Abstract

The shot-noise limited sensitivity of Michelson-type laser interferometers with Fabry-Perot arm cavities can be increased by the so-called power-recycling technique. In such a scheme the power-recycling cavity is optically coupled with the interferometer’s arm cavities. A problem arises because the central coupling mirror transmits a rather high laser power and may show thermal lensing, thermo-refractive noise and photo-thermo-refractive noise. Cryogenic cooling of this mirror is also challenging, and thus thermal noise becomes a general problem. Here, we theoretically investigate an all-reflective coupling scheme of two optical cavities based on a 3-port diffraction grating. We show that power-recycling of a high-finesse arm cavity is possible without transmitting any laser power through a substrate material. The power splitting ratio of the three output ports of the grating is, surprisingly, noncritical.

© 2010 Optical Society of America

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2010 (1)

F. Bruckner, D. Friedrich, T. Clausnitzer, M. Britzger, O. Burmeister, K. Danzmann, E.-B. Kley, A. Tunnermann, and R. Schnabel, "Realization of a monolithic high-reflectivity cavity mirror from a single silicon crystal," Phys. Rev. Lett., accepted (2010).
[CrossRef] [PubMed]

2009 (3)

K. Arai et al., "Status of Japanese gravitational wave detectors," Class. Quantum Grav. 26, 204020 (2009).
[CrossRef]

B. P. Abbott,  et al., "LIGO: the Laser Interferometer Gravitational-Wave Observatory," Rep. Prog. Phys. 72, 076901 (2009).
[CrossRef]

A. Thuring, C. Graf, H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, "Broadband squeezing of quantum noise in a Michelson interferometer with Twin-Signal-Recycling," Opt. Lett. 34, 824-826 (2009).
[CrossRef] [PubMed]

2008 (1)

2007 (2)

A. Thuring, R. Schnabel, H. Luck, and K. Danzmann, "Detuned Twin-Signal-Recycling for ultrahigh-precision interferometers," Opt. Lett. 32, 985-987 (2007).
[CrossRef] [PubMed]

H. Vahlbruch, S. Chelkowski, K. Danzmann, and R. Schnabel, "Quantum engineering of squeezed states for quantum communication and metrology," New J. Phys. 9, 371 (2007).
[CrossRef]

2006 (2)

2005 (2)

A. Wicht, P. Huke, and R.-H. Rinkleff, "Advancing the Optical Feed Back Concept: Grating Enhanced External Cavity Diode Laser," Physica Scripta. T 118, 82-84 (2005).
[CrossRef]

A. Bunkowski, O. Burmeister, K. Danzmann, and R. Schnabel, "Input-output relations for a 3-port grating coupled Fabry-Perot cavity," Opt. Lett. 30, 1183-1185 (2005).
[CrossRef] [PubMed]

2004 (1)

2000 (1)

V. B. Braginsky, M. L. Gorodetsky, and S. P. Vyatchanin, "Thermo-refractive noise in gravitational wave antennae," Phys. Lett. A 271, 303-307 (2000).
[CrossRef]

1999 (1)

V. B. Braginsky, M. L. Gorodetsky and S. P. Vyatchanin, "Thermodynamical fluctuations and photo-thermal shot noise in gravitational wave antennae," Phys. Lett. A 264, 1-10 (1999).
[CrossRef]

1998 (2)

1997 (1)

V. B. Braginsky, M. L. Gorodetsky, and F. Ya. Khalili, "Optical bars in gravitational wave antennas," Phys. Lett. A 232, 340-348 (1997).
[CrossRef]

1996 (1)

G. Heinzel, J. Mizuno, R. Schilling, A. Rudiger, W. Winkler, and K. Danzmann, "An experimental demonstration of resonant sideband extraction for laser-interferometric gravitational wave detectors," Phys. Lett. A 217, 305-314 (1996).
[CrossRef]

1994 (1)

K. A. Strain, K. Danzmann, J. Mizuno, P. G. Nelson, A. R¨udiger, R. Schilling, and W. Winkler, "Thermal lensing in recycling interferometric gravitational-wave detectors," Phys. Lett. A 194, 124-132 (1994).
[CrossRef]

1993 (1)

J. Mizuno, K. A. Strain, P. G. Nelson, J. M. Chen, R. Schilling, A. Rudiger, W. Winkler, and K. Danzmann, "Resonant sideband extraction: a new configuration for interferometric gravitational wave detectors," Phys. Lett. A 175, 273-276 (1993).
[CrossRef]

1992 (1)

1988 (1)

B. J. Meers, "Recycling in laser-interferometric gravitational-wave detectors," Phys. Rev. D 38, 2317-2326 (1988).
[CrossRef]

1987 (1)

Abbott, B. P.

