Waveguide grating mirror in a fully suspended 10 meter Fabry-Perot cavity

Daniel Friedrich; Bryan W. Barr; Frank Brückner; Stefan Hild; John Nelson; John Macarthur; Michael V. Plissi; Matthew P. Edgar; Sabina H. Huttner; Borja Sorazu; Stefanie Kroker; Michael Britzger; Ernst-Bernhard Kley; Karsten Danzmann; Andreas Tünnermann; Ken A. Strain; Roman Schnabel

doi:10.1364/OE.19.014955

Waveguide grating mirror in a fully suspended 10 meter Fabry-Perot cavity

Daniel Friedrich, Bryan W. Barr, Frank Brückner, Stefan Hild, John Nelson, John Macarthur, Michael V. Plissi, Matthew P. Edgar, Sabina H. Huttner, Borja Sorazu, Stefanie Kroker, Michael Britzger, Ernst-Bernhard Kley, Karsten Danzmann, Andreas Tünnermann, Ken A. Strain, and Roman Schnabel

Optics Express
Vol. 19,
Issue 16,
pp. 14955-14963
(2011)
•https://doi.org/10.1364/OE.19.014955

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Daniel Friedrich, Bryan W. Barr, Frank Brückner, Stefan Hild, John Nelson, John Macarthur, Michael V. Plissi, Matthew P. Edgar, Sabina H. Huttner, Borja Sorazu, Stefanie Kroker, Michael Britzger, Ernst-Bernhard Kley, Karsten Danzmann, Andreas Tünnermann, Ken A. Strain, and Roman Schnabel, "Waveguide grating mirror in a fully suspended 10 meter Fabry-Perot cavity," Opt. Express 19, 14955-14963 (2011)

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Related Topics
Table of Contents Category
- Instrumentation, Measurement, and Metrology
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Gratings (050.2770)
- Subwavelength structures (050.6624)
- Fabry-Perot (120.2230)
- Mirrors (230.4040)

About this Article
History
- Original Manuscript: April 18, 2011
- Revised Manuscript: May 22, 2011
- Manuscript Accepted: May 22, 2011
- Published: July 20, 2011

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Open Access

Abstract

We report on the first demonstration of a fully suspended 10m Fabry-Perot cavity incorporating a waveguide grating as the coupling mirror. The cavity was kept on resonance by reading out the length fluctuations via the Pound-Drever-Hall method and employing feedback to the laser frequency. From the achieved finesse of 790 the grating reflectivity was determined to exceed 99.2% at the laser wavelength of 1064 nm, which is in good agreement with rigorous simulations. Our waveguide grating design was based on tantala and fused silica and included a ≈ 20 nm thin etch stop layer made of Al₂O₃ that allowed us to define the grating depth accurately and preserve the waveguide thickness during the fabrication process. Demonstrating stable operation of a waveguide grating featuring high reflectivity in a suspended low-noise cavity, our work paves the way for the potential application of waveguide gratings as mirrors in high-precision interferometry, for instance in future gravitational wave observatories.

