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Carl Gardner in Uncategorized | October 20, 2010
Legal advice at the police station – French and Scottish style
It used to be common in England to suggest that criminal justice was better in France or in Scotland. Well, each is another country, and they do things differently there. I've absolutely no doubt some of their laws are better. But as far as rights at the police station are concerned, they're struggling to keep up.
In Scotland, they're waiting for the Supreme Court's judgment next Tuesday in Cadder v H.M. Advocate on whether the questioning of a suspect without access to a lawyer is in breach of the right to a fair trial, Scottish legislation astonishingly allowing access to a lawyer only after the police have had six hours to interrogate you first. The answer seems fairly obvious, given European Court of Human Rights rulings on the point in cases like Salduz v Turkey, and Lord Advocate Elish Angiolini has anticipated it by issuing interim guidelines in the meantime. These still don't match English standards, though – in particular they don't set a clear enough test for exceptionally delaying access to a lawyer or interviewing a suspect without one – and Justice isn't satisfied with them. In the meantime, the prospect of the Cadder judgment is causing some concern in Scotland.
It's reform time in France too, where suspects' right to consult a lawyer are amazingly constrained, from an English perspective: once taken into custody for questioning (garde à vue), you're only allowed a 30 minute interview with a lawyer, who is not permitted to be present while you're interviewed. Under a new reference procedure the Conseil Constitutionnel decided in July that this breaches the constitution, but deferred the effect of its ruling to allow the legislature until next July to amend the Code of Criminal Procedure. Then France lost in the European Court of Human Rights earlier this month in Brusco v France (only available in French, I'm afraid), partly because of the absence of a lawyer during questioning. This week the Cour de Cassation followed up with a ruling that access to a lawyer cannot be delayed merely because of the category of offence a suspect is arrested for, as the current rules allow. Here's Le Monde's news story about that ruling.
So the justice minister Michèle Alliot-Marie has tabled draft legislation that would give a right to be accompanied by a lawyer when questioned. That's a major breakthrough, but the reform allows access to a lawyer to be delayed for up to 12 hours by written authorisation of a prosecutor for the purpose of permitting (my translation)
the smooth conduct of urgent inquiries with a view to collecting or preserving evidence
which doesn't strike me as a very high test.
Worse, while creating rights with one hand, the proposal takes them away on the other by allowing the police to question a suspect outside the police station in what's called an audition libre. The idea is that in future, instead of arresting suspects and taking them into custody for questioning (with the new, irritating right to have a lawyer there) the police will normally ask suspects to consent to be interviewed – perhaps on the spot upon arrest, or at home, or even perhaps at a police station – without any right to legal assistance at all, and without limit of time. Yes, you'll be able to stop questioning at any moment. But then you'll be detained. It's a system aimed at pressurising suspects into giving up their Convention rights, and in my view probably breaches the right to a fair hearing. No wonder Le Monde calls the proposals an "optical illusion" of reform.
To ministers in Edinburgh and Paris, I commend the Police and Criminal Evidence Act 1984, and Code C issued under it. Some things the English and Welsh do better. Which is why it was especially depressing to hear the new justice minister Jonathan Djanogly, interviewed by Joshua Rozenberg on Law in Action this summer, say in the context of reforming criminal legal aid that he does not think the personal presence of a solicitor is always needed at a police station, and that a telephone interview may sometimes suffice. (You can still listen to the programme – he talks about this at about 20'35").
Maybe, in a few cases. I worry though that his comment reflects a backsliding attitude on fundamental, practical rights. As Scotland and France move reluctantly forwards, England and Wales should not go willingly back.
McSporrans Criminal Lawyers Edinburgh
Indeed the concern caused by the criminal law case of Cadder turned out to be justified, particularly now that we've seen at least 900 cases being dropped directly as a result of the case. Those accused of crimes across Scotland are now being advised to seek legal advice and representation from criminal lawyers to determine whether any evidence they have given without a lawyer while under detention may in fact be inadmissible.
Best wishes, Criminal Defence Lawyers in Edinburgh


Zimbabwe's unity government, increasingly desperate for aid, meets this weekend at a resort retreat in Victoria Falls to draw up a 100-day plan for economic and political reform.
The retreat comes after Zimbabwe emerged from a Southern African Development Community (SADC) summit in Swaziland with only moral backing for its economic reform efforts and a fraction of the financial aid it needs to begin a real recovery.
At the weekend meeting Prime Minister Morgan Tsvangirai is expected to come up with a plan for dealing with the renewed farm violence that has emerged as the major hurdle to Zimbabwe accessing foreign support.
Ahead of the crucial meeting, Tsvangirai was quoted in the Zimbabwe media this week saying that the West needed to back Zimbabwe, however "imperfect" the unity agreement was.
"Zimbabweans should not have to pay a further price for their determination to stand by their democratic ideals because the new government does not meet or match the 'clean slate' or 'total victory' standards expected by the West," Tsvangirai said.
According to one senior Zimbabwe treasury official, one of the key outcomes of this weekend's retreat will be the establishment of a committee that will go on a "road show" across Europe and Asia to seek aid. The committee will be made up of members of both parties.
The group will also be dispatched to the US to lobby for the removal of legislation that bars Zimbabwe from accessing aid from lenders such as the International Monetary Fund and the World Bank.
The government realises it is unlikely to secure the kind of support it is after if it is seen not to be acting on farm violence. Among Zimbabwe's most urgent requirements is aid in the form of farm inputs to avoid another poor harvest. But the spurt in farm occupations appears to have dashed those hopes, government officials fear.
This week the Zimbabwe Human Rights NGO Forum said it had noted a rise in the number of reported rights abuses, while farmer unions said there had been no relief from new farm takeovers.
"The formation of the inclusive government did not bring an end to civic repression, as witnessed by the continued heavy-handedness with which police handled the protests that took place in the month of February," said the forum in a report this week.
Human Rights NGO Forum director Abel Chikomo said some of the recorded violence was in the form of retributive attacks by MDC supporters against former members of Zanu-PF.
Zimbabwe officials have been putting a brave face on the weak support given by SADC at its weekend summit in Swaziland.
Finance Minister Tendai Biti said he did not go to the summit with unrealistic expectations. Zimbabwe realised that most SADC countries cannot support Zimbabwe's economy, but the region's endorsement of its economic recovery plan was important in sending signals to international donors.
Mozambican President Armando Guebuza was quoted saying his country was not in a position to extend financial aid to Zimbabwe but would "reflect to see what we can do and how, within our possibilities".
South Africa pledged US$30-million to be disbursed in tranches of US$10-million in the next three months, in addition to mobilising a bigger package for infrastructural development that would be administered through financial institutions such as the Development Bank of Southern Africa.
Namibia and Zimbabwe already have a partnership under which the former is refurbishing the Hwange thermal power station in northwest Zimbabwe in exchange for power. It is unlikely that Namibia will announce further assistance, Zimbabwe government officials said.
SADC "urged the developed countries to lift all forms of sanctions against Zimbabwe as these sanctions will undermine the country's and SADC efforts to normalise the situation in that member state".


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Remembering Sean Connery: 1930-2020…
EPISODE 1.1 – THE GATHERING…
EPISODE 1.2 – FAMILY TREE…
HLWW: WHO WE ARE & WHAT WE DO…
HLWW MERCHANDISE…
HLWW: THE GATHERING
HLWW's John Mosby remembers the iconic actor behind Juan Sánchez-Villalobos Ramírez and James Bond...
For many, the iconic Sir Sean Connery, who passed away on Friday (30th October) at the age of 90, will always be James Bond. He was the first man to portray Ian Fleming's 007 on the big screen (in 1962's Dr. No) and though the role went to a range of familiar actors over the decades that followed, it's Connery who began the franchise and created the standard by which all others are judged. He would appear in six further 007 outings, Dr. From Russia With Love (1963), Goldfinger (1964), Thunderball (1965) and You Only Live Twice (1967), Diamonds Are Forever (1971) lucratively stretching beyond his original contract and then returned for an unofficial Bond movie Never Say Never Again for Warner Bros. in 1983. But though the franchise made him an icon, he eventually sought to put distance between him and the character, which he did as he grew older.
He was born in Edinburgh on 25th August 1930 and would leave school just before his his 14th birthday, taking on a variety of mundane but practical jobs such as milkman, bricklayer and lifeguard. He was drafted into the Royal Navy and after leaving three years later he became a coffin polisher. But though he had little acting experience, he was alerted to an opportunity in the chorus-line of a touring production of South Pacific and he never looked back. A body-builder, he came third in the 1950's bout for Mr. Universe and did modelling as well, but when he had the opportunity to replace Jack Palance in Rod Serling's Requiem for a Heavyweight in the live 1957 production for the BBC, he started to get noticed and 1957's role in Anna Christie pushed him forward and films such as Another Time, Another Place (1958), Darby O'Gill and the Little People (1959) and The Frightened City (1961) followed. Though there were others in the possible frame for the role of James Bond, Cubby Broccoli noted his 'animal magnetism' and Connery was offered the initial $1 million, five-picture contract to play Bond.
But his career included a vast amount of quite different roles. He was the kind of actor that only comes along rarely – able to play characters that one could argue were only variants of the performer (Connery's roles were often portrayed as intimidating people with Scottish accents, whatever their background) but to do so with such conviction and charisma that audiences flocked to see him, whatever the genre or story.
During the 1970 he appeared in the likes of The Anderson Tapes (1971), The Offence (1973), Murder on the Orient Express (1974), the bizarre Zardoz (1974), John Huston's The Man Who Would Be King (1975), The Wind and the Lion (1975), Robin and Marian (1976) and A Bridge Too Far (1977). The 1980s were no less interesting with Connery moving into the mentor role in many films. He enjoyed himself in Terry Gilliam's Time Bandits (1981) and went to space for the same year's Outland. He was Jones Snr. in Indiana Jones and the Last Crusade in 1989. He also earned an Oscar for his role in The Untouchables (1987). Between those films he famously essayed the role of Juan Sánchez-Villalobos Ramírez, the 'Spanish peacock' swordmaster who mentors Christopher Lambert's Connor MacLeod in Highlander. The film might be just a bullet-point in Connery's vast career – and forever ironic for having a Scot play an Egyptian and a Frenchman playing a Scot – but the film became a cult hit and a beloved fantasy-tinged movie that endures to this day. Connery was contractually obliged to return for a sequel that didn't fare or age anywhere near as well.
"I think its much more interesting for the Europeans, rather than the Americans," he told me. "I think that we had The Name of the Rose it was at Twentieth Century Fox and the entire American takings was about $2 million. It must have cost twice that amount to publicise it and put out the prints. However, it made $63 million in Europe! It made more in Germany in eight days than it did in its whole run in America," Connery muses. " It's kind of encouraging to see that the response to Braveheart and Rob Roy was better, but I don't think that America has quite the same appreciation for this kind of movie as Europe does."
When I spoke to Highlander director Russell Mulcahy for Fearful Symmetry, he remembered how intimidating it was to meet the cinematic icon: "I remember the first day I went to the Savoy Hotel in London to meet Sean. I was there knocking on his door and thinking 'Oh My God… I'm going to meet James Bond!' This tall gentleman opens the door and there he was. We had the most wonderful meeting and loved the script. He was the perfect anchor for so many reasons. One, he's such a gentleman and also such a professional and a well-grounded, experienced actor."
Cinematographer Gerry Fisher also noted: "Connery didn't need to do (Highlander)… he doesn't need to do anything. Even then he could have lived quite happily and not have to do another film… but if he wanted to do something, he'd put a time on it and a price on it!"
The Nineties brought roles such as the possibly-defecting submarine captain in The Hunt for Red October (1990), the senior investigator helping Wesley Snipes in Rising Sun (1993) and as King Arthur in First Knight (1995).
I asked Connery during press for the film about the attraction to swashbuckling films and if there was some innate attraction to them over the years…
"I think its much more interesting for the Europeans, rather than the Americans. I think that we had the Name of the Rose (1986) it was at Twentieth Century Fox and the entire American takings was about $2 million. It must have cost twice that amount to publicise it and put out the prints. However, it made $63 million in Europe! It made more in Germany in eight days than it did in its whole run in America. It's kind of encouraging to see that the response to Braveheart and Rob Roy was better, but I don't think that America has quite the same appreciation for this kind of movie as Europe does."
He voiced Draco the dragon for the CGI-creature in Rob Cohen's Dragonheart (1996). The same year he appeared alongside Nic Cage in The Rock (1996). The Avengers (1998) based on the classic British tv series, was a rare spectacular disaster with Warner Bros. so furious over reports of a problematic shoot that they sent out a blanket fax to all publications warning them of negative coverage before release. That backfired spectacularly and the film misfired, flopping spectacularly though it was hardly Connery's fault. Entrapment (1999) alongside Catherine Zeta-Jones fared better, if only by comparison, but Finding Forrester (2000) was well-received.
There were controversies: a comment he made about hitting women drew condemnation and scorn from many and his campaign for an independent Scotland was counterbalanced by his decision to live abroad paying less tax. He still accepted a knighthood from the Queen in 2000 and became SIR Sean Connery describing it as one of the proudest days of his life.
Rumoured to have turned down key roles in The Matrix and The Lord of the Rings, Connery's last major film was 2003's The League of Extraordinary Gentlemen… a film that probably made The Avengers look calm. The film was based around Alan Moore's comics/graphic novels depicting a group of Victorian characters (such as Allan Quatermain, Mina Harker, Dorian Gray, Captain Nemo and Dr. Jekyll) combining together to fight a common foe. The infamous film saw Connery falling out spectacularly with the film's director Stephen Norrington and he had such a bad time that he later told The Times that it was a key reason in his decision to give up acting: "It was a nightmare. The experience had a great influence on me, it made me think about showbiz. I get fed up dealing with idiots…" (The film also frustrated co-creator Alan Moore who believed the film bared little resemblance to his work and who was furious when Twentieth Century Fox settled a lawsuit that claimed plagiarism to the extent he has subsequently refused personal credit or co-operation with any adaptation of his works).
Connery had by now enough money to never work again if he didn't want to and so took that option, confirming his retirement when he received the American Film Institute's Lifetime Achievement Award on in June,2006. he stuck to his guns and even refused a potential return to the Indiana Jones franchise, saying that "…retirement is just too much damned fun". Technically he had one last screen role as the voice of the title character in Sir Billi the Vet in 2012, a largely-missable small-budget CGI project .
Connery passed away on Friday at his home in the Bahamas after a long illness. He is survived by his second wife Micheline Roquebrune and his son, actor and director Jason Connery.
by	John Mosby	- May 13, 2020
Now is the time of the Quarantine: HL cast are Holy Ground-ed!
This week, Davis-Panzer's Kareem Dimashkie helped organise a virtual get-together for some familiar faces. HLWW's own John Mosby moderated the ZOOM meeting that featured Adrian Paul, Peter Wingfield, Jim...
SXP Watch Party! Anthony De Longis joins 'Highlander' ep viewing…
Get ready "Highlander" Fans! Join along as 2-time Guest Star ANTHONY DE LONGIS shares insights, stories and trivia in real time about his appearance on the iconic Davis-Panzer TV...
Stan Kirsch (Highlander's Richie Ryan) passes away…
Highlander Worldwide is sad to confirm the passing of our friend Stan Kirsch, who played Richie Ryan on Highlander. Born in 1968, acting was something that interested him from...
Dante's Infernal? Adrian Paul shoots to villainy with 'Arrow' debut…
One of the ongoing mysteries the CW show 'Arrow' appeared to be solved in this week's episode – the identity of the mysterious, nefarious and some might even say...
Copyright©2017 Highlander Worldwide

