OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 16, Iss. 7 — Jul. 1, 1999
  • pp: 1730–1739

Method for analyzing multiple-mirror coupled optical systems

Jun Mizuno and Ichirou Yamaguchi  »View Author Affiliations

JOSA A, Vol. 16, Issue 7, pp. 1730-1739 (1999)

View Full Text Article

Enhanced HTML    Acrobat PDF (320 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



A general method of analyzing Michelson-based, multiple-mirror optical systems, such as interferometers designed as gravitational-wave detectors, is described. Assuming that phase-modulated light is injected, signals that will be generated by demodulation of the photocurrent anywhere in the system, together with their frequency dependences, can be evaluated. Since this method is based on the manipulation of matrices, it is appropriate for computerization.

© 1999 Optical Society of America

OCIS Codes
(000.3860) General : Mathematical methods in physics
(120.2230) Instrumentation, measurement, and metrology : Fabry-Perot
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(120.4570) Instrumentation, measurement, and metrology : Optical design of instruments
(120.5050) Instrumentation, measurement, and metrology : Phase measurement
(120.5060) Instrumentation, measurement, and metrology : Phase modulation

Original Manuscript: October 22, 1998
Revised Manuscript: February 24, 1999
Manuscript Accepted: February 24, 1999
Published: July 1, 1999