B. P. Abbott,  et al., "LIGO: the Laser Interferometer Gravitational-Wave Observatory," Rep. Prog. Phys. 72, 076901 (2009).
[CrossRef]

Acernese, F.

F. Acernese et al., "The Virgo status," Class. Quantum Grav. 23, S635-S642 (2006).
[CrossRef]

Arai, K.

K. Arai et al., "Status of Japanese gravitational wave detectors," Class. Quantum Grav. 26, 204020 (2009).
[CrossRef]

Beyersdorf, P.

Braginsky, V. B.

V. B. Braginsky, M. L. Gorodetsky, and S. P. Vyatchanin, "Thermo-refractive noise in gravitational wave antennae," Phys. Lett. A 271, 303-307 (2000).
[CrossRef]

V. B. Braginsky, M. L. Gorodetsky and S. P. Vyatchanin, "Thermodynamical fluctuations and photo-thermal shot noise in gravitational wave antennae," Phys. Lett. A 264, 1-10 (1999).
[CrossRef]

V. B. Braginsky, M. L. Gorodetsky, and F. Ya. Khalili, "Optical bars in gravitational wave antennas," Phys. Lett. A 232, 340-348 (1997).
[CrossRef]

Britzger, M.

F. Bruckner, D. Friedrich, T. Clausnitzer, M. Britzger, O. Burmeister, K. Danzmann, E.-B. Kley, A. Tunnermann, and R. Schnabel, "Realization of a monolithic high-reflectivity cavity mirror from a single silicon crystal," Phys. Rev. Lett., accepted (2010).
[CrossRef] [PubMed]

Bruckner, F.

F. Bruckner, D. Friedrich, T. Clausnitzer, M. Britzger, O. Burmeister, K. Danzmann, E.-B. Kley, A. Tunnermann, and R. Schnabel, "Realization of a monolithic high-reflectivity cavity mirror from a single silicon crystal," Phys. Rev. Lett., accepted (2010).
[CrossRef] [PubMed]

Bunkowski, A.

Burmeister, O.

Byer, R. L.

Chelkowski, S.

H. Vahlbruch, S. Chelkowski, K. Danzmann, and R. Schnabel, "Quantum engineering of squeezed states for quantum communication and metrology," New J. Phys. 9, 371 (2007).
[CrossRef]

Chen, J. M.

J. Mizuno, K. A. Strain, P. G. Nelson, J. M. Chen, R. Schilling, A. Rudiger, W. Winkler, and K. Danzmann, "Resonant sideband extraction: a new configuration for interferometric gravitational wave detectors," Phys. Lett. A 175, 273-276 (1993).
[CrossRef]

Clausnitzer, T.

Dahmani, B.

Danzmann, K.

F. Bruckner, D. Friedrich, T. Clausnitzer, M. Britzger, O. Burmeister, K. Danzmann, E.-B. Kley, A. Tunnermann, and R. Schnabel, "Realization of a monolithic high-reflectivity cavity mirror from a single silicon crystal," Phys. Rev. Lett., accepted (2010).
[CrossRef] [PubMed]

A. Thuring, C. Graf, H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, "Broadband squeezing of quantum noise in a Michelson interferometer with Twin-Signal-Recycling," Opt. Lett. 34, 824-826 (2009).
[CrossRef] [PubMed]

D. Friedrich, O. Burmeister, A. Bunkowski, T. Clausnitzer, S. Fahr, E.-B. Kley, A. T¨unnermann, K. Danzmann, and R. Schnabel, "Diffractive beam splitter characterization via a power-recycled interferometer," Opt. Lett. 33, 101-103 (2008).
[CrossRef] [PubMed]

A. Thuring, R. Schnabel, H. Luck, and K. Danzmann, "Detuned Twin-Signal-Recycling for ultrahigh-precision interferometers," Opt. Lett. 32, 985-987 (2007).
[CrossRef] [PubMed]

