue lakes, huge valleys, traditional villages and, mountain huts. In addition, it traverses below the summits of 10 of the highest peaks in the Alps, and crosses several high passes. You'll see that it gets more exciting day after day. Its stunning mountain scenery makes this iconic trekking route one of the most beautiful walking holidays in Europe.
If you are in a good shape, it shouldn't be too difficult to complete. However, keep in mind that you'll need to walk for about seven to eight hours a day, cross 2,900 meter high passes, and walk along rugged terrains. The path is well marked, so you won't really need a guide. Despite that, I recommend that you bring a good GPS device or download a GPS app on your phone. Thus, you'll avoid getting lost on certain points.
Need more info on how to be prepared to hike day in and day out? Check out the ultimate guide to hiking.
Image courtesy of Miguel of Travel Sauro.
Find solitude in Romania's Carpathian Mountains
by Andra of Our World to Wander
If you want to experience the beauty of the Romanian Carpathians then you definitely need to try a trek. And one of the most beautiful trails in Europe is the one leading to the Mălăiești Hut, a beautiful hut situated in the Bucegi mountains. It spoils its visitors with incredible views over the surrounding landscape.
What is great about trekking in the Bucegi mountains is their accessibility. From Romania's capital, Bucharest, you simply need to take an hour and a half train ride all the way to Bușteni, the starting point to a lot of treks. If you start really early you can even do the trek in one day, but it's worth it to stay in the hut for a night. But it would be a pity to miss the sunset and the sunrise from the hut.
So the best option is to start from Gura Diham Hut and follow the path leading to Mălăiești. It will take you approximately five hours to get there and you don't need a guide. The trail is not that difficult, but keep in mind that the area is populated by brown bears. And also make sure to book a place in the hut as it is a very popular place, especially during the summer. And it will only cost you around USD 10 per person per night. Enjoy the nature in all its splendor.
Image by Andra of Our World to Travel
Trek along the iconic Tour du Mount Blanc in France
by Mags of The Family Freestylers
The Tour du Mont Blanc is one of the world's classic high altitude multi-day treks and arguably the most iconic walking holiday in Europe. The route circles Mont Blanc, the highest mountain in Western Europe and crosses three countries – France, Italy, and Switzerland. This famous 170km (105 miles) trek boasts magnificent alpine scenery with breath taking views over the snow capped Alps. Overall, the trail is considered moderate to strenuous so come prepared to earn some of the most spectacular mountain views in Europe.
The trek is grueling in places with an accumulated height gain and loss of approximately 10,000m climbing over 10 mountain cols (mountain passes) and descending into ten valley floors. Accommodation along the way is a mix of hotels or B&B's in towns and villages on valley floors or in high altitude mountain refuges accessible only on foot along the route. It's also possible to have meals at the refuges and camp close by.
The entire hike is normally undertaken in 11 days but many people choose to hike half of it one year, returning the following year to finish the trail. This trek can be hiked self guided – the trail is well marked, but note that the high mountain passes require a degree of mountain safety and solid navigation skills (in the event of bad weather). There are many companies which guide groups and also luggage handlers who are able to transport your luggage from point to point on the trail.
Become a strong hiker, with these tips for tackling your first peak.
Image by Mags of The Family Freestylers
For an epic, multi-day European trek filled with history and sea views, check out the Lycian Way in Turkey.
The best Trek in Europe to experience the Gods: Mount Olympus, Greece
by Helena of Just for One Summer
Mount Olympus, home to the twelve ancient Greek goods, is one of the most iconic mountains in the world and an absolute must for any hiker visiting Greece. There are few different routes to the summits of Mount Olympus but the easiest and most popular ascent starts at Litochoro and passes through Enipeas canyon and Prionia with an overnight stay at Spilios Agapitos Refuge. You'll need two days to complete this iconic European trek. Expect to hike eight to nine hours each day.