redpajama_set_name: RedPajamaCommonCrawl


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"/>We look to Lionel Logue, King George VI's speech therapist, whose manner may have been relaxed but whose dress sense was anything but.
This Week We're Channeling: Lionel Logue of The King's Speech
We look to Lionel Logue, King George VI's speech therapist, whose manner may have been relaxed but whose dress sense was anything but.
by Anna Prendergast
Geoffrey Rush as Lionel Logue in The King's Speech, 2010.
The royals have a habit of being just the right amount of stylish, which makes them irresistible subjects for film-makers, but as a result of their elevated position, the people around them often get overlooked. In The King's Speech, speech therapist Lionel Logue is as integral a character as Colin Firth's King George VI, and the film explores the relationship between the two. Portrayed by Geoffrey Rush, whose perfectly-timed humour, relaxed attitude and Australian patriotism balances Firth's tightly-wound British stuffiness, Logue appears to be as committed to his clothes as he is to the seemingly impossible task at hand. Despite his unorthodox approach to therapy, his tailoring is traditional and distinguished; costume designer Jenny Beavan intentionally gave him a "working wardrobe" of four suits, typical of a professional gentleman in the 1930s, and used photographs from the Logue family archives.
White Pin Collar Shirt
A pinstriped navy double-breasted two-piece lends extra authority and authenticity to his character, with generous belly, roped shoulders, a high gorge and no vents. Considering the era restrictions, the suit is cut well for Rush's figure, but an updated version would include more flair in the skirt and more definition in the silhouette. Cifonelli's sweeping lapels sit a few inches higher, and a narrower pinstripe is easier to wear. Pair it with a tall pointed collar and motif-printed tie in vintage silk, elegant cufflinks and a perfectly curled silver cowlick. More importantly, conduct yourself with Logue's empathy, kindness and good humour, and you'll be channelling him whatever you wear.
Burgundy Spot Silk Pocket Square
Anna Prendergast
AHHPrendergast
annahhp
Read more from Anna Prendergast
Lionel Logue
This Week We're Channelling


Sculpting stone exposes my raw essence as the stone allows my inner expression to come forth. It is a slow process and I am forced to see my inner workings as they evolve in the emerging sculptures. This process is always humbling and often embarrassing and yet, I am grateful for this journey of creative expression.


After shooting thousands and thousands of swimwear images in the Caribbean and the Bahamas, and thousands of swimwear images for major swimwear retailers, we can certainly capture your swimwear photography needs as well. Come on in! The water is just right!
At Positive Negatives, we know swimwear photography!


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Napster, Inc.
RIP Napster?
Yesterday US District Judge Marilyn Hall Patel issued a temporary restraining order, directing Napster not to facilitate or permit infringement of copyrighted material. Because the immensely popular person-to-person music trading service does not have the capability to separate copyrighted materials from non-copyrighted, this ruling will effectively shut down Napster at midnight tonight, pending...
Whither Napster?
On Monday a three-member panel from the Ninth US Circuit Court of Appeals in San Francisco ruled that a lower court decision requiring Napster to immediately halt music swapping was "overbroad." This week's decision was anything but a victory for Napster, however, as the panel instructed US District Judge Marilyn Hall Patel to craft a narrower injunction that would still require Napster to block...
Battle of the Bands: Metallica Sues Napster, Universities
It was only a matter of time before musicians filed lawsuits against Napster for listing downloadable audio files of their records on Napster's cooperative database. Metal band Metallica was the first to announce a lawsuit on April 13, 2000, alleging the company "violated the law by committing continuing copyright infringements, unlawful use of digital audio interface device, and violations of the...
https://scout.wisc.edu/report/be/2000/0420
The Digital Dilemma: Intellectual Property in the Information Age
This online edition of a book published by the National Academy of Sciences is an example of its subject matter as it explores the issue of intellectual property in a digital age. The book examines the question of what happens to copyright issues when innumerable numbers of people can download a book (or a song, e.g., through Napster) from one copy on the Web? (Apparently, the National Academy of...
https://www.nap.edu/read/9601/chapter/1
The Recording Industry and Napster, Inc. May Reach a Settlement
A federal judge's order, made public Wednesday, January 30, 2002, puts the recording industry's copyright infringement suit against Internet song-downloading service Napster, Inc. on hold until February 17, 2002. The case was put on hold for a month after both sides submitted requests to seek a possible settlement. Capitol Records, Inc. and Virgin Records America were the only two recording labels...


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( na numeração romana) foi um ano comum do século VIII, do Calendário Juliano, da Era de Cristo, teve início e fim em um sábado, com a letra dominical B

Falecimentos
 18 de Setembro - Papa João VII.
 Mommu, 42º imperador do Japão.

redpajama_set_name: RedPajamaWikipedia


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Research Data Management and Open Access Publishing
Seminar date
28 maggio - 23 giugno 2020
"ALL YOU NEED TO KNOW ABOUT RESEARCH DATA MANAGEMENT AND OPEN ACCESS PUBLISHING"
A cycle of 5 seminars for PhD students
Open Science is increasingly referred to as the "Science for the Future" and the "Future of Science". By making science more accessible - via Open Access publishing and proper Research Data Management - Open Science makes the scientific process more inclusive and the outputs of science more readily available and relevant for society. Consequently, more and more funding bodies, governments, research institutions and other agencies have emphasized the value and importance of open science practices and introduced policies on open access publications and data management and sharing.
Open Access (OA) aims at providing free of charge and unhindered access to research and its publications without copyright restrictions. Research Data Management (RDM) is a term that describes the organization, storage, preservation, and sharing of data collected and used in a research project. It involves the everyday management of research data during the lifetime of a research project. It also involves decisions about how data will be preserved and shared once the project is completed, for example depositing the data in a repository for long-term archiving and access. Good RDM and OA are crucial for reproducible and robust scientific research.
Completing this cycle of seminars will allow you, as a PhD candidate, to learn about
The importance of RDM in the general context as well as for your own work
How to write data management plans (DMP) for grant proposals as well as for your own research
Managing privacy, consent, confidentiality and IP issues with respect to to data management
RDM compliance requirements and how to increase funding chances for grant proposals such as H2020
The application of Open Access principles to scholarly publishing; understanding copyright and licenses in relation to publishing agreements; using POLITO tools that support Open Access publishing; how to find suitable OA journals/platforms and how to bypass predatory journals
Final evaluation will be carried through the assessment of the DMP plan as well as short assignments with feedback and multiple choice quiz at the end of the course.
15 contact hours and about 5 hours of individual study
The course will be taught in English and will be held remotely.
Sign up using this registration form.
For further info: federica.cappelluti@polito.it
Scarica la locandina (322,28 kB)


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redpajama_set_name: RedPajamaGithub


You have 7 heroes waiting for you with weapons and magic powers. While you are fighting against the 57 enemies, you can also enjoy incredible atmospheric background music. Choose random arms and various pets as you like.
Make full use of 16 unique maps and have fights with different bosses. Share the game with friends and family via sorts of social medias. With the newest update, you should experience this smooth game right now. Welcome to follow the developer at www.twitter.com/mooffgames. Download and have fun!


Sarah lives in Ottawa, Canada with her four boys and husband. Her creative hobbies include art journaling scrapbooking, stamping, and both paper and mixed media collage. She has exhibited her art at the North Bay Gallery in Ontario, Canada.


Go to My account and sign into your account using your email address and password. Click "Account Details" and update, remember to "Save Changes". For security purposes, we may request further information to verify your identity.
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S2K's patented One-Size-Fits-All Box Sign fits securely over our 4 feet tall mounting stakes. These box signs can be placed anywhere in your exterior zones, whether in your drive-thru, front lawns, parking areas or your entrance.
Market your promotions, products, or messaging at any angle! Our box sign's unique configuration provides 360-degree messaging.
Reflective graphics, led lights and 3-D graphics are available as add-ons.


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Clean energy is the order of the day. Describe briefly India's changing policy towards climate change in various international fora in the context of geopolitics.
General Studies Paper - 2, Topic - Green Energy
India has been actively engaged in international efforts to address climate change and has made significant progress in recent years in developing and implementing clean energy policies.
One key aspect of India's changing policy towards climate change has been its increasing focus on renewable energy sources. India has set ambitious targets for increasing the share of renewable energy in its total energy mix, and has made significant progress in developing and expanding its renewable energy infrastructure. For example, the country has set a target of achieving 175 GW of renewable energy capacity by 2022, and has made significant progress towards this goal, with around 113 GW of renewable energy capacity installed as of 2021.
In addition to promoting the use of renewable energy within its own borders, India has also been active in international efforts to address climate change through various fora, including the United Nations Framework Convention on Climate Change (UNFCCC) and the International Solar Alliance (ISA). India has played a key role in negotiations under the UNFCCC, and has taken a leadership role in the ISA, which is an international organization dedicated to promoting the use of solar energy as a means of addressing climate change.
Overall, India's changing policy towards climate change reflects the country's recognition of the need to take action to address this global challenge, as well as its commitment to playing a leadership role in international efforts to address it.


In order to address malnutrition and unemployment in semi-rural Cameroon, we cultivate mushrooms and train subsistence farmers as well people living with HIV. To accomodate the increasing pressure from communities to learn mushroom cultivation we are expanding our farm and training center. Not only are mushrooms a cheap source of protein and an environmentally friendly agriculture product, there is also a strong mushroom market that presents opportunities for employment in an impoverished area.
The population of Kumbo and the surrounding villages relies on subsistence farming of corn and potatoes for survival. As such, the diets of most people are based on starches with no environmentally friendly source of protein. In addition, population growth has pushed farmers to work the land at an unsustainable rate, leading to erosion and soil degredation. Women in particular suffer from this stress, as they are principally responsible for farming and providing food for the family.
Through our small mushroom cultivation and seed production center, we have already trained over 150 farmers, including women and people living with HIV in mushroom cultivation. In order to expand our own mushroom farm, which has become the first year-round source of mushrooms for local people, and to increase our capacity for training and following up with the women's groups and individuals who request training, we will build Shisong's first Mushroom Cultivation and Training Center.
The building will enable us to train and effectively follow up with over 200 subsistence farmers per year. Not only will income from mushrooms support the healthcare and education of their families, but mushroom cultivation will provide a sustainable source of protein for the households. We will increase our rate of cultivation from 10kg to 40 kg of dried mushroom per month which will enable us to offer employment opportunities for youth and women raising money for their education and families.


>> place. I'm not sure I understand the need for committing it to svn?
>> Also, where is the svn repo located and who will have commit rights?
> Just sending this back to the list - don't know the answer, sorry!
through moonmoon, there is only one admin account.
Web team, and you would of course get this right.
the planets (there are several, not only English one): up to you.


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\section{Introduction}
\label{sec-1}

When light is incident on a semiconductor, its energetic photons can excite electrons
from a lower valence band to a higher conduction band, thereby creating many electron-hole pairs
in the system\,\cite{add8,add26,ref1}.  Simultaneously, its electric-field component
is able to push away the negatively (positively) charged electrons (holes) in opposite
directions.  In this case, the excited electrons and holes will also exert a
back action on the incident light, resulting from the induced optical polarization
as a collection of local dipole moments from many displaced electrons and
holes\,\cite{add9,add29,new}. This polarization field can further scatter incident
photons resonantly\,\cite{ref1,add12,swk}. Therefore, the quantum nature of Dirac
electrons\,\cite{add1,add2,add3,add4,special,chapter} will be revealed in this electron
back action on the incident light.
\medskip

