This work was funded by the Science and Technology Facilities Council (STFC).

<ol>
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Authors have nothing to disclose.

The output beams of most lasers used in high-precision measurements are designed to have a shape well described as a fundamental Gaussian mode. This particular beam geometry combines low diffraction with a spherical wave front. While the low diffraction is one of the key advantages of laser light, the spherical wave front is equally important, as it allows the low-loss transformation of the laser beam by standard optical components with spherical surfaces. Different beam shapes can be created as well, and recently Laguerre-Gauss beams have become of interest for their potential application in high-precision interferometry.
In this paper we demonstrated the experimental procedure to create higher-order Laguerre-Gauss modes with 95% purity for high-power, ultra stable laser beams. To achieve this, we have combined standard techniques from different aspects of optical research, namely diffractive phase plates and laser pre-stabilization to mode cleaner cavities. Our experiment provides a simple, modular and very reliable method to create high power beams in user defined higher-order modes. A commercial ultra-stable laser is used as the light source. Its output is injected to a diffractive phase plate, which can convert up to 75% of the light into the desired Laguerre-Gauss mode. This light is then injected to a small optical cavity and an electronic feedback loop is used to stabilize the laser frequency of the laser to the cavity length. The beam transmitted by the cavity is to 95% in the desired mode and, like the fundamental mode beam at the origin of the setup, has very good frequency stability at audio frequencies. All the parts represent standard components in modern optical experiments. We have successfully demonstrated this technique for laser powers up to 80 W pure Laguerre-Gauss 33 mode.
It could be possible to achieve similar results by replacing the phase plate with another mode-converting element (for example, other diffractive elements or astigmatic mode converters). Alternatively a laser could be setup with an optical resonator tuned for the desired Laguerre-Gauss modes, using for example, an amplitude mask. Finally the laser frequency stabilization to the reference optical cavity could be exchanged with a similar scheme that uses an atomic reference. The need for an electronic feedback system is probably the main disadvantage, but this is inevitable for any light source used for precision interferometer.
However, we believe that the method demonstrated in this paper provides a simple and modular scheme which can be scaled to all ranges of required laser frequency, power, or shape and thus presents a powerful and versatile method. Each part, the laser source, the diffractive element, as well as the optical cavity can be changed or optimized individually, which means that also existing laser injection systems can be upgraded to use Laguerre-Gauss modes.

