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Tag: United Republic of Tanzania
Uganda: Joint Press Release – Court Case against Uganda and Tanzania over the EACOP Project to be heard (11.11.2022)
Author nilspeacePosted on November 11, 2022 November 11, 2022 Categories Africa, Business, Development, Governance, Government, Law, Leadership, PoliticsTags Albertine Region, Chinese National Offshore Oil Company, CNOC, CNOOC Uganda Limited, Crude Oil, Crude Oil Pipeline Development, Crude Oil Pipeline Rates, EAC, EACOP, East Africa, East African Community, East African Crude Oil, East African Crude Oil Pipeline, East African Protocol, George M. Masaju, Government of Tanzania, Government of the United Republic of Tanzania, Hoima-Lokichar-Lamu oil pipeline, Hon. Irene Muloni, Hon. Samia Suluhu Hassan, Irene Muloni, Lake Albert, Lamu, Lamu Port, Least-Cost Operation, Lt. Gen. Yoweri Kaguta Museveni., Manoah Esipisu, Mwesigwa Rukutana, President Hassan, President Museveni, Production Sharing Agreement, Samia Suluhu Hassan, Tanga, Tanzania, Tanzania State House, Tanzanian Foreign Affairs Ministry, Tanzanian Government, Total E&P Uganda, Uganda National Oil Company, United Republic of Tanzania, UNOC, Yoweri Kaguta Museveni, Yoweri Museveni, Yoweri Tibuhaburwa Kaguta MuseveniLeave a comment on Uganda: Joint Press Release – Court Case against Uganda and Tanzania over the EACOP Project to be heard (11.11.2022)
A look into Mzee's 60th Independence Speech [a story about coffee and the pipeline]
For the 35th time or the 36th time that President Yoweri Tibuhurwa Kaguta Museveni has addressed the nation in the Independence Day. This year its the 60th year since Uganda got independent from the United Kingdom. That day should have been held with fanfare and celebration across the Republic. However, the most vital and important place to be was at Kololo in Kampala. Where Museveni together with foreign dignitaries was at a bash. Which lasted and lasted.
President Museveni knows how to hold lectures and long one too. So, I will only take a few paragraphs here and there. That's the ones that stood out to me. Because there is a lot of flesh and bone in a two hour speech and nobody with a sane mind wants to fact-check that on a whim. Therefore, I am only looking at snippets and addressing them as I see fit.
Like this one: "These achievements, have been in spite of the stiff opposition we have been facing from the parasites and their foreign backers, such as the Monitor Newspaper. To take an example, this paper, on the 22nd of May, 2022,it said that: "Coffee deal stinks; but key culprit will not be punished". In other words, Museveni is committing a crime by trying to add value to the Ugandan Coffee" (Museveni, 09.10.2022).
By saying this we know his the Hawk Limited, which wasn't addressed in the Parliament report. As the Government of Uganda and Uganda Vinci Coffee Company Limited has made agreements, which is practically creating a coffee-bean monopoly to a company that isn't really built yet. Neither is the coffee factory built or ready for the levels of production either. That's why this whole deals stinks and could easily destroy the agricultural output with the marginalized agreements, which isn't benefiting the free-market or the spirit of producing it either. This is not an incentive or an adjustment anyone asked for, but a problem and bottleneck, which the President himself has created.
While on the subject. This paragraph is just amazing:
"After that, I engaged the Nestle group in Davos. They made it clear that they already had their factory for coffee processing in the UK and would not build a new one. Hence, we had to remain as Raw-Material producers, getting only 5% of the value of the coffee we grow. They, moreover, try to muddy the water by claiming that "blending" of the different coffees can only take place in Europe etc. We told them that Uganda had all the varieties and the blending would be done here. That has been the war, I have been involved in until I got the Vinci Company to help me. You heard how some of the MPs were talking in that war. Mwalimu Julius Nyerere, had built a coffee processing factory in Bukoba, in Tanzania, near the border of Uganda. When I heard of that, I linked our people with that effort. They were producing an end product known as "Star Coffee". Unknown to me, the officials of Uganda, allowed that effort to die out or get stunted. What blindness and betrayal this is!!" (Museveni, 09.10.2022).
It is interesting that Museveni mentions Nestle, which is only this year opening up a Coffee Processing Plant in Mexico. So, this means that Museveni is lying about the fact or that's so long ago that the Nestle company has changed their perspective. Nevertheless, I don't believe Coffee can only be processed in Europe. That is deliberately false and a lie. Just to malign Europe and direct his displeasure with Europeans in general. In this regard, there are already coffee processing plants in Uganda. Just to mention one other, which has been operating since 2015 is in Bushenyi district and run by APCU Union. This plant alone is sufficiently supporting 6,000 small-coffee growers in the Bushenyi district. So, the UVCC agreement with the monopoly is dismantling and destroying for instance that processing plant, which is already in operation since 2015. It just shows how his not interested in other visions than his own. His actually betraying the ones that already has started and is operating, which was also described earlier this in a Parliamentary Report. A report that the President don't mention or even cares to assess. It's his way or the highway…
On another note, he speaks of the petroleum pipeline:
"Recently, there was yet another manifestation of Imperialistic arrogance and hegemonism by elements in the European Union Parliament whereby they decreed from Brussels that the East African Crude Oil Pipeline should not be constructed, until those arrogant actors permit us to do so. Parasitic elements from that part of the World, have been causing problems for Africa and the World for the last 500 years" (Museveni, 09.10.2022).
It is really astonishing the ways that he puts this. That's why you know this is "his oil" and he feels tampered with. He isn't describing or explaining the grievances of the EU Parliament Motion. Only describing them the worst way possible and putting them in sinister political context. Instead, of trying to defend with proper arguments for the continuation of it. While again maligning his detractors with language that isn't fitting for a grown man of his stature, but we know that is common practice from him.
Well, enough for this piece, will write one more about his speech. Since it was over 30 pages and there was one more tale that needs to be addressed. It's coming on the page as soon as it is finished. Peace.
Author nilspeacePosted on October 9, 2022 October 9, 2022 Categories Africa, Army, Business, Development, Europe, Governance, Government, Law, Leadership, PoliticsTags Albertine Region, Chinese National Offshore Oil Company, CNOC, CNOOC Uganda Limited, Crude Oil, Crude Oil Pipeline Development, Crude Oil Pipeline Rates, EAC, EACOP, East Africa, East African Community, East African Crude Oil, East African Crude Oil Pipeline, East African Protocol, Enrica Pinetti, George M. Masaju, Government of Tanzania, Government of the United Republic of Tanzania, Government of Uganda, Hawk Limited, Hoima-Lokichar-Lamu oil pipeline, Hon. Irene Muloni, Hon. Samia Suluhu Hassan, Irene Muloni, Keith Muhakanizi, Lake Albert, Lamu, Lamu Port, Least-Cost Operation, Lt. Gen. Yoweri Kaguta Museveni., Manoah Esipisu, Mwesigwa Rukutana, National Resistance Movement, NRM, President Hassan, President Museveni, Production Sharing Agreement, Ramathan Ggoobi, Roko, Samia Suluhu Hassan, Tanga, Tanzania, Tanzania State House, Tanzanian Foreign Affairs Ministry, Tanzanian Government, Total E&P Uganda, Uganda, Uganda National Oil Company, Uganda Vinci Coffee Company, United Republic of Tanzania, UNOC, UVCCL, Yoweri Kaguta Museveni, Yoweri Museveni, Yoweri Tibuhaburwa Kaguta MuseveniLeave a comment on A look into Mzee's 60th Independence Speech [a story about coffee and the pipeline]
Opinion: Museveni's deflection on the EACOP…
"I spent 12 years studying English and one of the words I picked up in those years is insufferable. Some of these people(EU MPs) are insufferable. You(EU MPs) need to control yourself not to explode.(They are) so shallow, so egocentric and so wrong that they think they know everything broadcasting their ignorance all over the place but they should calm down. This is a wrong battleground for them" (…) "We don't take kindly to arrogance and that's why we spent so many years sorting out issues with arrogant people. The plan will be implemented according to schedule. I hope our partners join us firmly and advise them. When you go to the European Union parliament these are just young girls. A young girl is lecturing me what to do in Uganda! I think somebody should counsel those young people" – President Yoweri Kaguta Museveni (27.09.2022)
No one should expect that President Museveni would held himself accountable or be concerned with the grievances in the European Union Parliament Motion "JOINT MOTION FOR A RESOLUTION on violations of human rights in Uganda and Tanzania linked to investments in fossil fuels projects" which has caused issues since 14th September 2022.
We know that the East African Crude Oil Pipeline (EACOP) and the exploiting of petroleum in the Lake Albertine basin is key for the President. This industrial enterprise and agreements are a life-line for his reign. The currency coming from this has to be a slush-fund and direct funds to ensure longevity at this point. Since, he has no direct support or direct donors who will fund him in the same of which they did in 1990s or early 2000s. That's why the National Resistance Movement (NRM) and the government have a debt deficit cycle, which is destroying the economy. Only Petro-Dollar can save it and possibly get enough revenue to stop the downward spiral created over the years.
However, no one should believe that the Petro-Dollars or the EACOP will be beneficial for the Republic. If it has been sincere and honestly operating, the whole enterprise, the businesses and the agreements would have been transparent. Secondly, the grievances which was reflected in the motion haven't been discussed or addressed by the President. That's the arrogance of Museveni himself showing.
President Museveni isn't talking about the environmental damages or problems the EACOP is creating. Neither is he talking about the forced evictions or the human rights violations either. That doesn't concern him. We know this all about "his oil" and the oil development to make his bank-account pregnant. The final heist and the biggest slush-fund ever to President for Life.
The ones believing President Museveni is building and developing the petroleum industry out of the kindness of his heart. I hate to tell you, but this is all for selfish reasons. His doing it to earn money himself and is indifferent about the citizens itself. He will have a buck-load of cash and the costs will be burdened on the next generation. Museveni isn't in this to do good and he never was. That's why other industries are depleted, privatized and destroyed over the years in his reign.
It is interesting that his pinning the blame on the EU Parliament. When he doesn't even see what they said and what the Motion is stating. It's like that doesn't matter and like Museveni isn't reading the room. There are already banks backing off and the IMF/World Bank isn't interesting in it either. The timing of building the EACOP is wrong. He should have initiated this a decade earlier or so. However, he was busy with other schemes at the time…
President Museveni isn't saving face here. His just deflecting and not taking into account the Motion itself. That's apparently beneath him, but he should be concerned by this. They are just proving what sort of man he is and is not willing to see it. He rather rule with his ruthless acts and hope no one every tests him. However, the EU Parliament isn't beholden to him or have to live by his words. They are independent that way and therefore, he just have to act this way. He can rebuke them… but that doesn't help his case either. Total is European and French, so that company is bound by the laws and regulations in Europe. Therefore, Museveni cannot just deny the EU Parliament, because they are the ones that are the biggest developers in it too.
That's why the next moves of Total will be important and if they will defy the EU Parliament itself. This will be followed and we don't know how that one will go. What we do know… is that its harder for them and banks are not willing to loan development funds to it. In this regard, the Ugandan President should be more humble and be more friendly. However, he prefer being on the warpath, than being a person who reflects on the reality and adjust to it. Peace.
Author nilspeacePosted on September 28, 2022 September 28, 2022 Categories Africa, Business, Development, Governance, Government, Industry, Law, Leadership, PoliticsTags Albertine Region, Chinese National Offshore Oil Company, CNOC, CNOOC Uganda Limited, Crude Oil, Crude Oil Pipeline Development, Crude Oil Pipeline Rates, EAC, EACOP, East Africa, East African Community, East African Crude Oil, East African Crude Oil Pipeline, East African Protocol, George M. Masaju, Government of Tanzania, Government of the United Republic of Tanzania, Hoima-Lokichar-Lamu oil pipeline, Hon. Irene Muloni, Hon. Samia Suluhu Hassan, Irene Muloni, Lake Albert, Lamu, Lamu Port, Least-Cost Operation, Lt. Gen. Yoweri Kaguta Museveni., Manoah Esipisu, Mwesigwa Rukutana, President Hassan, President Museveni, Production Sharing Agreement, Samia Suluhu Hassan, Tanga, Tanzania, Tanzania State House, Tanzanian Foreign Affairs Ministry, Tanzanian Government, Total E&P Uganda, Uganda National Oil Company, United Republic of Tanzania, UNOC, Yoweri Kaguta Museveni, Yoweri Museveni, Yoweri Tibuhaburwa Kaguta MuseveniLeave a comment on Opinion: Museveni's deflection on the EACOP…
Uganda: Over Hundred CSOs Express Concern over Islamic Development Bank's Financing of EACOP (22.09.2022)
Author nilspeacePosted on September 23, 2022 September 23, 2022 Categories Africa, Business, Development, Governance, Government, Industry, Law, Leadership, PoliticsTags Albertine Region, Chinese National Offshore Oil Company, CNOC, CNOOC Uganda Limited, Crude Oil, Crude Oil Pipeline Development, Crude Oil Pipeline Rates, EAC, EACOP, East Africa, East African Community, East African Crude Oil, East African Crude Oil Pipeline, East African Protocol, George M. Masaju, Government of Tanzania, Government of the United Republic of Tanzania, H. E. Yoweri Kaguta Museveni, H.E. Uhuru Kenyatta, H.E. Yoweri Museveni, Hoima-Lokichar-Lamu oil pipeline, Hon. Irene Muloni, Hon. Samia Suluhu Hassan, Irene Muloni, Lake Albert, Lamu, Lamu Port, Least-Cost Operation, Lt. Gen. Yoweri Kaguta Museveni., Manoah Esipisu, Mwesigwa Rukutana, President Hassan, President Museveni, Production Sharing Agreement, Samia Suluhu Hassan, Tanga, Tanzania, Tanzania State House, Tanzanian Foreign Affairs Ministry, Tanzanian Government, Total E&P Uganda, Uganda National Oil Company, United Republic of Tanzania, UNOC, Yoweri Kaguta Museveni, Yoweri MuseveniLeave a comment on Uganda: Over Hundred CSOs Express Concern over Islamic Development Bank's Financing of EACOP (22.09.2022)
National Unity Platform (NUP): Robert Kyagulanyi aka Bobi Wine – Statement on the Resolution of the European Parliament on the Violation of Human Rights linked to Investments in the Oil Pipeline (21.09.2022)
Author nilspeacePosted on September 22, 2022 September 22, 2022 Format ImageCategories Africa, Business, Development, Economy, Governance, Government, Industry, Law, Leadership, PoliticsTags Albertine Region, Chinese National Offshore Oil Company, CNOC, CNOOC Uganda Limited, Crude Oil, Crude Oil Pipeline Development, Crude Oil Pipeline Rates, EAC, EACOP, East Africa, East African Community, East African Crude Oil, East African Crude Oil Pipeline, East African Protocol, George M. Masaju, Government of Tanzania, Government of the United Republic of Tanzania, H. E. Yoweri Kaguta Museveni, H.E. Uhuru Kenyatta, H.E. Yoweri Museveni, Hoima-Lokichar-Lamu oil pipeline, Hon. Irene Muloni, Hon. Samia Suluhu Hassan, Irene Muloni, Lake Albert, Lamu, Lamu Port, Least-Cost Operation, Lt. Gen. Yoweri Kaguta Museveni., Manoah Esipisu, Mwesigwa Rukutana, President Hassan, President Museveni, Production Sharing Agreement, Samia Suluhu Hassan, Tanga, Tanzania, Tanzania State House, Tanzanian Foreign Affairs Ministry, Tanzanian Government, Total E&P Uganda, Uganda National Oil Company, United Republic of Tanzania, UNOC, Yoweri Kaguta Museveni, Yoweri MuseveniLeave a comment on National Unity Platform (NUP): Robert Kyagulanyi aka Bobi Wine – Statement on the Resolution of the European Parliament on the Violation of Human Rights linked to Investments in the Oil Pipeline (21.09.2022)
National Environment Management Authority (NEMA): Clarification on information concerning the Environment and Social Impact Assessment (EISA) Reports for Oil and Gas Projects in Uganda (19.09.2022)
Author nilspeacePosted on September 19, 2022 September 19, 2022 Categories Africa, Business, Development, Governance, Government, Industry, Law, Leadership, PoliticsTags Albertine Region, Chinese National Offshore Oil Company, CNOC, CNOOC Uganda Limited, Crude Oil, Crude Oil Pipeline Development, Crude Oil Pipeline Rates, EAC, EACOP, East Africa, East African Community, East African Crude Oil, East African Crude Oil Pipeline, East African Protocol, George M. Masaju, Government of Tanzania, Government of the United Republic of Tanzania, H. E. Yoweri Kaguta Museveni, H.E. Uhuru Kenyatta, H.E. Yoweri Museveni, Hoima-Lokichar-Lamu oil pipeline, Hon. Irene Muloni, Hon. Samia Suluhu Hassan, Irene Muloni, Lake Albert, Lamu, Lamu Port, Least-Cost Operation, Lt. Gen. Yoweri Kaguta Museveni., Manoah Esipisu, Mwesigwa Rukutana, President Hassan, President Museveni, Production Sharing Agreement, Samia Suluhu Hassan, Tanga, Tanzania, Tanzania State House, Tanzanian Foreign Affairs Ministry, Tanzanian Government, Total E&P Uganda, Uganda National Oil Company, United Republic of Tanzania, UNOC, Yoweri Kaguta Museveni, Yoweri MuseveniLeave a comment on National Environment Management Authority (NEMA): Clarification on information concerning the Environment and Social Impact Assessment (EISA) Reports for Oil and Gas Projects in Uganda (19.09.2022)
Opinion: The EACOP is no bargain
This week there been new talks of the East African Crude Oil Pipe Line (EACOP). That comes after the European Union Parliament came with a Motion, which reflect dire concerns about it. The EACOP pipeline will go from the Lake Albert Basin down to Port of Tanga in Tanzania. It is a joint venture deal, but the one who has the most leverage is the French Company Total. While EACOP itself is a United Kingdom based corporation.
The others with smaller stakes are the Uganda National Oil Corporation (UNOC) and the Tanzania Petroleum Development Corporation (TPDC). Not to forget the even smaller ownership of the China National Offshore Oil Corporation. So, it is not like the EACOP or the drilling of petroleum is done by solely Ugandan ownership either. Not like Tanzania has a huge stake in it either.