Litochoro at the entrance to the Enipeas Canyon is an ideal base for trips to Mount Olympus. It has plenty of accommodation, taverns, and shops and can be easily reached from Athens by car, bus or train. The trail to the top starts at the upper side of Litochoro and roughly follows the Enipeas river to Prionia passing beautiful pools and waterfalls multiple times. It takes around six hours to complete the traverse and is much harder than it looks! From Prionia you'll need another three hours of a steady climb before settling down at the refuge for the night.
For the final ascent to the top wake up early to ensure clear skies and enough time for the return. It takes at least three hours to reach the first of the summits, Skala. From here, you can either embark on a hair rising scramble to the highest peak Mytikas or hike to the slightly lower but much easier Skolio. Either way, your efforts will be rewarded with breathtaking views of this legendary mountain along one of the best treks in Europe!
Image courtesy of Helena of Just for One Summer
Discover ancient Rome in England's Hadrians Wall Walk
by Carol of Wayfaring Views
The Hadrian's Wall Walk is a slow stroll through England's ancient Roman history. The wall represents the old Roman border between England and Scotland. Hadrian had an ongoing beef with the Scots and in 122 AD he built the wall and staffed it with 11,000 Roman soldiers in order to "keep out the barbarians."
In 2003, a site management scheme was point into place that would both protect the ruins and also allow hikers along the ancient pathway. The walking path runs along what remains of the wall and its remarkably well-preserved ruins and offers an excellent historical walking holiday in Europe.
The Hadrian's Wall Walk is a coast to coast path running eighty-four miles from Newscastle-Upon-Tyne to Bowness-on-Solway. However, you can do the best bits of the wall with a four-day itinerary between Hexam and Gilsland. On this section, you'll hike forty miles through rolling countryside and sheep pasture with plenty of time to get a history lesson at Chester's Fort, Housesteads Fort, Birdoswald fort and many of the milecastles along the way. Enjoy one of Europe's best walking holidays along the Hadrians Wall Walk.
Image courtesy of Carol at Wayfaring Views
Trekking and hiking the Făgărași Mountains Ridge, Romania
by Iuliana of Authentic Travels
Situated in the middle of the country, the Făgărași Mountains feature the highest peaks in Romania (Moldoveanu – 2544m and Negoiu – 2,535m). It's a matter of național pride to climb these peaks. If you aren't from this wonderful nation, the stunning scenery will captivate any hiker, so check out one of the best treks in Europe.
Totaling almost 70 kilometers, the main ridge of Făgărași can be hiked within a minimum of a five-six-day trek (six to eight hours of hiking per day). The trail is of moderate difficulty and a few short sections are equipped with cables to help you through the steep bits. Many people (especially foreigners) trek solo. On this iconic European trek, you will meet lots of people and groups of trekkers from all over the world. If you are not a regular trekker, but still want to trek the Făgărași ridge, be sure to train beforehand and get a guide. The trail is demanding and requires a good physical condition to be completed.
There are two ways to hike the ridge of the Făgărași Mountains:
Start in Turnu Roșu village (the western edge of the main ridge), hike the whole ridge in five to six days and then climb down in Rudărița (the eastern edge of the main ridge); or the other way round, but the ascent from Rudărița is long and monotonous.
Take the cable car to Bâlea Lake (2034m). Bâlea is located exactly at the middle of the ridge. From there, you can hike the eastern half of the ridge (Bâlea – Sâmbăta/ Rudărița – Moldoveanu summit is on this section) or the western half of the ridge (Bâlea – Turnu Roșu – Negoiu summit is included in this section).
Prefer to day hike? Here's a look at Europe's most breath-taking hikes.
Image courtesy of Iuliana of Authentic Travels.
Island hop by foot in Ios, Greece
by Cristina of Greece Logue
The hike from Mylopotas to Klima as a 1.5 to 2 hour hike on the island of Ios, Greece. You can stay anywhere on the island, but the port area or the Chora area are the best as there are plenty of accommodation options and there are also bus stations.
The hike starts at the end of Mylopotas beach. This walking holiday in Europe can be done both guided and unguided. Since they were re-doing the markings on the trail (last year, n.a. 2017) a guide is recommended, since the signage isn't always clear.
You won't need any special gear, except for a good pair of walking shoes, a water bottle, and a bandana. If you do the hike during summer, leave early to avoid the mid-day sun. Make your trek extra special by packing a picnic lunch to enjoy by the beach.