\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{FIG1}
\caption{(Color online) Schematics for a thick (semi-infinite) conductor in the region
$z < 0$ and having a frequency-dependent local dielectric function $\epsilon_M (\omega)$.
Two graphene layers at $ z = z_0$ and $z = z_1$ above the  surface of the conductor are embedded
within a semi-infinite dielectric having a dielectric constant $\epsilon_d$ in the
region $z>0$. These two-dimensional sheets are coupled to each other and also to
the semi-infinite conductor by an electromagnetic interaction.}
\label{f1}
\end{figure}
For the hybrid system shown schematically in Fig.\,\ref{f1}, we are faced with both radiative field
modes, such as photons and polaritons\,\cite{oe1,oe2,oe3,oe4,add28}, and evanescent field
modes, e.g., surface and graphene plasmons\,\cite{ritchie,add5,add6,add7}. Research on  the
optical response of graphene electrons has been previously reported\,\cite{new,add7,add10}, but
most of those studies have been concerned with the effect due to  radiation or grating-deflection
field coupling. In contrast to the plane-wave-like light field, we examine the role of coupling by a
surface-plasmon-polariton (SPP) near field\,\cite{add11,add15,add27} to graphene
electrons with a different dispersion relation from the usual linear one, i.e.,
$\omega=qc$, for light in free-space. In this paper, double graphene layers are
placed very close to the surface of a conducting substrate so that radiative and
evanescent fields are hybridized effectively\,\cite{add24,pan2}. Consequently, the
non-dispersive surface-plasmon (SP) mode can hybridize successfully with radiative
photon and polariton modes\,\cite{oe1,oe2}, as well as with the spatially-localized
graphene plasmon (G-P) mode\,\cite{pan2,add14}, as illustrated in Fig.\,\ref{f2}.
This is quite different from three-dimensional photonic lattices\,\cite{shawn,shawn2},
where electrons interact with quantized multi-subband photons in the first
Brillouin zone.
\medskip

\begin{figure}
\centering
\includegraphics[width=0.4\textwidth]{FIG2}
\caption{(Color online) Schematic representation of the energy dispersion relations
for radiative and evanescent light as well as field modes for the hybrid structure depicted
in Fig.\,\ref{f1}. These include photons, polaritons, surface-plasmon polaritons (SPPs),
two non-degenerate graphene plasmons (G-Ps), and surface plasmons (SPs).}
\label{f2}
\end{figure}
Such a unique dispersion relation of the hybrid light-plasmon modes should be verified experimentally
by optical spectroscopy\,\cite{add18,add25,add31,add32}. The effective scattering
matrix\,\cite{new,add19,add30} for such a coupled system predicts distinctive features
neither from the graphene sheets nor from the conductor on their own, and it retains the properties
of a longitudinal electromagnetic interaction\,\cite{ref1,new,add12} between electrons
in double-layer graphene and a conductor. This scattering matrix can also be employed for
constructing an effective-medium theory\,\cite{add20,add21,add22,add23} used for investigating
the optical properties of inserted biomolecules and metamaterials between the graphene
sheets and the surface of the conductor. As a whole, a local environmentally sensitive
super-resolution near-field imaging\,\cite{xzhang} (beyond the diffraction limit)
should be possible for functionalized biomolecules bounded with either metallic
nanodots and nanorods or carbon atoms of graphene\,\cite{add13,ieee}.
\medskip

The remainder of the paper is arranged as follows. In Sec.\,\ref{sec-2}, we present the
Green's function formalism for the hybridized system depicted in Fig.\,\ref{f1}. By using
Green's functions, an integral equation is established from Maxwell's equations
by treating double-layer graphene as a localized polarization-field source. In
Sec.\,\ref{sec-3}, the optical-response function of Dirac electrons in gapped
graphene is obtained after explicitly calculating the density-density correlation
function at low temperatures. Based on linear-response theory for the localized
graphene polarization field, we present in Sec.\,\ref{sec-4} a self-consistent
equation for the total electric field after combining the integral equation
derived in Sec.\,\ref{sec-2} with the optical-response function calculated in
Sec.\,\ref{sec-3}, from which a dispersion relation is obtained for the hybridized
G-P and SP modes. In addition, a local effective scattering matrix, as well as a
spatial distribution for the scattering field, are derived in Sec.\,\ref{sec-4},
which are further accompanied by an optical absorption spectrum calculated for
hybridized G-P and SP modes. Finally, conclusions and some remarks are given in
Sec.\,\ref{sec-5}.

\section{Green's Function for Hybridized Systems}
\label{sec-2}

Considering steady states, then from one of  Maxwell's equations we can write down the following
equation\,\cite{ref1,new,add11} for two semi-infinite non-magnetic media
in position-frequency space as

\begin{equation}
\mbox{\boldmath$\nabla$$\times$\boldmath$\nabla$$\times$\boldmath$E$}({\mbox{\boldmath$r$}};\omega)-\epsilon_{b}(x_3;\omega)\,\frac{\omega^2}{c^2}\,\mbox{\boldmath$E$}({\mbox{\boldmath$r$}};\omega)
=\frac{\omega^2}{\epsilon_0c^2}\,\mbox{\boldmath${\cal P}$}^{\rm loc}({\mbox{\boldmath$r$}};\omega)\ ,
\label{e2.1}
\end{equation}
where $\mbox{\boldmath$E$}({\mbox{\boldmath$r$}};\omega)$ represents the electric component
of an electromagnetic field,
${\mbox{\boldmath$r$}}=({\mbox{\boldmath$r$}}_\|,x_3)=(x_1,x_2,x_3)$ is a three-dimensional
position vector, $\omega$ is the angular frequency of the incident light. In addition,
$\mbox{\boldmath$H$}({\mbox{\boldmath$r$}};\omega)=\left(1/i\omega\mu_0\right)\mbox{\boldmath$\nabla$}\times\mbox{\boldmath$E$}({\mbox{\boldmath$r$}};\omega)$ represents the magnetic component
of the electromagnetic field, $\epsilon_0$, $\mu_0$ and $c$ are the permittivity, permeability
and speed of light in vacuum, respectively. Furthermore,
$\mbox{\boldmath${\cal P}$}^{\rm loc}({\mbox{\boldmath$r$}};\omega)$ is a local polarization
field produced by optical transitions of electrons in graphene sheets embedded in a semi-infinite dielectric, as schematically shown in Fig.\,\ref{f1}, and the position-dependent dielectric
function takes the form

\begin{equation}
\epsilon_{b}(x_3;\omega)=\left\{\begin{array}{ll}
\epsilon_{d}\ , & \mbox{for $x_3>0$}\\
\epsilon_{M}(\omega)\ , & \mbox{for $x_3<0$}
\end{array}\right.\ .
\label{e2.2}
\end{equation}
In Eq.\,(\ref{e2.2}), $\epsilon_{d}$ characterizes the semi-infinite dielectric
material in the region $x_3>0$, while $\epsilon_M (\omega)=\epsilon_s-\Omega_p^2/[\omega(\omega+i0^+)]$
is the dielectric function of the semi-infinite conducting material in the region $x_3<0$.
For the Maxwell equation in Eq.\,(\ref{e2.1}), we have introduced the corresponding Green's
function ${\cal G}_{\mu\nu}({\mbox{\boldmath$r$}},{\mbox{\boldmath$r$}}^\prime;\omega)$ that
satisfies the equation\,\cite{add11}

\begin{equation}
\sum\limits_{\mu}\left[\epsilon_{b}(x_3;\omega)\,\frac{\omega^2}{c^2}\,\delta_{\lambda\mu}-\frac{\partial^2}{\partial x_{\lambda}\partial x_{\mu}}+\delta_{\lambda\mu}\,\nabla_{\bf r}^2\right]{\cal G}_{\mu\nu}({\mbox{\boldmath$r$}},{\mbox{\boldmath$r$}}^\prime;\omega)=\delta_{\lambda\nu}\,\delta({\mbox{\boldmath$r$}}-{\mbox{\boldmath$r$}}^\prime)\ ,
\label{e2.3}
\end{equation}
where $\nabla_{\bf r}^2=\sum\limits_{\mu}\,\partial^2/\partial x^2_{\mu}$ is the Laplace
operator, $\delta_{\lambda\mu}$ is the  Kronecker delta, and the indices $\lambda,\,\mu=1,\,2,\,3$
specify three spatial directions. Using the Green's function determined by Eq.\,(\ref{e2.3}),
we can rewrite the Maxwell equation in Eq.\,(\ref{e2.1}) in integral form\,\cite{ref1,new}

\begin{equation}
E_{\mu}({\mbox{\boldmath$r$}};\omega)=E^{\rm inc}_{\mu}({\mbox{\boldmath$r$}};\omega)+\frac{\omega^2}{\epsilon_0c^2}\sum\limits_{\nu}\int d^3{\mbox{\boldmath$r$}}^\prime\,
{\cal G}_{\mu\nu}({\mbox{\boldmath$r$}},{\mbox{\boldmath$r$}}^\prime;\omega)\,
{\cal P}_{\nu}^{\rm loc}({\mbox{\boldmath$r$}}^\prime;\omega)\ ,
\label{e2.4}
\end{equation}
where $E^{\rm inc}_{\mu}({\bf r};\omega)$ stands for a solution for the following homogeneous
equation\,\cite{add11}

\begin{equation}
\sum\limits_{\nu}\left[\epsilon_{b}(x_3;\omega)\,\frac{\omega^2}{c^2}\,\delta_{\mu\nu}-\frac{\partial^2}{\partial x_{\mu}\partial x_{\nu}}
+\delta_{\mu\nu}\,\nabla^2_{\bf r}\right]E^{\rm inc}_{\nu}({\mbox{\boldmath$r$}};\omega)=0\ .
\label{e2.5}
\end{equation}
The source term ${\cal P}_{\nu}^{\rm loc}({\bf r}^\prime;\omega)$ in Eq.\,(\ref{e2.4})
usually depends linearly on the total electric field (assuming a weak field) and can be related
to the optical response function\,\cite{ref1,new} of an electronic system.
\medskip

Specifically, for a translationally invariant hybrid semi-infinite system, the Green's
function can be expressed in terms of its two-dimensional (2D) Fourier transform for the $(x_1,x_2)$-plane

\begin{equation}
{\cal G}_{\mu\nu}({\mbox{\boldmath$r$}},{\mbox{\boldmath$r$}}^\prime;\omega)=\int
\frac{d^2{\mbox{\boldmath$q$}}_{\|}}{(2\pi)^2}\,e^{i{\bf q}_{\|}\cdot({\bf r}_{\|}-{\bf r}^\prime_{\|})}\,g_{\mu\nu}({\mbox{\boldmath$q$}}_{\|},\omega\vert x_3,x_3^\prime)\ ,
\label{e2.6}
\end{equation}
where we have introduced a 2D wave vector $\mbox{\boldmath$q$}_{\|}=(q_1,q_2)$.
Substituting Eq.\,(\ref{e2.6}) into Eq.\,(\ref{e2.3}), we obtain a set of coupled
differential equations

\begin{equation}
{\footnotesize
\left[\begin{array}{ccc}
\displaystyle{\epsilon_b\,\frac{\omega^2}{c^2}-q_2^2+\frac{d^2}{dx_3^2}}
& q_1q_2 & \displaystyle{-iq_1\frac{d}{dx_3}}\\
q_1q_2 & \displaystyle{\epsilon_b\,\frac{\omega^2}{c^2}-q_1^2+\frac{d^2}{dx_3^2}}
& \displaystyle{-iq_2\frac{d}{dx_3}}\\
\displaystyle{-iq_1\frac{d}{dx_3}} & \displaystyle{-iq_2\frac{d}{dx_3}}
& \displaystyle{\epsilon_b\,\frac{\omega^2}{c^2}-q_{\|}^2}
\end{array}\right]\,\left[\begin{array}{ccc}
g_{11} & g_{12} & g_{13}\\
g_{21} & g_{22} & g_{23}\\
g_{31} & g_{32} & g_{33}
\end{array}\right]
=\delta(x_3-x_3^\prime)\,\left[\begin{array}{ccc}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1
\end{array}\right]}\ .
\label{e2.7}
\end{equation}
After introducing a rotational transformation\,\cite{add11} in $\mbox{\boldmath$q$}_{\|}$-space

\begin{equation}
f_{\mu\nu}(q_{\|},\omega\vert x_3,x_3^\prime)
=\sum\limits_{\mu^\prime,\nu^\prime}\,{\cal S}_{\mu\mu^\prime}({\mbox{\boldmath$q$}}_{\|})\,{\cal S}_{\nu\nu^\prime}({\mbox{\boldmath$q$}}_{\|})\,g_{\mu^\prime\nu^\prime}({\mbox{\boldmath$q$}}_{\|},\omega\vert x_3,x_3^\prime)\ ,
\label{e2.8}
\end{equation}
where the rotational matrix ${\cal S}({\mbox{\boldmath$q$}}_{\|})$ is

\begin{equation}
{\cal S}({\mbox{\boldmath$k$}}_{\|})=\frac{1}{q_{\|}}\,\left[\begin{array}{ccc}
q_1 & q_2 & 0\\
-q_2 & q_1 & 0\\
0 & 0 & q_{\|}
\end{array}\right]\ ,
\label{e2.9}
\end{equation}
we get an equivalent but simple expression for Eq.\,(\ref{e2.7}) as

\begin{equation}
\left[\begin{array}{ccc}
\displaystyle{\epsilon_b\,\frac{\omega^2}{c^2}+\frac{d^2}{dx_3^2}} & 0
& \displaystyle{-iq_{\|}\frac{d}{dx_3}}\\
0 & \displaystyle{\epsilon_b\,\frac{\omega^2}{c^2}-q_{\|}^2+\frac{d^2}{dx_3^2}} & 0\\
\displaystyle{-iq_{\|}\frac{d}{dx_3}} & 0 & \displaystyle{\epsilon_b\,\frac{\omega^2}{c^2}-q_{\|}^2}
\end{array}\right]\,\left[\begin{array}{ccc}
f_{11} & f_{12} & f_{13}\\
f_{21} & f_{22} & f_{23}\\
f_{31} & f_{32} & f_{33}
\end{array}\right]
=\delta(x_3-x_3^\prime)\,\left[\begin{array}{ccc}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1
\end{array}\right]\ .
\label{e2.10}
\end{equation}
In order to acquire the solution for Eq.\,(\ref{e2.10}), we have to apply both the
finite-value boundary condition at $x_3^\prime=\pm\infty$ as well as the continuity
boundary condition at the $x_3=0$ interface.
This leads to the following
five nonzero $f_{\mu\nu}(q_{\|},\omega\vert x_3,x_3^\prime)$ functions\,\cite{add11,add15}
for Eq.\,(\ref{e2.8}), i.e.,

\begin{eqnarray}
&&f_{22}(q_{\|},\omega\vert x_3,x_3^\prime)
\nonumber\\
&=&\left\{\begin{array}{llll}
\displaystyle{-\left(\frac{i}{2p}\right)\frac{2p}{p_{d}+p}\
e^{ip_{d}x_3-ipx_3^\prime}}\ , & x_3>0\ ,\,x_3^\prime<0\\
\\
\displaystyle{-\left(\frac{i}{2p}\right)\left[e^{ip|x_3-x_3^\prime|}-
\frac{p_{d}-p}{p_{d}+p}\ e^{-ip(x_3+x_3^\prime)}\right]}\ , & x_3<0\ ,\,x_3^\prime<0\\
\\
\displaystyle{-\left(\frac{i}{2p_{d}}\right)\left[e^{ip_{d}|x_3-x_3^\prime|}+
\frac{p_{d}-p}{p_{d}+p}\ e^{ip_{d}(x_3+x_3^\prime)}\right]}\ , & x_3>0\ ,\,x_3^\prime>0\\
\\
\displaystyle{-\left(\frac{i}{2p_{d}}\right)\frac{2p_{d}}{p_{d}+p}\ e^{-ip(x_3-x_3^\prime)}}\ ,
& x_3<0\ ,\,x_3^\prime>0
\end{array}\right.\ ,\ \
\label{e2.11}
\end{eqnarray}