H. Vahlbruch, S. Chelkowski, K. Danzmann, and R. Schnabel, "Quantum engineering of squeezed states for quantum communication and metrology," New J. Phys. 9, 371 (2007).
[CrossRef]

A. Bunkowski, O. Burmeister, K. Danzmann, R. Schnabel, T. Clausnitzer, E.-B. Kley, and A. T¨unnermann, "Optical characterization of ultra-high diffraction efficiency gratings," Appl. Opt. 45, 5795-5799 (2006).
[CrossRef] [PubMed]

A. Bunkowski, O. Burmeister, K. Danzmann, and R. Schnabel, "Input-output relations for a 3-port grating coupled Fabry-Perot cavity," Opt. Lett. 30, 1183-1185 (2005).
[CrossRef] [PubMed]

A. Bunkowski, O. Burmeister, P. Beyersdorf, K. Danzmann, R. Schnabel, T. Clausnitzer, E.-B. Kley, and A. T¨unnermann "Low-loss grating for coupling to a high-finesse cavity," Opt. Lett. 29, 2342-2344 (2004).
[CrossRef] [PubMed]

G. Heinzel, J. Mizuno, R. Schilling, A. Rudiger, W. Winkler, and K. Danzmann, "An experimental demonstration of resonant sideband extraction for laser-interferometric gravitational wave detectors," Phys. Lett. A 217, 305-314 (1996).
[CrossRef]

K. A. Strain, K. Danzmann, J. Mizuno, P. G. Nelson, A. R¨udiger, R. Schilling, and W. Winkler, "Thermal lensing in recycling interferometric gravitational-wave detectors," Phys. Lett. A 194, 124-132 (1994).
[CrossRef]

J. Mizuno, K. A. Strain, P. G. Nelson, J. M. Chen, R. Schilling, A. Rudiger, W. Winkler, and K. Danzmann, "Resonant sideband extraction: a new configuration for interferometric gravitational wave detectors," Phys. Lett. A 175, 273-276 (1993).
[CrossRef]

Drullinger, R.

Fahr, S.

Friedrich, D.

F. Bruckner, D. Friedrich, T. Clausnitzer, M. Britzger, O. Burmeister, K. Danzmann, E.-B. Kley, A. Tunnermann, and R. Schnabel, "Realization of a monolithic high-reflectivity cavity mirror from a single silicon crystal," Phys. Rev. Lett., accepted (2010).
[CrossRef] [PubMed]

D. Friedrich, O. Burmeister, A. Bunkowski, T. Clausnitzer, S. Fahr, E.-B. Kley, A. T¨unnermann, K. Danzmann, and R. Schnabel, "Diffractive beam splitter characterization via a power-recycled interferometer," Opt. Lett. 33, 101-103 (2008).
[CrossRef] [PubMed]

Fritschel, P.

Gorodetsky, M. L.

V. B. Braginsky, M. L. Gorodetsky, and S. P. Vyatchanin, "Thermo-refractive noise in gravitational wave antennae," Phys. Lett. A 271, 303-307 (2000).
[CrossRef]

V. B. Braginsky, M. L. Gorodetsky and S. P. Vyatchanin, "Thermodynamical fluctuations and photo-thermal shot noise in gravitational wave antennae," Phys. Lett. A 264, 1-10 (1999).
[CrossRef]

V. B. Braginsky, M. L. Gorodetsky, and F. Ya. Khalili, "Optical bars in gravitational wave antennas," Phys. Lett. A 232, 340-348 (1997).
[CrossRef]

Graf, C.

Heinzel, G.

G. Heinzel, J. Mizuno, R. Schilling, A. Rudiger, W. Winkler, and K. Danzmann, "An experimental demonstration of resonant sideband extraction for laser-interferometric gravitational wave detectors," Phys. Lett. A 217, 305-314 (1996).
[CrossRef]

Hollberg, L.

Huke, P.

A. Wicht, P. Huke, and R.-H. Rinkleff, "Advancing the Optical Feed Back Concept: Grating Enhanced External Cavity Diode Laser," Physica Scripta. T 118, 82-84 (2005).
[CrossRef]

Khalili, F. Ya.