Durante as últimas décadas experimentos de alta precisão interferometria foram empurrados em direção a um regime sensibilidade final, quando os efeitos quânticos estão começando a desempenhar um papel decisivo. Nestes experimentos em curso e futuras, tais como resfriamento a laser de osciladores mecânicos 1, armadilhas ópticas para espelhos 2, a geração de massas de teste emaranhados 3, quantum não-demolição de interferometria 4, estabilização de frequência de lasers com cavidades rígidas 5 e 6 de detecção de ondas gravitacionais , 7, 8, os investigadores enfrentam uma multiplicidade de fontes de limitação fundamental e técnica de ruído. Um dos problemas mais graves é o ruído térmico dos espelhos da cavidade das configurações interferométrico, que é causada pela excitação térmica dos átomos que formam os substratos de espelho e o espelho de revestimentos anti-reflexo 7, 8, 9. Este efeito, também chamado movimento browniano, vai causar uma incerteza na fase dea luz refletida a partir de qualquer massa de teste e, portanto, irá se manifestar como uma limitação do ruído fundamental na saída interferômetro. Por exemplo, a sensibilidade do projeto previsto de avançado gravitacional onda antenas, tais como Advanced LIGO, Advanced Virgo, eo Telescópio Einstein, é limitado por este tipo de ruído na região mais sensível da banda de freqüência de observação de 10, 11, 12. Físicos experimentais da comunidade trabalhar duro em um esforço contínuo para minimizar essas contribuições de ruído e melhorar a sensibilidade de seus instrumentos. No caso particular de espelho Browniano ruído, um método para atenuação é empregar um feixe de maior tamanho da mancha utilizado actualmente norma fundamental HG feixe 00 sobre as superfícies de ensaio em massa, uma vez que a maior média de feixe de forma mais eficaz sobre os movimentos da superfície aleatórios 13, 14. A densidade espectral de potência do ruído térmico espelho tenha sido mostrado com a escalao inverso do tamanho do feixe Gaussiano espelho para o substrato e com o inverso do quadrado da superfície do espelho 9. No entanto, como os pontos de feixe são feitos maior, uma maior fracção da energia de luz é perdida ao longo da borda da superfície reflectora. Se é utilizada uma viga com uma distribuição de intensidade radial mais homogénea do que o normalmente usado HG feixe 00 (ver por exemplo Figura 1), o nível de ruído térmico Browniano pode ser reduzido sem aumentar este tipo de perda. Entre todos os tipos de feixe mais homogêneos que foram sugeridas por novas versões de alta precisão interferometria, por exemplo vigas Mesa ou modos cônicas 13, 14, os mais promissores são de maior ordem LG vigas devido à sua compatibilidade com o potencial esférico usado atualmente superfícies de espelho 15. Por exemplo, a taxa de detecção de estrela de nêutrons binário em sistemas de espiral - que são consideradas as fontes astrofísicas mais promissores para uma primeira GW detectarião - poderia ser melhorado em cerca de um factor de 2 ou mais 16 com o custo de uma quantidade mínima de modificações no desenho de interferómetros de segunda geração em construção 10, 11. Em adição aos benefícios de ruído térmico, as distribuições de intensidade mais amplas de ordem superior LG vigas (ver por exemplo, Figura 2) foram mostrados para atenuar a amplitude das aberrações ópticas térmicas de dentro dos interferómetros. Isso reduziria o grau em que os sistemas de compensação térmica são invocados em experimentos futuros para atingir as sensibilidades de design 19. Investigámos e demonstrado a viabilidade de gerar LG vigas aos níveis de pureza e de estabilidade necessário para operar com êxito interferómetros GW ao melhor à sua sensibilidade de 16, 18, ​​19, 20, 21, 22. O método proposto combina técnicas e competências desenvolvidas em diversas áreas da física e da ótica sucessoh como a geração de elevada estabilidade, baixo ruído único modo de feixes de laser 23, a utilização de moduladores espaciais de luz e os elementos ópticos de difracção para a manipulação dos perfis espaciais de feixes de luz 18, 22, 24, 25, 26, e a utilização de técnicas avançadas para a detecção, de controlo e estabilização das cavidades ópticas de ressonância 27 tendo em vista uma mais purificação e estabilização da luz do laser. Este método tem sido demonstrado com sucesso em experiências laboratoriais, exportada para testes em interferómetros protótipo em larga escala de 20, e para a geração de modos LG em potências laser de alta até 80 W 21. Neste artigo, apresentamos os detalhes do método de geração de ordem superior LG vigas e discutir uma metodologia para a caracterização e validação do feixe resultante. Além disso, no Passo 4 um método para investigações numéricos de cavidades com espelhos não-perfeito 19 está delineado.

