Abstract

The exponential decay and phase shift reflectivity measurement methods are examined for a real optical cavity, i.e., one that includes mechanical vibrations and light source fluctuations in wavelength and amplitude. We compare the two methods and examine the problems inherent in each and present methods for overcoming these. Both methods are shown to be excellent for measuring high reflectivity cavity losses (particularly for the free electron laser case) although suitable precautions should be taken.

© 1991 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. M. Herbelin, J. A. McKay, M. A. Kwok, R. H. Ueunten, D. S. Urevig, D. J. Spencer, D. J. Benard, “Sensitive Measurement of Photon Lifetime and True Reflectances in an Optical Cavity by a Phase-Shift Method,” Appl. Opt. 19, 144–147 (1980).
    [CrossRef] [PubMed]
  2. N. A. Vinokurov, U. N. Litvinenko, “Method for Measuring Reflection Coefficients Near Unity,” Preprint INP 79-24, Institute of Nuclear Physics 630090, Novosibirsk, U.S.S.R.
  3. M. Billardon et al., “Recent Results of the ACO Storage Ring FEL Experiment,” J. de Physique 44, 29–71 (1983).
  4. P. Elleaume, M. Velghe, M. Billardon, J. M. Ortega, “Diagnostic Techniques and UV-Induced Degradation of the Mirrors Used in the Orsay Storage Ring Free-Electron Laser,” Appl. Opt. 24, 2762–2770 (1985).
    [CrossRef] [PubMed]
  5. See, for example, Ref. 3 and, for general properties of spontaneous emission of undulators, D. F. Alferov, Yu. A. Bashmakov, E. G. Bessonov, “Theory of the Undulator Radiation,” Sov. Phys. Tech. Phys. 18, 1336 (1974); W. B. Colson, Thesis (unpublished), Stanford U. (1977); S. Krinsky, “An Undulator for the 700 MeV VUV Ring of the National Synchrotron Light Source,” Nucl Instrum. Methods 172, 73 (1980); D. J. Thomson, W. Poole, Eds., ESRF, Supplement II, The Machine (European Science Foundation, Strasbourg, France, 1979); J. M. Ortega, M. Billardon, G. Jezequel, P. Thiry, Y. Petroff, “Vacuum Ultraviolet Emission of a Permanent Magnet Undulator on the ACO Storage Ring in Orsay,” J. Phys. Paris 45, 1883 (1984).
    [CrossRef]

1985 (1)

1983 (1)

M. Billardon et al., “Recent Results of the ACO Storage Ring FEL Experiment,” J. de Physique 44, 29–71 (1983).

1980 (1)

1974 (1)

See, for example, Ref. 3 and, for general properties of spontaneous emission of undulators, D. F. Alferov, Yu. A. Bashmakov, E. G. Bessonov, “Theory of the Undulator Radiation,” Sov. Phys. Tech. Phys. 18, 1336 (1974); W. B. Colson, Thesis (unpublished), Stanford U. (1977); S. Krinsky, “An Undulator for the 700 MeV VUV Ring of the National Synchrotron Light Source,” Nucl Instrum. Methods 172, 73 (1980); D. J. Thomson, W. Poole, Eds., ESRF, Supplement II, The Machine (European Science Foundation, Strasbourg, France, 1979); J. M. Ortega, M. Billardon, G. Jezequel, P. Thiry, Y. Petroff, “Vacuum Ultraviolet Emission of a Permanent Magnet Undulator on the ACO Storage Ring in Orsay,” J. Phys. Paris 45, 1883 (1984).
[CrossRef]

Alferov, D. F.

See, for example, Ref. 3 and, for general properties of spontaneous emission of undulators, D. F. Alferov, Yu. A. Bashmakov, E. G. Bessonov, “Theory of the Undulator Radiation,” Sov. Phys. Tech. Phys. 18, 1336 (1974); W. B. Colson, Thesis (unpublished), Stanford U. (1977); S. Krinsky, “An Undulator for the 700 MeV VUV Ring of the National Synchrotron Light Source,” Nucl Instrum. Methods 172, 73 (1980); D. J. Thomson, W. Poole, Eds., ESRF, Supplement II, The Machine (European Science Foundation, Strasbourg, France, 1979); J. M. Ortega, M. Billardon, G. Jezequel, P. Thiry, Y. Petroff, “Vacuum Ultraviolet Emission of a Permanent Magnet Undulator on the ACO Storage Ring in Orsay,” J. Phys. Paris 45, 1883 (1984).
[CrossRef]

Bashmakov, Yu. A.

See, for example, Ref. 3 and, for general properties of spontaneous emission of undulators, D. F. Alferov, Yu. A. Bashmakov, E. G. Bessonov, “Theory of the Undulator Radiation,” Sov. Phys. Tech. Phys. 18, 1336 (1974); W. B. Colson, Thesis (unpublished), Stanford U. (1977); S. Krinsky, “An Undulator for the 700 MeV VUV Ring of the National Synchrotron Light Source,” Nucl Instrum. Methods 172, 73 (1980); D. J. Thomson, W. Poole, Eds., ESRF, Supplement II, The Machine (European Science Foundation, Strasbourg, France, 1979); J. M. Ortega, M. Billardon, G. Jezequel, P. Thiry, Y. Petroff, “Vacuum Ultraviolet Emission of a Permanent Magnet Undulator on the ACO Storage Ring in Orsay,” J. Phys. Paris 45, 1883 (1984).
[CrossRef]

Benard, D. J.