\begin{eqnarray}
&&f_{13}(q_{\|},\omega\vert x_3,x_3^\prime)
\nonumber\\
&=&\left\{\begin{array}{llll}
\displaystyle{\frac{iq_{\|}c^2}{2\epsilon_{M}(\omega)\omega^2}
\left[\frac{2\epsilon_{M}(\omega)p_{d}}{\epsilon_{M}(\omega)p_{d}+\epsilon_{d}\,p}\right]
\,e^{ip_{d}x_3-ipx_3^\prime}}\ , & x_3>0\ ,\,x_3^\prime<0\\
\\
\displaystyle{\frac{iq_{\|}c^2}{2\epsilon_{M}(\omega)\omega^2}\left[e^{ip|x_3-x_3^\prime|}\,{\rm sgn}(x_3-x_3^\prime)+\frac{\epsilon_{M}(\omega)p_{d}-\epsilon_{d}\,p}{\epsilon_{M}(\omega)p_{d}+\epsilon_{d}\,p}\ e^{-ip(x_3+x_3^\prime)}\right]}\ , & x_3<0\ ,\,x_3^\prime<0\\
\\
\displaystyle{\frac{iq_{\|}c^2}{2\epsilon_{d}\,\omega^2}\left[e^{ip_{d}|x_3-x_3^\prime|}\,{\rm sgn}(x_3-x_3^\prime)+\frac{\epsilon_{M}(\omega)p_{d}-\epsilon_{d}\,p}{\epsilon_{M}(\omega)p_{d}+\epsilon_{d}\,p}\
e^{ip_{d}(x_3+x_3^\prime)}\right]}\ , & x_3>0\ ,\,x_3^\prime>0\\
\\
\displaystyle{-\frac{iq_{\|}c^2}{2\epsilon_{d}\,\omega^2}\left[\frac{2\epsilon_{d}\,p}{\epsilon_{M}(\omega)p_{d}+\epsilon_{d}\,p}\right]\,e^{-ipx_3+ip_{d}x_3^\prime}}\ , & x_3<0\ ,\,x_3^\prime>0
\end{array}\right.
\label{e2.12}
\end{eqnarray}

\begin{eqnarray}
&&f_{33}(q_{\|},\omega\vert x_3,x_3^\prime)
\nonumber\\
&=&\left\{\begin{array}{llll}
\displaystyle{-\frac{ik^2_{\|}c^2}{\omega^2}\left[
\frac{1}{\epsilon_{M}(\omega)p_{d}+\epsilon_{d}\,p}\right]\,e^{ip_{d}x_3-ipx_3^\prime}}\ ,
& x_3>0\ ,\,x_3^\prime<0\\
\\
\displaystyle{\frac{c^2}{\epsilon_{M}(\omega)\omega^2}\,\delta(x_3-x_3^\prime)-
\frac{ik^2_{\|}c^2}{2p\,\epsilon_{M}(\omega)\omega^2}}\\
\displaystyle{\times\left[e^{ip|x_3-x_3^\prime|}-\frac{\epsilon_{M}(\omega)p_{d}-\epsilon_{d}\,p}
{\epsilon_{M}(\omega)p_{d}+\epsilon_{d}\,p}\ e^{-ip(x_3+x_3^\prime)}\right]}\ , & x_3<0\ ,\,x_3^\prime<0\\
\\
\displaystyle{\frac{c^2}{\epsilon_{d}\,\omega^2}\,\delta(x_3-x_3^\prime)-\frac{iq_{\|}^2c^2}{2p_{d}\epsilon_{d}\,\omega^2}}\\
\displaystyle{\times\left[e^{ip_{d}|x_3-x_3^\prime|}+\frac{\epsilon_{M}(\omega)p_{d}-\epsilon_{d}\,p}{\epsilon_{M}(\omega)p_{d}+\epsilon_{d}\,p}\ e^{ip_{d}(x_3+x_3^\prime)}\right]}\ , & x_3>0\ ,\,x_3^\prime>0\\
\\
\displaystyle{-\frac{ik^2_{\|}c^2}{\omega^2}\left[\frac{1}{\epsilon_{M}(\omega)p_{d}+\epsilon_{d}\,p}\right]\,e^{-ipx_3+ip_{d}x_3^\prime}}\ , & x_3<0\ ,\,x_3^\prime>0
\end{array}\right.
\label{e2.13}
\end{eqnarray}

\begin{eqnarray}
&&f_{11}(q_{\|},\omega\vert x_3,x_3^\prime)
\nonumber\\
&=&\left\{\begin{array}{llll}
\displaystyle{-\frac{ip_{d}p\,c^2}{\omega^2}\left[\frac{1}{\epsilon_{M}(\omega)p_{d}+\epsilon_{d}\,p}\right]\,e^{ip_{d}x_3-ipx_3^\prime}}\ , & x_3>0\ ,\,x_3^\prime<0\\
\\
\displaystyle{-\frac{ip\,c^2}{2\epsilon_{M}(\omega)\omega^2}\left[e^{ip|x_3-x_3^\prime|}+\frac{\epsilon_{M}(\omega)p_{d}-\epsilon_{d}\,p}{\epsilon_{M}(\omega)p_{d}+\epsilon_{d}\,p}\ e^{-ip(x_3+x_3^\prime)}\right]}\ , & x_3<0\ ,\,x_3^\prime<0\\
\\
\displaystyle{-\frac{ip_{d}c^2}{2\epsilon_{d}\,\omega^2}\left[e^{ip_{d}|x_3-x_3^\prime|}-\frac{\epsilon_{M}(\omega)p_{d}-\epsilon_{d}\,p}{\epsilon_{M}(\omega)p_{d}+\epsilon_{d}\,p}\ e^{ip_{d}(x_3+x_3^\prime)}\right]}\ , & x_3>0\ ,\,x_3^\prime>0\\
\\
\displaystyle{-\frac{ip_{d}c^2}{2\epsilon_{d}\,\omega^2}\left[\frac{2\epsilon_{d}p}{\epsilon_{M}(\omega)p_{d}+\epsilon_{d}\,p}\right]\,e^{-ipx_3+ip_{d}x_3^\prime}}\ , & x_3<0\ ,\,x_3^\prime>0
\end{array}\right.
\label{e2.14}
\end{eqnarray}

\begin{eqnarray}
&&f_{31}(q_{\|},\omega\vert x_3,x_3^\prime)
\nonumber\\
&=&\left\{\begin{array}{llll}
\displaystyle{\frac{iq_{\|}c^2}{\omega^2}\left[\frac{p}{\epsilon_{M}(\omega)p_{d}+\epsilon_{d}\,p}\right]\,e^{ip_{d}x_3-ipx_3^\prime}}\ , & x_3>0\ ,\,x_3^\prime<0\\
\\
\displaystyle{\frac{iq_{\|}c^2}{2\epsilon_{M}(\omega)\omega^2}\left[e^{ip|x_3-x_3^\prime|}\,{\rm sgn}(x_3-x_3^\prime)-\frac{\epsilon_{M}(\omega)p_{d}-\epsilon_{d}\,p}{\epsilon_{M}(\omega)p_{d}+\epsilon_{d}\,p}\ e^{-ip(x_3+x_3^\prime)}\right]}\ , & x_3<0\ ,\,x_3^\prime<0\\
\\
\displaystyle{\frac{iq_{\|}c^2}{2\epsilon_{d}\,\omega^2}\left[e^{ip_{d}|x_3-x_3^\prime|}\,{\rm sgn}(x_3-x_3^\prime)-\frac{\epsilon_{M}(\omega)p_{d}-\epsilon_{d}\,p}{\epsilon_{M}(\omega)p_{d}+\epsilon_{d}\,p}\
e^{ip_{d}(x_3+x_3^\prime)}\right]}\ , & x_3>0\ ,\,x_3^\prime>0\\
\\
\displaystyle{-\frac{iq_{\|}c^2}{\omega^2}\left[\frac{p_{d}}{\epsilon_{M}(\omega)p_{d}+\epsilon_{d\,}p}\right]\,e^{-ipx_3+ip_{d}x_3^\prime}}\ , & x_3<0\ ,\,x_3^\prime>0
\end{array}\right.
\label{e2.15}
\end{eqnarray}
where ${\rm sgn}(x)$ is the sign function,

\begin{equation}
p_{d}(q_{\|},\omega)=\sqrt{\epsilon_{d}\,\frac{\omega^2}{c^2}-q_{\|}^2}\ ,
\label{e2.16}
\end{equation}

\begin{equation}
p(q_{\|},\omega)=\sqrt{\epsilon_{M}(\omega)\,\frac{\omega^2}{c^2}-q_{\|}^2}\ ,
\label{e2.17}
\end{equation}
${\rm Im}[p_{d}(q_{\|},\omega)]\geq 0$, and ${\rm Im}[p(q_{\|},\omega)]\geq 0$.
From these five nonzero $f_{\mu\nu}(q_{\|},\omega\vert x_3,x_3^\prime)$ functions, we arrive at

\begin{equation}
g_{\mu\nu}({\mbox{\boldmath$q$}}_{\|},\omega\vert x_3,x_3^\prime)
=\sum\limits_{\mu^\prime,\nu^\prime}\,f_{\mu^\prime\nu^\prime}(q_{\|},\omega\vert x_3,x_3^\prime)\,{\cal S}_{\mu^\prime\mu}({\mbox{\boldmath$q$}}_{\|})\,{\cal S}_{\nu^\prime\nu}({\mbox{\boldmath$q$}}_{\|})\ ,
\label{e2.18}
\end{equation}
which can be substituted into Eq.\,(\ref{e2.6}) to obtain the Green's function
${\cal G}_{\mu\nu}({\mbox{\boldmath$r$}},{\mbox{\boldmath$r$}}^\prime;\omega)$ in
position space. However, in  our model system depicted in Fig.\,\ref{f1},  we only
consider the case when $x_3,\,x'_3> 0$.

\section{Optical Response Function for Graphene}
\label{sec-3}

For an embedded 2D graphene sheet, the optical response function
for Dirac electrons is found to be\,\cite{ref3}

\begin{equation}
\chi^{(0)}_{\rm s}(p_\|,\omega)=\left(\frac{e^2}{\epsilon_0\,p_\|^2}\right)
\Pi^{(0)}_{\rm s}(p_\|,\omega)\ ,
\label{b1}
\end{equation}
where ${\mbox{\boldmath$p$}}_\|=(p_1,p_2)$ stands for the in-plane electron
wave vector, and $\Pi^{(0)}_{\rm s}(p_\|,\,\omega)$ represents the density-density
response function for Dirac electrons within the graphene sheet which is given by\,\cite{add16}

\[
\Pi^{(0)}_{\rm s}(p_\|,\omega)=\frac{4}{{\cal A}}\sum_{n_1,n_2=\pm 1}
\sum_{{\bf k}_\|}\left|<n_1,{\mbox{\boldmath$k$}}_\|\vert e^{-i{\bf p}_\|
\cdot{\bf r}_\|}\vert n_2,{\mbox{\boldmath$k$}}_\|+{\mbox{\boldmath$p$}}_\|>\right|^2
\]
\begin{equation}
\times\frac{f_0(\varepsilon_{n_1,k_\|})-f_0(\varepsilon_{n_2,k_\|+p_\|})}{\varepsilon_{n_2,k_\|+p_\|}-\varepsilon_{n_1,k_\|}-\hbar(\omega+i0^+)}\ .
\label{e3.20}
\end{equation}
In Eq.\,(\ref{e3.20}), ${\cal A}$ is the normalization area for graphene,  ,
$\varepsilon_{\pm,k_\|}=\pm\sqrt{\hbar^2v_{F}^2k_\|^2+\varepsilon^2_{G}/4}$ are the
kinetic energies for the upper ($+$, electrons) and lower ($-$, holes) Dirac cones,
$v_{F}$ is the Fermi velocity of graphene electrons, $\varepsilon_{G}$ is the induced
energy gap of the graphene sheet, and $f_0(x)$ represents  the Fermi-Dirac distribution
function for thermal-equilibrium electrons. At very low temperatures, we have
$f_0(\varepsilon_{n,k_\|})\approx\Theta(E_{F}-\varepsilon_{n,k_\|})$, where
$E_{F}$ is the Fermi energy of doped electrons and $\Theta(x)$ is the unit step function.
\medskip

After a lengthy calculation, from Eq.\,(\ref{e3.20}) we obtain an analytic
expression for a gapped graphene sheet at $T\approx 0$\,K as follows\,\cite{add16}:

\begin{eqnarray}
&&\Pi^{(0)}_{\rm s}(p_\|,\omega)=\frac{2E_{F}}{\pi\hbar^2v_F^2}-
\frac{p_\|^2}{4\pi\hbar\sqrt{|v_F^2p_\|^2-\omega^2|}}
\nonumber\\
&\times& \left\{i\left[G_>(x_{1,-})-G_>(x_{1,+})\right]{\cal Q}_{1_<}(x_{2,-})+
\left[G_<(x_{1,-})+iG_>(x_{1,+})\right]{\cal Q}_{2_<}(x_{2,-},\,x_{2,+})\right.
\nonumber\\
&+& \left[G_<(x_{1,+})+G_<(x_{1,-})\right]{\cal Q}_{3_<}(x_{2,-})+
\left[G_<(x_{1,-})-G_<(x_{1,+})\right]{\cal Q}_{4_<}(x_{2,+})
\nonumber\\
&+& \left[G_>(x_{1,+})-G_>(x_{1,-})\right]{\cal Q}_{1_>}(x_{2,-},\,x_3)
+\left[G_>(x_{1,+})+iG_<(x_{1,-})\right]{\cal Q}_{2_>}(x_{2,-},\,x_{2,+})
\nonumber\\
&+& \left[G_>(x_{1,+})-G_>(-x_{1.-})-i\pi[2-x_0^2]\right]{\cal Q}_{3_>}(x_{2,+})
\nonumber\\
&+& \left[G_>(-x_{1,-})+G_>(x_{1,+})-i\pi[2-x_0^2]\right]{\cal Q}_{4_>}(x_{2,-},\,x_3)
\nonumber\\
&+&\left.\left[G_0(x_{1,+})-G_0(x_{1,-})\right]{\cal Q}_{5_>}(x_3)\right\}\ ,
\label{b2}
\end{eqnarray}
where $E_{F}=\sqrt{(\hbar v_{F}k_{F})^2+(\varepsilon_{G}/2)^2}-\varepsilon_{G}/2$ with
respect to the zero-energy point at $k_\|=0$, and $k_{F}=\sqrt{(E_{F}+
\varepsilon_{G}/2)^2-(\varepsilon_{G}/2)^2}/\hbar v_{F}$ is the Fermi wave number.
\medskip