<table border="1">
	<tbody>
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			<td>The experimental apparatus discussed in this paper requires the following types of instruments:</td>
		</tr>
		<tr>
			<td></td>
			<td></td>
			<td></td>
			<td>Instrument</td>
		</tr>
		<tr>
			<td>Solid state Laser source, Nd:YAG 1064 nm CW laser</td>
			<td></td>
			<td></td>
			<td>Quantity: 1</td>
		</tr>
		<tr>
			<td>Faraday Isolator</td>
			<td></td>
			<td></td>
			<td>Quantity: 1</td>
		</tr>
		<tr>
			<td>Electro-Optic Modulator (EOM)</td>
			<td></td>
			<td></td>
			<td>Quantity: 1</td>
		</tr>
		<tr>
			<td>CCDcamera beam profiler</td>
			<td></td>
			<td></td>
			<td>Quantity: 1</td>
		</tr>
		<tr>
			<td>Lenses</td>
			<td></td>
			<td></td>
			<td>Quantity: depending on apparatus design</td>
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			<td>Steering Mirrors</td>
			<td></td>
			<td></td>
			<td>Quantity: depending on apparatus design</td>
		</tr>
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			<td>Aperture</td>
			<td></td>
			<td></td>
			<td>Quantity: 1</td>
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			<td>High reflectivity mirrors (for normal incidence)</td>
			<td></td>
			<td></td>
			<td>Quantity: 2</td>
		</tr>
		<tr>
			<td>Piezoelectric ring</td>
			<td></td>
			<td></td>
			<td>Quantity: 1</td>
		</tr>
		<tr>
			<td>Cavity spacer</td>
			<td></td>
			<td></td>
			<td>Quantity: 1</td>
		</tr>
		<tr>
			<td>Photodiodes and related control electronics</td>
			<td></td>
			<td></td>
			<td>Quantity: 1 or more, depending on apparatus design</td>
		</tr>
		<tr>
			<td>Spatial light modulator</td>
			<td></td>
			<td></td>
			<td>Quantity: 1 
			Holoeye LCR-2500</td>
		</tr>
		<tr>
			<td></td>
			<td></td>
			<td></td>
			<td>All the above instruments are commercially available and no particular specification is required. We leave the choice of the most suitable instruments to the experimenter's discretion.</td>
		</tr>
		<tr>
			<td></td>
			<td></td>
			<td></td>
			<td>For the interest of the experimenter interested in reproducing the protocol, we recommend the following tools used in our experiment:</td>
		</tr>
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			<td></td>
			<td></td>
			<td></td>
			<td>Tools</td>
		</tr>
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			<td>Innolight OEM 300NE, 1064 nm, 300 mW</td>
			<td></td>
			<td></td>
			<td>Laser Source:</td>
		</tr>
		<tr>
			<td>SIMTOOLs</td>
			<td></td>
			<td></td>
			<td>Software for data analysis, available at <a href="http://www.gwoptics.org/simtools/" target="_blank">www.gwoptics.org/simtools/</a></td>
		</tr>
		<tr>
			<td>FINESSE</td>
			<td></td>
			<td></td>
			<td>Software for optical simulations, <a href="http://www.gwoptics.org/finesse/" target="_blank"> www.gwoptics.org/finesse/</a></td>
		</tr>
		<tr>
			<td></td>
			<td></td>
			<td></td>
			<td>Finally, the phase plate employed in the present experiment was manufactured by Jenoptik GmbH, based on a custom design provided by the Authors.</td>
		</tr>
	</tbody>
</table>

the generation of higher-order laguerre-gauss optical beams for high-precision interferometry

Ruído térmico em espelhos de alta refletividade é um grande impedimento para vários tipos de alta precisão interferométrico experimentos que visam atingir o limite quântico padrão ou para resfriar os sistemas mecânicos para o seu estado fundamental quântico. Isto é, por exemplo, espera-se o caso de futuras observatórios de ondas gravitacionais, cuja sensibilidade aos sinais de ondas gravitacionais para ser limitado na banda de frequência mais sensível, por vibração atómica das suas massas espelho. Uma abordagem promissora a ser prosseguido para ultrapassar esta limitação é empregar de ordem superior Laguerre-Gauss feixes ópticos (LG), em lugar do modo fundamental utilizado convencionalmente. Devido à sua distribuição de intensidade de luz mais homogénea dessas vigas em média, de forma mais eficaz sobre as flutuações accionado termicamente da superfície de espelho, o que por sua vez reduz a incerteza sobre a posição do espelho detectada por a luz laser. Nós demonstramos um método promissor para gerarde maior ordem LG vigas por moldar um feixe fundamentais Gaussian com a ajuda de elementos ópticos difrativos. Mostra-se que, com a detecção convencionais e técnicas de controlo, que são conhecidos por estabilizar raios laser fundamentais, os modos de ordem mais elevada LG pode ser purificado e estabilizado tão bem a um nível comparativamente alto. Um conjunto de ferramentas de diagnóstico nos permite controlar e adequar as propriedades dos feixes gerados LG. Isto permitiu-nos produzir um feixe LG com a mais alta pureza relatado até o momento. A compatibilidade demonstrada de modos de ordem superior LG com técnicas de interferometria padrão e com o uso da óptica esféricos padrão torna um candidato ideal para a aplicação de uma geração futura de alta precisão interferometria.