Bessonov, E. G.

See, for example, Ref. 3 and, for general properties of spontaneous emission of undulators, D. F. Alferov, Yu. A. Bashmakov, E. G. Bessonov, “Theory of the Undulator Radiation,” Sov. Phys. Tech. Phys. 18, 1336 (1974); W. B. Colson, Thesis (unpublished), Stanford U. (1977); S. Krinsky, “An Undulator for the 700 MeV VUV Ring of the National Synchrotron Light Source,” Nucl Instrum. Methods 172, 73 (1980); D. J. Thomson, W. Poole, Eds., ESRF, Supplement II, The Machine (European Science Foundation, Strasbourg, France, 1979); J. M. Ortega, M. Billardon, G. Jezequel, P. Thiry, Y. Petroff, “Vacuum Ultraviolet Emission of a Permanent Magnet Undulator on the ACO Storage Ring in Orsay,” J. Phys. Paris 45, 1883 (1984).
[CrossRef]

Billardon, M.

Elleaume, P.

Herbelin, J. M.

Kwok, M. A.

Litvinenko, U. N.

N. A. Vinokurov, U. N. Litvinenko, “Method for Measuring Reflection Coefficients Near Unity,” Preprint INP 79-24, Institute of Nuclear Physics 630090, Novosibirsk, U.S.S.R.

McKay, J. A.

Ortega, J. M.

Spencer, D. J.

Ueunten, R. H.

Urevig, D. S.

Velghe, M.

Vinokurov, N. A.

N. A. Vinokurov, U. N. Litvinenko, “Method for Measuring Reflection Coefficients Near Unity,” Preprint INP 79-24, Institute of Nuclear Physics 630090, Novosibirsk, U.S.S.R.

Appl. Opt. (2)

J. de Physique (1)

M. Billardon et al., “Recent Results of the ACO Storage Ring FEL Experiment,” J. de Physique 44, 29–71 (1983).

Sov. Phys. Tech. Phys. (1)

See, for example, Ref. 3 and, for general properties of spontaneous emission of undulators, D. F. Alferov, Yu. A. Bashmakov, E. G. Bessonov, “Theory of the Undulator Radiation,” Sov. Phys. Tech. Phys. 18, 1336 (1974); W. B. Colson, Thesis (unpublished), Stanford U. (1977); S. Krinsky, “An Undulator for the 700 MeV VUV Ring of the National Synchrotron Light Source,” Nucl Instrum. Methods 172, 73 (1980); D. J. Thomson, W. Poole, Eds., ESRF, Supplement II, The Machine (European Science Foundation, Strasbourg, France, 1979); J. M. Ortega, M. Billardon, G. Jezequel, P. Thiry, Y. Petroff, “Vacuum Ultraviolet Emission of a Permanent Magnet Undulator on the ACO Storage Ring in Orsay,” J. Phys. Paris 45, 1883 (1984).
[CrossRef]

Other (1)

N. A. Vinokurov, U. N. Litvinenko, “Method for Measuring Reflection Coefficients Near Unity,” Preprint INP 79-24, Institute of Nuclear Physics 630090, Novosibirsk, U.S.S.R.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Exponential decay time measurement.

Fig. 2
Fig. 2

Experimental setup for instantaneous phase shift measurement. For real loss measurements the lock-in exit filter is tuned over 1 s, and the calculator averaged separate signals Asin and Acos during 30 s before the phase calculation corresponding to Eq. (14).

Fig. 3
Fig. 3

Examples of temporal evolution of the Fabry-Perot exit signal using the experimental setup with a 2-mW He–Ne laser: (A) Asin and Acos signals are simultaneously recorded with a double trace oscilloscope; (B) evolution of the instantaneous phase for another recording and comparison with the averaging corresponding to Eq. (14).

Fig. 4
Fig. 4

Spectrum of the 6328-Å He–Ne line: (A) spectrum with a time scan of 50 ms; (B) corresponding phase distribution over one 300-MHz free spectral range for cavity length L = 50 cm; (C) envelope of 256 scans illustrating the jitter of laser modes during 13 s; (D) corresponding averaged phase distribution over one free spectral range.

Fig. 5
Fig. 5

(I) Example of calculated F(n) functions [Eq. (23)] for the time range 0 < t < 2τ after switching the light source: (a) mirror reflectivity R = 1 − 6 × 10−3, λ = 5000 Å, linewidth σω/ω0 = 1 × 10−6, mirror velocity υ/c = 1 × 10−7; (b) R = 1–3 × 10−2, λ = 5000 Å, σω/ω0 = 1 × 10−7, υ/c = 1 × 10−8; (II) Experimental decay measurements: (c) detuned optical cavity; (d) and (e) two successive decays with the optical cavity tuned.