V. B. Braginsky, M. L. Gorodetsky, and F. Ya. Khalili, "Optical bars in gravitational wave antennas," Phys. Lett. A 232, 340-348 (1997).
[CrossRef]

Kley, E.-B.

Levin, Y.

Y. Levin, "Internal thermal noise in the LIGO test masses: A direct approach," Phys. Rev. D 57, 659-663 (1998).
[CrossRef]

Luck, H.

Meers, B. J.

B. J. Meers, "Recycling in laser-interferometric gravitational-wave detectors," Phys. Rev. D 38, 2317-2326 (1988).
[CrossRef]

Mehmet, M.

Mizuno, J.

G. Heinzel, J. Mizuno, R. Schilling, A. Rudiger, W. Winkler, and K. Danzmann, "An experimental demonstration of resonant sideband extraction for laser-interferometric gravitational wave detectors," Phys. Lett. A 217, 305-314 (1996).
[CrossRef]

K. A. Strain, K. Danzmann, J. Mizuno, P. G. Nelson, A. R¨udiger, R. Schilling, and W. Winkler, "Thermal lensing in recycling interferometric gravitational-wave detectors," Phys. Lett. A 194, 124-132 (1994).
[CrossRef]

J. Mizuno, K. A. Strain, P. G. Nelson, J. M. Chen, R. Schilling, A. Rudiger, W. Winkler, and K. Danzmann, "Resonant sideband extraction: a new configuration for interferometric gravitational wave detectors," Phys. Lett. A 175, 273-276 (1993).
[CrossRef]

Nelson, P. G.

K. A. Strain, K. Danzmann, J. Mizuno, P. G. Nelson, A. R¨udiger, R. Schilling, and W. Winkler, "Thermal lensing in recycling interferometric gravitational-wave detectors," Phys. Lett. A 194, 124-132 (1994).
[CrossRef]

J. Mizuno, K. A. Strain, P. G. Nelson, J. M. Chen, R. Schilling, A. Rudiger, W. Winkler, and K. Danzmann, "Resonant sideband extraction: a new configuration for interferometric gravitational wave detectors," Phys. Lett. A 175, 273-276 (1993).
[CrossRef]

R¨udiger, A.

K. A. Strain, K. Danzmann, J. Mizuno, P. G. Nelson, A. R¨udiger, R. Schilling, and W. Winkler, "Thermal lensing in recycling interferometric gravitational-wave detectors," Phys. Lett. A 194, 124-132 (1994).
[CrossRef]

Rinkleff, R.-H.

A. Wicht, P. Huke, and R.-H. Rinkleff, "Advancing the Optical Feed Back Concept: Grating Enhanced External Cavity Diode Laser," Physica Scripta. T 118, 82-84 (2005).
[CrossRef]

Rudiger, A.

G. Heinzel, J. Mizuno, R. Schilling, A. Rudiger, W. Winkler, and K. Danzmann, "An experimental demonstration of resonant sideband extraction for laser-interferometric gravitational wave detectors," Phys. Lett. A 217, 305-314 (1996).
[CrossRef]

J. Mizuno, K. A. Strain, P. G. Nelson, J. M. Chen, R. Schilling, A. Rudiger, W. Winkler, and K. Danzmann, "Resonant sideband extraction: a new configuration for interferometric gravitational wave detectors," Phys. Lett. A 175, 273-276 (1993).
[CrossRef]

Schilling, R.

G. Heinzel, J. Mizuno, R. Schilling, A. Rudiger, W. Winkler, and K. Danzmann, "An experimental demonstration of resonant sideband extraction for laser-interferometric gravitational wave detectors," Phys. Lett. A 217, 305-314 (1996).
[CrossRef]

K. A. Strain, K. Danzmann, J. Mizuno, P. G. Nelson, A. R¨udiger, R. Schilling, and W. Winkler, "Thermal lensing in recycling interferometric gravitational-wave detectors," Phys. Lett. A 194, 124-132 (1994).
[CrossRef]

J. Mizuno, K. A. Strain, P. G. Nelson, J. M. Chen, R. Schilling, A. Rudiger, W. Winkler, and K. Danzmann, "Resonant sideband extraction: a new configuration for interferometric gravitational wave detectors," Phys. Lett. A 175, 273-276 (1993).
[CrossRef]

Schnabel, R.