In Eq.\,(\ref{b2}), we have introduced three self-defined functions which are given by

\begin{eqnarray}
&&G_<(x)=x\sqrt{x_0^2-x^2}-\left(2-x_0^2\right)\,\cos^{-1}\left(\frac{x}{x_0}\right)\ , \label{b3}\\
&&G_>(x)=x\sqrt{x^2-x_0^2}-\left(2-x_0^2\right)\,\cosh^{-1}\left(\frac{x}{x_0}\right)\ , \label{b4}\\
&&G_0(x)=x\sqrt{x^2-x_0^2}-\left(2-x_0^2\right)\,\sinh^{-1}\left(\frac{x}{\sqrt{-x_0^2}}\right)\ .
\label{b5}
\end{eqnarray}
Moreover, nine region functions employed in Eq.\,(\ref{b2}) are defined by

\begin{eqnarray}
&&{\cal Q}_{1_<}(x_{2,-})=\Theta(E_{F}-x_{2,-}-\hbar\omega)\ ,
\nonumber\\
&&{\cal Q}_{2_<}(x_{2,-},\,x_{2,+})=\Theta(-\hbar\omega-E_{F}+x_{2,-})\,\Theta(\hbar\omega+E_{F}-x_{2,-})\,\Theta(E_{F}+x_{2,+}-\hbar\omega)\ ,
\nonumber\\
&&{\cal Q}_{3_<}(x_{2,-})=\Theta(-E_{F}+x_{2,-}-\hbar\omega)\ ,
\nonumber\\
&&{\cal Q}_{4_<}(x_{2,+})=\Theta(\hbar\omega+E_{F}-x_{2,+})\,\Theta(\hbar v_{F}p_\|-\hbar\omega)\ ,
\nonumber\\
&&{\cal Q}_{1_>}(x_{2,-},\,x_3)=\Theta(2k_{F}-p_\|)\,\Theta(\hbar\omega-x_3)\,\Theta(E_{F}+x_{2,-}-\hbar\omega)\ ,
\nonumber\\
&&{\cal Q}_{2_>}(x_{2,-},\,x_{2,+})=\Theta(\hbar\omega-E_{F}-x_{2,-})\,\Theta(E_{F}+x_{2,+}-\hbar\omega)\ ,
\nonumber\\
&&{\cal Q}_{3_>}(x_{2,+})=\Theta(\hbar\omega-E_{F}-x_{2,+})\ ,
\nonumber\\
&&{\cal Q}_{4_>}(x_{2,-},\,x_3)=\Theta(p_\|-2k_{F})\,\Theta(\hbar\omega-x_3)\,\Theta(E_{F}+x_{2,-}-\hbar\omega)\ ,
\nonumber\\
&&{\cal Q}_{5_>}(x_3)=\Theta(\hbar\omega-\hbar v_{F}p_\|)\,\Theta(x_3-\hbar\omega)\ .
\end{eqnarray}
Finally, we have defined six
variables $x_0,\,x_{1,\pm},\,x_{2,\pm}$ and $x_3$ in region functions through

\begin{eqnarray}
&&x_0=\sqrt{1+\frac{\varepsilon_{G}^2}{\hbar^2v_{F}^2p_\|^2-\hbar^2\omega^2}}\ ,
\nonumber\\
&&x_{1,\pm}=\frac{2E_{F}\pm\hbar\omega}{\hbar v_{F}p_\|}\ ,
\nonumber\\
&&x_{2,\pm}=\sqrt{\hbar^2v_{F}^2(p_\|\pm k_{F})^2+\varepsilon_{G}^2/4}\ ,
\nonumber\\
&&x_3=\sqrt{\hbar^2v_{F}^2p_\|^2+\varepsilon_{G}^2}\ .
\end{eqnarray}
\medskip

For the gapless graphene sheet with $\varepsilon_{G}=0$, Eq.\,(\ref{b2}) reduces to\,\cite{zero}

\[
\Pi^{(0)}_{\rm s}(p_\|,\omega)=i\pi\,\frac{F(p_\|,\omega)}{\hbar^2v_{F}^2}+
\frac{2E_{F}}{\pi\hbar^2v_{F}^2}
-\frac{F(p_\|,\omega)}{\hbar^2v_{F}^2}\left\{G\left(\frac{\hbar
\omega+2E_{F}}{\hbar v_{F}p_\|}\right)-\Theta\left(\frac{2E_{F}-
\hbar\omega}{\hbar v_{F}p_\|}-1\right)\right.
\]
\begin{equation}
\left.\times\left[G\left(\frac{2E_{F}-\hbar\omega}{\hbar v_{F}p_\|}\right)
-i\pi\right]-\Theta\left(\frac{\hbar\omega-2E_{F}}{\hbar v_{F}p_\|}+1\right)
G\left(\frac{\hbar\omega-2E_{F}}{\hbar v_{F}p_\|}\right)\right\}\ ,
\label{b19}
\end{equation}
where another two self-defined functions are

\begin{equation}
F(p_\|,\omega)=\frac{1}{4\pi}\,\frac{\hbar v_{F}^2p_\|^2}{\sqrt{\omega^2-v_{F}^2p_\|^2}}\ ,
\label{b20}
\end{equation}

\begin{equation}
G(z)=z\sqrt{z^2-1}-\ln\left(z+\sqrt{z^2-1}\right)\ .
\label{b21}
\end{equation}

\section{Hybridized Modes for Double-Layer Graphene}
\label{sec-4}

We would like to emphasize that our model system, illustrated in Fig.\,\ref{f1},
consists of a semi-infinite conducting  substrate along with a dielectric material
with an embedded double-layer graphene above the conductor surface.
A surface-plasmon (SP) field is locally excited through a surface grating by normally-incident
light outside the graphene region. This surface-propagating SP field further excites
Dirac electrons in the off-surface coupled pair of graphene sheets. As a result, the induced
optical-polarization field from the excited Dirac electrons constitutes local resonant scattering
sources to the Maxwell equation for the propagating SP field in the system.\,\cite{ref1,add12}
\medskip

Making use of the Green's function
${\cal G}_{\mu\nu}({\mbox{\boldmath$r$}},{\mbox{\boldmath$r$}}^\prime;\omega)$ in
Eq.\,(\ref{e2.6}), we have converted the Maxwell equation for the electric field
$\mbox{\boldmath$E$}({\mbox{\boldmath$r$}};\omega)$ into a three-dimensional integral
equation, as presented by Eq.\,(\ref{e2.4}) in which
$\mbox{\boldmath$E$}^{\rm inc}({\mbox{\boldmath$r$}};\omega)$ represents the external
SP near field in the region defined as  $x_3>0$, given explicitly by\,\cite{ref1,new}

\begin{equation}
\mbox{\boldmath$E$}^{\rm inc}({\mbox{\boldmath$r$}};\omega)=E_0\,
e^{i{\bf q}_0(\omega)\cdot{\bf D}_0}\,\frac{c}{\omega}
\left[i\hat{\mbox{\boldmath$q$}}_0\beta_3(q_0,\omega)-
\hat{\mbox{\boldmath$x$}}_3q_0(\omega)\right]\,e^{i{\bf q}_0(\omega)
\cdot{\bf r}_\|}\,e^{-\beta_3(q_0,\,\omega)x_3}\ .
\label{e2}
\end{equation}
In Eq.\,(\ref{e2}), $\hat{\mbox{\boldmath$q$}}_0$ and $\hat{\mbox{\boldmath$x$}}_3$
are unit vectors along the directions of the in-plane SP wave vector
$\mbox{\boldmath$q$}_0(\omega)=q_0(\omega)(\cos\theta_0,\,\sin\theta_0)$
and $x_3$, $E_0$ is the field amplitude, $\omega$ is the field frequency,
$\theta_0$ is the angle of the incident SP field with respect to the
$x_1$ direction, $\mbox{\boldmath$D$}_0$ indicates the position vector
of the surface grating, and the introduced in-plane and out-of-plane wave
numbers are given, respectively, by

\begin{equation}
q_0(\omega)=\frac{\omega}{c}\sqrt{\frac{\epsilon_{d}\,
\epsilon_{M}(\omega)}{\epsilon_{d}+\epsilon_{M}(\omega)}}\ ,
\label{e3}
\end{equation}

\begin{equation}
\beta_3(q_0,\omega)=\sqrt{q^2_0(\omega)-\frac{\omega^2}{c^2}}\ ,
\label{e4}
\end{equation}
where ${\rm Re}[q_0(\omega)]\geq 0$ and ${\rm Re}[\beta_3(q_0,\omega)]\geq 0$.
As $q_0\rightarrow\infty$, from Eq.\,(\ref{e3}) we know that $\epsilon_{d}+
\epsilon_{M}(\omega)=0$, which gives rise to the uncoupled SP energy\,\cite{sp}
$\hbar\Omega_r=\hbar\Omega_p/\sqrt{\epsilon_s+\epsilon_d}$.
Therefore, Eq.\,(\ref{e2}) represents the SP-like near field in the limit of
$q_0\rightarrow\infty$, while it becomes a light-like radiation field in the limit
of $q_0\rightarrow 0$. The complex $\epsilon_{M}(\omega)$ in Eq.\,(\ref{e3}) implies
an in-plane propagation loss for the SP field.
\medskip

For the two-dimensional graphene sheets, we can simply write down
$\mbox{\boldmath${\cal P}$}^{\rm loc}({\mbox{\boldmath$r$}}^\prime;\omega)
=\sum\limits_{j=0,1}\,
\mbox{\boldmath${\cal P}$}^{\rm s}({\mbox{\boldmath$r$}}^\prime_\|;\omega
\vert z_j)\,\delta(x_3^\prime-z_j)$ with
$z_j$ labeling the positions of two graphene sheets in the $x_3$ direction. Therefore,
from Eq.\,(\ref{e2.4}) we obtain

\begin{equation}
E_{\mu}({\mbox{\boldmath$r$}}_\|;\omega\vert x_3)=
E^{\rm inc}_{\mu}({\mbox{\boldmath$r$}}_\|;\omega\vert x_3)+\frac{\omega^2}{\epsilon_0c^2}
\sum\limits_{\nu=1}^{3}\,\sum_{j=0}^{1}\int d^2{\mbox{\boldmath$r$}}_\|^\prime\,
{\cal G}_{\mu\nu}({\mbox{\boldmath$r$}}_\|,{\mbox{\boldmath$r$}}_\|^\prime;\omega
\vert x_3,z_j)\,{\cal P}_{\nu}^{\rm s}({\mbox{\boldmath$r$}}_\|^\prime;\omega\vert z_j)\ ,
\label{e5}
\end{equation}
where ${\cal G}_{\mu\nu}({\mbox{\boldmath$r$}}_\|,{\mbox{\boldmath$r$}}_\|^\prime;\omega\vert x_3,z_j)$ represents ${\cal G}_{\mu\nu}({\mbox{\boldmath$r$}},{\mbox{\boldmath$r$}}^\prime;\omega)$ evaluated at ${\mbox{\boldmath$r$}}=({\mbox{\boldmath$r$}}_\|,x_3)$ $\&$
${\mbox{\boldmath$r$}}^\prime=({\mbox{\boldmath$r$}}^\prime_\|,z_j)$, and $\mbox{\boldmath$E$}({\mbox{\boldmath$r$}}_\|;\omega\vert x_3)$ is simply $\mbox{\boldmath$E$}({\mbox{\boldmath$r$}};\omega)$ at
${\mbox{\boldmath$r$}}=({\mbox{\boldmath$r$}}_\|,x_3)$.
\medskip

After performing a Fourier transformation on the Green's function, as given by Eq.\,(\ref{e2.6}),
for the translationally invariant semi-infinite hybrid conductor system within
the $(x_1,x_2)$-plane, we can rewrite Eq.\,(\ref{e5}) as

\[
E_{\mu}({\mbox{\boldmath$r$}}_\|;\omega\vert x_3)=E^{\rm inc}_{\mu}({\mbox{\boldmath$r$}}_\|;\omega\vert x_3)
\]
\begin{equation}
+\frac{\omega^2}{\epsilon_0c^2}\sum\limits_{\nu=1}^{3}\,\sum_{j=0}^{1}
\int \frac{d^2{\mbox{\boldmath$q$}}_{\|}}{(2\pi)^2}\,
e^{i{\bf q}_\|\cdot{\bf r}_\|}\,g_{\mu\nu}({\mbox{\boldmath$q$}}_\|,\omega\vert x_3,z_j)\,{\cal P}_{\nu}^{\rm s}({\mbox{\boldmath$q$}}_\|,\omega\vert z_j)\ ,
\label{e7}
\end{equation}
where we have introduced the Fourier transformed polarization field

\begin{equation}
\mbox{\boldmath${\cal P}$}^{\rm s}({\mbox{\boldmath$q$}}_\|,\omega\vert z_j)=\int d^2{\mbox{\boldmath$r$}}^\prime_\|\,e^{-i{\bf q}_\|\cdot{\bf r}^\prime_\|}\,\mbox{\boldmath${\cal P}$}^{\rm s}({\mbox{\boldmath$r$}}^\prime_\|;\omega\vert z_j)\ .
\label{e8}
\end{equation}
Using linear response theory\,\cite{new,ref2} for translationally-invariant monolayer
graphene sheets within the $(x_1,x_2)$-plane, we obtain

\begin{equation}
{\cal P}_{\nu}^{\rm s}({\mbox{\boldmath$q$}}_\|,\omega\vert z_j)=\epsilon_0\chi^{(0)}_j(q_\|,\omega)\left(1-\delta_{\nu 3}\right)\int d^2{\mbox{\boldmath$r$}}^\prime_\|\,e^{-i{\bf q}_\|\cdot{\bf r}^\prime_\|}\,
E_{\nu}({\mbox{\boldmath$r$}}^\prime_\|;\omega\vert z_j)\ ,
\label{e9}
\end{equation}
where the optical polarization of graphene is limited within each
sheet, $\chi^{(0)}_j(q_\|,\omega)$ is the optical-response function for
Dirac electrons within the $j$th graphene sheet and is given,
under the condition $q_\|>\omega/c$, by Eqs.\,(\ref{b1}) and (\ref{e3.20}).
\medskip

