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

<ol>
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Quantum Grav. 30, 035004 (2013).</li><li>Carbone, L., Bogan, C., Fulda, P., Freise, A., Willke, W. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Generation+of+High-Purity+Higher-Order+Laguerre-Gauss+Beams+at+High+Laser+Power.">Generation of High-Purity Higher-Order Laguerre-Gauss Beams at High Laser Power.</a> Physical Review Letters. 110, 251101 (2013).</li><li>Fulda, P., et al. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Phaseplate+design+for+Laguerre-Gauss+mode+conversion.">Phaseplate design for Laguerre-Gauss mode conversion.</a> , (2013).</li><li>Willke, B. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Stabilized+lasers+for+advanced+gravitational+wave+detectors.">Stabilized lasers for advanced gravitational wave detectors.</a> Laser &amp; Photonics Rev. , 1-15 (2010).</li><li>Granata, M., Buy, C., Ward, R., Barsuglia, M. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Higher-Order+Laguerre-Gauss+Mode+Generation+and+Interferometry+for+Gravitational+Wave+Detectors.">Higher-Order Laguerre-Gauss Mode Generation and Interferometry for Gravitational Wave Detectors.</a> Physical Review Letters. 105, 231102 (2010).</li><li>Matsumoto, N., Ando, T., Inoue, T., Ohtake, Y., Fukuchi, N., Hara, T. <a target="_blank" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed&cmd=Search&doptcmdl=Citation&defaultField=Title+Word&term=Generation+of+high-quality+higher-order+Laguerre+Gaussian+beams+using+liquid-crystal-on-silicon+spatial+light+modulators.">Generation of high-quality higher-order Laguerre Gaussian beams using liquid-crystal-on-silicon spatial light modulators.</a> J. 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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.

過去十年の間に高精度干渉実験は量子効果が決定的な役割を果たし始めている究極の感度体制に向けてプッシュされた。このような機械的なオシレーター1、ミラー2、もつれテスト大衆の第3世代、量子非解体干渉4、リジッド空洞5、および重力波検出6レーザーの周波数安定化のための光学トラップのレーザー冷却としてこれらの継続と将来の実験では、 、7、8、研究者は、基本的かつ技術的なノイズ源を制限する多数に直面している。最も深刻な問題の一つは、ミラー基板とミラー反射コーティング7、図8、図9を構成する原子の熱励起によって引き起こされる干渉セットアップのキャビティミラーの熱雑音である。またブラウン運動と呼ばれるこの効果は、位相の不確定性が発生します光は任意のテスト大衆から反射と干渉計出力の基本的なノイズの制限として、したがってマニフェストます。例えば、そのような高度なLIGO、アドバンストVIRGO、およびアインシュタインの望遠鏡などの高度な重力波アンテナ、の投影設計感度は、観測周波数帯域10、11、12の最も敏感な領域におけるこの種のノイズによって制限される。 コミュニティの実験物理学者は、これらのノイズの寄与を最小化するために、それらの機器の感度を向上させるために継続的な努力に懸命に働く。ミラーブラウンノイズの特定のケースでは、緩和のための一つの方法は、表面のランダムな動き上でより効果的に拡大ビームの平均ので、試験質量面上で現在使用される標準的な基本波HG 00ビームの大きいビームスポットサイズを使用することである13、14。ミラー熱雑音のパワースペクトル密度は、スケールとすることが示されているミラー基板用と鏡面9逆二乗のガウスビームサイズの逆数。ビームスポットを大きくしているようにしかし、光パワーの大きな部分は、反射面の縁の上に失われます。一方、一般的に使用されるHG 00ビーム（例えば図1を参照）よりも、より均一な半径方向強度分布を有するビームを使用する場合、 ブラウン熱雑音レベルが損失のこのタイプを増加させることなく低減することができる。高精度な干渉の新しいバージョンのために提案されているすべてのより均一ビームの種類の中で、例えばメサビームまたは円錐形のモード13、14のために、最も有望である高次のLGは現在使用されて球状との潜在的な互換性のためにビームミラー面15。例えば、スパイラルシステムにおけるバイナリ中性子星の検出率 - 最初のGWのために最も有望な天体物理学の源と考えられている検出イオンは-現在建設10、11下第二世代干渉計の設計変更の最小限のコストで2つ以上の16の約倍に向上させることができた。熱雑音の利点に加えて、より高次のLGビームの広い強度分布は、（例えば、 図2に参照）内の干渉光学系の熱収差の大きさを軽減することが示されている。これは、熱補償システムが設計感度19に到達する将来の実験に依拠している程度を減少させる。 我々は、調査し、正常純度と正常に感度16、18、19、20、21、22のせいぜいGW干渉計を動作させるために必要な安定性のレベルでLGビームを生成する可能性を実証した。提案手法では、物理学と光学成功の多様な分野で開発された技術と専門知識を兼ね備えています高い安定性の生成時間として、低ノイズシングルモードレーザビーム23を 、光ビームの空間プロファイル18の操作、22、24、25、26、との使用のための空間光変調器と回折光学素子の使用検出のための高度な技術、レーザ光 ​​のさらなる精製および安定化を目指し共振型光キャビティ27の制御および安定化。この方法は、正常に大規模なプロトタイプ干渉計内のテスト20、80 21 Wに高いレーザーパワーでLGモードを生成するためにエクスポートされ、室内実験で実証されている。この記事では、高次LGビームを生成する方法の詳細を提示し、その結果、ビームの特性評価および検証のための方法論を議論する。さらに、ステップ4で非完璧なミラー19と空洞の数値調査するための方法が概説されています

<table border="1">
	<tbody>
		<tr>
			<td></td>
			<td></td>
			<td></td>
			<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>
		</tr>
		<tr>
			<td>Steering Mirrors</td>
			<td></td>
			<td></td>
			<td>Quantity: depending on apparatus design</td>
		</tr>
		<tr>
			<td>Aperture</td>
			<td></td>
			<td></td>
			<td>Quantity: 1</td>
		</tr>
		<tr>
			<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>
		<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

高反射ミラーにおける熱雑音は、標準量子限界に到達するか、またはそれらの量子基底状態に機械システムを冷却することを目指した高精度の干渉実験の数種類の大きな障害である。これは、例えば重力波信号に対する感度それらの鏡質量の原子の振動によって、最も感度の高い周波数帯に限定されると予想される将来の重力波の観測の場合である。この制限を克服するために追求されて一つの有望なアプローチは、従来用いられて基本モードの代わりに、高次ラゲールガウス（LG）光ビームを使用することである。そのおかげで、より均一な光強度分布にこれらのビームは、ミラー表面の熱駆動変動をより効果的に平均化する上で、順番にレーザ光によって感知されたミラー位置の不確実性を減少させる。 私たちは、生成するための有望な方法を実証回折光学素子の助けを借りて、基本的なガウシアンビームを成形することにより、高次のLGビーム。我々は、基本波レーザ光を安定化させるために知られている従来の感知および制御技術を用いて、高次のLGモードを精製することができ、比較的ハイレベルでも同様に安定化されたことを示している。診断ツールのセットは、私たちは、生成されたLG電子ビームの特性を制御し、オーダーメイドすることができます。これは私たちがこれまでに報告された最高純度のLGビームを生成することができました。標準的な干渉技術と、標準的な球状光学を利用した高次のLGモードの互換性が実証され、それらの高精度な干渉の将来世代への適用に理想的な候補になります。

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.

高精度の干渉のための高次ラゲールガウス光ビームの生成

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 ビュー.  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.