F. Bruckner, D. Friedrich, T. Clausnitzer, M. Britzger, O. Burmeister, K. Danzmann, E.-B. Kley, A. Tunnermann, and R. Schnabel, "Realization of a monolithic high-reflectivity cavity mirror from a single silicon crystal," Phys. Rev. Lett., accepted (2010).
[CrossRef] [PubMed]

A. Thuring, C. Graf, H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, "Broadband squeezing of quantum noise in a Michelson interferometer with Twin-Signal-Recycling," Opt. Lett. 34, 824-826 (2009).
[CrossRef] [PubMed]

D. Friedrich, O. Burmeister, A. Bunkowski, T. Clausnitzer, S. Fahr, E.-B. Kley, A. T¨unnermann, K. Danzmann, and R. Schnabel, "Diffractive beam splitter characterization via a power-recycled interferometer," Opt. Lett. 33, 101-103 (2008).
[CrossRef] [PubMed]

A. Thuring, R. Schnabel, H. Luck, and K. Danzmann, "Detuned Twin-Signal-Recycling for ultrahigh-precision interferometers," Opt. Lett. 32, 985-987 (2007).
[CrossRef] [PubMed]

H. Vahlbruch, S. Chelkowski, K. Danzmann, and R. Schnabel, "Quantum engineering of squeezed states for quantum communication and metrology," New J. Phys. 9, 371 (2007).
[CrossRef]

A. Bunkowski, O. Burmeister, K. Danzmann, R. Schnabel, T. Clausnitzer, E.-B. Kley, and A. T¨unnermann, "Optical characterization of ultra-high diffraction efficiency gratings," Appl. Opt. 45, 5795-5799 (2006).
[CrossRef] [PubMed]

A. Bunkowski, O. Burmeister, K. Danzmann, and R. Schnabel, "Input-output relations for a 3-port grating coupled Fabry-Perot cavity," Opt. Lett. 30, 1183-1185 (2005).
[CrossRef] [PubMed]

A. Bunkowski, O. Burmeister, P. Beyersdorf, K. Danzmann, R. Schnabel, T. Clausnitzer, E.-B. Kley, and A. T¨unnermann "Low-loss grating for coupling to a high-finesse cavity," Opt. Lett. 29, 2342-2344 (2004).
[CrossRef] [PubMed]

Shoemaker, D.

Strain, K. A.

K. A. Strain, K. Danzmann, J. Mizuno, P. G. Nelson, A. R¨udiger, R. Schilling, and W. Winkler, "Thermal lensing in recycling interferometric gravitational-wave detectors," Phys. Lett. A 194, 124-132 (1994).
[CrossRef]

J. Mizuno, K. A. Strain, P. G. Nelson, J. M. Chen, R. Schilling, A. Rudiger, W. Winkler, and K. Danzmann, "Resonant sideband extraction: a new configuration for interferometric gravitational wave detectors," Phys. Lett. A 175, 273-276 (1993).
[CrossRef]

Sun, K.-X.

Thuring, A.

Tunnermann, A.

Vahlbruch, H.

A. Thuring, C. Graf, H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, "Broadband squeezing of quantum noise in a Michelson interferometer with Twin-Signal-Recycling," Opt. Lett. 34, 824-826 (2009).
[CrossRef] [PubMed]

H. Vahlbruch, S. Chelkowski, K. Danzmann, and R. Schnabel, "Quantum engineering of squeezed states for quantum communication and metrology," New J. Phys. 9, 371 (2007).
[CrossRef]

Vyatchanin, S. P.

V. B. Braginsky, M. L. Gorodetsky, and S. P. Vyatchanin, "Thermo-refractive noise in gravitational wave antennae," Phys. Lett. A 271, 303-307 (2000).
[CrossRef]

V. B. Braginsky, M. L. Gorodetsky and S. P. Vyatchanin, "Thermodynamical fluctuations and photo-thermal shot noise in gravitational wave antennae," Phys. Lett. A 264, 1-10 (1999).
[CrossRef]

Weiss, R.

Wicht, A.

A. Wicht, P. Huke, and R.-H. Rinkleff, "Advancing the Optical Feed Back Concept: Grating Enhanced External Cavity Diode Laser," Physica Scripta. T 118, 82-84 (2005).
[CrossRef]

Winkler, W.