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References

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[Crossref]
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[Crossref]
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S. Hild, S. Chelkowski, A. Freise, J. Franc, N. Morgado, R. Flaminio, and R. DeSalvo, “A xylophone configuration for a third-generation gravitational wave detector,” Class. and Quantum Grav., 27, 015003 (2010).
[Crossref]
Y. Levin, “Internal thermal noise in the LIGO test masses: A direct approach,” Phys. Rev. D 57, 659–663 (1998).
[Crossref]
G. M. Harry, A. M Gretarsson, P. R. Saulson, S. E. Kittelberger, S. D. Penn, W. J. Startin, S Rowan, M. M. Fejer, D. R. M. Crooks, G. Cagnoli, J. Hough, and N Nakagawa, “Thermal noise in interferometric gravitational wave detectors due to dielectric optical coatings,” Class. Quantum Grav. 19, 897–917 (2002).
[Crossref]
R. Nawrodt, S. Rowan, J. Hough, M. Punturo, F. Ricci, and J.-Y. Vinet, “Challenges in thermal noise for 3rd generation of gravitational wave detectors,” Gen Relativ Gravit 43, 593–622 (2011).
[Crossref]
A. Bunkowski, O. Burmeister, D. Friedrich, K. Danzmann, and R. Schnabel, “High reflectivity grating waveguide coatings for 1064 nm,” Class. Quantum Grav. 23, 7297–7303 (2006).
[Crossref]
F. Brückner, T. Clausnitzer, O. Burmeister, D. Friedrich, E.-B. Kley, K. Danzmann, A. Tünnermann, and R. Schnabel, “Monolithic dielectric surfaces as new low-loss light-matter interfaces,” Opt. Lett. 33, 264–266 (2008).
[Crossref] [PubMed]
F. Brückner, D. Friedrich, T. Clausnitzer, O. Burmeister, M. Britzger, E.-B. Kley, K. Danzmann, A. Tünnermann, and R. Schnabel, “Demonstration of a cavity coupler based on a resonant waveguide grating,” Opt. Express 17, 163–169 (2009).
[Crossref] [PubMed]
F. Brückner, D. Friedrich, T. Clausnitzer, M. Britzger, O. Burmeister, K. Danzmann, E.-B. Kley, A. Tünnermann, and R. Schnabel, “Realization of a Monolithic High-Reflectivity Cavity Mirror from a Single Silicon Crystal,” Phys. Rev. Lett. 104, 163903 (2010).
[Crossref] [PubMed]
M. P. Edgar, B. W. Barr, J. Nelson, M. V. Plissi, K. A. Strain, O. Burmeister, M. Britzger, K. Danzmann, R. Schnabel, T. Clausnitzer, F. Brückner, E.-B. Kley, and A. Tünnermann, “Experimental demonstration of a suspended diffractively coupled optical cavity,” Opt. Lett. 343184–3186 (2009).
[Crossref] [PubMed]
B. W. Barr, S. H. Huttner, J. R. Taylor, B. Sorazu, M. V. Plissi, and K. A. Strain, “Optical modulation techniques for length sensing and control of optical cavities,” Appl. Opt. 467739–7745 (2007).
[Crossref] [PubMed]
S. H. Huttner, B. W. Barr, M. V. Plissi, J. R. Taylor, B. Sorazu, and K. A. Strain, “Novel sensing and control schemes for a three-mirror coupled cavity,” Class. and Quantum Grav., 24, 3825 (2007).
[Crossref]
A. Hessel and A. A. Oliner, “A New Theory of Wood’s Anomalies on Optical Gratings,” Appl. Opt. 4, 1275–1297 (1965).
[Crossref]
L. Mashev and E. Popov, “Diffraction efficiency anomalies of multicoated dielectric gratings,” Opt. Commun. 51, 131–136 (1984).
[Crossref]
L. Mashev and E. Popov, “Zero Order Anomaly of Dielectric Coated Gratings,” Opt. Commun. 55, 377–380 (1985).
[Crossref]
G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, and A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron 15, 886–887 (1985).
[Crossref]
R. Magnusson and S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
[Crossref]
C. F. R. Mateus, M. C. Y. Huang, L. Chen, C. J. Chang-Hasnain, and Y. Suzuki, “Broad-Band Mirror (1.12–1.62) μm Using a Subwavelength Grating),” IEEE Photonics Technology Letters 16, 1676–1678 (2004).
[Crossref]
R. Magnusson and M. Shokooh-Saremi, “Physical basis for wideband resonant reflectors,” Opt. Express 16, 3456–3462 (2008).
[Crossref] [PubMed]
F. Brücker, D. Friedrich, M. Britzger, T. Clausnitzer, O. Burmeister, E.-B. Kley, K. Danzmann, A. Tünnermann, and R. Schnabel, “Encapsulated subwavelength grating as a quasi-monolithic resonant reflector,” Opt. Express 17, 24334–24341 (2009).
[Crossref]
M. Shokooh-Saremi and R. Magnusson, “Leaky-mode resonant reflectors with extreme bandwidths,” Opt. Lett. 35, 1121–1123 (2010).
[Crossref] [PubMed]
A. Sharon, D. Rosenblatt, and A. A. Friesem, “Resonant grating-waveguide structures for visible and near-infrared radiation,” J. Opt. Soc. Am. A 14, 2985–2993 (1997).
[Crossref]
M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
[Crossref]
M. V. Plissi, C. I. Torrie, M. E. Husman, N. A. Robertson, K. A. Strain, H. Ward, H. Lück, and J. Hough, “GEO 600 triple pendulum suspension system: Seismic isolation and control,” Rev. Sci. Instrum. 71, 2539–2545 (2000).
[Crossref]
R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser Phase and Frequency Stabilization Using an Optical Resonator,” Appl. Phys. B 31, 97–105 (1983).
[Crossref]
A. E. Dangor and S. J. Fielding, “The response of the Fabry-Perot interferometer to rapid changes in optical length,” J. Phys. D: Appl. Phys. 3, 413–421 (1970).
[Crossref]

