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

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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.

지난 수십 년 동안 고정밀 간섭 실험은 양자 효과에 결정적인 역할을 시작하는 최고의 감도 정권을 향해 밀고 있었다. 기계적 오실레이터 1, 미러 2, 얽혀 시험 질량의 3 세대, 양자 비 철거 간섭 4, 엄격한 공동 5와 레이저의 주파수 안정화 및 중력파 검출 6 광학 트랩의 레이저 냉각 등이 진행 중이거나 향후 실험에서 , 7, 8, 연구자들은 기본적인 기술 노이즈 소스를 제한하는 군중을 직면하고있다. 가장 심각한 문제 중 하나는 미러 기판과 거울 반사 코팅 7, 8, 9를 구성하는 원자의 열 자극에 의해 발생되는 간섭 설정의 공동 거울의 열 잡음이다. 또한 브라운 운동 불리는이 효과는의 단계에서 불확실성의 원인이됩니다빛은 모든 시험 질량에서 반사 간섭계의 출력에 근본적인 소음 제한과 같은 것 때문에 매니페스트. 예를 들어, 고급 LIGO, 고급 VIRGO, 그리고 아인슈타인 망원경과 같은 고급 중력파 안테나,의 예상 디자인 감도는 관측 주파수 대역 10의 가장 민감한 지역, 11, 12의 노이즈의이 유형에 의해 제한됩니다. 사회의 실험 물리학 자들은 이러한 잡음 기여를 최소화하고 자신의 악기의 감도를 개선하기 위해 지속적인 노력에 최선을 다하고 있습니다. 거울 브라운 노이즈의 특별한 경우 완화를위한 하나의 방법은 표면의 임의의 움직임을 통해보다 효율적으로 큰 빔 평균 때문에, 시험 물질 표면에 현재 사용되는 표준 기본 HG 00 빔의 큰 빔 스폿의 크기를 고용하는 것입니다 13, 14. 미러 열 잡음의 전력 스펙트럼 밀도는로 확장하기 위해 표시되었습니다미러 기판과 거울 표면 9 역 광장 가우시안 빔의 크기의 역수. 빔 스팟이 큰 만들어진 그러나, 가벼운 힘의 더 큰 부분은 반사 표면의 가장자리에 손실됩니다. 하나는 일반적으로 사용되는 HG 00 빔 (예 : 그림 1 참조)보다 더 균일 한 반경 강도 분포와 빔을 사용하는 경우, 브라운 열 잡음 레벨은 손실의 유형을 증가하지 않고 감소 될 수있다. 고정밀 간섭계의 새 버전에 대해 제안 된 모든 더 균일 한 빔 타입 사이에, 예를 들어 메사 빔 또는 원뿔 형태 13, 14, 가장 유망한 높은 순서 LG는 현재 사용중인 구형 자신의 잠재적 인 호환성으로 인해 광선입니다 거울 표면 15. 예를 들어, 나선형 시스템의 바이너리 중성자 별의 검출 율 - 첫 GW에 가장 유망한 천체 물리학 소스로 간주됩니다 검색이온 - 현재 건설 10, 11에 따라 2 세대 간섭계의 설계 변경의 최소 금액의 비용을 2 개 이상 16 약 요인에 의해 강화 될 수있다. 간섭계에서 광학 열 수차의 크기를 줄이기 위해 표시되었습니다 열 잡음 혜택뿐만 아니라, 고차원 LG 빔의 폭 넓은 강도 분포는 (예를 들어 그림 2는 다음과 같이 참조). 이 열 보상 시스템을 설계 민감도에게 19에 도달하기 위해 향후 실험에 의존되는 범위를 줄일 수 있습니다. 우리는 연구하고 성공적으로 순결과 성공적으로 감도 16, 18, ​​19, 20, 21, 22의 최상의 GW 간섭계를 작동하는 데 필요한 안정성 수준에서 LG 빔을 생성의 가능성을 증명하고있다. 제안 된 방법은 물리학 및 광학 SUC의 다양한 분야에서 개발 된 기술과 전문 지식을 결합하여높은 안정성의 생성과 같은 H, 저잡음 단일 모드 레이저는 23 광선, 광선의 공간 프로파일 (18)의 조작, 22, 24, 25, 26, 그리고 사용 공간 광 변조기 및 회절 광학 요소의 사용 감지, 제어 및 레이저 빛의 더 정화 및 안정화를 목표로 공진 광학 공동 27 안정화를위한 고급 기술. 이 방법은 성공적으로 대규모 프로토 타입 간섭계 20 시험을 위해, 그리고 W 21 80 높은 레이저 파워에서 LG 모드를 생성하기위한 수출, 실험실 실험에서 입증되었습니다. 이 문서에서 우리는 더 높은 순서 LG 빔을 생성하는 방법의 세부 사항을 제시하고 그 결과 빔의 특성화 및 검증 방법을 설명합니다. 또한, 4 단계에서 비의 완벽한 거울 19와 충치의 수치 조사하는 방법이 설명되어 있습니다.