Throughout the hike, you have the sea to your right (as you go from Mylopotas to Klima). The views are spectacular, and you can watch the ferries as they come and go from the island. Do the hike in May and you'll have Klima beach for yourself.
Image courtesy of Cristina of Greece Logue.
Ditch winter and head to the beach on the Fisherman's Trail in Portugal
by James of the Portugalist
The Fisherman's Trail is a 125 km trail that stretches from the Cabo de São Vicente in the Southwest tip of the Algarve to Porto Covo in the Alentejo. The route hugs the Western Coast of Portugal, is exclusively pedestrian, and is an incredibly photogenic route that crosses over some of the most scenic beaches in Southern Portugal.
The trail is broken up into five stages and four complementary stages. Each stage is no more than 25 km. Some people walk the entire stretch in one go, while many others just complete a section or two so. Depending on the length of your trip, you can either treat it as a day hike or a multi-day hike.
The Portuguese tourism board have carefully developed the route and there is accommodation available along each section. There are also plenty of restaurants and even luggage transfer services – how is that for a trekking route in Europe?! There is no need for a guide or special equipment, although normal hiking gear is recommended.
Because the Algarve can reach temperatures of 40°C and more during the summer months, it's best to attempt this walk in spring, autumn, or even parts of winter.
Image courtesy of James of the Portugalist.
Trek through the green scenery of Scotland's West Highland Way
by Gemma at Two Scots Abroad
The West Highland Way is one of the many treks in Scotland but this one differs from others because it is a popular multi-day hike which takes you past lochs, up hills and into valleys making it one of the most scenic walking holidays in Europe. You can do it solo as it is well-signposted or as part of a group.
The WHW starts at Milngavie just outsides of Glasgow and ends in Fort William. Wayers can choose how many days to trek for but usually, they take one week to complete the route. To keep costs down there are camping spots as well as B&Bs and hotels for those looking to ensure a dry night's sleep! The hardest part of the hike is said to be the Devil's Staircase. However, many actually find it to be a breeze in comparison to the challenge of hiking over multiple days.
Image courtesy of Gemma at Two Scots Abroad
From beaches to grassy highlands, iconic ruins to dramatic mountain vistas there is a walking holiday in Europe for every kind of hiker. Discover the best treks in Europe and hike to your heart's content. Where will you trek next?


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WHFD Practices Aerial Ladder Operations
Watch Hill Firefighters continue to take advantage of the sunny but cool weather to practice aerial ladder operations at the Watch Hill Motor Court.
Firefighters have visited the complex several times in the past two months to practice positioning the ladder truck in a single location to fulfill multiple target objectives on the various buildings within the motel complex.
As always, we appreciate the willingness of our local businesses to allow us to practice on their properties during the quieter "off" season.
Ladder 104 reaches the peak of the roof on one of the buildings located on a hill in the Watch Hill Motor Court Complex
Ladder 104 reaches the corner of a flat roof on one of the buildings located in the Watch Hill Motor Court Complex
Ladder 104 reaches the middle of a peaked roof on one of the buildings located in the Watch Hill Motor Court Complex
Ladder 104 is positioned in a location that allows full aerial ladder access to the roofs of four of the five buildings located in the Watch Hill Motor Court Complex
Ladder 104 reaches the for the peak of a roof on one of the buildings located in the Watch Hill Motor Court Complex
Ladder 104 is extended towards the roof of one of the buildings located in the Watch Hill Motor Court Complex


About the book: Bright Not Broken sheds new light on this vibrant population by identifying who twice exceptional children are and taking an unflinching look at why they're stuck. The first work to boldly examine the widespread misdiagnosis and controversies that arise from our current diagnostic system, it serves as a call to action call for parents and professionals to question whyour mental health and education systems continue to fail some of our brightest children. Too often, these kids are caught in an endless cycle of chasing diagnostic labels, being prescribed dangerous medications, and are never given the tools to fully develop their gifts and talents.