2011 (1)

R. Nawrodt, S. Rowan, J. Hough, M. Punturo, F. Ricci, and J.-Y. Vinet, “Challenges in thermal noise for 3rd generation of gravitational wave detectors,” Gen Relativ Gravit 43, 593–622 (2011).
[Crossref]

2010 (5)

F. Brückner, D. Friedrich, T. Clausnitzer, M. Britzger, O. Burmeister, K. Danzmann, E.-B. Kley, A. Tünnermann, and R. Schnabel, “Realization of a Monolithic High-Reflectivity Cavity Mirror from a Single Silicon Crystal,” Phys. Rev. Lett. 104, 163903 (2010).
[Crossref] [PubMed]

S. Hild, S. Chelkowski, A. Freise, J. Franc, N. Morgado, R. Flaminio, and R. DeSalvo, “A xylophone configuration for a third-generation gravitational wave detector,” Class. and Quantum Grav., 27, 015003 (2010).
[Crossref]

G. M. Harry (for the LIGO Scientific Collaboration), “Advanced LIGO: the next generation of gravitational wave detectors,” Class. Quantum Grav. 27, 084006 (2010).
[Crossref]

K. Kuroda and LCGT Collaboration, “Status of LCGT,” Class. Quantum Grav. 27, 084004 (2010).
[Crossref]

M. Shokooh-Saremi and R. Magnusson, “Leaky-mode resonant reflectors with extreme bandwidths,” Opt. Lett. 35, 1121–1123 (2010).
[Crossref] [PubMed]

2009 (3)

F. Brückner, D. Friedrich, T. Clausnitzer, O. Burmeister, M. Britzger, E.-B. Kley, K. Danzmann, A. Tünnermann, and R. Schnabel, “Demonstration of a cavity coupler based on a resonant waveguide grating,” Opt. Express 17, 163–169 (2009).
[Crossref] [PubMed]

M. P. Edgar, B. W. Barr, J. Nelson, M. V. Plissi, K. A. Strain, O. Burmeister, M. Britzger, K. Danzmann, R. Schnabel, T. Clausnitzer, F. Brückner, E.-B. Kley, and A. Tünnermann, “Experimental demonstration of a suspended diffractively coupled optical cavity,” Opt. Lett. 343184–3186 (2009).
[Crossref] [PubMed]

F. Brücker, D. Friedrich, M. Britzger, T. Clausnitzer, O. Burmeister, E.-B. Kley, K. Danzmann, A. Tünnermann, and R. Schnabel, “Encapsulated subwavelength grating as a quasi-monolithic resonant reflector,” Opt. Express 17, 24334–24341 (2009).
[Crossref]

2008 (2)

F. Brückner, T. Clausnitzer, O. Burmeister, D. Friedrich, E.-B. Kley, K. Danzmann, A. Tünnermann, and R. Schnabel, “Monolithic dielectric surfaces as new low-loss light-matter interfaces,” Opt. Lett. 33, 264–266 (2008).
[Crossref] [PubMed]

R. Magnusson and M. Shokooh-Saremi, “Physical basis for wideband resonant reflectors,” Opt. Express 16, 3456–3462 (2008).
[Crossref] [PubMed]

2007 (2)

S. H. Huttner, B. W. Barr, M. V. Plissi, J. R. Taylor, B. Sorazu, and K. A. Strain, “Novel sensing and control schemes for a three-mirror coupled cavity,” Class. and Quantum Grav., 24, 3825 (2007).
[Crossref]