Yes, CNOOC and Total will pay taxes and transport fees of sorts, which is normal for a pipeline to the respective nations. However, they are still able and licenced to drill or exploit the oil fields in the Lake Albert Basin. They are doing that and then later transporting the petroleum from Bunyoro to the coast of Tanzania, which is a long pipeline over vast different environments, which could cause harm or endanger it even. That's only one thing in this enterprise.
The drilling and exploitation of the petroleum in Lake Albert basin and parts of it is done in Murchison Falls National Park. Meaning the ones with the licence to drill will do so in areas, which has already been facilitated to reserve the nature and the habitats of the species living there. In beautiful surroundings and unique as well. Therefore, the whole petroleum industry could cast a spell on the area as a whole. Especially, if there was something going wrong or an "accident" would occur, which could cause massive damage to the areas.
The EU Parliament Motion has been called "colonial" and "Supremacist" also said to block industrial development in East Africa. They are calling it undermining and saying the Europeans should mine their own business. Which is very ironic, when Total, the French company is the main investor and company involved. The lions share of it run by Total and not by anyone else. Before that UK businesses Tullow and Heritage had vast control of the licences to drill in Lake Albert Basin. So, it's very weird to call one thing "colonial" and yet allowing the same "colonial" enterprises to run business there.
The Ugandans speaking ill of it. Should first ask for Good Governance and proper due diligence over the projects. The development done and the deals should be looked into. Because, little to none is public. Only the diplomatic papers and some shareholding or ownership details are known. The licencing deals for drilling is "secret" and the transactional parts of also under wraps. The same with registration and practically ownership of the UNOC and TPDC. They are not entities who is very open or transparent about.
That's why when the minority isn't transparent, and the Total company isn't either. The grand public cannot know what is at stake. Who is getting a cut and who is really earning on it. Not like the monies or the credible profits cannot be siphoned. It is not like it is in the domain of the public or it has a knowledge about that.
That's the making of the President and his Office. They have kept it all hidden for a reason. It is his business and not the citizens one. It is his "oil" and he can do what he likes. He don't like nosy people and don't want to be bothered with real questions. That's why its been kept secret… and it's deliberately so. Peace.
Author nilspeacePosted on September 18, 2022 September 18, 2022 Categories Africa, Business, Development, Governance, Government, Law, Leadership, PoliticsTags Albertine Region, Chinese National Offshore Oil Company, CNOC, CNOOC Uganda Limited, Crude Oil, Crude Oil Pipeline Development, Crude Oil Pipeline Rates, EAC, EACOP, East Africa, East African Community, East African Crude Oil, East African Crude Oil Pipeline, East African Protocol, George M. Masaju, Government of Tanzania, Government of the United Republic of Tanzania, H. E. Yoweri Kaguta Museveni, H.E. Uhuru Kenyatta, H.E. Yoweri Museveni, Hoima-Lokichar-Lamu oil pipeline, Hon. Irene Muloni, Hon. Samia Suluhu Hassan, Irene Muloni, Lake Albert, Lamu, Lamu Port, Least-Cost Operation, Lt. Gen. Yoweri Kaguta Museveni., Manoah Esipisu, Mwesigwa Rukutana, President Hassan, President Museveni, Production Sharing Agreement, Samia Suluhu Hassan, Tanga, Tanzania, Tanzania State House, Tanzanian Foreign Affairs Ministry, Tanzanian Government, Total E&P Uganda, Uganda National Oil Company, United Republic of Tanzania, UNOC, Yoweri Kaguta Museveni, Yoweri MuseveniLeave a comment on Opinion: The EACOP is no bargain
The EACOP Pipeline might have a harder time getting funding after EU Parliament Motion condemning it…
The East African Crude Oil Export Pipeline (EACOP) could face a new hurdle and delays. As there is not only a collective of activists and organizations working to stifle the options of funding the EACOP. The EACOP needs funding to be able to be built. The EACOP has to get funding from either Corporate Banks or Multi-National Institutions. The World Bank said in 2019 that it wouldn't support it and neither will International Monetary Fund (IMF). Therefore, the EACOP needs to get $5 billion elsewhere to fund the building of it.
Because of the Motion today made by the European Parliament. I had to go back and find more new information into the EACOP. The EACOP haven't been a steady operation or a quick fix. No, this has been a slow train moving. There has been slowly getting things in order and the it was only last year both Uganda and Tanzania had settle their negotiations. This is why it haven't gotten further. The slow approach is also a reason why the EACOP is lacking funding and not being operational. The deadlines of the production, the exports and a possibly a refinery in Uganda has also been on hold. Therefore, the EU motion could only get into existence because of the slow movements from the Government of Uganda.
Take a look here…
EACOP Ownership:
"The Tilenga oilfield, to the north of Lake Albert, will include operations within the Murchison Falls National Park, and is operated and owned 56.67% by TotalEnergies. The Kingfisher oilfield, at the southern end of the lake, is being developed by CNOOC which owns 28.33%, and Uganda's UNOC, which has a 15% stake. The two projects are expected to start producing oil in 2025 and reach a peak production of 230,000 barrels per day, which will rank the Lake Albert oil fields as one of Africa's top 10 oil projects. The shareholders in the pipeline are TotalEnergies (62%), UNOC (15%), Tanzania Petroleum Development Corporation TPDC (15%) and CNOOC (8%)" (Tonderayi Mukeredzi – 'Controversial East Africa oil pipeline moves one step closer to construction' 25.03.2022, ChinaDialogue,net).
EACOP Funding:
"The East African Crude Oil Export Pipeline, which will have a daily capacity of 216,000 barrels a day, will be funded on a 40% to 60% equity-debt ratio, according to UNOC, a partner in the project. The link is designed to move land-locked Uganda's oil to international markets. The Islamic Development Bank became the first lender to commit funding to the project when it approved $100 million for EACOP at the weekend. TotalEnergies SE is leading development of the project with a 62% stake in the cross-border pipeline. UNOC and Tanzania Petroleum Development Corp. each have a 15% interest, and the rest is owned by China's Cnooc Ltd" (Fred Ojambo – 'Uganda expects all pledged funding for oil pipeline by end of November' 12.09.2022, WorldOil.com)
EU Parliament Motion:
"Calls for the EU and the international community to exert maximum pressure on Ugandan and Tanzanian authorities, as well as the project promoters and stakeholders, to protect the environment and to put an end to the extractive activities in protected and sensitive ecosystems, including the shores of Lake Albert, and commit to using the best available means to preserve the culture, health, and future of the communities affected and to explore alternatives in line with international climate and biodiversity commitments; calls on the promoters of the EACOP project in Uganda and Tanzania to resolve all disputes that should have been resolved prior to the launch of the project, and to take into account all the above-mentioned risks, threatening this project; urges TotalEnergies to take one year before launching the project to study the feasibility of an alternative route to better safeguard protected and sensitive ecosystems and the water resources of Uganda and Tanzania, limiting the vulnerability of the watersheds in the African Great Lakes region, which is a critical resource for the region, and to explore alternative projects based on renewable energies for better economic development" (European Parliament – 'JOINT MOTION FOR A RESOLUTION on violations of human rights in Uganda and Tanzania linked to investments in fossil fuels projects' 14.09.2022).
We know the EU Parliament Motion is causing a stir in Kampala and Dodoma. There will be words and possible ramifications of this. The EACOP is mostly owned and run by the Total Energie and that's French. So, the French or France, which is a Member State in the EU could suddenly apply more pressure on the French Petroleum Company. That would further delay and possibly stop the EACOP pipeline. Since, the pipeline isn't majority owned by the governments of Uganda or Tanzania. This is why the condemnation of the EU Parliament matters.
Just like the activists and NGOs who is working on overtime pressuring banks and lenders to the EACOP. That in combination of the new policies of greener technology or following climate change programs within the World Bank (WB) or International Monetary Fund (IMF). The EACOP certainly needs lots of funding and fiscal funds. That's why the EU Parliament condemnation can alter a lot and add pressure, which is even stronger than the activists or NGOs who is already running a marathon with it.
This pipeline is contested, not only because of the parties involved and the respective republic's. No, it is because of the environments that the oil is drilled and where the petroleum pipeline route goes as well. Peace.
Author nilspeacePosted on September 15, 2022 September 15, 2022 Categories UncategorizedTags Albertine Region, Chinese National Offshore Oil Company, CNOC, CNOOC Uganda Limited, Crude Oil, Crude Oil Pipeline Development, Crude Oil Pipeline Rates, EAC, EACOP, East Africa, East African Community, East African Crude Oil, East African Crude Oil Pipeline, East African Protocol, George M. Masaju, Government of Tanzania, Government of the United Republic of Tanzania, H. E. Yoweri Kaguta Museveni, H.E. Uhuru Kenyatta, H.E. Yoweri Museveni, Hoima-Lokichar-Lamu oil pipeline, Hon. Irene Muloni, Hon. Samia Suluhu Hassan, Irene Muloni, Lake Albert, Lamu, Lamu Port, Least-Cost Operation, Lt. Gen. Yoweri Kaguta Museveni., Manoah Esipisu, Mwesigwa Rukutana, President Hassan, President Museveni, Production Sharing Agreement, Samia Suluhu Hassan, Tanga, Tanzania, Tanzania State House, Tanzanian Foreign Affairs Ministry, Tanzanian Government, Total E&P Uganda, Uganda National Oil Company, United Republic of Tanzania, UNOC, Yoweri Kaguta Museveni, Yoweri MuseveniLeave a comment on The EACOP Pipeline might have a harder time getting funding after EU Parliament Motion condemning it…
Tanzania: Loliondo land-grab – Hon. Masauni claims it is Kenyan migrants fault…
"Engineer Masauni challenged immigration officers at all exit and entry points, to be extra vigilant in curbing such illegal entries into Tanzania. "The immigration department should come up with effective strategies to control such an increasing trend, which is likely to jeopardise our peace and security," directed Masauni, shortly before inspecting the ongoing exercise of demarcating the 1,500 square kilometers of the Loliondo Game Controlled Area (LGCA)" (…) "The Minister's directive comes in the wake of reports of immigrants from neighboring countries who take advantage of long porous borders to enter and operate illegally in Tanzania. Engineer Masauni said that there have been reports of illegal immigrants entering the country, singling out the Loliondo area as a worst case scenario, where hundreds herds of cattle have been found grazing within the LGCA, threatening the Serengeti-Ngorongoro ecosystem" (Edward Qorro – 'Tanzania: Immigration Tasked to Intensify Border Security' 15.06.2022, Tanzania Daily News).
Let's be perfectly clear… this is excuses. The main culprit is the Tanzanian state and the authorities who has sold land to a foreign company, which wants to use that for trophy hunting and game-drives. That is the Tanzanian government trading away ancestral territories for a Middle Eastern Company. This is what is happening to the Masaai land and the Loliondo areas.
Where the state has brutally displaced people and used violent means to do so. People are arrested and taken away for defending their rights to live on the land. Now, the authorities and the Minister blaming immigration and making up a false problem. Where the Tanzanian government claims the Kenyan government is allowing illegal migration and undocumented people from their bordering territories. That is really saying something…
The Chama Cha Mapinduzi (CCM) should self-reflect on their own actions and what they are really doing. It is their interests over a historical and indigenous land, which is their responsibility to protect. Instead they are serving the interests of the tourism industry and one company. That is really compelling and telling. The interest is the common good, but a quick buck on selling out traditions and exploiting that.
The Tanzanian government cannot come with excuses like this. That should get pushback from Nairobi and the Jubilee Party. There is a need for President Uhuru Kenyatta to call President Samia Suluhu. Especially, when the Hon. Hamad Masauni, the Minister of Home Office has been saying this to the media. He is coming with allegations, which is blaming Kenya for the trouble in the border region of Tanzania.
However, everyone with some common sense. It is the land-grabbing and the internal conflicts, which is the reason. If the Tanzanian government had respected the indigenous land and not sold it. That's the reason for the aftermath and all the pushback. This is very easy, but the CCM is making it very hard.
Hon. Masauni is speaking out of line. Since their actions is actually causing migrations from Tanzania to Maasai territory on the other side of border into Kenya. Therefore, it would be fair for Kenya to speak of the refugees and fleeing Maasai brothers who fled the brutal eviction in the Loliondo area. That's saying something and the Tanzanian government isn't the victim.
The Maasai people of the Loliondo area are the victims and the hurt ones here. The CCM is the one who profits on the misery of the Maasai people. Just so a Middle Eastern company can get territories to show the beauty of Tanzania. A beauty that was destroyed for the ones who actually resided there… Peace.
Author nilspeacePosted on June 16, 2022 June 17, 2022 Categories UncategorizedTags ACT Wazalendo, Alliance for Change and Transparency, CCM, CHADEMA, Chama cha Demokrasia na Maendeleo, Chama Cha Mapinduzi, Freeman Mbowe, Loliondo, Maasai, Ngorongoro District, Ortello Business Company, President Suluhu, Samia Suluhu, Tanzania, Tundu Lissu, Twaha Mwaipany, United Republic of Tanzania, Zitto KabweLeave a comment on Tanzania: Loliondo land-grab – Hon. Masauni claims it is Kenyan migrants fault…
Tanzania: Loliondo land-grab – When tourism beats the ownership of indigenous land…
"The Tanzanian government is intending to resume the eviction of about 70,000 Maasai community in Loliondo to make room for trophy hunting and elite tourism. The move, which will involve dispossessing over 70,000 Maasai pastoralists of 1500sq km of their ancestral land. This is aimed at creating room for Ortello Business Company, a hunting firm owned by the United Arab Emirates' royal family, as part of investment for the government" (Western Nyota TV, 13.06.2022).
The Chama Cha Mapundzi (CCM) don't care and isn't concerned with the rule of law. As the commercial interests of the tourism industry and a foreign investors involvement matters more. That's why Maasai territories are now possibly evicted for the sole purpose of an investor making a game-drive. So, that the animals can be seen by foreigners visiting and ensuring profits to the state coffers.
That's done without the consent of the Masaai people or their ancestral rights to territories in question. This is why they have been demonstrating, because the state is violently and brutally evicting them from their homeland. Just so one investor can have a safari company and game-drive in the area. That's really not developing, but ceasing territory for man's pleasure and his company.
"According to reliable information received by IWGIA, the Tanzanian state is currently moving ahead to grab 1,500 km2 of ancestral and legally registered village land belonging to the Maasai Indigenous people in the Loliondo division of Ngorongoro District in northern Tanzania. This land grab will forcefully evict the Maasai from their land. This is happening despite the fact that the East African Court of Justice is expected to issue a ruling on this matter on 22 June 2022. On 7 June 2022, a paramilitary group of around 700 people (mainly police, park rangers, military and other security forces) arrived in Loliondo to demarcate the 1,500 km2 area of land as a Game Reserve1. This land is the ancestral land of the Maasai people who have formalized land tenure rights to this land as per the Village Land Act No.5 of 1999. The demarcation of this land as a Game Reserve will effectively dispossess the Maasai people of their ancestral land since people and livestock will not be allowed to live in a Game Reserve" (International Work Group for Indigenous Affairs (IWGIA) – 'Urgent Alert: New serious threats towards the Maasai people of Loliondo in Tanzania' 13.06.2022).
"Over 100,000 Maasai are facing a forcefully eviction from their ancestral land in Loliondo and Ngorongoro, Tanzania Maasai have been living in Lolondo and Ngorongoro before Tanzania gained independence. The eviction is purely to making way for Trophy Hunting" (Tanzania Leaks, 13.06.2022).
This here shouldn't be accepted and people should speak out on the matter. It isn't righteous and fair. Clearly it is injustice, pure and simple. Tanzanian authorities and state is doing practically a land-grab and with violent means to get their way. This eviction isn't done with measures of fairness or compensation. No, these are people who is violently removed and evicted from their territories, which have been theirs for generations.
That the Tanzanian authorities are doing this and using all sorts of violent ways. Just shows the world how unjust this is. The CCM and the President knows this as they are just serving a Middle Eastern company "prime-real-estate" for safari and a game-drive. That's why they are doing this and showing the lack of care or concern for the people who belongs in this territory.
If CCM and the President wanted a favourable look. This wasn't it. They are just displaying brutality and what they do for some cheap coins. Peace.
Author nilspeacePosted on June 13, 2022 June 13, 2022 Categories Africa, Business, Development, Economy, Governance, Government, Law, Leadership, PoliticsTags ACT Wazalendo, Alliance for Change and Transparency, CCM, CHADEMA, Chama cha Demokrasia na Maendeleo, Chama Cha Mapinduzi, Freeman Mbowe, Loliondo, Maasai, Ngorongoro District, Ortello Business Company, President Suluhu, Samia Suluhu, Tanzania, Tundu Lissu, Twaha Mwaipany, United Republic of Tanzania, Zitto KabweLeave a comment on Tanzania: Loliondo land-grab – When tourism beats the ownership of indigenous land…