Setting $x_3=z_j$ in Eq.\,(\ref{e7}) and meanwhile using Eq.\,(\ref{e9}),
we arrive at the following two coupled self-consistent equations for the total electric
field $\mbox{\boldmath$E$}({\mbox{\boldmath$q$}}_\|,\omega\vert z_j)$ on the graphene
sheets, i.e.,

\[
E_{\mu}({\mbox{\boldmath$q$}}_\|,\omega\vert z_j)=E^{\rm inc}_{\mu}({\mbox{\boldmath$q$}}_\|,\omega\vert z_j)
\]
\begin{equation}
+\frac{\omega^2}{c^2}\sum\limits_{\nu=1}^{3}\,\sum_{j'=0}^{1}
g_{\mu\nu}({\mbox{\boldmath$q$}}_\|,\omega\vert z_j,z_{j'})\left(1-\delta_{\nu 3}\right)\chi^{(0)}_{j'}(q_\|,\omega)\,E_{\nu}({\mbox{\boldmath$q$}}_\|,\omega\vert z_{j'})\ ,
\label{e12}
\end{equation}
where $j=0,\,1$ and we have used a Fourier transform for the electric field
$\mbox{\boldmath$E$}({\mbox{\boldmath$r$}}_\|;\omega\vert z_j)$ as

\begin{equation}
\mbox{\boldmath$E$}({\mbox{\boldmath$q$}}_\|,\omega\vert z_j)=\int d^2{\mbox{\boldmath$r$}}_\|\,e^{-i{\bf q}_\|\cdot{\bf r}_\|}\,\mbox{\boldmath$E$}({\mbox{\boldmath$r$}}_\|;\omega\vert z_j)\ .
\label{e13}
\end{equation}
Moreover, $\mbox{\boldmath$E$}^{\rm inc}({\mbox{\boldmath$q$}}_{\|},\omega\vert z_j)$
in Eq.\,(\ref{e12}) can be calculated directly from Eq.\,(\ref{e2}) as

\[
\mbox{\boldmath$E$}^{\rm inc}({\mbox{\boldmath$q$}}_{\|},\omega\vert z_j)=\delta({\mbox{\boldmath$q$}}_{\|}-{\mbox{\boldmath$q$}}_0)\,E_0\,e^{i{\bf q}_0\cdot{\bf D}_0}\,\frac{(2\pi)^2c}{\omega}
\left(i\hat{\mbox{\boldmath$q$}}_0\beta_3-\hat{\mbox{\boldmath$x$}}_3q_0\right)\,e^{-\beta_3z_j}
\]
\begin{equation}
\equiv(2\pi)^2\delta({\mbox{\boldmath$q$}}_{\|}-{\mbox{\boldmath$q$}}_0)\,\mbox{\boldmath$A$}({\mbox{\boldmath$q$}}_0,\omega\vert z_j)\,E_0\,e^{i{\bf q}_0\cdot{\bf D}_0}\ .
\label{e14}
\end{equation}
where $A_{\mu}({\mbox{\boldmath$q$}}_0,\omega\vert z_j)$ for $\mu=1,\,2,\,3$ represents
the field enhancement factors.
\medskip

If we set $\mbox{\boldmath$E$}^{\rm inc}({\mbox{\boldmath$q$}}_{\|},\omega\vert z_j)=0$
in Eq.\,(\ref{e12}), we are able to obtain the following dispersion equation
for the self-sustained density oscillations
within two graphene sheets, and the resulting dispersion relation
$\omega=\Omega_{\rm g-sp}({\mbox{\boldmath$q$}}_\|)$ for the hybrid
graphene-surface plasmon modes is determined by the following secular equation\,\cite{ref2-1}

\begin{equation}
{\cal D}et\left[\delta_{\mu\nu}\,\delta_{jj'}-\frac{\omega^2}{c^2}\,
g_{\mu\nu}({\mbox{\boldmath$q$}}_\|,\omega\vert z_j,z_{j'})\left(1-\delta_{\nu 3}\right)\chi^{(0)}_{j'}(q_\|,\omega)\right]\equiv{\cal D}et\left[\tilde{\cal C}^{jj'}_{\mu\nu}({\mbox{\boldmath$q$}}_\|,\omega)\right]=0\ ,
\label{e15}
\end{equation}
where $\mu,\,\nu=1,\,2,\,3$, $j,j'=0,\,1$, and the $2\times 2$ block (or $6\times 6$)
coefficient matrix $\tilde{\cal C}^{jj'}_{\mu\nu}({\mbox{\boldmath$q$}}_\|,\omega)$ is given by

\begin{equation}
{\footnotesize
\left[\begin{array}{cc}
\displaystyle{\delta_{\mu\nu}-\frac{\omega^2}{c^2}g_{\mu\nu}({\mbox{\boldmath$q$}}_\|,\omega\vert z_0,z_0)\left(1-\delta_{\nu 3}\right)\chi^{(0)}_{0}(q_\|,\omega)} & \displaystyle{-\frac{\omega^2}{c^2}
g_{\mu\nu}({\mbox{\boldmath$q$}}_\|,\omega\vert z_0,z_1)\left(1-\delta_{\nu 3}\right)\chi^{(0)}_{1}(q_\|,\omega)}\\
\displaystyle{-\frac{\omega^2}{c^2}g_{\mu\nu}({\mbox{\boldmath$q$}}_\|,\omega\vert z_1,z_0)\left(1-\delta_{\nu 3}\right)\chi^{(0)}_{0}(q_\|,\omega)} & \displaystyle{\delta_{\mu\nu}-\frac{\omega^2}{c^2}
g_{\mu\nu}({\mbox{\boldmath$q$}}_\|,\omega\vert z_1,z_1)\left(1-\delta_{\nu 3}\right)\chi^{(0)}_{1}(q_\|,\omega)}
\end{array}\right]}\ .
\label{e16}
\end{equation}
The $z_j$ position dependence in Eq.\,(\ref{e15}) reflects the distinctive near-field
coupling\,\cite{ref1,new} between the surface plasmons and Dirac electrons in graphene.
Here, the factor $g_{\mu\nu}({\mbox{\boldmath$q$}}_\|,\omega\vert z_j,z_{j'})$ comes
from the surface-plasmon response, while the other factor $\chi^{(0)}_{j}(q_\|,\omega)$
corresponds to the graphene optical response. Therefore, their product in
Eq.\,(\ref{e15}) represents contributions to the hybrid graphene-surface plasmon modes.
The uncoupled surface-plasmon dispersion relation is included through
$g_{\mu\nu}({\mbox{\boldmath$q$}}_\|,\omega\vert z_j,z_j)$ in an inexplicit way
for $q_\|\rightarrow\infty$. Moreover, the interlayer graphene coupling becomes
insignificant as $q_\||z_1-z_0|\gg 1$. Graphically, the dispersion relation of such
hybrid graphene-surface plasmon modes can be shown with the sign switching in the
density plot for the real part of
$1/{\cal D}et\left[\tilde{\cal C}^{jj'}_{\mu\nu}({\mbox{\boldmath$q$}}_\|,\omega)\right]$ within the ($\omega,q_\|$)-plane.
\medskip

By using the $6\times 6$ inverted coefficient matrix
$\{\tilde{\cal C}^{jj'}_{\mu\nu}({\mbox{\boldmath$q$}}_\|,\omega)\}^{-1}$
calculated from Eq.\,(\ref{e16}), we can further compute the distribution of
the total electric field (i.e., incident field plus the scattering field) by

\[
E_{\mu}({\mbox{\boldmath$q$}}_\|,\omega\vert x_3)=E^{\rm inc}_{\mu}({\mbox{\boldmath$q$}}_\|,\omega\vert x_3)+\frac{\omega^2}{c^2}\sum_{j=0}^{1}\,\chi^{(0)}_{j}(q_\|,\omega)
\]
\begin{equation}
\times\sum\limits_{\nu=1}^{3}\,\left\{g_{\mu\nu}({\mbox{\boldmath$q$}}_\|,\omega\vert x_3,z_j)\left(1-\delta_{\nu 3}\right)
\left[\sum_{\mu^\prime=1}^{3}\,\sum_{j'=0}^{1}\,\{\tilde{\cal C}^{jj'}_{\nu\mu^\prime}({\mbox{\boldmath$q$}}_\|,\omega)\}^{-1}\,E^{\rm inc}_{\mu^\prime}({\mbox{\boldmath$q$}}_\|,\omega\vert z_{j'})\right]\right\}\ ,
\label{e17}
\end{equation}
where the second term represents the contribution from the scattering field\,\cite{new}.
\medskip

It is clear from Eq.\,(\ref{e17}) that in the absence of a graphene sheet,
i.e. $\chi^{(0)}_{j}(q_\|,\omega)=0$, the semi-infinite dielectric (with a
relative dielectric constant $\epsilon_{d}$) is static, uniform and isotropic in
the upper half space with an effective scattering matrix\,\cite{new}
$\alpha_{\mu\nu}^{\rm eff}({\mbox{\boldmath$q$}}_\|,\omega\vert x_3,x_3^\prime)=0$.
In the presence of the graphene sheets, on the other hand, the induced local polarization
fields from the excited Dirac  electrons within the graphene sheets are introduced.
In addition, the retarded coulomb coupling between plasmon excitations in the graphene
and in the semi-infinite conductor is also introduced into the system at the same time.
Consequently, the effective scattering matrix
$\alpha_{\mu\nu}^{\rm eff}({\mbox{\boldmath$q$}}_\|,\omega\vert x_3,x_3^\prime)$
becomes finite, dynamical, non-uniform and anisotropic in the space, and is
calculated from Eq.\,(\ref{e17}) as

\begin{eqnarray}
&&\alpha_{\mu\nu}^{\rm eff}({\mbox{\boldmath$q$}}_\|,\omega\vert x_3,x_3^\prime)\equiv
\frac{\partial}{\partial E^{\rm inc}_{\nu}({\mbox{\boldmath$q$}}_\|,\omega\vert x'_3)}
\left[E_{\mu}({\mbox{\boldmath$q$}}_\|,\omega\vert x_3)-E^{\rm inc}_{\mu}({\mbox{\boldmath$q$}}_\|,\omega\vert x_3)\right]
\nonumber
\\
&=&\frac{\omega^2}{c^2}\,\sum_{j'=0}^{1}\,\delta(x_3^\prime-z_{j'})\,\sum_{j=0}^{1}\,
\chi^{(0)}_{j}(q_\|,\omega)\sum\limits_{\nu^\prime=1}^{3}\,g_{\mu\nu^\prime}({\mbox{\boldmath$q$}}_\|,\omega\vert x_3,z_j)\left(1-\delta_{\nu^\prime 3}\right)
\{\tilde{\cal C}^{jj'}_{\nu^\prime\nu}({\mbox{\boldmath$q$}}_\|,\omega)\}^{-1}\ .\ \ \ \
\label{eff-1}
\end{eqnarray}
In Eq.\,(\ref{eff-1}), ${\mbox{\boldmath$q$}}_\|$ is a real vector, the single
factor $\chi^{(0)}_{j}(q_\|,\omega)$
represents the contribution from the resonant excitation of Dirac electrons within
the graphene sheet, while the combined factor
$g_{\mu\nu^\prime}({\mbox{\boldmath$q$}}_\|,\omega\vert x_3,z_j)\left(1-\delta_{\nu^\prime 3}\right)\{\tilde{\cal C}^{jj'}_{\nu^\prime\nu}({\mbox{\boldmath$q$}}_\|,\omega)\}^{-1}$
corresponds to the electromagnetic coupling between the semi-infinite conductor and the
graphene sheet. Using Eq.\,(\ref{eff-1}), we can define a local effective scattering matrix
through

\begin{eqnarray}
&&\alpha_{\mu\nu}^{\rm eff}({\mbox{\boldmath$q$}}_\|,\omega\vert x_3)=\int\limits_{0}^{\infty} dx_3^\prime\,\alpha_{\mu\nu}^{\rm eff}({\mbox{\boldmath$q$}}_\|,\omega\vert x_3,x_3^\prime)
\nonumber
\\
&=&\frac{\omega^2}{c^2}\,\sum_{j=0}^{1}\,
\chi^{(0)}_{j}(q_\|,\omega)\sum\limits_{\nu^\prime=1}^{3}\,g_{\mu\nu^\prime}({\mbox{\boldmath$q$}}_\|,\omega\vert x_3,z_j)\left(1-\delta_{\nu^\prime 3}\right)\,
\sum_{j'=0}^{1}\,\{\tilde{\cal C}^{jj'}_{\nu^\prime\nu}({\mbox{\boldmath$q$}}_\|,\omega)\}^{-1}\ .\ \ \ \
\label{eff-2}
\end{eqnarray}
which displays two peaks at $x_3=z_0,\,z_1$, and the broadening of the peak is determined
by the exponential decay\,\cite{add11,add15} of the Green's function.
This implies that the dielectric constant in the region
between graphene sheets and the surface of the semi-infinite conductor will be
modified significantly only if the graphene sheets stay very close to the surface
of the semi-infinite conductor, i.e., the SP wavelength is required to be larger
than the sheet separation from the surface.
\medskip

