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

<ol>
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J., Levin, Y., Matsko, A. B., Thorne, K. S., Vyatchanin, S. 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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.

Nel corso degli ultimi decenni, esperimenti di alta precisione interferometrici sono stati spinti verso un regime di sensibilità finale dove gli effetti quantistici stanno iniziando a giocare un ruolo decisivo. In questi esperimenti in corso e futuri, come il raffreddamento laser di oscillatori meccanici 1, trappole ottiche per specchi 2, generazione di masse di prova entangled 3, quantistica non-demolizione interferometria 4, stabilizzazione frequenza di laser con cavità rigide 5, e la rilevazione delle onde gravitazionali 6 , 7, 8, i ricercatori si trovano ad affrontare una moltitudine di limitare le sorgenti di rumore fondamentali e tecnici. Uno dei problemi più gravi è il rumore termico degli specchi cavità delle configurazioni interferometriche, che è causato dall&#39;eccitazione termica degli atomi che compongono i substrati specchio e lo specchio rivestimenti riflettenti 7, 8, 9. Questo effetto, chiamato anche moto browniano, causerà una incertezza nella fase dila luce riflessa da eventuali masse di test e sarà quindi manifestarsi come una limitazione fondamentale rumore nell&#39;output interferometro. Per esempio, la sensibilità proiettata progettazione di avanzate gravitazionale onda antenne, quali Advanced Ligo, Advanced VIRGO, e il telescopio Einstein, è limitato da questo tipo di rumore nella regione più sensibile della banda di frequenza di osservazione di 10, 11, 12. Fisici sperimentali nella comunità lavorano duro in un continuo sforzo per ridurre al minimo questi contributi di rumore e di migliorare la sensibilità dei loro strumenti. Nel caso particolare di rumore specchio browniano, un metodo per la mitigazione è di impiegare una trave maggiore dimensione del punto del attualmente utilizzata norma fondamentale HG 00 fascio sulle superfici di massa di test poiché una più grande medie fascio più efficacemente sui moti casuali della superficie 13, 14. La densità spettrale di potenza del rumore termico specchio ha dimostrato di scala conl&#39;inverso della dimensione del fascio gaussiano per il substrato specchio e con il quadrato per la superficie dello specchio 9. Tuttavia, poiché i punti di fasci vengono fatti più grandi, una parte maggiore del potere della luce si perde oltre il bordo della superficie riflettente. Se si utilizza un fascio con una distribuzione più omogenea intensità radiale rispetto al comunemente usato HG trave 00 (vedi ad esempio la Figura 1), il livello di rumore browniano termico può essere ridotto senza aumentare questo tipo di perdita. Tra tutti i tipi di raggi più omogenei che sono state proposte per le nuove versioni di alta precisione interferometria, per esempio Mesa travi o modi conici 13, 14, i più promettenti sono di ordine superiore LG travi causa della loro compatibilità con il potenziale sferica attualmente utilizzato superfici a specchio 15. Per esempio, il tasso di rilevamento di stella di neutroni in sistemi binari a spirale - che sono considerate le sorgenti astrofisiche più promettenti per un primo GW rilevareion - potrebbe essere migliorato di circa un fattore di 2 o più 16 a costo di una minima quantità di modifiche nella progettazione di interferometri di seconda generazione in costruzione 10, 11. Oltre ai vantaggi rumore termico, le più ampie distribuzioni di intensità dei fasci LG ordine superiore (vedi come esempio la figura 2) hanno dimostrato di ridurre la grandezza di aberrazioni termiche di ottica all&#39;interno degli interferometri. Ciò ridurrebbe la misura in cui i sistemi di compensazione termica sono invocati in futuri esperimenti per raggiungere la sensibilità di progettazione 19. Abbiamo studiato e dimostrato la fattibilità di generare fasci LG ai livelli di purezza e stabilità necessarie per operare con successo interferometri GW al meglio delle loro sensibilità 16, 18, ​​19, 20, 21, 22. Il metodo proposto combina tecniche e le competenze sviluppate in diverse aree della fisica e dell&#39;ottica such come la generazione di alta stabilità, singolo laser a basso rumore di modo travi 23, l&#39;utilizzo di modulatori spaziali di luce ed elementi ottici diffrattivi per la manipolazione dei profili spaziali di fasci di luce 18, 22, 24, 25, 26, e l&#39;uso di tecniche avanzate per il rilevamento, controllo e stabilizzazione della cavità ottiche risonanti 27 volte ad un ulteriore purificazione e stabilizzazione della luce laser. Questo metodo è stato dimostrato con successo negli esperimenti di laboratorio, esportato per prove in grande scala prototipo interferometri 20, e per generare i modi LG ad alte potenze laser fino a 80 W 21. In questo articolo presentiamo i dettagli del metodo di generazione di LG fasci di ordine superiore e discutiamo di una metodologia per la caratterizzazione e la validazione del fascio risultante. Inoltre, al punto 4 viene illustrato un metodo per le indagini numeriche della cavità con specchi non perfette 19.

<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

Rumore termico in specchi ad alta riflettività è uno dei principali ostacoli per i diversi tipi di alta precisione interferometrico esperimenti che mirano a raggiungere il limite quantistico standard o per raffreddare i sistemi meccanici per il loro stato fondamentale quantistica. Questo è per esempio il caso si prevede di futuri osservatori di onde gravitazionali, la cui sensibilità ai segnali di onde gravitazionali essere limitati nella banda di frequenza più sensibile, dalla vibrazione atomica delle loro masse specchio. Un approccio promettente perseguito per superare questo limite è di impiegare ordine superiore Laguerre-Gauss (LG) fasci ottici al posto del modo fondamentale usato convenzionalmente. Grazie alla loro distribuzione dell&#39;intensità luminosa più omogenea media più efficacemente negli fluttuazioni alimentato termicamente della superficie dello specchio, che a sua volta riduce l&#39;incertezza nella posizione specchio rilevata dalla luce laser queste travi. Dimostriamo un metodo promettente per generareordine superiore LG travi sagomando un fascio gaussiano fondamentale con l&#39;aiuto di elementi ottici diffrattivi. Mostriamo che con il rilevamento convenzionali e tecniche di controllo che sono noti per stabilizzare fasci laser fondamentali, LG modi di ordine superiore possono essere purificati e stabilizzati altrettanto bene ad un livello relativamente elevato. Una serie di strumenti diagnostici ci permette di controllare e personalizzare le proprietà dei raggi LG generati. Questo ci ha permesso di produrre un fascio di LG con la massima purezza segnalato fino ad oggi. La compatibilità dimostrato di modalità LG ordine superiore con tecniche di interferometria standard e con l&#39;uso di ottiche sferiche standard di loro un candidato ideale per l&#39;applicazione in una futura generazione di alta precisione interferometria fa.

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.

La generazione di ordine superiore Laguerre-Gauss fasci ottici ad alta precisione 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 Views.  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.