us. In your precious name we pray. Amen.
and many of you are dressed for the fun. I hope you will all stay for the fun and fellowship.
by saying that I am walking in faith.
All Lives Matter. For me, there is a commonality – a common humanity – that underlies all.
agree that Black Lives Matter too. Black Lives Matter also.
need. To hope and dream and ask for what we need.
issue. It is complicated. When are you in power, when are you marginalized?
sometimes to throw up our hands and walk away. But what would that solve?
on, then my life, my community is impacted.
to engage in active listening rather than active shooting?
Book of Common Prayer Baptismal Rite.
dead, but he stood merciful in the face of fear.
Officer Davis. I know my story is not your story, but that's why we need to hear each other.
succeed 100% of the time.
of the parable is now up to us. Amen.
On Religion in Healthcare by Deepak N. Kotecha, O.D.


text:
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Around the world, Australia & New Zealand, Health, Politics
Get Involved: Medicinal Cannabis for Health Professionals
One of the major problems facing Australia's current medicinal cannabis market is the minute number of health professionals that actually have any research-based knowledge of the plant and its medical applications.
This issue has been clearly identified to me by a variety of different sources, including most recently Dr Teresa Towpik of MediHuanna. She informed me of the severe lack of education pertaining to the field of medicinal cannabis, and the express need for health professionals to be trained in this field of medicine.
Some of the forces combating this re-education problem include Sharlene & Paul Mavor, founders of the cannabis-related charity Medicinal Cannabis Research Australia (MCRA). The MCRA was established in December 2016 and serves the purpose of educating patients and caregivers about cannabis' plethora of remedial properties. One of their major focuses involves re-educating health professionals on the benefits of cannabis by delivering a balanced and scientific argument.
From the 12th to the 22nd of March, Medical Cannabis Research Australia will be hosting a number of seminars for Health Professionals which will describe the cannabis plant and its medicinal uses in great detail. The event will be run in Darwin, Brisbane, Sydney, Canberra, Hobart, Melbourne, Adelaide and Perth, with tickets available from their website, MCRA.org.au
Sharlene's medical science background, majoring in Microbiology, Virology and Biotechnology, couples perfectly with Paul's role as a pharmacist. Back in May 2016, Paul Mavor was actually the first person to gain the legal right to import medicinal cannabis to Australia for the purposes of patient prescription, proving him to be a pivotal player in the budding domestic medical cannabis industry.
In a recent interview, the pair expressed to me that their background in the medical sciences and an observable lack of research-based information on the topic was what inspired them to start the MCRA. The pair travelled to Canada and Israel, as well as the United States, where they got to gain valuable insight into the current legal medical markets, which could later be utilized to further development of the Australian Industry.
The underlying goal of these seminars is re-education, and the pair feel that their health professional counterparts should feel obligated to get involved, as their main role is, after all, to assist the patient to the best of their abilities. In our consultation, Paul discussed the need for a medically controlled and tested product to ensure consistency and maximum results.
For the Australian Medicinal Cannabis industry to be successful in delivering medicine to patients, the doctors need to be the ones with the information and skills necessary to prescribe it. Gaining import licenses is a crucial first step in this task as it supplies doctors and medical institutes with the product necessary to conduct the aforementioned research.
The Mavor's selfless efforts in organising these events has certainly not been in vain, with a number of powerful speakers agreeing to give presentations at the seminars. Their website notes that the speakers will consist of Pharmacists, Prescribers, Scientists/Researchers, Patient Advocates and Health Professionals.
A number of notable presenters include; Rhys Cohen – Senior Project Officer at The Lambert Initiative (Sydney University), several scientists from the Lambert Initiative, Dr Teresa Towpik – General Practitioner and author of "An Introduction to Medicinal Cannabis"' as well as Dr Ian Brighthope of Nutrition Care Pharmaceuticals.
Whether you're looking to further your knowledge on the medical applications of the cannabis plant, or are simply interested in hearing a doctor's perspective on the issue, check out if there's an event happening close to you!
Catch the Medicinal Cannabis Research Australia team at the upcoming seminars for Health Professionals in March 2018 and grab yourself some tickets at their website, MCRA.org.au
Share your views with our community in the comments section below.
Matt Buchanan
Cannabis advocate, activist and enthusiast. I've been studying Cannabis and its counterculture extensively since 2011, and am extremely passionate about medicinal cannabis due to my family's relationship with cancer. My role at Greendorphin involves interviewing members of the Canna-community and sharing their story, as well as writing some personal pieces. When I'm not blogging you can find me playing basketball, reading books and listening to East Coast Hip Hop. My favourite food is pizza, my favourite strain is Sour Diesel.
Latest posts by Matt Buchanan (see all)
Dr John Teh & PlantMed Medicinal Cannabis Clinic: Greendorphin News Podcast #1 - February 7, 2019
Anatomy of the Cannabis Plant: Grow with Greendorphin - December 6, 2018
High Hemp Organic Wraps Review CBD+ [VIDEO] - November 26, 2018
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redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Форматирование чисел в JS. 01 вместо 1 Нужна помощь, в cursera задание по js, нужно чтобы выводилось корректное время, при добавлении интервала.
module.exports = function(hours, minutes, interval) {
  if (hours >= 0, hours <= 23,
    minutes >= 0, minutes <= 59) {
    if (minutes + interval >= 60) {
      hours = hours + 1;
      minutes = (minutes +
        interval - 60)
    } else {
      minutes = minutes + interval
    };
    if (minutes.length = 1) {
      miutes = '0' + minutes
    };
    return hours + ':' + minutes
  } else {};
};