Using Eq.\,(\ref{e17}), we also get the total electric field in the real space, yielding

\[
E_{\mu}({\mbox{\boldmath$r$}}_\|,\omega\vert x_3)=E^{\rm inc}_{\mu}({\mbox{\boldmath$r$}}_\|,\omega\vert x_3)+\frac{\omega^2}{c^2}\int \frac{d^2{\mbox{\boldmath$q$}}_\|}{(2\pi)^2}\,
e^{i{\bf q}_\|\cdot{\bf r}_\|}\sum_{j=0}^{1}\,\chi^{(0)}_{j}(q_\|,\omega)
\]
\begin{equation}
\times\sum\limits_{\nu=1}^{3}\,\left\{g_{\mu\nu}({\mbox{\boldmath$q$}}_\|,\omega\vert x_3,z_j)\left(1-\delta_{\nu 3}\right)
\left[\sum_{\mu^\prime=1}^{3}\,\sum_{j'=0}^{1}\,\{\tilde{\cal C}^{jj'}_{\nu\mu^\prime}({\mbox{\boldmath$q$}}_\|,\omega)\}^{-1}\,E^{\rm inc}_{\mu^{\prime}}({\mbox{\boldmath$q$}}_\|,\omega\vert z_{j'})\right]\right\}\ ,
\label{e17r}
\end{equation}
where ${\mbox{\boldmath$E$}}^{\rm inc}({\mbox{\boldmath$r$}}_\|,\omega\vert x_3)$
has already been given by Eq.\,(\ref{e2}), and
${\mbox{\boldmath$E$}}^s({\mbox{\boldmath$r$}}_\|,x_3\vert\omega)={\mbox{\boldmath$E$}}({\mbox{\boldmath$r$}}_\|,\omega\vert x_3)-{\mbox{\boldmath$E$}}^{\rm inc}({\mbox{\boldmath$r$}}_\|,\omega\vert x_3)$
stands for the spatial distribution of the scattering field.
\medskip

Furthermore, by employing the calculated electric field
$\mbox{\boldmath$E$}({\mbox{\boldmath$q$}}_\|,\omega\vert z_j)$
on the graphene sheets from Eq.\,(\ref{e12}), the optical-absorption coefficient
$\beta_{\rm abs}(\omega)$ for the SP field by Dirac electrons is expressed
as\,\cite{abs,lorentz,abs2,abs3}

\begin{equation}
\beta_{\rm abs}(\omega)=\frac{\omega\sqrt{\epsilon_{d}}}{n_r(\omega)\,c}\left[\frac{1}{\exp(\hbar\omega/k_{\rm
B}T)-1}+1\right]\,{\rm Im}\left[\alpha_{L}(\omega)\right]\ ,
\label{e18}
\end{equation}
where $\alpha_{L}(\omega)$ is the complex Lorentz function given by

\[
\alpha_{L}(\omega)=\left(\frac{2\pi e^2}{\epsilon_0\epsilon_{r}q_0^2}\right)\left(q_0^2-\epsilon_{d}\,\frac{\omega^2}{c^2}\right)^{1/2}\,\sum_{j=0}^{1}\,
\left[\Pi^{(0)}_{j}(q_0,\,\omega)+\{\Pi^{(0)}_{j}(q_0,\,-\omega)\}^\ast\right]
\]
\begin{equation}
\times\left|\sum_{\mu=1}^3\,\hat{e}_{\mu}\,\sum_{\nu=1}^3\,\sum_{j'=0}^{1}\,
\{\tilde{\cal C}^{jj'}_{\mu\nu}({\mbox{\boldmath$q$}}_0,\omega)\}^{-1}A_{\nu}({\mbox{\boldmath$q$}}_0,\omega\vert z_{j'})\right|\ ,
\label{e19}
\end{equation}
$\epsilon_r$ is the average dielectric constant of graphene embedded in the dielectric host.
In Eq.\,(\ref{e19}), ${\mbox{\boldmath$q$}}_0\equiv{\rm Re}[q_0(\omega)]\,\hat{\mbox{\boldmath$q$}}_0$ is a real vector,
$\hat{\mbox{\boldmath$e$}}^{\rm inc}=(\hat{e}_1,\hat{e}_2,\hat{e}_3)$ represents the unit polarization vector for the propagating SP field,
and the scaled refractive index function $n_r(\omega)$
in Eq.\,(\ref{e18}) is\,\cite{abs,lorentz,abs2,abs3}

\begin{equation}
n_r(\omega)=
\frac{1}{\sqrt{2}}\left(1+{\rm Re}\left[\alpha_{L}(\omega)\right]
+\sqrt{\left\{1+{\rm Re}\left[\alpha_{L}(\omega)\right]\right\}^2+\left\{{\rm Im}\left[\alpha_{L}
(\omega)\right]\right\}^2}\,\right)^{1/2}\ .
\label{e20}
\end{equation}
We now turn to presenting and discussing our numerical results in the next section.

\section{Results and Discussions}
\label{sec-5}

In our numerical calculations,  we use the Fermi wave vector $k_{F}=\sqrt{\pi n_0}$
as the scale for wave numbers, $1/k_{F}$ for lengths, and $E_{F}=\hbar v_{F}k_{F}$ for
energies. The direction of propagation of the SP field is chosen as
$\hat{\mbox{\boldmath$q$}}_0=(1,0,0)$ for convenience, and we also
set $\epsilon_{s}=13.3$, $\epsilon_{d}=\epsilon_{r}=2.4$,
$v_{F}=1\times 10^8\,$cm/s, and $n_0=5\times 10^{11}\,$cm$^{-2}$ for the
doping density in graphene. Moreover, the SP energy $\hbar\Omega_r$ and the half
bandgap $\Delta=\varepsilon_G/2$ will be given directly in figure captions.
\medskip

\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{FIG3}
\caption{(Color online)    Results for hybrid plasmon modes in different ranges of
wavenumber. Density plots are presented for the real part of
$\mathcal{D}^{-1}({\mbox{\boldmath$q$}}_\|, \omega \,
\vert \, z_0, z_1)= 1/{\cal D}et\left[\tilde{\cal C}^{jj'}_{\mu\nu}(q_x,\omega)\right]$
using Eq.\,(\ref{e16})  with $q_x^{\rm max}/k_{F} = 0.03,\,0.075,\,2.5$ and
$6.0$. The hybrid-plasmon dispersions initially appear
as strong variations between positive (red) and negative (violet)
peaks. The layer separations from the conductor surface are $z_0=0.01\,k_{F}^{-1}$ and
$z_1=0.011\,k_{F}^{-1}$. The surface-plasmon energy is equal to $\hbar\Omega_r =1.78\,E_{F,1}$.
Each graphene sheet is equally doped up to a Fermi energy $E_{F,1}=E_{F,2}=E_F$ and
acquires a zero bandgap, i.e.,  $\Delta_1=\Delta_2=0$.}
\label{f3}
\end{figure}
For a retarded interaction between light and graphene electrons, both radiative and
evanescent modes must be considered in the hybrid system. The radiative modes include
photons and polaritons, while the evanescent (localized) modes appear as surface-plasmon
polaritons (SPPs), graphene plasmons (G-Ps), and surface plasmons (SPs).
Figure\ \ref{f3} displays the real part of
$\mathcal{D}^{-1}({\bf q}_\parallel, \omega \, \vert \, z_1, z_2)
= 1/{\cal D}et\left[\tilde{\cal C}^{jj'}_{\mu\nu}(q_x,\omega)\right]$
for four different ranges of $q_x$. As Fig.\,\ref{f3}($a$) shows, in addition to the SPP mode,
the hybridizations of both radiative photon and polariton modes with localized
SPs (illustrated in Fig.\,\ref{f2}) appear in this very small $q_x$ range.
As the $q_x$ range slightly expands in Fig.\,\ref{f3}($b$), the SPP mode in
Fig.\,\ref{f3}($a$) is fully developed, which is accompanied by
two degenerate acoustic-like G-P modes at very low energies. As the $q_x$ range
further increases in Figs.\,\ref{f3}($c$) and \ref{f3}($d$), the G-P energy exceeds
that of the SPP. Consequently, a single anticrossing of the G-P with SPP appears.
\medskip

\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{FIG4}
\caption{(Color online)  Density plots for the real part of
$\mathcal{D}^{-1}({\mbox{\boldmath$q$}}_\parallel, \omega \, \vert \, z_0, z_1)$
with $q_x^{\rm max}/k_{F} = 0.03,\,0.075,\,2.5$ and $6.0$.  We set $z_1=0.01\,k_{F}^{-1}$,
$z_2=0.011\,k_{F}^{-1}$ and $\hbar \Omega_r = 1.78\,E_{F}$.
Each graphene sheet has the same Fermi energy $E_{F,1}=E_{F,2}=E_F$ but has
different bandgaps $\Delta_1=0$ and
$\Delta_2 = 0.5\,E_{F}$.}
\label{f4}
\end{figure}
In Fig.\,\ref{f4}, a finite bandgap  parameter $\Delta_2=0.5\,E_F$ is introduced to the top graphene
layer, and two G-P modes become non-degenerate. By comparing with Fig.\,\ref{f3},
no changes in Figs.\,\ref{f4}($a$) and \ref{f4}($b$) are found for photon and polariton
modes in smaller ranges of $q_x$. However,
as the $q_x$ range is increased in Figs.\,\ref{f4}($c$) and \ref{f4}($d$),
the splitting of the two acoustic-like G-P modes and two optical-like SPP modes become
visible in Fig.\,\ref{f4}($c$), where the lower (higher) energy G-P mode is associated
with the top (bottom) layer. Moreover, the top-layer G-P mode after the second
anticrossing in Fig.\,\ref{f4}($d$) is enhanced by reducing the Landau damping due
to a finite bandgap.
\medskip

\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{FIG5}
\caption{(Color online)  Density plots for the real part of
$\mathcal{D}^{-1}({\mbox{\boldmath$q$}}_\parallel, \omega \, \vert \, z_0, z_1)$
with $q_x^{\rm max}/k_{F} = 0.03,\,0.075,\,2.5$ and $6.0$. We chose $z_1=0.01\,k_{F}^{-1}$,
$z_2=0.011\,k_{F}^{-1}$ and $\hbar \Omega_r = 1.78\,E_{F}$.
Each graphene layer has zero bandgap, i.e.,  $\Delta_1=\Delta_2=0$ but has different
Fermi energies $E_{F,1}=E_F$ and $E_{F,2}=2E_F$.}
\label{f5}
\end{figure}
In Fig.\,\ref{f5}, doping in the top layer is increased, thereby leading to two
non-degenerate acoustic-like G-P modes, where the top layer has a higher G-P energy.
Compared with Fig.\,\ref{f3}, we find small but visible change in
Figs.\,\ref{f5}($a$) and \ref{f5}($b$) for the optical-like SPP mode in
shorter ranges for  $q_x$ since the graphene Fermi velocity is independent of doping.
For larger $q_x$ ranges in Figs.\,\ref{f5}($c$) and \ref{f5}($d$), the anticrossing
gap is greatly increased due to  an enhanced retarded Coulomb interaction between the graphene
layers and the conducting substrate for higher doping in the top layer. Meanwhile,
the energy of the SPP mode is pushed up  significantly, which is attributed to the increased
slope of the optical-like SPP mode by higher doping in the top layer.
However, the anticrossing is still dominated by the bottom layer G-P mode since the
Landau damping of the top layer G-P mode becomes large due to its higher doping level.
as shown in Fig.\,\ref{f5}($d$).
\medskip

\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{FIG6}
\caption{(Color online)   Density plots for the real part of
$\mathcal{D}^{-1}({\mbox{\boldmath$q$}}_\parallel, \omega \, \vert \, z_0, z_1)$
with $q_x^{\rm max}/k_{F} = 0.03,\,0.075,\,2.5$ and $6.0$. We chose $z_1=0.01\,k_{F}^{-1}$,
$z_2=0.011\,k_{F}^{-1}$ and $\hbar \Omega_r = 1.78\,E_{F}$.
Each graphene layer has a different Fermi energy given by
 $E_{F,1}=E_F$ and $E_{F,2}=2E_F$ as well as
different bandgaps $\Delta_1=0$ and $\Delta_2=E_{F}$.}
\label{f6}
\end{figure}
The splitting of G-P modes in Fig.\,\ref{f4} with $\Delta_2=E_F/2$ becomes much more clear in Fig.\,\ref{f6} after we bring into a bandgap $\Delta_2=E_F$ to the top graphene layer in Fig.\,\ref{f5}.
We find no changes about photon and polariton modes in Figs.\,\ref{f6}($a$) and \ref{f6}($b$) for smaller $q_x$ ranges in comparison with Fig.\,\ref{f5}.
On the other hand, as the $q_x$ range is enlarged in Figs.\,\ref{f6}($c$) and \ref{f6}($d$), two non-degenerate acoustic-like G-P modes occur clearly in the anticrossing region, similar to Figs.\,\ref{f4}($c$) and \ref{f4}($d$).
Here, the increased doping level in the top layer pushes up the energy of optical-like SPP mode and expands the anticrossing gap, while the increased bandgap of the same layer splits the acoustic-like G-P mode into two
at the same time, in comparisons with Figs.\,\ref{f4} and \ref{f5}, respectively.
As a result, two successive plasmon-mode hybridizations can been see very clearly in Figs.\,\ref{f6}($c$) within the anticrossing region.
\medskip