G. Heinzel, J. Mizuno, R. Schilling, A. Rudiger, W. Winkler, and K. Danzmann, "An experimental demonstration of resonant sideband extraction for laser-interferometric gravitational wave detectors," Phys. Lett. A 217, 305-314 (1996).
[CrossRef]

J. Mizuno, K. A. Strain, P. G. Nelson, J. M. Chen, R. Schilling, A. Rudiger, W. Winkler, and K. Danzmann, "Resonant sideband extraction: a new configuration for interferometric gravitational wave detectors," Phys. Lett. A 175, 273-276 (1993).
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Appl. Opt. (2)

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New J. Phys. (1)

H. Vahlbruch, S. Chelkowski, K. Danzmann, and R. Schnabel, "Quantum engineering of squeezed states for quantum communication and metrology," New J. Phys. 9, 371 (2007).
[CrossRef]

Opt. Lett. (7)

Phys. Lett. A (6)

K. A. Strain, K. Danzmann, J. Mizuno, P. G. Nelson, A. R¨udiger, R. Schilling, and W. Winkler, "Thermal lensing in recycling interferometric gravitational-wave detectors," Phys. Lett. A 194, 124-132 (1994).
[CrossRef]

V. B. Braginsky, M. L. Gorodetsky and S. P. Vyatchanin, "Thermodynamical fluctuations and photo-thermal shot noise in gravitational wave antennae," Phys. Lett. A 264, 1-10 (1999).
[CrossRef]

V. B. Braginsky, M. L. Gorodetsky, and S. P. Vyatchanin, "Thermo-refractive noise in gravitational wave antennae," Phys. Lett. A 271, 303-307 (2000).
[CrossRef]

V. B. Braginsky, M. L. Gorodetsky, and F. Ya. Khalili, "Optical bars in gravitational wave antennas," Phys. Lett. A 232, 340-348 (1997).
[CrossRef]

J. Mizuno, K. A. Strain, P. G. Nelson, J. M. Chen, R. Schilling, A. Rudiger, W. Winkler, and K. Danzmann, "Resonant sideband extraction: a new configuration for interferometric gravitational wave detectors," Phys. Lett. A 175, 273-276 (1993).
[CrossRef]

G. Heinzel, J. Mizuno, R. Schilling, A. Rudiger, W. Winkler, and K. Danzmann, "An experimental demonstration of resonant sideband extraction for laser-interferometric gravitational wave detectors," Phys. Lett. A 217, 305-314 (1996).
[CrossRef]

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Physica Scripta. T (1)

A. Wicht, P. Huke, and R.-H. Rinkleff, "Advancing the Optical Feed Back Concept: Grating Enhanced External Cavity Diode Laser," Physica Scripta. T 118, 82-84 (2005).
[CrossRef]

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Figures (12)

Fig. 1.
Fig. 1.

(a) In laser-interferometric gravitational wave detectors two coupled cavities are formed by the two mirrors of an arm cavity and the power-recycling mirror (PRM). (b) The transmissive coupling mirror can be replaced by an all-reflective 3-port grating. Without laser transmission thermal lensing and substrate thermal noise sources can be substantially suppressed.

Fig. 2.
Fig. 2.

Labeling of the optical fields at a linear three-mirror cavity. Every mirror mi has two ports with pairs of in- and output field amplitudes denoted by the vectors (a i ,b i ) and (a′ i ,b′ i ), respectively, and the amplitude transmission and reflection coefficients ρi and τi . Thus two optically coupled cavities with the macroscopic length L i and the microscopic detuning parameter Φ i are formed. For most considerations in this paper we choose ρ 1 2 = 0.7, ρ 2 2 = 0.8, ρ 3 2 = 0.9, and b3 = 0. For possible applications in high-power laser interferometry we also discuss parameter sets with ρ 3-values close to unity.

Fig. 3.
Fig. 3.

Transmitted power at a loss-less three-mirror cavity with ρ 1 2 = 0.7,ρ 2 2 = 0.8 and ρ 3 2 = 0.9 as a function of the detunings Φ1 and Φ2. Here, on peak resonance all power is transmitted. The lines in the zoomed-in figure on the right represent the resonance branches of the individual cavities given by Eqs. (12) (dashed) and (14) (dotted), respectively.