Суть в том, что я не могу понять, как сделать, чтобы вместо 1, на вывод приходило 01. На сайте была подсказка:
Для выполнения этого задания могут понадобиться методы глобального объекта Math. Например, метод floor.

A: String.prototype.padStart() заполняет текущую строку другой сторокой (несколько раз, если нужно) так, что итоговая строка достигает заданной длины. Заполнение осуществляется в начале (слева) текущей строки.
Вот готовая функция:
function padNum(var num) {
    return num.toString().padStart(2,0);
}

A: Можно использовать такую функцию:
function MakeTwoDigits(num) {
  if( num >= 0 && num <= 9) {
    return "0" + num;
  }
  else {
    return "" + num;
  }
}

A:

console.log([0, 1, 9, 10, 59].map(x => ("0"+x).slice(-2)).join(" "))

redpajama_set_name: RedPajamaStackExchange


My personality and my dreams: I am a very active girl, my friends say I'm always positive-charged and often ask what I do to be so happy and joyful. I see this world in bright colors, I love nature and traveling, I love the movement in any form – dancing, running, cycling, evening walks. At the same time I love to read books on psychology, so I know that relationship is like "building a house" where both partners must consider each other requests, be attentive and hear each other.


What does it feel like to leave one's home behind and start a whole new life somewhere else?
In the German Emigration Center, you can follow the fate of an emigrant: the farewell from home; the passage and arrival in the New World. Elaborate reconstructions and multi-media stagings breathe life into the emigration.
Since the prize-winning experience museum's expansion, you can now travel further into the interior of the USA. In a replica of New York Central Train Station, you can discover how emigrants settled down as immigrants in their new homeland, built up a home for themselves and where they found work.
In the adjacent annex building you can set off on a personal search for traces and, amidst everyday utensils, get to know 15 moving family stories of immigrants who had come to Germany since the 17th century.
Immigrants and emigrants united in the best European museum 2007.
The German Shipping Museum is located directly by the Weser dyke promenade with the museum harbour. The collection comprises more than 500 ship models: be it magnificent sailing yachts, modern day tankers or the oldest boat on earth - in the area surrounding the German Shipping Museum, every seafarer can set off on a discovery tour!
With the unique specialisation in Nordic fauna, the theme zoo is among the most modern facilities in the whole of Europe. The location, directly on the Weser dyke allows unique integration of the large, nature-like zoo facilities into the natural ocean seascape.
The North Sea aquarium delights visitors with its marine diversity - immerse yourself in a vividly-coloured underwater world!
Visit the public viewing platform and discover the unique view across the maritime city of Bremerhaven!
With an altitude of 147 metres, ATLANTIC Hotel Sail City is one of the highest buildings along the German North Sea coast. At a height of 86 metres, you are afforded an exclusive view across Bremerhaven and the surrounding area from the publicly accessible viewing platform.


text:
Tara Lynne Barr (born October 2, 1993) is an American actress. She is best known for portraying Roxy in the dark comedy film God Bless America (2011), which earned her a Young Artist Award nomination for Best Leading Young Actress in a Feature Film, and Laura Meyers in the Hulu comedy-drama series Casual.

Early life
Barr was born in Orange County, California, on October 2, 1993, and was raised in Fountain Valley, California. The daughter of long-time employees of Southern California Edison Company, her parents enrolled Barr and her older sister in the before- and after-school care program of the Boys & Girls Clubs of Huntington Valley. When the club began its Clubhouse Theater program, Barr's sister performed in several children's theater productions, inspiring Barr to take up acting. She was seven years old. Barr attended Fountain Valley public schools, and graduated from Fountain Valley High School in 2011. She was accepted to Chapman University's College of Performing Arts on the Dean's Scholarship, but left the school after a year to focus solely on her career.

Career
Barr performed in local children's theater productions until, at the age of 10, she began to express interest in film and television work. Shortly thereafter she obtained an agent and started working professionally. At the age of 11, she starred in the short dark comedy film Road Kill, which won the Audience Choice Award at the 2005 New York City Horror Film Festival. While still not yet in her teens, she appeared as the host of Encore Channel's Fear Fest: 31 Nights of Horror, as well as in television series such as Joan of Arcadia, Crossing Jordan, Drake & Josh, Zoey 101, and The Suite Life of Zack & Cody.

After her high school graduation in 2011, Barr starred in the dark comedy film God Bless America, her break out role. Following this, Barr starred as the title role in the short film Dawn, Rose McGowan's directorial debut. This was followed by a major recurring role on the NBC series, Aquarius, starring David Duchovny, in which Barr portrayed the character Katie, who was based on real-life Manson Family member Patricia Krenwinkel. Barr left the series before its second season to star in the Hulu original series, Casual, as Laura Meyers. Casual aired for four seasons from 2015 to 2018. Barr was praised for her performance.

In 2020, Barr was cast in the AMC series Dispatches From Elsewhere to portray the young version of star Sally Field's character.

Filmography

Film

Television

Video games

Audio

Awards and nominations

References

External links

1993 births
Actresses from Orange County, California
American child actresses
American film actresses
American musical theatre actresses
American television actresses
Living people
21st-century American actresses

redpajama_set_name: RedPajamaWikipedia


Its hard to believe this is the eighth year of this event. What started out as an , has now become the show of the year. With over $600K raised for medical teams, creating many amazing relationships, this event just keeps getting better and better.
We were honored to brand this event with a fresh new look with a focus on music and art. Keeping it fresh is always a challenge, but we always get inspired from the supporting artists and mainly the cause.


Authentic MYRIAM SCHAEFER Joyce Bicolor Small Satchel. Black Saffiano leather with contrast box calfskin leather and silver tone hardware. Navy arrow tote handles with perforated details; removable fixed length shoulder strap. Two way domed zip top and sides; designer foiled logo at upper front center, hanging snap charm with serial number. Gussets keep items secure; navy leather feet protect the base. Saffron leather lining with two open slip and one wall zip pocket. Great pre-owned condition; light wear consistent with gentle use. The exterior leather shows the faintest rubbing to the bottom corners and feet. Navy handles have minor creasing with a small faint scuff; tiny nick to a front arrow tip that has lifted the navy surface (photo 7). The interior leather has some light marks with a few pen marks as shown in photos. Please review all photos as they are considered part of the description. Includes duster.
Measures approx: 12" L x 9.5" H x 6.5" D Handle Drop: 4" Strap Drop: 21"


This 100% Pure Estate Grown Kona Coffee is grown on the Big Island of Hawaii.
This 100% Pure Estate Grown Kona Coffee is grown on the Big Island of Hawaii. Pure Kona is one one of the top grades of Kona Coffee and has a rich aroma, sweet taste, and medium body.
This coffee is given a medium roast.