B. W. Barr, S. H. Huttner, J. R. Taylor, B. Sorazu, M. V. Plissi, and K. A. Strain, “Optical modulation techniques for length sensing and control of optical cavities,” Appl. Opt. 467739–7745 (2007).
[Crossref] [PubMed]

2006 (2)

A. Bunkowski, O. Burmeister, D. Friedrich, K. Danzmann, and R. Schnabel, “High reflectivity grating waveguide coatings for 1064 nm,” Class. Quantum Grav. 23, 7297–7303 (2006).
[Crossref]

B. Willke, P. Ajith, B. Allen, P. Aufmuth, C. Aulbert, S. Babak, R. Balasubramanian, B. W. Barr, S. Berukoff, A. Bunkowski, G. Cagnoli, C. A. Cantley, M. M. Casey, S. Chelkowski, Y. Chen, D. Churches, T. Cokelaer, C. N. Colacino, D. R. M. Crooks, C. Cutler, K. Danzmann, R. J. Dupuis, E. Elliffe, C. Fallnich, A. Franzen, A. Freise, I. Gholami, S. Goßler, A. Grant, H. Grote, S. Grunewald, J. Harms, B. Hage, G. Heinzel, I. S. Heng, A. Hepstonstall, M. Heurs, M. Hewitson, S. Hild, J. Hough, Y. Itoh, G. Jones, R. Jones, S. H. Huttner, K. Kötter, B. Krishnan, P. Kwee, H. Lück, M. Luna, B. Machenschalk, M. Malec, R. A. Mercer, T. Meier, C. Messenger, S. Mohanty, K. Mossavi, S. Mukherjee, P. Murray, G. P. Newton, M. A. Papa, M. Perreur-Lloyd, M. Pitkin, M. V. Plissi, R. Prix, V. Quetschke, V. Re, T. Regimbau, H. Rehbein, S. Reid, L. Ribichini, D. I. Robertson, N. A. Robertson, C. Robinson, J. D. Romano, S. Rowan, A. Rüdiger, B. S. Sathyaprakash, R. Schilling, R. Schnabel, B. F. Schutz, F. Seifert, A. M. Sintes, J. R. Smith, P. H. Sneddon, K. A. Strain, I. Taylor, R. Taylor, A. Thüring, C. Ungarelli, H. Vahlbruch, A. Vecchio, J. Veitch, H. Ward, U. Weiland, H. Welling, L. Wen, P. Williams, W. Winkler, G. Woan, and R. Zhu, “The GEO-HF project,” Class. Quantum Grav. 23, S207–S214 (2006).
[Crossref]

2004 (1)

C. F. R. Mateus, M. C. Y. Huang, L. Chen, C. J. Chang-Hasnain, and Y. Suzuki, “Broad-Band Mirror (1.12–1.62) μm Using a Subwavelength Grating),” IEEE Photonics Technology Letters 16, 1676–1678 (2004).
[Crossref]

2002 (1)

G. M. Harry, A. M Gretarsson, P. R. Saulson, S. E. Kittelberger, S. D. Penn, W. J. Startin, S Rowan, M. M. Fejer, D. R. M. Crooks, G. Cagnoli, J. Hough, and N Nakagawa, “Thermal noise in interferometric gravitational wave detectors due to dielectric optical coatings,” Class. Quantum Grav. 19, 897–917 (2002).
[Crossref]

2000 (1)

M. V. Plissi, C. I. Torrie, M. E. Husman, N. A. Robertson, K. A. Strain, H. Ward, H. Lück, and J. Hough, “GEO 600 triple pendulum suspension system: Seismic isolation and control,” Rev. Sci. Instrum. 71, 2539–2545 (2000).
[Crossref]

1998 (1)

Y. Levin, “Internal thermal noise in the LIGO test masses: A direct approach,” Phys. Rev. D 57, 659–663 (1998).
[Crossref]

1997 (1)

A. Sharon, D. Rosenblatt, and A. A. Friesem, “Resonant grating-waveguide structures for visible and near-infrared radiation,” J. Opt. Soc. Am. A 14, 2985–2993 (1997).
[Crossref]

1992 (1)

R. Magnusson and S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
[Crossref]

1985 (2)