\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{FIG7}
\caption{(Color online)   Optical absorption spectra $\beta(\omega)$ (in units of
$k_{F,1}$) as a function of scaled photon energy $\hbar\omega/E_{F,1}$.
Panels $(a)$ and $(b)$ are for two graphene layers at $z_0=0.01\,k^{-1}_{F}$, $z_1=1.1\,z_0$
and having $E_{F,1}=E_{F,2}=E_F$ (all but ($e$)),
$\Delta_1=\Delta_2=\Delta$ (all but ($f$)) with
different SP energies $\hbar\Omega_r/E_F =0.25,\,1.78$, respectively.
The red, black, blue and green curves correspond to
$\Delta/E_{F}=0,\,0.3,\,0.5,\,0.7$, respectively.
Inset $(i1)$ shows the dependence of $\beta(\omega)$ on $z_0$ for
$\omega/\Omega_r= 0.4,\,0.5,\,0.6,\,0.7$.
All the other panels have $\hbar\Omega_r = 1.78\,E_{F,1}$.
Panel $(c)$ shows the red, black, blue, green and orange curves for
$z_0/k^{-1}_{F}= 1,\,5,\,10,\,25,\,50$ with $z_1=2\,z_0$ and $\Delta=0$.
Plot $(d)$ displays the red, black, blue, green and orange curves for
$z_1/z_0=1.1,\,10,\,50,\,100,\,200$ with $z_0=k^{-1}_{F}$ and $\Delta=0$.
Panel $(e)$ presents the red, black, blue and green curves for
$E_{F,2}/E_{F,1}=1,\,1.5,\,2$ and
$2.5$ with $\Delta/E_{F,1}=0.3$, $z_0 = 0.01\,k_{F,1}^{-1}$ and $z_2 = 1.1\,z_0$.
Plot $(f)$ displays the red, black, blue and green curves relate for
$\Delta_2/E_F=0,\,0.3,\,0.5,\,0.7$ with $z_0 = 0.01\,k_{F}^{-1}$, $z_1
= 1.1\,z_0$ and $\Delta_1=0$.}
\label{f7}
\end{figure}
The $z_j$ dependence in the secular equation
${\cal D}et\left[\tilde{\cal C}^{jj'}_{\mu\nu}(q_x,\omega)\right]=0$
reflects the distinctive evanescent coupling between SPs and G-Ps.
By moving the graphene sheet a bit further away from the surface of the conductor
(increasing $z_j$), the anticrossing gap will shrink due to
decreased retarded coupling between them. Meanwhile, the strengths of all
the plasmon, polariton and photon modes will increase due to loss suppression
of these modes to the conducting substrate. The incident SP field suffers not
only Ohmic loss during its propagation along the conductor surface, but also
absorption loss by its coupling to G-Ps.
\medskip

Figures\ \ref{f7}($a$) and \ref{f7}($b$) compare the absorption spectra
$\beta_{\rm abs}(\omega)$ for $\hbar\Omega_r/E_F=0.25$ and $1.78$. When
$\hbar\Omega_r$ is increased, the decay length of the SPP field becomes shorter,
as indicated by the inset of Fig.\,\ref{f7}($b$). Consequently, the SPP field
will concentrate more within the region close to the conducting surface, and the
overall absorptions of various plasmon modes look much stronger in
Fig.\,\ref{f7}($b$). Here, the highest sharp peak in Fig.\,\ref{f7}($b$) is
associated with the optical-like G-P mode which is hybridized with the SP mode.
The deep trough on its left-hand side results from the anticrossing gap.
Another peak with $\Delta\neq 0$ below this trough is attributed to the
acoustic-like G-P mode which is accompanied by a hybrid SP peak on its right-hand
side for $\Delta=0$ case. Finally, the rounded shoulder below this acoustic-like
G-P peak comes from the SPP mode. As the bandgap $\Delta$ is increased, the hybrid
SP peak is quickly suppressed, and  the acoustic-like G-P peak slightly moves
down in energy from the trough side. Meanwhile, the SPP round peak is also
reduced with increasing bandgap $\Delta$.
\medskip

When both graphene layers are moved further  away from the surface of the conductor
in Fig.\,\ref{f7}($c$),  there is little change in the highest optical-like
G-P peak. On the other hand, the
acoustic-like G-P peak is completely suppressed for a larger layer separation from the surface,
leading to a single sharp hybrid SP peak below the trough. This is further accompanied by
the dramatic reduction of the SPP round peak. If only the interlayer separation is increased
but the bottom layer is fixed, we find from Fig.\,\ref{f7}($d$) a very similar effect as
that in Fig.\,\ref{f7}($c$). However, unresolved weak absorption from the bottom graphene
layer still exists in this case.-
\medskip

In Fig.\,\ref{f7}($e$), we compare our results for double gapped-graphene layers having
different doping levels in the top layer. The increased doping in the upper layer
has no effect on the highest optical-like G-P peak and trough. Although the hybrid SP peak
is suppressed by increasing the doping, the acoustic-like G-P peak is enhanced. More importantly,
the rounded SPP peak increases greatly in this case by a large retarded Coulomb coupling to
the conducting surface due to a higher doping level. If only the bandgap of the top layer is
increased from zero, while that of the bottom layer is kept zero, we find a similar effect
in Fig.\,\ref{f7}($f$) in comparison to that in Fig.\,\ref{f7}($b$), where the bandgaps of
both layers are the same and increased from zero.
\medskip

\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{FIG8}
\caption{(Color online)  3D plots for
$[\text{Re}\{\alpha^{\rm eff}_{11} (q_x, \omega \, \vert \, x_3)\}]^{1/5}$
with $ \omega /\Omega_r= 0.7\, (a)$, $0.8\,(b)$, $0.9\,(c)$ and $1.0\,(d)$,
where $E_{F,1}=E_{F,2}=E_F$, $\Delta_1=\Delta_2=0$, $z_0 = 5.0\,k^{-1}_{F}$, $z_1 = 1.2\,z_0$,
and $\hbar \Omega_r / E_F = 1.78$.}
\label{f8}
\end{figure}
In addition to  optical absorption by the G-Ps, SPs and SPPs in Fig.\,\ref{f7},
resonant scattering of the SP from double-layer G-Ps also appears, as given by
Eq.\,(\ref{eff-2}). Figure\ \ref{f8} presents 3D plots for
$[{\rm Re}\{\alpha^{\rm eff}_{11}(q_x,\omega\vert x_3)\}]^{1/5}$ with four
$\omega$ values, where the two graphene sheets are placed relatively close to the
surface. The scattering matrix is defined by $\alpha^{\rm eff}_{\mu\nu}\equiv\
delta(E_{\mu}-{\cal E}_{\mu}^{\rm inc})/\delta{\cal E}^{\rm inc}_{\nu}$, and therefore,
its signs correspond to an enhanced ($+$) or weakened ($-$) SPP field after the scattering
with G-Ps. If both $q_x$ and $x_3$ are sufficiently large, such scattering is
significantly  suppressed,  leaving only a wide and flat basin in the upper-right corners of
Figs.\,\ref{f8}($a$)-\ref{f8}($d$). If $q_x$ is very small, the photon and SPP
 radiative modes dominate, and then,
${\rm Re}\{\alpha^{\rm eff}_{11}(q_x,\omega\vert x_3)\}$ remains negative and
becomes independent of $x_3$.When $q_x$ is intermediate, the SPP evanescent modes
start entering in with increasing $\omega$ up to $\Omega_{r}$.
In this case, the positive-peak strength is reduced and its peak coverage is squeezed
into a smaller $x_3$ region where the localization of the SPP field is still insignificant.
In addition, the positive peak is broken into two islands in Fig.\,\ref{f8}($c$), and it
switches to a negative peak followed by a negative constant in Fig.\,\ref{f8}($d$).
On the other hand, when $q_x$ becomes very large for a strongly-localized SPP field,
its scattering by double-layer G-Ps becomes very small except for the resonance region
very close to the surface as shown by the sharp negative edges in the lower-right corners
of Figs.\,\ref{f8}($a$)-\ref{f8}($d$).
With increasing $\omega/\Omega_r$ in Fig.\,\ref{f7}($b$)-\ref{f7}($d$), it is
interesting to note that this deep negative edge is pushed up to a large $q_x$ region
due to SP resonance, and the $V$-shape feature at $q_x/k_F=2$
is sharpened simultaneously due to enhanced localization of the SPP field.
\medskip

\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{FIG9}
\caption{(Color online)  Complex amplitudes $\tilde{E}_1^s(\omega\vert z_0)/E_0$ of
a scattering field, calculated from Eq.\,(\ref{e17r}), as functions of incident
photon energy $\hbar\omega/E_F$ for $x_3=z_0$, $\hbar\Omega_r/E_F= 1.78$ [($a$), ($b$)]
and $\hbar\Omega_r/E_F= 0.25$ [($c$), ($d$)]. The real [($a$), ($c$)] and
imaginary [($b$), ($d$)] parts of this complex amplitude are displayed
for $\Delta/E_F=0$ and $0.8$ for each panel.  Here, we write the scattering field
as: $\left.E^s_1(x_1,x_3\vert\omega)\right|_{x_3=z_0}\equiv\tilde{E}^s_1(\omega\vert z_0)\,\exp(i{\rm Re}[q_0(\omega)]x_1)$.
Moreover, we assume $E_{F,1}=E_{F,2}=E_F$, $\Delta_1=\Delta_2=\Delta$, $z_0 = 0.01\,k^{-1}_{F}$, and $z_1 = 200\,z_0$.}
\label{f9}
\end{figure}
In Figs.\,\ref{f9}($a$)-\ref{f9}($d$), we display the real and imaginary parts
of calculated complex amplitudes $\tilde{E}_1^s(\omega\vert z_0)/E_0$ from Eq.\,(\ref{e17r})
as functions of $\hbar\omega/E_F$. When $\hbar\Omega_r/E_F=1.78$ in \ref{f9}($a$), we
find a broad peak at $\omega=\Omega_r$ for the real part of $\tilde{E}_1^s(\omega\vert z_0)$
due to field scattering by the lower-energy SPP branch, which is further accompanied by a
very (weak) strong plasmon resonance associated with field scattering by the higher-energy
(acoustic-like) optical-like G-P branch. Similar peak and dual-plasmon-resonance features
are also observed for the imaginary part of $\tilde{E}_1^s(\omega\vert z_0)$ in \ref{f9}($b$)
but with an out-of-phase plasmon resonance for the optical-like G-Ps.
Moreover, these unique scattering features in Figs.\,\ref{f9}($a$) and \ref{f9}($b$)
are enhanced significantly with increased graphene bandgap $\Delta$.
\medskip

Clearly, the peak and dual-plasmon-resonance features observed in
Figs.\,\ref{f9}($a$) and \ref{f9}($b$) can be very well correlated to the
absorption peaks in Fig.\,\ref{f7}. When the value of $\hbar\Omega_r/E_F$ is
reduced from $1.78$ to $0.25$ in Figs.\,\ref{f9}($c$) and \ref{f9}($d$), on the
other hand, the lowest broad SPP peak is greatly reduced for both real and
imaginary parts of the complex amplitude $\tilde{E}_1^s(\omega\vert z_0)$.
Meanwhile, the dual-plasmon-resonance (anticrossing-gap) region is shrunk dramatically,
although the double scattering peaks by optical-like (right) and acoustic-like
(left) G-P branches are still clearly visible. Furthermore, the increasing
graphene bandgap $\Delta$ suppresses the SPP peak while it slightly enhances the
dual-plasmon-resonance peaks at the same time. It is interesting to point out that the real
and imaginary parts of the complex amplitude $\tilde{E}_1^s(\omega\vert z_0)$ in this
case becomes in-phase in \ref{f9}($c$) and \ref{f9}($d$) for the
right-most plasmon resonance associated with the optical-like G-P branch.

\section{Conclusions and Remarks}
\label{sec-6}

The effect of electron back action on the hybridization of radiative and evanescent
fields has been demonstrated by using a retarded interaction,
which is seen as hybrid dispersions for both radiative (small $q_x$ range) and
evanescent (large $q_x$ range) field modes. Instead of a reaction force in Newtonian
mechanics, the back action in this electro-optical study is an induced polarization
field from the double-layer Dirac plasmons, which redistributes an incident
surface-plasmon-polariton field by resonant scattering. The localization
characteristics of such a retarded interaction ensures high sensitivity
to dielectric environments surrounding and between the graphene sheets,
including variations in the conducting substrate, cladding layer,
electronic properties of embedded graphene by a bandgap, as well as the graphene
distance from the surface of the conductor. This provides a unique advantage in
wavelength-selective optical scrutinizing for chemically-active molecules or
proteins bounded with carbon atoms in graphene.
\medskip

The tools for optical probing which we discussed in this paper include either scattering
or optical  absorption of an incident evanescent electromagnetic field. In the case of
evanescent-field  scattering, we computed the spatial dependence of a Fourier transformed
scattering matrix, which demonstrates the scattering enhancement,reduction and even suppression
as functions of graphene separations ($z_0$) from the surface of the conductor as well as between
themselves ($z_1-z_0$) and the wave numbers ($q_x$) of the evanescent surface-plasmon-polariton
field at several frequencies close to the localized surface-plasmon resonance.
This derived scattering matrix lays the foundation for constructing an effective-medium
theory commonly employed in finite-difference time-domain methods\,\cite{add17,mit} for
solving Maxwell's equations numerically. Furthermore, the calculated full spatial
dependence for the scattering electromagnetic field shows unique features in three
different regions, including ones below, between and above two graphene sheets.
\medskip

For optical absorption, on the other hand, the triple peaks corresponding to the lower
acoustic-like graphene plasmon, the middle surface-plasmon and the higher optical-like graphene
plasmon modes  are seen to dominate the variable hybridization features at high conductor
plasma frequencies. However, the rounded peak associated with the surface-plasmon-polariton
mode at the lowest energy is found to be dominant at low plasma frequencies. In addition,
this rounded peak further demonstrates that localized modes can be enhanced significantly
when two graphene layers are placed closer to the conductor surface. These unique features in
resonant absorption enable the selective excitation of radiative polariton modes for their
condensation and a threshold-free laser afterwards.
\medskip

We would like to emphasize that the use of linear response theory\,\cite{ref2}
for calculating the optical-response function in Eq.\,(\ref{e9}) only applies to a weak electromagnetic
field. On the other hand, if the total electric field is strong, we must calculate the
induced polarization field using the quantum-kinetic equations\,\cite{add8,add26,ref1,add9,add29}.
In this case, the populations of electrons and holes in a density matrix become extreme
non-equilibrium functions of wave vector for these photo-generated carriers. Moreover, the
polarization field is determined by summing the light-induced coherence in the density
matrix for all occupied states of photo-carriers. Furthermore, if the electric field is
extremely strong, we expect an opening of energy gaps due to electron-photon dressing
effects\,\cite{ref1,dressing,dressing2,kibis}.
The theory for graphene-plasmon hybridization in this paper can be easily generalized to other 2D materials, such as silicene, germanene, molybdenum disulfide, etc.

\begin{acknowledgments}
D.H. would like to thank the support from the Air Force Office of Scientific Research (AFOSR).
\end{acknowledgments}

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