MWO: Forums - Recruiting Scandinavian Volunteer Moderator!
We're currently looking to recruit a Volunteer Moderator who can speak both English and Norwegian.
This Volunteer Moderator would communicate with us (The Devs.) on current situations and discussions on the Scandinavian forums as well as moderate and possibly even run small forum events for this community, such as participation and contest type events!
Please post on this thread if you're interested!
I might be interested, but my real-life schedule is rather crammed at the moment, so it would depend on how much time I would be required to commit towards moderator duties.
Edited by Garegaupa, 09 April 2015 - 05:59 AM.
The Space Pope is up for the task and he is currently a language moderator on another forum of another game developer (and would be willing to provide a link or further information by PM if required), which he feels provides him with a solid background for a similar task on these forums.
He is also quite active and generally considers himself to be a neutral fellow on the forums.
Edited by The True Space Pope, 11 April 2015 - 04:02 PM.
I think it might be hard to breathe some life into the Scandinavian community but it is a nice initiative!
But I hope that I am wrong though.
Is it restricted to norwegian only?
I think she really want someone who speak Swedish - Nordic.
Men som du ser så är det här forumet helt stendött.
Så brukar det tyvärr vara, vi Skandinavier är ofta rätt så internationella av oss.


Case Taping Packaging testing is another term use for adhesive testing. The self-adhesives are used to ensure the quality of the products that are packed inside the products. Adhesives and tapes are responsible for maintaining and to ruin the quality of the goods. If the quality of the adhesives and tapes is not up to the mark, it can spoil the quality of the content which is packed inside the packet and vice versa. To ensure the quality of the packaging products, it is necessary first to judge the quality of the case taping with the help of high-quality of adhesives or case taping packaging testing instruments.
Presto Stantest, the foremost testing instruments manufacturer, is all set to manufactures a wide range of packaging testing instruments that are recommended by the distinguished companies. Also to fulfill different testing needs, we at Presto plan, develop and execute a wide range of Case Taping Packaging Testing machines that completely matches with global standards like ISO, ASTM and JIS.
To measure the adhesion strength of the packaging products, adhesives and case tapings, Presto offers a wide range of testing instruments which includes, Peel/ Seal Adhesion tester, Heat Sealer, Seal integrity tester and many more. For more information on these instruments, consult our experts.
Not only does the company bring itself a leading manufacturer of Laboratory testing instruments all across the globe, but also this means fully to turn its potential clients into regular customers. Since the company stepped in the market, this has been actually leading to run well by the best quality practices in producing lab and other testing instruments.
For more information, consult our technocrats.


Jesse Roberson from Scheller's Fitness and Cycling introduces us to the 2012 Trek 7.3 FX Hybrid Bicycle. This bike is great for short rides or for longer trips. Designed for fitness with flat handlebars. All around tires can even take you off the pavement.


text:
\section{Introduction}

Recently, there has been a renewed and avid interest in studying a class of materials known as the
left-handed materials (LHMs). These materials have simultaneously negative real parts of dielectric permittivity $\rho$ and
magnetic permeability $\mu$, so that their refractive index is negative. The properties of such meterials were investigated first by Veselago in 1967 \cite{Veselago}.
As shown by Veselago, LHMs exhibit some peculiar electromagnetic properties such as negative index of
refraction and  wave vector, $\textbf{k}$, and Poynting vector, $\textbf{S}$, having opposite
directions. Veselago realized that a slab of LHM would act as a lens.

Due to the absence of naturally ocurring materials possessing both negative permittivity and
negative permeability, Veselago's predictions did not receive much attention until recently,
when a material with both negative permittivity and negative permeability at microwave frequencies was built \cite{Smith}.
Subsequently, the properties of LHMs were analyzed by many authors.

According to Abbe's diffraction limit, conventional lenses based on positive index materials with curved surfaces
are not able to resolve an object's fine details that are smaller than half of the light wavelength $\lambda$.
The limitation occurs because the waves with transverse wave numbers larger than $2 \pi n/\lambda$, which carry
information about the fine sub-$\lambda$ details of the object, decay exponentially in free space.
In a negative index material slab, however, the evanescent wave components can grow exponentially
and thus compensate for the exponential decay. Therefore, under ideal conditions, all Fourier components from
the object can be recovered at the image plane producing a resolution far below the diffraction limit \cite{Pendrytwo}.

Milton \textit{et al.} proved superlensing in the quasistatic
regime (where the wavelength is much larger than the object), and discused limitations of superlenses in this
regime due to anomalous localized resonance. If the source being imaged responds to an applied field, it must lie
outside the resonant regions to be successfully imaged \cite{Milton}.

In the electrostatic limit, the magnetic and electric fields decouple, and the requirement for superlensing of
transverse-magnetic waves is reduced to only $\rho = - \rho_h$, where $\rho_h$ is the permittivity of the host medium
interfacing the lens \cite{Pendrytwo}. An example of such near field superlens is a slab of silver in air illuminated
at its surface plasmon resonance (where $\rho = -1$). Experiments with silver slabs have already shown rapid growth
of evanescent waves \cite{Liu}, submicron imaging \cite{Melville}, and imaging beyond the diffraction limit \cite{Fang}.
A major draw-back of such near-field superlenses based on bulk metals is that they can operate only at a single frequency
$\omega$ satisfying the lens condition $\rho (\omega) = - \rho_h$. Shalaev \textit{et al.} proposed a "tunable"
near-field superlens made of metal-dielectric composites that can operate at any desired visible or near-infrared wavelength
with the frequency controlled by the metal filling factor of the composite (here the inhomogeneities are assumed to be much
smaller than the wavelength) \cite{Shalaev}.

It was shown by Efros \textit{et al.} that a two dimensional photonic crystal made from a non-magnetic dielectric
has negative values of both the electric permittivity and the magnetic permeability in some frequency range
\cite{Efros_one}, and the photonic crystal behaves like a real LHM with respect to the propagating modes only.
 If amplification of evanescents waves occurs, it is due to some other reason; for example, excitation of surface waves by
the evanescent waves. Such an amplification may provide an improvement of the image in the near-field region, but
it does not affect the image near the far-field focal point \cite{Efros_two}.

The physical principles that allow negative refraction in photonic crystals arise from the dispersion
characteristics of wave propagation in periodic media and are very different from those in LHMs. They also do not
require both negative electric permittivity and magnetic permeability \cite{Kosaka,Notomi}.
The negative refraction of beams can be described by analyzing the equifrequency surface of the band
structures \cite{Kosaka,Notomi,Luo}. If the constant-frequency contour is everywhere convex, an
incoming plane wave from air will couple to a single mode that propagates into the crystal on the negative side of
the boundary, and thus negative refraction in the first band is realized.  Luo \textit{et al.} have shown all-angle
negative refraction could be achieved at the lowest band of two-dimensional photonic crystals in the case of
$\textbf{S} \cdot\textbf{k} > 0$ \cite{Luo}. Such all-angle refraction is essential for superlens application.
The photonic crystal not only focuses all propagating waves without limitation of finite aperture, but also
amplifies at least some evanescent waves, and the unconventional imaging effects are due to the
presence of additional near-field light. A perfect lens, made of left-handed materials, focuses all
propagating waves and all evanescent waves. The important difference for superlensing with a
photonic crystals is that only finite number of evanescent waves is amplified. This is a consequence of Bragg
scattering of light to leaky photon modes \cite{Luotwo}. The resolution of a photonic-crystal superlens at a single
frequency is only limited by its surface period instead of the wavelength
\cite{Luotwo}.

More recently Huang \textit{et al.} proposed an
alternative approach to all-angle negative refraction in two-dimensional photonic crystals. By
applying appropriate modifications with surface grating to the flat photonic lens, he is able to focus large
and/or far way objects \cite{Huang}.

Inspired by the current research in structures that produce sub-wavelength focusing, we use derivative-based
minimization techniques to produce structures that will provide sub-wavelength focus with non-magnetic materials and without the need for negative
permittivities.
Rather than restricting to designs based on photonic crystal structures,
we allow as admissible {\em any} periodic composite structure (with
fixed period) whose refractive index is bounded above and below by
fixed constants.  And rather than performing parametric optimization over
a small number of variables describing the structure, we use techniques
of ``topology optimization'' in which material distribution is completely
arbitrary. We are able to obtain structured that focus a point source
that is far away from the lens, and also we can obtain structures that give an
image at a chosen distance from the lens.  Since structures incorporating
gratings are included in our admissible class, such designs
will naturally arise through the optimization process if they
produce the best possible image.

For simplicity, only the case of ``two-dimensional'' structures
in $E$-parallel polarization is considered.  The ideas here should
extend to the other polarization case
and the full three-dimensional problem, although there are some
technical hurdles.

The paper proceeds as follows. In Section 2 we describe the model problem and review a variational formulation
of the Helmholtz equation in a periodic geometry.  The inclusion of a
small amount of energy absorption in the medium allows a uniform
upper bound on the norm of the electric field, independent of the
particular admissible structure (and thereby preventing resonances).
 In Section 3 we present the optimization problem and derive the
optimality conditions. In Section 4 we depict the numerical experiments and show structures that have produced sub-wavelength
focus.

\section{Model problem}
In this paper we consider time-harmonic electromagnetic wave propagation through nonmagnetic $(\mu = 1)$
heterogeneous media for which the dielectric coefficient is constant in one direction, i.e.
$\epsilon(x,y,z)= \rho(x,y)$. Assuming that the electric field vector $E=(0,0,u)$, Maxwell's equations
reduce to the Helmholtz equation
\begin{equation} \label{helmholtz}
\triangle u + \omega^2 \rho u = 0, \quad \mbox{in ${\mathbb R}^2$},
\end{equation}
where $\omega$ represents the frequency, and
$\rho \in L^\infty({\mathbb R}^2)$
is the dielectric coefficient.

\subsection{Periodic structure}
Assume that the dielectric coefficient $\rho(x,y)$ is
periodic in the $x$ variable
\[ \rho (x,y) = \rho (x+2\pi,y), \quad
                             \mbox{for all $(x,y) \in {\mathbb R}^2$}.
\]
Taking the period to be $2 \pi$ imposes no loss of generality since
any other period can be obtained by rescaling $\omega$.

Assume that the regions $\{ y > 0 \}$, and
$\{ y < -b \}$ are  homogeneous, for some fixed constant $b > 0$.
In particular, assume for $y > 0$ and $y < -b$, that
$\rho(x,y) = 1$.  The slab $-b < y < 0$ may
contain inhomogeneous material.