All the experimental results so far described in the text and shown in the figures constitute a representative example of a successful execution of the beam conversion protocol. The most representative result is the purity of the generated beam: a successful beam conversion should lead to a beam purity on the order of 95% or above. A good example of successful beam conversion is the measurement of the intensity profile of an 82.8 Watts, 96% pure LG33 beam obtained in 21 and here shown in Figure 12.
Similarly, as discussed in protocol sections, the mode conversion efficiencies of the phase plate and of the overall experimental setup are a good indicator of the successful design of the experimental apparatus, including the phase plate and the mode cleaner cavity. Values of order 50% to 60% and above are generally considered a good value for the mode conversion efficiency. The highest conversion efficiency reported so far with this type of setup is about 70% 21.
The simulation investigation described in Protocol Sec 4 should result in numbers for beam purity with realistic mirrors, suggested mirror specifications and the resulting beam purity when these specifications are adopted. An example of the results you can expect with realistic mirror maps are shown in 19 where an original LG33 purity of 89% is obtained, compared to a purity of &gt;99% for HG00. A purity of &gt;99% for the LG33 mode is achieved using specific mirror requirements with a major reduction of astigmatism in the mirror surface.
<img alt="Figure 1" src="/files/ftp_upload/50564/50564fig1.jpg" /> 
Figure 1. Intensity patterns for Hermite-Gauss (HG) modes up to order 6. The intensity patterns are normalized to have the same peak intensity, for visibility.
<img alt="Figure 2" src="/files/ftp_upload/50564/50564fig2.jpg" /> 
Figure 2. Intensity patterns for helical LG modes up to order 9. The intensity patterns are normalized to have the same peak intensity, for visibility.
<img alt="Figure 3" fo:conten-width="5in" fo:src="/files/ftp_upload/50564/50564fig3highres.jpg" src="/files/ftp_upload/50564/50564fig3.jpg" /> 
Figure 3. Sketch of a conventional setup for production and stabilization of HG00 beams.
<img alt="Figure 4" fo:conten-width="5in" fo:src="/files/ftp_upload/50564/50564fig4highres.jpg" src="/files/ftp_upload/50564/50564fig4.jpg" /> 
Figure 4. A sketch of the experimental setup discussed in this paper. The HG00 beam is first mode-matched to a desired waist size via a telescope then injected on the phase plate. The main diffracted beam is separated from the higher diffraction orders with an aperture and then sent to the Mode Cleaner cavity. A photodiode is used to extract the error signal for controlling the cavity length. The beam intensity is analyzed by a CCD camera.
<img alt="Figure 5" src="/files/ftp_upload/50564/50564fig5.jpg" /> 
Figure 5. Phase modulation profile to convert a HG00 mode to LG33 mode.
<img alt="Figure 6" src="/files/ftp_upload/50564/50564fig6.jpg" /> 
Figure 6. Example of blazed phase modulation profiles for generating LG33 modes.
<img alt="Figure 7" src="/files/ftp_upload/50564/50564fig7.jpg" /> 
Figure 7. Comparison between the intensity distribution of the composite beam generated by the phase plate (left) and the theoretical intensity distribution for a pure LG33 beam with same parameters.
<img alt="Figure 8" src="/files/ftp_upload/50564/50564fig8.jpg" /> 
Figure 8. Example of beam intensity profile fitting applied to a real LG33 beam transmitted from a phase plate (left) compared to fit results (center) and residuals of fit (right). <a href="https://www.jove.com/files/ftp_upload/50564/50564fig8large.jpg" target="_blank"> Click here to view larger figure</a>.
<img alt="Figure 9" fo:conten-width="4.5in" fo:src="/files/ftp_upload/50564/50564fig9highres.jpg" src="/files/ftp_upload/50564/50564fig9.jpg" /> 
Figure 9. Profile of an LG33 beam with Gaussian fit shown for comparison.