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, and A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron 15, 886–887 (1985).
[Crossref]

L. Mashev and E. Popov, “Zero Order Anomaly of Dielectric Coated Gratings,” Opt. Commun. 55, 377–380 (1985).
[Crossref]

1984 (1)

L. Mashev and E. Popov, “Diffraction efficiency anomalies of multicoated dielectric gratings,” Opt. Commun. 51, 131–136 (1984).
[Crossref]

1983 (1)

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser Phase and Frequency Stabilization Using an Optical Resonator,” Appl. Phys. B 31, 97–105 (1983).
[Crossref]

1981 (1)

M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
[Crossref]

1970 (1)

A. E. Dangor and S. J. Fielding, “The response of the Fabry-Perot interferometer to rapid changes in optical length,” J. Phys. D: Appl. Phys. 3, 413–421 (1970).
[Crossref]

1965 (1)

A. Hessel and A. A. Oliner, “A New Theory of Wood’s Anomalies on Optical Gratings,” Appl. Opt. 4, 1275–1297 (1965).
[Crossref]

1940 (1)

M. Punturo, M. Abernathy, F. Acernese, B. Allen, N. Andersson, K. Arun, F. Barone, B. Barr, M. Barsuglia, M. Beker, N. Beveridge, S. Birindelli, S. Bose, L. Bosi, S. Braccini, C. Bradaschia, T. Bulik, E. Calloni, G. Cella, E. Chassande Mottin, S. Chelkowski, A. Chincarini, J. Clark, E. Coccia, C. Colacino, J. Colas, A. Cumming, L. Cunningham, E. Cuoco, S. Danilishin, K. Danzmann, G. De Luca, R. De Salvo, T. Dent, R. De Rosa, L. Di Fiore, A. Di Virgilio, M. Doets, V. Fafone, P. Falferi, R. Flaminio, J. Franc, F. Frasconi, A. Freise, P. Fulda, J. Gair, G. Gemme, A. Gennai, A. Giazotto, K. Glampedakis, M. Granata, H. Grote, G. Guidi, G. Hammond, M. Hannam, J. Harms, D. Heinert, M. Hendry, I. Heng, E. Hennes, S. Hild, J. Hough, S. Husa, S. Huttner, G. Jones, F. Khalili, K. Kokeyama, K. Kokkotas, B. Krishnan, M. Lorenzini, H. Lück, E. Majorana, I. Mandel, V. Mandic, I. Martin, C. Michel, Y. Minenkov, N. Morgado, S. Mosca, B. Mours, H. Müller-Ebhardt, P. Murray, R. Nawrodt, J. Nelson, R. Oshaughnessy, C. D. Ott, C. Palomba, A. Paoli, G. Parguez, A. Pasqualetti, R. Passaquieti, D. Passuello, L. Pinard, R. Poggiani, P. Popolizio, M. Prato, P. Puppo, D. Rabeling, P. Rapagnani, J. Read, T. Regimbau, H. Rehbein, S. Reid, L. Rezzolla, F. Ricci, F. Richard, A. Rocchi, S. Rowan, A. Rüdiger, B. Sassolas, B. Sathyaprakash, R. Schnabel, C. Schwarz, P. Seidel, A. Sintes, K. Somiya, F. Speirits, K. Strain, S. Strigin, P. Sutton, S. Tarabrin, A. Thüring, J. van den Brand, C. van Leewen, M. van Vegge, C. van den Broeck, A. Vecchio, J. Veitch, F. Vetrano, A. Vicere, S. Vyatchanin, B. Willke, G. Woan, P. Wolfango, and K. Yamamoto, “The Einstein Telescope: a third-generation gravitational wave observatory,” Class. and Quantum Grav., 27, 194002 (2010).

Abernathy, M.

Acernese, F.

Ajith, P.

Allen, B.

Andersson, N.

Arun, K.

Aufmuth, P.

Aulbert, C.

Babak, S.

Balasubramanian, R.

Barone, F.

Barr, B.

Barr, B. W.

Barsuglia, M.

Beker, M.

Berukoff, S.

Beveridge, N.

Birindelli, S.

Bose, S.

Bosi, L.

Braccini, S.

Bradaschia, C.