Suppose a point source is placed above the slab at the point
$(0,h)$, which generates the incident field
$u_i(x,y) = H_0^{(1)}(\omega r)$, where
$r = \sqrt{x^2 + (h - y)^2}$ is the distance from the source,
and $H_0^{(1)}$ is the Hankel function.  For $y < h$, we have
the representation
\[ u_i(x,y) = \frac{1}{\pi} \int_\mathbb R \frac{1}{\beta(\xi)}
e^{i \xi x - i \beta(\xi) (y - h)} \, d\xi,
\]
where $\beta(\xi) = \sqrt{\omega^2 - \xi^2}$ whenever
the argument is positive,
and $\beta(\xi) = i \sqrt{\xi^2 - \omega^2}$ otherwise \cite{Morse}.
It follows that
\[ f(x) \equiv u_i(x,0) =
\frac{1}{\pi} \int_\mathbb R \frac{e^{i\beta(\xi)h}}{\beta(\xi)}
e^{i \xi x} \, d\xi,
\]
and
\[ g(x) \equiv \frac{\partial u_i}{\partial y}(x,0) =
\frac{-i}{\pi} \int_\mathbb R e^{i\beta(\xi)h} e^{i \xi x} \, d\xi.
\]
We can rewrite
\[ f(x) = \int_{-1/2}^{1/2} f_\alpha(x) e^{i\alpha x} \, d\alpha,
\quad \mbox{where } \;\;
f_\alpha(x) = \frac{1}{\pi} \sum_{n \in {\mathbb Z}}
\frac{e^{i\beta(n+\alpha )h}}{\beta(n + \alpha )} e^{inx},
\]
and
\[ g(x) = \int_{-1/2}^{1/2} g_\alpha(x) e^{i\alpha x} \, d\alpha,
\quad \mbox{where } \;\;
g_\alpha(x) = \frac{-i}{\pi} \sum_{n \in {\mathbb Z}}
   {e^{i\beta(n+\alpha )h}} e^{inx}.
\]
($f_\alpha$ may fail to converge, but only at the isolated
values of $\alpha$ for which $(n+\alpha)^2 = \omega^2$ for some $n$.
$g_\alpha$ always converges due to the exponential decay in $n$.)

Above the slab $y > 0$, we separate the solution $u$
to (\ref{helmholtz})
into the incident and scattered field: $u = u_i + u_s$.
The scattered field $u_s$ can also be separated by the Fourier
transform in $x$,
\[ u_s(x,y) = \int_\mathbb R \hat{u}_s(\xi,y) e^{i\xi x}\, d\xi.
\]
Plugging this representation back into (\ref{helmholtz}),
and solving for each frequency $\xi$ separately, we find
that
$\hat{u}(\xi,y) = a(\xi)e^{i\beta(\xi) y} + b(\xi) e^{-i\beta(\xi) y}$.
The second term on the right corresponds to an {\em incoming} wave,
which we insist must be zero since we want the scattered field to
consist only of outgoing waves.  Then
$\hat{u}_s(\xi,y) = a(\xi) e^{i\beta(\xi)y}$.
It follows that $u_s(x,0) = \int a(\xi) e^{i\xi x} \, d\xi$,
and
\begin{eqnarray*}  \frac{\partial u_s}{\partial y}(x,0) &=&
 \int i\beta(\xi) a(\xi) e^{i\xi x} \, d\xi \\
 &=& \int i\beta(\xi) \hat{u}_s(\xi,0) e^{i\xi x} \, d\xi
 \equiv (Tu_s)(x).
\end{eqnarray*}
The linear operator $T$ (Dirichlet-to-Neumann map)
then defines the relationship between the traces
$u_s|_{\{y=0\}}$ and $\partial_y u_s|_{\{y = 0\}}$:
$T(u_s|_{\{y = 0\}}) = (\partial_y u_s)|_{\{y=0\}}$.
On the boundary $\{ y = 0\}$, the solution
$u = u_i + u_s$ should then satisfy
\begin{eqnarray*}
 \partial_y u - Tu &=& \partial_y u_i - Tu_i + \partial_y u_s -Tu_s
 = g -Tf  \\
 &=& 2g.
\end{eqnarray*}

Define the periodic domain (circle)
\[ \mathcal S = \mathbb R/2 \pi \mathbb Z. \]
Define the first Brillouin zone $K=[-\frac{1}{2}, \frac{1}{2}]$. To reduce the problem (\ref{helmholtz}) over ${\mathbb R}^2$
to a family of problems over $\mathcal S \times \mathbb R$, we define for $g \in L^2({\mathbb R}^2)$ the Floquet transform
$\mathcal F$ by
\[\mathcal F (g) = e^{-i\alpha x} \sum_{n \in Z} g(x-2 \pi n,y)e^{i 2 \pi \alpha n}, \quad \alpha \in K. \]
The sum can be considered as a Fourier series in the quasi-momentum variable $\alpha$, with values in
$L^2(\mathcal S \times \mathbb R)$. The map $g \mapsto \mathcal F g$ is an isomorphism from $L^2({\mathbb R}^2)$ to the direct
product space $\int^\oplus_K L^2(\mathcal S \times \mathbb R)$ \cite{Kuchment}.
Floquet theory assures that the solution
$u$ can be written
\begin{equation} \label{uexp}
 u(x,y) = \int_{-1/2}^{1/2} u_\alpha(x,y)e^{i \alpha x} \, d\alpha,
\end{equation}
where each function $u_\alpha$ is $2\pi$-periodic in the $x$
variable, and satisfies the equation
\[ \triangle_\alpha u_\alpha + \omega^2 \rho u_\alpha = 0,
\]
where $\triangle_\alpha = \triangle + 2 i \alpha \partial_1 - |\alpha|^2$.
The boundary condition
$\partial_y u - T u = 2g$ on $\{y = 0\}$ translates to
\[
 \partial_y u_\alpha  - T_\alpha u_\alpha = 2 g_\alpha, \quad \mbox{on $\{y = 0\}$},
\]
where
\begin{equation} \label{Tdef}
T_\alpha u_\alpha = \sum_{n\in {\mathbb Z}}
 i \beta(n + \alpha) \hat{u}_\alpha(n) e^{inx},
\end{equation}
(here $\hat{u}_\alpha(n)$ are the Fourier series coefficients of $u_\alpha$ with respect to the $x$-variable).
Similar considerations apply at the lower boundary of the slab
$\{ y = -b \}$, where (assuming there is no incoming
wave coming from below) we find
$\partial_y u_\alpha + T_\alpha u_\alpha = 0$.

\subsection{Existence and uniqueness of solutions}
Let $\Omega = \mathcal S \times (0, -b)$, $\Gamma_0 = \{y = 0 \}$, $\Gamma_b = \{y = -b \}$.
Define an admissible class of dielectric coefficients
\[ \mathcal{A} = \{ \rho = \rho_r + i\rho_{i} \in L^\infty(\Omega):
  \rho_{r_0} \leq \rho_r(x) \leq \rho_{r_1}
  \mbox{ and }\rho_{{i}_0} \leq \rho_{i}(x) \leq \rho_{{i}_1} \; \mbox{a.e.}\},
\]
where $\rho_{{r}_0}$, $\rho_{{i}_0} > 0$.
Given the incident wave $u_i$ generated by the point source
at $(0,h)$, we must solve the family of problems
\begin{align} \label{feqn}
\triangle_\alpha u_\alpha + \omega^2 \rho_r u_\alpha + i\omega^2 \rho_{i} u_\alpha & =  0,
                                       &\mbox{in $\Omega$}\\
\nonumber
(\frac{\partial}{\partial y} - T_\alpha) u_\alpha & = 2g_\alpha,
                        &\mbox{ on $\Gamma_0$} \\
\nonumber
(\frac{\partial}{\partial y} + T_\alpha) u_\alpha & = 0,
                        &\mbox{ on $\Gamma_b $},
\end{align}
for all $\alpha \in [-\frac{1}{2}, \frac{1}{2}]$.
Existence and uniqueness of weak solutions, with a uniform bound, may be
obtained for $\rho_{i} > 0$.

\begin{lemma} \label{Solution}
For each $\rho \in  {\mathcal A}$ with $\rho_{i} > 0$ and $\alpha \in [-\frac{1}{2}, \frac{1}{2}]$, problem  (\ref{feqn})
admits a unique weak solution $u_\alpha \in H^2(\Omega)$.  Furthermore,
there exists a constant $C$ depending on $\rho_{i}$, ${\mathcal A}$,
such that $\| u_\alpha \|_{H^2(\Omega)} \leq C$,
independent of $\rho \in  {\mathcal A}$ and $\alpha$.
\end{lemma}
\begin{proof}
For convenience we drop the subscript $\alpha$ on solutions.
Define for $u$, $v \in H^1(\Omega)$
\[ a(u,v) = \int_\Omega \nabla u \cdot
               \overline{\nabla v} -
               \omega^2\int_\Omega \rho u \overline{v} + \alpha^2\int_\Omega  u \overline{v} - 2i\alpha \int_\Omega \partial_x {u} \overline{v} -
	        \int_{{\Gamma}_0} (T_{\alpha} u) \overline{v} - \int_{{\Gamma}_b} (T_{\alpha} u) \overline{v},
\]
and
\[ b(v) =   2\int_{{\Gamma}_0} g_{\alpha} \overline{v} .
\]
It is straightforward to show
that $a(u,v)$ defines a bounded sesquilinear form over
$H^1(\Omega) \times H^1(\Omega)$, and that
$b(v)$ is a bounded linear functional on $H^1(\Omega)$.
Weak solutions $u \in H^1(\Omega)$ of (\ref{feqn}) solve the
variational problem
\begin{equation} \label{varprob}
a(u,v) = b(v) \quad \mbox{for all $v \in H^1(\Omega)$}.
\end{equation}
The sesquilinear form $a$ uniquely defines a linear operator $A: H^1(\Omega)
\rightarrow H^1(\Omega)$
such that $a(u,v) = \langle Au, v \rangle_{H^1(\Omega)}$,
and the functional $b(v)$ is uniquely
identified with an  element $b \in H^1(\Omega)$ such
that $b(v) = \langle b, v \rangle$
by reflexivity and an abuse of notation.  Problem (\ref{varprob}) is
then equivalently stated
\begin{equation} \label{abst_varprob}
A u = b.
\end{equation}

We intend to show that $a$ is coercive by establishing a bound
$|a(u,u)| \geq c > 0$
for all $u \in H^1(\Omega)$ with $\| u \|_{H^1(\Omega)} = 1$.

Integrating by parts in $x$, we see by periodicity that
 \[ \int_\Omega (\partial_x {u}) \overline{u} = 0 \]