Huh. Thank you. I am familiar with "spirited child" – I have two. I knew that one was more "extreme" than the other, and…I think this may be the extra bit I've been looking for. Heavens, parenting is exhausting.
Umm, yeah…on more of a regular basis than I'd like to admit!


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Explore St. Paul
About Main Street
Lyric Project
Donate to the Lyric Project
Things are happening in St. Paul and we want you to be a part of the experience! New restaurants, shops, and a hotel are bringing people to town for a visit and maybe an extended stay. The Clinch River and Spearhead Trails have much to offer to the outdoor enthusiast, but we want more. We want to see live music and live theatre on the stage of the Lyric Theater in downtown St. Paul.
People want to live where there is cultural arts and entertainment. Businesses want to locate where there is a thriving downtown. The new design will have much needed conference space. The Lyric Theater can once again be the center of our small town experience. Your donation will help revitalize the Lyric Theater and make it a downtown destination for locals, the people in neighboring towns, and visitors far and wide.
We are happy to report these achievements:
• $300,000 grant from Appalachian Regional Commission
• $100,000 from Town of St. Paul
• $53,555 in Capital Campaign to date
• $$35,000 in general fundraising to date
• $40,000 grant from Virginia Coalfield Economic Development Authority
• Additional $200,000 from Appalachian Regional Commission
• Architect hired; designs approved by Department of Historic Resources
• Phase I construction bids will be opened on December 18
Tax-deductible donations may be made to: The Lyric Project P.O. Box 375 St. Paul, VA 24283 or at Carter Bank & Trust, 3054 Fourth Avenue, St. Paul, VA, or online at www.stpaulmainstreet.org,
A donation receipt will be mailed to you.
Every donation is vitally important and very much appreciated. Your gift will live on for future generations.
Sincerely, Ad Hoc Committee for Lyric Theater Revitalization
FrazierFeas_sm.pdf
CORECON.pdf
St. Paul Main Street
EMAIL: stpaulmainstreet (at) gmail (dot) com
ADDRESS: 16612 Broad Street and Market Square, St. Paul, VA 24283
© 2023 by St. Paul Main Street


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What We're Watching: Trump's motives, Hong Kong drama, big birthday surprise
Trump's Added Motivation to Win – The US criminal code sets the federal statute of limitations on obstruction of justice at five years. That appears to mean that if Donald Trump wins reelection next November, he will have presidential immunity from prosecution until it's too late ever to indict him for alleged crimes detailed in the report from former special counsel Robert Mueller. Legal experts can weigh in on the possibility of an asterisk here, but it certainly looks like voters, those who back Trump and those who want him out, will be asked to weigh this factor as they go to the polls next year.
Hong Kong – The drama in the streets of Hong Kong has become even more frightening in recent days. This week, we saw thugs armed with clubs attacking defenseless protesters. After video surfaced of a pro-government lawmaker shaking hands with some of these suspected gang members, demonstrators trashed his office. That legislator then posted social media death threats against a pro-democracy political rival. Officials in Beijing have now explicitly warned that they can send Chinese army troops into Hong Kong, and protesters say they won't be intimidated and that demonstrations will continue. In short, the stakes in Hong Kong are rising, and the escalation of promises and threats continues.
Uganda – A 37 year-old Ugandan pop star, opposition lawmaker, and self-proclaimed "Ghetto President" named Bobi Wine announced this week that he will run for president in 2021 against Yoweri Museveni, the man who has held power in Uganda since Wine was four years old. The singer, whose real name is Robert Kyagulanyi, was arrested for treason last year, and he claims (credibly) that he was tortured while in custody. There's a lot that could go wrong here.
Alexander Kliment – Signal's own Alex Kliment is celebrating a big birthday today. We won't throw out a number, but let's call it the "gateway to early middle age." We're watching to see how Alex, creator of Puppet Regime, marks the occasion, because he's one of the funniest and most creative people in the universe, and because he's our friend. #SignalSalute
French "perverts in the bushes" – We admit it…we only read the headline: "Perverts in bushes are ruining nude zone in Paris park, say naturists." We're ignoring this story because we just don't want to know any more about that.