Britzger, M.

Brücker, F.

Brückner, F.

Bulik, T.

Bunkowski, A.

A. Bunkowski, O. Burmeister, D. Friedrich, K. Danzmann, and R. Schnabel, “High reflectivity grating waveguide coatings for 1064 nm,” Class. Quantum Grav. 23, 7297–7303 (2006).
[Crossref]

Burmeister, O.

A. Bunkowski, O. Burmeister, D. Friedrich, K. Danzmann, and R. Schnabel, “High reflectivity grating waveguide coatings for 1064 nm,” Class. Quantum Grav. 23, 7297–7303 (2006).
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Class. and Quantum Grav. (3)

Class. Quantum Grav. (5)

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Gen Relativ Gravit (1)

IEEE Photonics Technology Letters (1)

J. Opt. Soc. Am. (1)

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J. Opt. Soc. Am. A (1)

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J. Phys. D: Appl. Phys. (1)

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Rev. Sci. Instrum. (1)

Sov. J. Quantum Electron (1)

Other (1)

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Fig. 1 (a) Principle of a waveguide grating mirror under normal incidence in a ray picture. The incoupled first order diffracted rays are guided via total internal reflection at the substrate with index of refraction n _L < n _H. The outcoupling at the grating structure is a constructive interference if all grating parameter are designed properly, hence, providing 100% reflectivity. (b) Equivalent waveguide grating architecture realized in this work, including an Al₂O₃ layer as etch stop in order to define the grating depth and waveguide layer thickness in the fabrication process.

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Fig. 2 (a) First order diffraction efficiency from a tantala grating (n _H = 2.186, g = 390 nm, f = b/d = 0.38, TE-polarization) into a material with varying index of refraction n _L, illustrating the range of the grating period that allows for resonant excitation as predicted by Eq. (1). (b) If an etch stop layer with index of refraction n _M = 1.66 is implemented, the first diffraction orders can propagate through the etch stop layer for d ≥ λ ₀/n _M independent of its thickness s.

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Fig. 3 (a) SEM image of a fabricated waveguide grating structure. Except of the fill factor all parameters are the same for the sample investigated here. (b) Calculated reflectivity (RCWA) of a waveguide grating under normal incidence for the parameters marked in the SEM image and a fill factor of f = 0.38. The white lines mark an area of reflectivity ≥ 99%.

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Fig. 4 (a) Schematic of the prototype facility including the laser bench, vacuum system and waveguide cavity having a length of ≈ 10m. The cavity was stabilized using the Pound-Drever-Hall scheme also depicted at the laser bench. (b) Intermediate and lower stage of the triple suspension system used for the cavity mirrors. Behind the main test mass a second triple suspension carries the so-called reaction mass, which is used to act on the main test mass. (c) Test mass with the waveguide grating mirror (area of 10 × 15mm) attached.

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Fig. 5 Cavity scan via tuning of the laser frequency. Reflected power (red trace) and Pound-Drever-Hall error-signal (green trace) detected with the photodiodes PD_R and PD_E, respectively. The reflected signal for a stabilized cavity (blue trace) indicates a visibility of ≥ 0.6.

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Fig. 6 Typical measurements of the transmitted light of a swept cavity resonance for different mirror velocities v (red trace). Theoretical results for the transmitted power |a _T|² are based on Eq. (3) with r ₁ r ₂ = 0.996 (black line), which corresponds to a cavity finesse of 790. Hence, a lower boundary for the waveguide grating power reflectivity is |r ₁|² ≥ 0.992.

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Equations (4)

Equations on this page are rendered with MathJax. Learn more.

487 nm = λ_{0} / n_{H} \leq d \leq λ_{0} / n_{L} = 734 nm,

v = \frac{2 Ω}{FSR} \frac{λ}{2} \frac{1}{Δ τ},

a_{T} (τ) = a_{0} \sum_{m = 1}^{\infty} t_{1} t_{2} {(r_{1} r_{2})}^{m - 1} exp (i ϕ_{m} (τ)),

ϕ_{m} (τ) \approx (2 m - 1) \frac{2 π L (τ)}{λ} - m (m - 1) \frac{2 L_{0}}{c} v \frac{2 π}{λ},