 Let $\Lambda^+ (\alpha) = \lbrace n \in {\mathbb Z} \colon Im (\beta_n) = 0 \rbrace$ and $\Lambda^-(\alpha) = {\mathbb Z} - \Lambda^+ (\alpha)$.
 Each $\Lambda^+ (\alpha)$ is a finite set and $0 \in \Lambda^+ (\alpha)$.
  We can write
 \[ - \int_{{\Gamma}_j} (T_{\alpha} u) \overline{u} =
 - \int_{\Gamma_j} \sum_{n \in \Lambda^+ (\alpha)} i \beta_n(\alpha)\hat{u}(n) e^{i nx} \overline{u}
 - \int_{\Gamma_j} \sum_{n \in \Lambda^- (\alpha)} i \beta_n(\alpha)\hat{u}(n)
 e^{ inx} \overline{u},  \]
for $j = 0$, $b$. Notice that all $\beta_n(\alpha) \in \Lambda^- (\alpha)$ satisfy $- i \beta_n \geq 0$,
so the second term on the right hand side of the equation above is real and non-negative.
  We then have
\begin{eqnarray*}
a(u,u)   =  \int_\Omega |\nabla u|^2 + \alpha^2 \int_\Omega |u|^2 +
\int_{\Gamma_j} \sum_{n \in \Lambda^- (\alpha)} -i\beta_n(\alpha)\hat{u}(n) e^{inx} \overline{u}
 - \omega^2\int_\Omega \rho_r |u|^2\\
 - i\int_{\Gamma_j} \sum_{n \in \Lambda^+ (\alpha)} \beta_n(\alpha)\hat{u}(n) e^{i nx} \overline{u}
 -i\omega^2\rho_{i} \int_\Omega |u|^2.
    \end{eqnarray*}
Assuming $\|u\|_{H^1(\Omega)}^2 =
    \int_\Omega |\nabla u|^2 +\int_\Omega |u|^2  = 1$,
and noticing that
the first four terms on the right-hand side are purely
real and the last two terms
are purely imaginary, we find
\begin{eqnarray*}
2| a(u,u) |   \geq  \left| 1
+ \int_{\Gamma_j} \sum_{n \in \Lambda^- (\alpha)} -i\beta_n(\alpha)\hat{u}(n) e^{inx} \overline{u}
                   - \int_\Omega (1 + \omega^2 \rho_r - {\alpha}^2 ) |u|^2 \right| \\
         + \left| \int_{\Gamma_j} \sum_{n \in \Lambda^+ (\alpha)} \beta_n(\alpha)\hat{u}(n) e^{i nx} \overline{u} +
\omega^2 \rho_{i} \int_\Omega |u|^2 \right|.
\end{eqnarray*}
For convenience, write
$t = \int_{\Gamma_j} \sum_{n \in \Lambda^- (\alpha)} -i\beta_n(\alpha)\hat{u}(n) e^{inx} \overline{u} $,
$r = \int_\Omega(1 + \omega^2 \rho_r \}) |u|^2$, and
$s = \int_\Omega |u|^2$.
Obviously $t,r$, and $s$ are nonnegative real numbers which depend on $u$ (and
$\rho$ in the case of $r$).
Although $t$ and $s$ are essentially independent, $r$ must satisfy
\begin{equation} \label{rbounds}
(1 + \rho_{r_0})s \leq r \leq (1 + \rho_{r_1}) s.
\end{equation}
With this notation,
\[ 2|a(u,u)| \geq |1 + t +{\alpha}^2 s - r| + \omega^2 \rho_{i} s.
\]
Note that in the case $s \geq  \frac{1}{2(1+\rho_{r_1})}$,  we have
$|a(u,u)| \geq \frac{1}{2}\omega^2 \rho_{i} s \geq \frac{\omega^2 \rho_{i}}{4(1+\rho_{r_1})}$.
Otherwise, $s <  \frac{1}{2(1+\rho_{r_1})}$ so that $r < \frac{1}{2}$, and
$|a(u,u)| \geq \frac{1}{2}|1 + t - r|  > \frac{1}{4}$.
Hence, for all $s, t \geq 0$, and all $r$ satisfying (\ref{rbounds}),
\[ |a(u,u)| \geq  c =
\min\left\{\frac{\omega^2 \rho_{i}}{4(1+\rho_{r_1})}, \frac{1}{4} \right\}.
\]
The bound thus holds for every $u$ with $\| u \|_{H^1(\Omega)} = 1$
and for every
$\rho \in {\mathcal A}$ with $\rho_{i} > 0$.
Given this coercivity bound, direct application of the Lax-Milgram
Theorem yields
existence of the bounded  solution operator
$A^{-1}$ for problem (\ref{abst_varprob})
such that $\| A^{-1} \| \leq 1/c$.
Thus $\| u \|_{H^1(\Omega)} \leq \| b \|_{H^1(\Omega)}/c$.

Given the  bound on $\| u \|_{H^1(\Omega)}$, a uniform $H^2(\Omega)$ bound
follows easily, since $\triangle_{\alpha} u = - \omega^2 \rho u$
is uniformly bounded in $L^2(\Omega)$.
\end{proof}

The complete solution $u$
to the original problem (\ref{helmholtz}) can then be
reconstructed from (\ref{uexp}).

\section{Optimal design}

The goal of the optimization is to make a perfect image of
the incident field $u_i=H_0^{(1)}(\omega r)$ on the opposite side of the slab
$\{ y < -b \}$.  A ``mirror image'' to the incident field
converging at the point $(0, -(b+h_1))$ would look like
$H_0^{(2)}(\omega\sqrt{x^2 + (y+b+h_1)^2})$, where
$H_0^{(2)} = \overline{H_0^{(1)}}$
is the conjugate Hankel function.  Thus we want the trace
\[ u(x,-b) = H_0^{(2)}(\omega\sqrt{x^2 + h_1^2}) \equiv q(x).\]

\begin{figure}
\centerline{
\includegraphics[width=70mm]{modelproblem1.eps}
}
\label{figure1}
\caption{Model problem.  A time-harmonic wave from a point source
is incident from above.  We wish to determine the unknown periodic
medium such that a focus is attained below.}
\end{figure}

The Bloch representations of $u$ and $f$ allows us to see that by setting
\begin{equation} \label{d_cond}
 u_\alpha(x,-b) = q_\alpha(x),
\end{equation}
with $q_\alpha$ defined similarly to $f_\alpha$, we get $u(x,-b) = q(x)$.

Problem (\ref{feqn}) together with the additional boundary
condition (\ref{d_cond}) is overposed.  However, by allowing
$\rho$ to vary as a design variable, it may be possible to
make (\ref{d_cond}) hold approximately for each $\alpha$.

\subsection{Problem Formulation}

Let $F(\rho,\alpha) = u_\alpha|_{\Gamma_b}$,
where $u_\alpha \in H^1(\Omega)$ is the
weak solution to problem (\ref{feqn}).

Consider the minimization
\begin{equation} \label{minprob}
 \inf_{\rho \in \mathcal{A}} J(\rho) =
           \frac{1}{2} \int_{-1/2}^{1/2}
\| F(\rho,\alpha) - q_\alpha \|^2_2 \, d\alpha.
\end{equation}

\begin{theorem} \label{Optimization}
The optimization problem has a solution.
\end{theorem}
\begin{proof}
The proof follows the well-known direct method in the calculus of varations.
$\mathcal A$ is weak $\star$ $\ L^{\infty}$ compact. Consider a minimizing sequence $\lbrace \rho_n \rbrace$ with some subsequence
(still denoted by $\lbrace \rho_n \rbrace$) converging weak $\star$ $\ L^{\infty}$ to some $\rho \in \mathcal A$.
Denote by $u_n$ the solution to the boundary value problem corresponding to $\rho_n$. By Lemma (\ref{Solution}),
the sequence $\lbrace u_n \rbrace$ has has a weakly convergent
subsequence (still denoted $\lbrace u_n \rbrace$), $u_n \rightharpoonup u$ in $H^1(\Omega)$ for some $u \in H^1(\Omega)$.
Hence, $u_n \longrightarrow u$ strongly in $L^2(\Omega)$. We hold $v \in H^1(\Omega)$ fixed,
and we have $u_n\overline{v} \longrightarrow u \overline{v}$ strongly in $L^1(\Omega)$.
For $k$ fixed, $a_{\rho_k} (u_n,v) \longrightarrow a_{\rho_k} (u,v)$ as $n \longrightarrow \infty$.
Here we used the fact that $T^{\alpha}:H^{1/2}(\Gamma_j) \to H^{-1/ 2}(\Gamma_j)$ is continuous.
Since $u \overline{v} \in L^1(\Omega)$, $a_{\rho_k} (u,v) \longrightarrow a_{\rho} (u,v)$
as $\rho_k \longrightarrow \rho$ weak $\star$ $L^\infty$.
The trace map extends uniquely to a continuous linear operator $\tau:H^1(\Omega) \longrightarrow H^{1/2}(\Gamma)$.
Thus, the traces are also convergent:
$u_n|_{\Gamma_j} \rightharpoonup u|_{\Gamma_j}$ weakly in $H^{1/2}(\Gamma)$.
This implies $F(\rho_n, \alpha) \rightharpoonup F(\rho, \alpha)$ weakly in $H^{1 /2}(\Gamma)$,
and, hence, $F_\alpha: \mathcal A \longrightarrow H^{1/2}(\Gamma)$ is weak $\star$ $L^\infty$ continuous for fixed $\alpha$.
But our bound on the solution is independent of $\alpha$, and thus, the minimization problem has at least one solution $\rho \in \mathcal A$.
\end{proof}

\subsection{Adjoint-state derivatives}

Let $\delta \rho = \delta \rho_r + i \delta \rho_{i}$ be a "small" perturbation to the coefficient
$\rho$. We denote the
linearization of $J(\rho)$ with respect to $ \delta \rho$ by $DJ(\rho)\delta \rho$. We have
\begin{eqnarray*}
 DJ(\rho) =& \int_{-\frac{1}{2}}^{\frac{1}{2}} Re \langle DF_{\alpha}(\rho)\delta \rho, F(\rho,\alpha) -
q_{\alpha} \rangle_{L^2(\Gamma_b)} \, d\alpha \\
=& \int_{-\frac{1}{2}}^{\frac{1}{2}} Re \langle \delta \rho, DF_{\alpha}^{\star}(\rho) ( F(a,\alpha) -
 q_{\alpha} )\rangle_{L^2(\Omega)} \, d\alpha.
\end{eqnarray*}
For fixed $\alpha$, \[ DF_{\alpha}(\rho): L^2(\Omega)
\longrightarrow L^2(\Gamma_b) \]

and
\[DF_{\alpha}^{\star}(\rho): L^2(\Gamma_b)
\longrightarrow L^2(\Omega).\]

$DF_{\alpha}(\rho)(\delta \rho) = \delta u_{\alpha}\mid_{\Gamma_b}$ where $\delta u_{\alpha}$ solves the linearized problem:
\begin{align} \label{perturbprob}
\triangle_\alpha \delta u_{\alpha} + \rho \delta u_{\alpha} & =  -\delta \rho u,
                                       & \mbox{in $\Omega$}\\
\nonumber
(\frac{\partial}{\partial y} - T_\alpha) \deltau_\alpha & = 0,
                        & \mbox{on $\Gamma_0$} \\
\nonumber
(\frac{\partial}{\partial y} + T_\alpha) \deltau_\alpha & = 0,
                        & \mbox{on $\Gamma_b$},
\end{align}

The $L^2$ adjoint of the derivative $DF_{\alpha}(\rho)(\cdot)$ is the linear operator $DF_{\alpha}^{\star}(\cdot)$ such that
\[\langle DF_{\alpha}(\rho)(\delta \rho), \psi \rangle_{L^2(\Gamma_b)} = \langle \delta \rho, DF^{\star}(\rho)(\psi)\rangle_{L^2(\Omega)}.\]
For all $\psi \in L^2(\Gamma_b)$ let $w_{\alpha}$ solve

\begin{align} \label{wprob}
\triangle_\alpha w_{\alpha} + \omega^2\rho_r w_{\alpha} - i\omega^2 \rho_{i} w_{\alpha} & = 0,
                                       & \mbox{in $\Omega$}\\
\nonumber
(\frac{\partial}{\partial y} - T^{\star}_\alpha) w_{\alpha} & = 0,
                        & \mbox{on $\Gamma_0$} \\
\nonumber
(\frac{\partial}{\partial y} + T^{\star}_\alpha) w_{\alpha} & = -\psi,
                        & \mbox{on $\Gamma_b$},
\end{align}
where $T^{\star}_\alpha f = -\sum i \overline{\beta}_n \hat{f}(n) e^{inx}$.
An integration by parts argument shows that
\[ \int_{\Gamma_b} \delta u_{\alpha} = \int_{\Omega} \delta \rho u_{\alpha}
\overline {w}_{\alpha}.\]
The $L^2$ "gradient" of the functional $J(\rho)$ is the function $G(\rho) \in L^2(\Omega)$ for which
\[ DJ(\rho)(\delta \rho) = Re \left ( \int_{\Omega} \delta \rho G(\rho) \right ) \quad \mbox{and} \quad
G(\rho) = \int_{-\frac{1}{2}}^{\frac{1}{2}}u_{\alpha}\overline{w}_{\alpha} \, d\alpha \]
where $w_{\alpha}$ solves (\ref{wprob}) with $\psi = F (\rho,\alpha) - q_\alpha$.