Fig. 4.
Fig. 4.

Compound mirror reflectivities ∣ρ m2m1 2 (a) and ∣ρ m2m3 2 (b) as a function of Φ1 and Φ2, respectively. The maximum of the transmitted power occurs at the set of detunings where the compound mirrors matches the reflectivity of the respective single mirror ∣ρ m2m1 1 = ±25°)∣2 =ρ 3 2 =0.7 and ∣ρ m2m3 2 = ±6.5°)∣2 =ρ 1 2 = 0.9.

Fig. 5.
Fig. 5.

3-port grating coupler: (a) Labeling of amplitude diffraction coefficients for second-order Littrow configuration. (b) Amplitude diffraction coefficients and amplitude reflectivity for normal incidence.

Fig. 6.
Fig. 6.

Schematic for an all-reflective coupling of two optical cavities by a 3-port grating. The amplitudes of the fields impinging on the optical component with subscript i are denoted as a i , b i , and c i . The output fields are denoted as a′ i , b′ i , and c′ i . The grating g2 and mirror m1 form the so-called power-recycling cavity of macroscopic length L1 and microscopic detuning parameter Φ1. The grating and mirror m3 form the so-called arm cavity of macroscopic length L2 and microscopic detuning parameter Φ2.

Fig. 7.
Fig. 7.

Transmitted power at a power-recycled 3-port cavity with ρ 1 2 = 0.7,ρ 0 2 = 0.8, η 2 = η 2min and ρ 3 2 = 0.9 as a function of the detunings Φ1 and Φ2. A moderate power enhancement is only present around a maximum at Φ1 = 90° and Φ2 = 0°. The lines in the zoomed-in figure on the right represent the resonant detuning of the first cavity Φ1 res = -0.5arg[ρ g2m3 2)] (dashed) to compensate the phase shift due to reflection at the second cavity and the resonant detuning of the second cavity Φ2 res = -0.5arg[ρ g2m1 1)] (dotted), respectively.

Fig. 8.
Fig. 8.

Compound mirror power reflectivities ∣ρ g2m1 2 (a) and ∣ρ g2m3 2 (b) for the η 2min-configuration as a function of Φ1 and Φ2, respectively. The resonance of the second cavity Φ2 = 0° corresponds to a maximum of the compound mirror reflectivity of the cavity.

Fig. 9.
Fig. 9.

Transmitted power at a power-recycled 3-port cavity with ρ 1 2 = 0.7, ρ 0 2 = 0.8,η 2 = η 2max and ρ 3 2 = 0.9 as a function of the detunings Φ1 and Φ2. The maxima of the transmitted power are positioned at the crossings of the individual resonance branches, e.g. at Φ1 = 78°,Φ2 = -6.5° and Φ1 = 102°,Φ2 = 6.5°. At this operating point only about half of the light is transmitted. The lines in the zoomed-in figure on the right represent Φ1 res = -0.5arg[ρ g2m3 2)] (dashed) and Φ2 res = -0.5arg[ρ g2m1 1)] (dotted), respectively.

Fig. 10.
Fig. 10.

Compound mirror reflectivities ∣ρ g2m1 2 (a) and ∣ρ g2m3 2 (b) for the η 2max-configuration as a function of Φ1 and Φ2, respectively. The maximum of the transmitted power (see Figure 9, with Φ1 = 78° and Φ1 = 102°, respectively, and Φ2 = ±6°) does not correspond to the detuning where ∣ρ g2m3 2 = ρ 1 2 =0.7.

Fig. 11.
Fig. 11.

Transmitted power at a power-recycled 3-port cavity with ρ 1 2 = 0.7, ρ 0 2 = 0.8,η 2 2 = η 0 2 = 0.45 and ρ 3 2 = 0.9 as a function of the detunings Φ1 and Φ2. The operating point Φ1 = 113°, Φ2 = -3.3° corresponds to a maximum of the internal power in the arm cavity. The lines in the zoomed-in figure on the right represent Φ1 res = - 0.5arg[ρ g2m3 2)] (dashed) and Φ2 res = -0.5arg[ρ g2m1 1)] (dotted), respectively.

Fig. 12.
Fig. 12.