<img alt="Figure 10" fo:conten-width="4.5in" fo:src="/files/ftp_upload/50564/50564fig10highres.jpg" src="/files/ftp_upload/50564/50564fig10.jpg" /> 
Figure 10. Light power transmitted by a linear cavity as a function of the cavity length, when injecting a beam generated by the phase plate. The resonant peaks at 0 and 1 FSR correspond to the desired LG33 mode. A fit to this dominant mode is shown for comparison (blue line). The red curve shows the result of the numerical model, based on the modal content described in Table 2. Pictures of the unwanted beams to be filtered by the cavity are shown in the insets.
<img alt="Figure 11" fo:conten-width="5in" fo:src="/files/ftp_upload/50564/50564fig11highres.jpg" src="/files/ftp_upload/50564/50564fig11.jpg" /> 
Figure 11. An example of a mirror surface map for one of the Advanced LIGO optical mirrors 19.
<img alt="Figure 12" src="/files/ftp_upload/50564/50564fig12.jpg" /> 
Figure 12. Intensity profile of a 82.8 W LG33 beam transmitted by a linear cavity (left) compared with fit residuals (right).
<table border="1">
	<tbody>
		<tr>
			<td>p</td>
			<td>l</td>
			<td>0</td>
			<td>1</td>
			<td>2</td>
			<td>3</td>
			<td>4</td>
			<td>5</td>
			<td>6</td>
			<td>7</td>
			<td>8</td>
			<td>9</td>
		</tr>
		<tr>
			<td colspan="2">0</td>
			<td>1.0</td>
			<td>2.8</td>
			<td>1.7</td>
			<td>2.0</td>
			<td>2.2</td>
			<td>2.5</td>
			<td>2.6</td>
			<td>3.1</td>
			<td>3.0</td>
			<td>3.2</td>
		</tr>
		<tr>
			<td colspan="2">1</td>
			<td>1.7</td>
			<td>2.7</td>
			<td>2.2</td>
			<td>2.4</td>
			<td>2.6</td>
			<td>2.8</td>
			<td>2.9</td>
			<td>2.8</td>
			<td>&nbsp;</td>
			<td>&nbsp;</td>
		</tr>
		<tr>
			<td colspan="2">2</td>
			<td>2.2</td>
			<td>2.4</td>
			<td>2.5</td>
			<td>2.7</td>
			<td>2.9</td>
			<td>3.0</td>
			<td>&nbsp;</td>
			<td>&nbsp;</td>
			<td>&nbsp;</td>
			<td>&nbsp;</td>
		</tr>
		<tr>
			<td colspan="2">3</td>
			<td>2.5</td>
			<td>2.7</td>
			<td>2.8</td>
			<td>3.0</td>
			<td>&nbsp;</td>
			<td>&nbsp;</td>
			<td>&nbsp;</td>
			<td>&nbsp;</td>
			<td>&nbsp;</td>
			<td>&nbsp;</td>
		</tr>
		<tr>
			<td colspan="2">4</td>
			<td>2.9</td>
			<td>3.0</td>
			<td>&nbsp;</td>
			<td>&nbsp;</td>
			<td>&nbsp;</td>
			<td>&nbsp;</td>
			<td>&nbsp;</td>
			<td>&nbsp;</td>
			<td>&nbsp;</td>
			<td>&nbsp;</td>
		</tr>
	</tbody>
</table>
Table 1. Optimum ratio between input HG00 beam size and LGpl phase image beam size for LG modes up to the order 9.
<table border="1">
	<tbody>
		<tr>
			<td>Mode</td>
			<td>LG33</td>
			<td>LG63</td>
			<td>LG43</td>
			<td>LG53</td>
			<td>LG32</td>
			<td>LG62</td>
		</tr>
		<tr>
			<td>Power</td>
			<td>75%</td>
			<td>8%</td>
			<td>4%</td>
			<td>4%</td>
			<td>4%</td>
			<td>1%</td>
		</tr>
	</tbody>
</table>
Table 2. Mode content analysis described by the cavity scan shown in Figure 10.

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The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry

Large laser-interferometers are being constructed to create a new type of astronomy based on gravitational waves. Their sensitivities, as for many other high-precision experiments, are approaching fundamental noise limits such as the atomic vibration of their components. We are pioneering technologies to overcome these limits using novel laser beam shapes.

A geração de alta ordem Laguerre-Gauss feixes óticos para High-precision Interferometria

Thermal noise in high-reflectivity mirrors is a major impediment for several  types of high-precision interferometric experiments that aim to reach the  ...

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21.4K visualizações.  University of Birmingham. Large laser-interferometers are being constructed to create a new type of astronomy based on gravitational waves. Their sensitivities, as for many other high-precision experiments, are approaching fundamental noise limits such as the atomic vibration of their components. We are pioneering technologies to overcome these limits using novel laser beam shapes.