 Define the normal cones at the minimizer $\hat {\rho}= \hat{\rho_r} + i \hat{\rho_{i}}$:
\[ N(\hat{\rho_r}) = \left \lbrace \xi \in (L^\infty)': \int_\Omega (\hat{\rho_r} - \rho_r)d \xi \geq 0 \quad
\forall \rho_r \in \mathcal A \right \rbrace,\]
and
\[ N(\hat{\rho_{i}} )= \left \lbrace \zeta \in (L^\infty)':
\int_\Omega (\hat{\rho_{i}} - \rho_{i}) d \zeta \geq 0 \quad
\forall \rho_{i} \in \mathcal A \right \rbrace.\]
The gradient is normal to $\mathcal A$ at $\hat{\rho}$: $-G \bigcap N(\hat{\rho}) \not= 0$.
Thus, we have
\[ - \int_\Omega (\hat{\rho_r} - \rho_r) Re ( G(\rho)) dx \geq 0 \quad \forall \rho_r \in \mathcal A. \]
The continuity of the gradient is sufficient to reduce the above to pointwise optimality conditions. In
particular,
\begin{align*}
\hat{\rho_r} = \rho_{r_{0}} \quad  &\Rightarrow \quad Re(G(\rho)) > 0 \\
\rho_{r_{0}} < \hat{\rho_r} < \rho_{r_{1}} \quad  &\Rightarrow \quad Re(G(\rho)) = 0 \\
\hat{\rho_r} = \rho_{r_{1}} \quad  &\Rightarrow \quad Re(G(\rho)) < 0
\end{align*}
for almost every x in $\Omega$.

Similarly,
\[\int_\Omega (\hat{\rho_{i}} - \rho_{i}) Im ( G(\rho)) dx \geq 0 \quad \forall \rho_{i}
\in \mathcal A,\]
and the pointwise optimality conditions are:
\begin{align*}
\hat{\rho_{i}} = \rho_{i_{0}} \quad  &\Rightarrow \quad Im(G(\rho)) < 0 \\
\rho_{i_{0}} < \hat{\rho_{i}} < \rho_{i_{1}} \quad  &\Rightarrow \quad Im(G(\rho)) = 0 \\
\hat{\rho_{i}} = \rho_{i_{1}} \quad  &\Rightarrow \quad Im(G(\rho)) > 0
\end{align*}
for almost every x in $\Omega$.

In a similar optimal design problem involving waveguides, one can
use the weak continuation property to show
\cite{Dobson} that at an optimal $\rho$,
at least one of the four equality bound constraints
above on the real and imaginary parts of $\rho$ is attained at almost
every point $x \in \Omega$.  A similar argument for the present problem
is greatly complicated by the form of $\nabla J$, involving an integral
of a family of PDE solutions, rather than just a single background-adjoint
pair.  A precise a-priori characterization of optimal solutions is thus
difficult.

\section{Numerical Results}
Approximate solutions of (\ref{minprob}) are sought through numerical
discretization and optimization.  The variational problem
(\ref{varprob}) was discretized with a first-order finite element
method, using piecewise bilinear elements on a uniform, rectangular
grid.  The design variable $\rho$ was approximated by a piecewise
constant function on the same uniform grid.  The nonlocal boundary
operators $T_\alpha$ defined by (\ref{Tdef}) were approximated
by explicitly calculating the Fourier coefficients of the traces
of the finite element basis, then truncating the sum in (\ref{Tdef}).
The resulting finite element scheme can be shown to converge and to
conserve energy, provided
all the propagating terms are included in the sum \cite{Bao}.
This discretization leads to a large, sparse (except for the
boundary terms), non-Hermitian matrix problem, which for simplicity
is solved using the direct sparse solver in {\em Matlab}.

The integral in (\ref{minprob}) was approximated by a discrete
sum in $\alpha$.  By imposing $x$-axis symmetry in the designs,
it suffices to integrate only over positive $\alpha$.  In the
following examples, we used 20 equally-spaced positive values
of $\alpha$ to approximate the integral.

Despite the convenience of imposing a positive lower bound on
the imaginary part of $\rho$ in Lemma~2.1 for obtaining a uniform
upper bound on solutions, we found that the numerical experiments
were quite insensitive to small dissipations. Thus in most of
the examples below, we set $\rho_{i_0} = 0$.

After discretizing $J(\rho)$ through finite elements, optimization
was accomplished with a straightforward projected gradient descent
algorithm as in \cite{Dobson}, using the adjoint as derived in
Section~3 to calculate the gradient.
We performed a large number of numerical experiments with this
method, using different initial guesses for the design variable
$\rho$, and varying the frequency $\omega$, source and focus
locations $h$ and $h_1$,
and constraints $\rho_{r_0}, \rho_{r_1}, \rho_{i_0}, \rho_{i_1}$
on the real and imaginary parts of $\rho$.

Generally speaking, we found that the method was able to produce,
from almost any initial guess, a structure which produced
relatively high field intensity near the desired focus.  Some
parameter choices and initial guesses resulted in structures with
much better focusing properties than others.

In the first experiment we start with a purely real material and
allow the real part of the dielectric coefficient to vary between
$\rho_{r_0} = 1$ and $\rho_{r_1}=12$. The source is positioned at
$2.5$ units from the slab ($h=2.5$), and we are looking to obtain a focus
$2.5$ units on the opposite side of the slab ($h_1=2.5$). Through numerical optimization
we discover the structure shown in Figure \ref{example1} which gives a spot size
$0.424 \lambda$.
True subwavelength imaging is only possible if evanescent modes are present
at the interface between the structure and the transmission medium.
Figure~\ref{mode_plot} shows the evanescent modes for the generated image versus those of the
target, clearly showing that the structure produces evanescent modes
which approximate those of the objective.

In the second example we image a source far away from the lens ($h=90$), and we are looking for
the lens that will produce the best image ten units from the lens ($h_1=10$). Far away objects
are much more difficult to image, but through optimization of the structure, we are able to
obtain a spot of size $0.38 \lambda$.
In the third example we start with a photonic crystal and allow purely real structures
varying between $\rho_{r_0} = 1$ and $\rho_{r_1}=12$. The source is positioned at
$2.4$ units from the slab ($h=2.4$) and we are looking to obtain a focus
four units away on the opposite side of the slab ($h_1=4$). The optimized structure is shown in Figure \ref{example3},
and it gives a focus with spot size $0.395 \lambda$, which is much
better than the one obtained if we just use photonic crystal as suggested by Luo \textit{et al.}
which produced a focus with spot size $0.67 \lambda$ \cite{Luo}.

Example four allows both the real and imaginary parts of the dielectric coefficient to vary
($\rho_{r_0} = 1$, $\rho_{r_1}=12$, and $\rho_{i_0} = 0$, $\rho_{i_1}=1$).
The distance between the source and the lens and
the distance between the lens and the image are set to four units ($h=4$, $h_1=4$).
The optimized structure gives a focus with a
spot size $0.284 \lambda$ (Figure \ref{example4}), which is a significant improvement to those obtained by
structures described in the research literature so far, although as far as
we know, no real materials exist with these dielectric coefficients.

All of the numerical
experiments were computationally intensive.  Each iteration
required the solution of a family of diffraction problems, two for each
$\alpha$, and because of the crude optimization method employed,
many iterations were typically required.  Most of the examples
below took on the order of two days to run on a workstation.
Since our purpose here was simply to illustrate the feasibility
of designing such structures through mathematical optimization,
we did not devote much effort to improving the efficiency of the
numerical methods.  We believe that computation time could be
improved by at least an order of magnitude with existing, but
more sophisticated, numerical methods.

\begin{figure}
\centerline{
\begin{tabular}{c}
\includegraphics[width=60mm]{ex1_eps_initial.eps}
\hspace*{4mm}
\includegraphics[width=60mm]{ex1_epsilon.eps} \\
\includegraphics[width=60mm]{ex1_spot.eps}
\hspace*{4mm}
\includegraphics[width=60mm]{ex1_realu.eps}
\end{tabular}
}
\caption{Results from Example 1. Upper left: initial (real) $\rho$;
upper right: optimized solution; lower left: intensity cross section
(blue) versus point source (red).  Spot size is 0.424; lower right:
real part of $E$ field
within the solution box.  Eight periods are shown.}
\label{example1}
\end{figure}

\begin{figure} \label{mode_plot}
\centerline{
\includegraphics[width=60mm]{mode_plot.eps}
}
\caption{Magnitude of evanescent modes for Example 1 (blue), versus
those of the target (red).}
\end{figure}

\begin{figure}
\centerline{
\begin{tabular}{c}
\includegraphics[width=60mm]{ex6_eps_initial.eps}
\hspace*{4mm}
\includegraphics[width=60mm]{ex6_epsilon.eps} \\
\includegraphics[width=60mm]{ex6_spot.eps}
\hspace*{4mm}
\includegraphics[width=60mm]{ex6_realu.eps}
\end{tabular}
}
\caption{Results from Example 2. Upper left: initial (real) $\rho$;
upper right: optimized solution; lower left: intensity cross section.
Spot size is 0.380; lower right: real part of $E$ field
within the solution box.  Eight periods are shown.}
\label{example2}
\end{figure}

\begin{figure}
\centerline{
\begin{tabular}{c}
\includegraphics[width=60mm]{ex3_eps_initial.eps}
\hspace*{4mm}
\includegraphics[width=60mm]{ex3_epsilon.eps} \\
\includegraphics[width=60mm]{ex3_spot.eps}
\hspace*{4mm}
\includegraphics[width=60mm]{ex3_realu.eps}
\end{tabular}
}
\caption{Results from Example 3. Upper left: initial (real) $\rho$;
upper right: optimized solution; lower left: intensity cross section
(blue) versus point source (red).  Spot size is 0.395; lower right:
real part of $E$ field
within the solution box.  Eight periods are shown.}
\label{example3}
\end{figure}

\begin{figure}
\centerline{
\begin{tabular}{c}
\includegraphics[width=60mm]{ex4_eps_initial.eps}
\hspace*{4mm}
\includegraphics[width=60mm]{ex4_epsilon.eps} \\
\includegraphics[width=60mm]{ex4_spot.eps}
\hspace*{4mm}
\includegraphics[width=60mm]{ex4_realu.eps}
\end{tabular}
}
\caption{Results from Example 4. Upper left: initial (real) $\rho$;
upper right: optimized solution; lower left: intensity cross section
(blue) versus point source (red).  Spot size is 0.284; lower right:
real part of $E$ field
within the solution box.  Eight periods are shown.}
\label{example4}
\end{figure}

\section{Conclusions}
We have demonstrated the feasibility of designing periodic structures
with subwavelength focusing properties, via mathematical optimization.
The approach is mathematically sound and through numerical
discretization yields plausible, if somewhat non-intuitive, solutions.

We point out two weaknesses with the approach presented here, all
of which we believe could be improved through further work.
First, the performance of the
optimized structures tends to be very sensitive to small perturbations
in material parameters.  This fact combined with the relative
complexity of the solutions means that attempting to fabricate
these structures is at this point not an attractive idea.
Both the sensitivity and the complexity could be addressed through
adding appropriate constraints or penalties to the objective. For
example, optimization through a level-set approach would
necessarily yield structures composed of only two materials,
with no intermediate-index areas \cite{Santosa}.
This together with total variation penalties may
yield ``simpler'' solutions.  Of course such constraints
may incur a decrease in the performance of the structure.

Second, the designs created with this approach are not translationally
invariant.  Specifically, moving the structure laterally relative
to the point source may result in decrease of focus.  There are a few
ways to circumvent this problem.  In the simplest case, the distance
$h$ from the point source to the structure is large so that the
wavefront impinging on the structure is nearly planar.  Our experiments
have shown that in this case focusing is not dependent on the lateral
position of the point source (and is much less senstive to vertical
translations), although the focus does translate periodically with
the structure.  One can also optimize for structures whose
period is very small compared to the wavelength, although such
structures tend to have less focusing power.  The ultimate solution
to this problem would require building the translation invariance
of the focus (not of the medium) into the objective function.
Unfortunately this would increase the complexity of the computations
beyond the capability of the simplistic approach presented here,
but with further work and refinement it should be possible.

redpajama_set_name: RedPajamaArXiv