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			<td>The experimental apparatus discussed in this paper requires the following types of instruments:</td>
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			<td></td>
			<td></td>
			<td></td>
			<td>Instrument</td>
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			<td>Solid state Laser source, Nd:YAG 1064 nm CW laser</td>
			<td></td>
			<td></td>
			<td>Quantity: 1</td>
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			<td>Faraday Isolator</td>
			<td></td>
			<td></td>
			<td>Quantity: 1</td>
		</tr>
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			<td>Electro-Optic Modulator (EOM)</td>
			<td></td>
			<td></td>
			<td>Quantity: 1</td>
		</tr>
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			<td>CCDcamera beam profiler</td>
			<td></td>
			<td></td>
			<td>Quantity: 1</td>
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			<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>
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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>
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			<td>Piezoelectric ring</td>
			<td></td>
			<td></td>
			<td>Quantity: 1</td>
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			<td>Cavity spacer</td>
			<td></td>
			<td></td>
			<td>Quantity: 1</td>
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			<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>
		<tr>
			<td></td>
			<td></td>
			<td></td>
			<td>Tools</td>
		</tr>
		<tr>
			<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

고 반사 거울의 열 잡음은 표준 양자 제한에 도달하거나 자신의 양자 바닥 상태로 기계 시스템을 냉각하는 것을 목표로 고정밀 간섭 실험의 여러 유형에 대한 주요 장애물이다. 이 예를 들어 중력파 신호 감도 미래의 중력파 관측소의 경우는 자신의 거울 질량의 원자 진동, 가장 민감한 주파수 대역에서 제한 될 것으로 예상된다이다. 이 제한을 극복하기 위해 추진되고 하나의 유망한 접근 방법은 종래 사용되는 기본 모드 대신 고차원 Laguerre-가우스 (LG) 광 빔을 적용하는 것입니다. 때문에 그들보다 더 균일 한 광 강도 분포에이 빔은 다시 미러 위치의 불확실성이 레이저 빛에 의해 감지 감소 거울 표면의 열 구동 변동에보다 효과적으로 평균. 우리는 생성하는 유망한 방법을 보여회절 광학 소자의 도움으로 기본적인 가우시안 빔을 형성하여 고차원 LG 빔. 우리는 근본적인 레이저 빔을 안정화 알려진 기존의 감지 및 제어 기술로, 고차원 LG 모드는 정제 할 수 있으며, 비교적 높은 수준에서 단지뿐만 아니라 안정 것을 보여줍니다. 진단 도구의 집합은 우리가 생성 LG 빔의 속성을 제어하고 조정할 수 있습니다. 이것은 우리가 지금까지보고 된 가장 높은 순도와 LG 빔을 생성 할 수있었습니다. 표준 간섭 기술과 표준 구면 광학의 사용과 높은 순서 LG 모드 시연 호환성이 그들을 고정밀 간섭의 미래 세대 응용 프로그램에 대한 이상적인 후보합니다.

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.

Watch this Scientific Journal Video about The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry at JoVE.com

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.

고정밀 간섭에 대한 높은 순서 Laguerre-가우스 광학 빔의 생성

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 조회수.  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.