Jun Mizuno and Ichirou Yamaguchi, "Method for analyzing multiple-mirror coupled optical systems," J. Opt. Soc. Am. A 16, 1730-1739 (1999)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. Whitcomb, M. E. Zucker, “LIGO: The laser interferometer gravitational-wave observatory,” Science 256, 325–333 (1992). [CrossRef] [PubMed]
  2. C. Bradaschia, R. Del Fabbro, A. Di Virgilio, A. Giazotto, H. Kautzky, V. Montelatici, D. Passuello, A. Brillet, O. Cregut, P. Hello, C. N. Man, P. T. Manh, A. Marraud, D. Shoemaker, J. Y. Vinet, F. Barone, L. Di Fiore, L. Milano, G. Russo, J. M. Aguirregabiria, H. Bel, J. P. Duruisseau, G. Le Denmat, Ph. Tourrenc, M. Capozzi, M. Longo, M. Lops, I. Pinto, G. Rotoli, T. Damour, S. Bonazzola, J. A. Marck, Y. Gourghoulon, L. E. Holloway, F. Fuligni, V. Iafolla, G. Natale, “The VIRGO project: A wide band antenna for gravitational wave detection,” Nucl. Instrum. Methods Phys. Res. A 289, 518–525 (1990). [CrossRef]
  3. K. Danzmann, H. Lück, A. Rüdiger, R. Schilling, M. Schrempel, W. Winkler, J. Hough, G. P. Newton, N. A. Robertson, H. Ward, A. M. Campbell, J. E. Logan, D. I. Robertson, K. A. Strain, J. R. J. Bennett, V. Kose, M. Kühne, B. F. Schutz, D. Nicholson, J. Shuttleworth, H. Welling, P. Aufmuth, R. Rinkeff, A. Tünnermann, B. Willke, “GEO 600—A 600 m laser interferometric gravitational wave antenna,” in First Edoardo Amaldi Conference on Gravitational Wave Experiments, E. Coccia, G. Pizzella, F. Ronga, eds. (World Scientific, Singapore1995), pp. 100–111.
  4. J. Mizuno, K. A. Strain, P. G. Nelson, J. M. Chen, R. Schilling, A. Rüdiger, W. Winkler, K. Danzmann, “Resonant sideband extraction: a new configuration for interferometric gravitational wave detectors,” Phys. Lett. A 175, 273–276 (1993). [CrossRef]
  5. J. Mizuno, “Comparison of optical configurations for interferometric gravitational-wave detectors,” (Max-Planck-Institut für Quantenoptik, D–85740, Garching, Germany, 1995).
  6. In addition to the longitudinal position, the orientation of each optical component must be aligned, either passively or actively, to the proper optical axis. This is, however, a different (though related) problem that can be treated separately. See, e.g., Ref. 7, and references therein.
  7. Y. Hefetz, N. Mavalvala, D. Sigg, “Principles of calculating alignment signals in complex resonant optical interferometers,” J. Opt. Soc. Am. B 14, 1597–1605 (1997). [CrossRef]
  8. L. Schnupp, talk presented at the European Collaboration Meeting on Interferometric Detection of Gravitational Waves, Sorrento, Italy, 1988.
  9. M. W. Regehr, “Signal extraction and control for an interferometric gravitational wave detector,” Ph.D. dissertation (California Institute of Technology, Pasadena, Calif., 1994).
  10. M. W. Regehr, F. J. Raab, S. E. Whitcomb, “Demonstration of a power-recycled Michelson interferometer with Fabry-Perot arms using frontal modulation,” Opt. Lett. 20, 1507–1509 (1995). [CrossRef] [PubMed]
  11. R. Flaminio, H. Heitmann, “Longitudinal control of an interferometer for the detection of gravitational waves,” Phys. Lett. A 214, 112–122 (1996). [CrossRef]
  12. It is difficult to modulate the optical path lengths of an interferometric gravitational-wave detector for optical and mechanical reasons [see, e.g., Ref. 11].
  13. D. Schnier, J. Mizuno, G. Heinzel, H. Lück, A. Rüdiger, R. Schilling, M. Schrempel, W. Winkler, K. Danzmann, “Power recycling in the Garching 30 m prototype interferometer for gravitational-wave detection,” Phys. Lett. A 225, 210–216 (1997). [CrossRef]
  14. R. W. P. Drever, J. Hough, A. J. Munley, S.-A. Lee, R. Spero, S. E. Whitcomb, H. Ward, G. M. Ford, M. Hereld, N. A. Robertson, I. Kerr, J. R. Pugh, G. P. Newton, B. Meers, E. D. Brook, Y. Gürsel, “Gravitational wave detectors using laser interferometers and optical cavities: ideas, principles and prospects,” in Quantum Optics, Experimental Gravitation, and Measurement Theory, P. Meystre, M. O. Scully, eds. (Plenum, New York, 1983), pp. 503–514.
  15. B. J. Meers, “Recycling in laser-interferometric gravitational-wave detectors,” Phys. Rev. D 38, 2317–2326 (1988). [CrossRef]
  16. The use of the algorithm described in this paper, however, is not limited to these assumptions. For instance, if the east and south paths are regarded as the two arms and the north path is used for detection, the response of (non-Michelson-based) synchronous recycling (Ref. 14) can be analyzed.
  17. These matrices are often used in analyzing multilayer dielectric coatings, as can be found in, e.g., Ref. 18. We, however, adopted a slightly modified definition of Eq. (4) that does not distinguish the direction of incidence. (The results are identical except for constant offsets in the definition of detunings.)
  18. M. V. Klein, T. E. Furtak, Optics, 2nd ed. (Wiley, New York, 1986), Sec. 5.4.
  19. To be exact, this expression must be rewritten as “these sidebands work mainly as the local oscillators” because there is actually no clear distinction between the carrier and the local oscillator in such a system in which the two are modulated by the signal simultaneously (like the example discussed in this paper). The signal sidebands induced on the carrier light interfere with the local oscillator light, whereas the signal sidebands induced on the local oscillator light interfere with the carrier light in contributing the signal, as is described in Section 4. Nevertheless, we adopt the naming in the text in view of the “main” signal generation.
  20. P. Fritschel, D. Shoemaker, R. Weiss, “Demonstration of light recycling in a Michelson interferometer with Fabry–Perot cavities,” Appl. Opt. 31, 1412–1418 (1992). [CrossRef] [PubMed]
  21. Evaluating the frequency response of a coupled optical system was initially explored by Vinet et al. (Ref. 22) to obtain the detector’s response to gravitational waves (i.e., not for the control purpose). The analysis adopted in this paper is similar to theirs but has a slightly different viewpoint, and it is modified and extended for generality. In their approach the signal-induced sidebands are produced from the carrier in each transit inside a cavity, whereas in our approach they are produced from the summed amplitude of the carrier. The results should be identical, and the choice is a matter of preference.
  22. J.-Y. Vinet, B. Meers, C. N. Man, A. Brillet, “Optimization of long-baseline optical interferometers for gravitational-wave detection,” Phys. Rev. D 38, 433–447 (1988). [CrossRef]
  23. Actually there will be a slight difference due to the nonstationary shot-noise effect discussed in, e.g., Refs. 24 and 25. It is, in principle, possible to extend the present analysis to include such effects. In practice, however, the difference is less than a few decibels and can be ignored in almost any applications (excepting, probably, the most sensitive gravitational-wave readout).
  24. T. M. Niebauer, R. Schilling, K. Danzmann, A. Rüdiger, W. Winkler, “Nonstationary shot noise and its effect on the sensitivity of interferometers,” Phys. Rev. A 43, 5022–5029 (1991). [CrossRef] [PubMed]
  25. B. J. Meers, K. A. Strain, “Modulation, signal, and quantum noise in interferometers,” Phys. Rev. A 44, 4693–4703 (1991). [CrossRef] [PubMed]
  26. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B: Photophys. Laser Chem. 31, 97–105 (1983). [CrossRef]
  27. J. Hough, H. Ward, G. A. Kerr, N. L. Mackenzie, B. J. Meers, G. P. Newton, D. I. Robertson, N. A. Robertson, R. Schilling, “The stabilisation of lasers for interferometric gravitational wave detectors,” in The Detection of Gravitational Waves, D. G. Blair, ed. (Cambridge U. Press, Cambridge, UK, 1991), pp. 329–352.
  28. G. Heinzel, K. A. Strain, J. Mizuno, K. D. Skeldon, B. Willke, W. Winkler, R. Schilling, A. Rüdiger, K. Danzmann, “Experimental demonstration of a suspended dual recycling interferometer,” Phys. Rev. Lett. 81, 5493–5496 (1998). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


Fig. 1 Fig. 2 Fig. 3
Fig. 4

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited