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.

Geçen on yıllarda yüksek hassasiyetli İnterferometrik deneyler kuantum etkileri belirleyici bir rol oynamaya başlıyor nihai bir hassasiyet rejimine karşı itildiler. Bu mekanik osilatörler 1, aynalar 2, dolaşmış testi kitlelerin nesil 3, kuantum olmayan yıkım interferometri 4, sert boşlukları 5 ile lazerlerin frekans istikrar ve yerçekimi dalga algılama 6 optik tuzakları lazer soğutma bu devam eden ve gelecekteki deneyler, içinde , 7, 8, araştırmacılar temel ve teknik ses kaynakları sınırlayan çok sayıda karşı karşıya bulunmaktadır. En ciddi problemlerden biri, ayna substratlar ve ayna yansıtıcı kaplamalar 7, 8, 9 oluşturan atomlar arasında termal uyarma neden olduğu interferometrik düzeneklerinin boşluğun aynalar, termal gürültüsüdür. Ayrıca Brown hareketi olarak adlandırılan bu etki, aşamasında bir belirsizliğe yol açmaktadırIşık herhangi bir test kitleler yansıyan ve interferometre çıkış temel bir gürültü sınırlama gibi olacak bu nedenle tezahür. Mesela, Gelişmiş LIGO, Gelişmiş BAŞAK, ve Einstein Teleskobu gibi gelişmiş çekim dalga antenler, öngörülen tasarım duyarlılığı gözlem frekans bandı 10 en hassas bölgesi, 11, 12 gürültü bu tür ile sınırlıdır. Toplumda deneysel fizikçiler bu gürültü katkıları en aza indirmek için ve araçların duyarlılığı geliştirmek için sürekli bir çaba çalışıyoruz. Ayna Brown gürültü özel durumda, hafifletilmesi için bir yöntem yüzeyin rastgele hareketleri üzerinde daha etkili bir şekilde daha büyük bir kiriş ortalamalar, çünkü test kütlesi yüzeylerde şu anda kullanılan standart temel HG 00 kirişin daha büyük bir ışın spot büyüklüğü kullanmaktır 13, 14. Ayna ısıl gürültü gücü spektral yoğunluk ile büyütmek için gösterilmiştirayna yüzey ve ayna yüzey 9 için ters kare ile Gauss ışın boyutu ters. Kiriş noktalar daha büyük imal edilmiştir Ancak, ışık gücü daha büyük bir bölümü, burada yansıtıcı yüzey kenarına kaybolur. En sık kullanılan HG 00 kiriş (örneğin Şekil 1) bakın daha homojen bir radyal yoğunluk dağılımına sahip olan bir kiriş kullanıyorsa, Brownian ısıl gürültü seviyesi kaybı bu tür arttırmaksızın azaltılabilir. Yüksek hassasiyetli interferometri yeni sürümleri için önerilmiştir tüm daha homojen ışın türleri arasında, örneğin Mesa kiriş veya konik modları 13, 14, en umut verici üst düzey LG şu anda kullanılan küresel ile olası uyumluluk nedeniyle kirişler olan ayna yüzeyi 15. Örneğin, spiral sistemlerinde ikili nötron yıldızının tespit oranı - bir ilk GW için en umut verici astrofizik kaynakları göz önünde bulundurulduğunda tespitiyon - şu anda yapım 10, 11 altında ikinci nesil interferometreler tasarımında değişiklikler az miktarda pahasına 2 veya daha fazla 16 hakkında bir faktör tarafından geliştirilmiş olabilir. Interferometreler içindeki optik termal sapmaları büyüklüğünü azaltmak için gösterilmiş olan ısıl gürültü avantajlarına ek olarak, daha yüksek dereceden LG kirişlerin geniş yoğunluk dağılımları (örneğin Şekil 2, bakınız). Bu termal kompanzasyon sistemleri tasarımı hassasiyetleri 19 ulaşmak için gelecekte deneylerde güvenerek ne ölçüde azaltacaktır. Her iki grupta, başarılı bir saflık ve başarılı bir şekilde hassasiyeti, 16, 18, ​​19, 20, 21, 22 arasında en az GW interferometreler çalıştırmak için gerekli stabilite düzeyde LG demetlerinin uygulanabilirliğini ortaya koymuştur. Önerilen yöntem fizik ve optik suc çeşitli alanlarda geliştirilen teknikler ve uzmanlık birleştiriryüksek bir stabilite kuşak s, düşük gürültü, tek modlu 23 lazer ışınları, ışık ışınları mekansal Profiller 18 bir manipülasyon, 22, 24, 25, 26, ve kullanımı için uzamsal ışık modülatörleri, difraktif optik elemanların kullanımını algılama, kontrol ve lazer ışığı bir daha arıtılmadan ve stabilizasyon amaçlayan rezonans optik boşlukların 27 stabilizasyonu için gelişmiş yöntemler. Bu yöntem başarılı bir şekilde büyük ölçekli prototip interferometreler 20 testleri için, ve W 21 80 yüksek lazer güçler de LG modları oluşturmak için ihraç, laboratuvar deneylerinde gösterilmiştir. Bu yazıda daha yüksek dereceden LG kiriş üretme yöntemi ayrıntılarını ortaya çıkan ışının karakterizasyonu ve doğrulanması için bir yöntem tartışılmaktadır. Ayrıca, adım 4 mükemmel olmayan ayna 19 ile boşlukları sayısal araştırmalar için bir yöntem gösterilmiştir.

<table border="1">
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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>
		</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>
		<tr>
			<td>Aperture</td>
			<td></td>
			<td></td>
			<td>Quantity: 1</td>
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		<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>
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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

Yüksek yansıtma aynalar termal gürültü standart kuantum sınırına ulaşmak için ya da kuantum zemin durumuna mekanik sistemleri soğutmak amacı yüksek hassasiyetli İnterferometrik deney çeşitli için önemli bir engeldir. Bu, örneğin yerçekimi dalga sinyalleri duyarlılık gelecekte yerçekimi dalgası gözlem, söz konusu kendi ayna kitlelerin atomik titreşim, en hassas frekans bandında sınırlı olması bekleniyor olduğunu. Bu sınırlama üstesinden gelmek için sürdürülen bir umut verici bir yaklaşım geleneksel olarak kullanılan temel modunda yerine üst düzey Laguerre-Gauss (LG) optik kirişler istihdam etmektir. Sayesinde, daha homojen ışık yoğunluğu dağılımı için bu kirişler da aynaya konumda belirsizlik lazer ışığı tarafından algılanan azaltır ayna yüzeyi, en termal tahrik dalgalanmalar üzerinde daha fazla etkili ortalama. Biz üretmek için umut verici bir yöntem göstermekdifraktif optik elemanlar yardımıyla temel bir Gauss ışın şekillendirme tarafından üst düzey LG kirişler. Biz temel lazer ışınları istikrar için bilinen geleneksel algılama ve kontrol teknikleri ile, üst düzey LG modları saf olabilir ve bir nispeten yüksek bir seviyede gibi iyi stabilize olduğunu göstermektedir. Tanı araçları kümesi bize oluşturulan LG kirişlerin özelliklerini kontrol ve terzi sağlar. Bu bizim bugüne kadar bildirilen en yüksek saflıkta bir LG ışın üretmek için etkin. Standart interferometri tekniklerle ve standart küresel optik kullanımı ile üst düzey LG modları gösterdi uyumluluk onları yüksek hassasiyetli interferometri bir gelecek nesil uygulama için ideal bir aday yapar.

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.

Yüksek hassasiyetli İnterferometre için Yüksek mertebeden Laguerre-Gauss Optik Kirişlerin Üretimi

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 Görüntüleme.  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.