Compound mirror reflectivities ∣ρ g2m1 2 (a) and ∣ρ g2m3 2 (b) for η 2 2 = η 0 2 as a function of Φ1 and Φ2, respectively. Note that ∣ρ g2m3 2 is not symmetric around Φ1 = 0°. As a consequence no resonance doublets occur in Figure 11. The maximum of the transmitted power does not correspond to the detuning where |∣ρ g2m3 2 = ρ 3 2 = 0.9 and ∣ρ g2m1 2 = ρ 1 2 = 0.7, respectively.

Equations (29)

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S = ( ρ ρ ) ,
ρ 2 + τ 2 = 1 .
a 2 = i τ 2 b 2 + ρ 2 a 2 ,
b 2 = i τ 2 a 2 + ρ 2 b 2 ,
b 2 = ρ 3 e 2 i Φ 2 b 2 .
b 2 = i τ 2 1 ρ 2 ρ 3 e 2 i Φ 2 a 2 .
a 2 = ρ 2 ρ 3 e 2 i Φ 2 1 ρ 2 ρ 3 e 2 i Φ 2 a 2 .
ρ m 2 m 3 = ρ 2 ρ 3 e 2 i Φ 2 1 ρ 2 ρ 3 e 2 i Φ 2 .
b 1 = i τ 1 1 ρ 2 ρ m 2 m 3 e 2 i Φ 1 a 1 .
a 1 = ρ 1 ρ m 2 m 3 e 2 i Φ 1 1 ρ 1 ρ m 2 m 3 e 2 i Φ 1 a 1 ,
b 3 = τ 2 τ 3 e i Φ 2 1 ρ 2 ρ 3 e 2 i Φ 2 b 1 · e i Φ 1 .
Φ 1 res = 1 2 arg [ ρ m 2 m 3 ( Φ 2 ) ] .
ρ m 2 m 1 = ρ 2 ρ 1 e 2 i Φ 1 1 ρ 1 ρ 2 e 2 i Φ 1 ,
Φ 2 res = 1 2 arg [ ρ m 2 m 1 ( Φ 1 ) ] .
ρ m 2 m 3 ( Φ 2 ) = ρ 1 ,
ρ m 2 m 1 ( Φ 1 ) = ρ 3 ,
ρ 0 2 + 2 η 1 2 = 1 ,
η 0 2 + η 1 2 + η 2 2 = 1 .
η 0 min max = η 2 min max = 1 ± ρ 0 2 .
a 2 = η 2 e i ϕ 2 a 2 + η 1 e i ϕ 1 b 2 ,
b 2 = η 1 e i ϕ 1 a 2 + ρ 0 b 2 ,
b 2 = ρ 3 e 2 i Φ 2 b 2 .
b 2 = η 1 e i ϕ 1 1 ρ 0 ρ 3 e 2 i Φ 2 a 2 .
a 2 = ( ρ 3 η 1 2 e 2 i ( ϕ 1 + Φ 2 ) 1 ρ 0 ρ 3 e 2 i Φ 2 + η 2 e i ϕ 2 ) a 2 ρ g 2 m 3 .
b 1 = i τ 1 1 ρ 1 ρ g 2 m 3 e 2 i Φ 1 a 1 .
ρ g 2 m 1 = ρ 1 η 1 2 e 2 i ( ϕ 1 + Φ 1 ) 1 ρ 1 η 2 e i ( ϕ 2 + 2 Φ 1 ) + ρ 0 .
a 1 = ( ρ 1 τ 1 2 ρ g 2 m 3 e 2 i Φ 1 1 ρ 1 ρ g 2 m 3 e 2 i Φ 1 ) · a 1 ,
b 3 = τ 1 τ 3 η 1 e i ( ϕ 1 + Φ 1 + Φ 2 ) ( 1 ρ 0 ρ 3 e 2 i Φ 2 ) ( 1 ρ 1 ρ g 2 m 3 e 2 i Φ 1 ) · a 1 ,
c 2 = [ ( η 0 e i Φ 1 + ρ 3 η 1 2 e i ( 2 ϕ 1 + Φ 1 + 2 Φ 2 ) 1 ρ 0 ρ 3 e 2 i Φ 2 ) . i τ 1 1 ρ 1 ρ g 2 m 3 e 2 i Φ 1 ] · a 1 .

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