Fig. 6
Fig. 6

Standard deviation σF of the F(t) functions calculated for 0 < t < 2τ vs mirror velocity υ/c and linewidth σω/ω0. This calculation was performed for a simple Gaussian linewidth and mirror reflectivity R = 0.99.

Equations (39)

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

d I / d t = I p / Ө = I / τ ,
τ = 2 L c ( 1 R 2 ) ,
τ = 2 L c log R 2
tan ϕ = 2 π ν 0 τ
tan φ = 2 π ν o Ө R 2 1 R 2 2 π ν o τ R 2 log R 2 1 R 2
A in ( t ) = A o cos ( ω o t + φ o ) sin ω t ,
A out ( t ) = T r A o 2 k = 0 R k { sin [ ( ω + ω 0 ) ( t k Ө ) ] + sin [ ( ω ω 0 ) ( t k Ө ) ] } ,
A out ( t ) = T r A o 2 [ a sin ( ω + ω 0 ) t b cos ( ω + ω 0 ) t + c sin ( ω ω 0 ) t d cos ( ω ω 0 ) t 1 + 2 R cos ( ω ω 0 ) Ө + R 2 ] ,
a = 1 R cos ( ω + ω 0 ) Ө, b = R sin ( ω + ω 0 ) Ө, c = 1 + R cos ( ω ω 0 ) Ө , d = R sin ( ω ω 0 ) Ө ,
I out ( t ) = T r 2 A o 2 / 4 I o cos ( 2 ω 0 t ϕ ) ,
I o 2 = 4 [ 1 + R 2 2 R cos ( ω + ω 0 ) Ө ] [ 1 + R 2 2 R cos ( ω ω 0 ) Ө ] , tan ϕ = 2 R sin ω Ө cos ω 0 Ө R 2 sin 2 ω 0 Ө 1 2 R cos ω Ө cos ω 0 Ө + R 2 cos 2 ω 0 Ө .
I o = 1 ( 1 R ) 2 [ 1 + 4 R ( 1 R ) 2 sin 2 φ ] ,
tan ϕ = 2 ω o Ө R ( 1 R 2 sin 2 φ ) ( 1 R ) 2 [ 1 + 4 R ( 1 R ) 2 sin 2 φ ] .
I out ( t ) T r 2 A o 2 / 4 = ( 1 2 R cos 2 φ + R 2 ) cos 2 ω 0 t + 2 ω 0 Ө R ( cos 2 φ R ) sin 2 ω 0 t ( 1 2 R cos 2 φ + R 2 ) 2 .
φ 1 = ( k 1 2 ) π < φ < ( k + 1 2 ) π = φ 2 .
tan ϕ = 2 ω o Ө R φ 2 φ 1 φ 1 φ 2 cos 2 φ R 1 2 R cos 2 φ + R 2 d φ 0 .
I sin = 2 ω o Ө R φ 2 φ 1 φ 1 φ 2 cos 2 φ R ( 1 + R 2 2 R cos 2 φ ) 2 d φ , I sin = ω o Ө R φ 2 φ 1 | sin 2 φ ( 1 R 2 ) ( 1 + R 2 2 R cos 2 φ )
+ 2 R ( 1 R 2 ) 2 arctan [ ( 1 + R ) 2 tan φ ( 1 R 2 ) ] | φ 1 φ 2 , I cos = 1 φ 2 φ 1 φ 1 φ 2 d φ 1 + R 2 2 R cos 2 φ , I cos = 1 ( φ 2 φ 1 ) ( 1 R 2 ) | arctan [ ( 1 + R ) 2 tan φ ( 1 R 2 ) ] | φ 1 φ 2 .
tan φ = I sin I cos = 2 R 2 ω o Ө 1 R 2 ,
A n = R A n 1 .
d I / I = ( 1 R 2 ) d t / Ө .
A = A o ( 1 R ) m = 0 R m exp ( i m α ) .
A ( t ) = ( 1 R ) m = 0 A ( t m Ө ) R m exp ( i m α ) ,
A n = A o ( 1 R ) m = n R m exp ( i m α ) = A o ( 1 R ) R n exp ( i n α ) 1 R exp ( i α ) ,
I n = I o ( 1 R ) 2 ( 1 + R 2 2 R cos α ) R 2 n .
A in ( t ) = k A k ( t ) ,
A ( t ) = ( 1 R ) k m = 0 A k ( t m Ө ) R m exp ( i m α k ) ,
A ( n ) = ( 1 R ) k R n exp ( i n α k ) m = 0 A k ( t = m Ө ) R m exp ( i m α k ) .
I n = R 2 n ( 1 R ) 2 k [ m = 0 A k ( t = m Ө ) R m exp ( i m α k ) ] 2 * .
ω r = β · ω i = 1 υ / c 1 + υ / c ω i ( 1 2 υ / c ) ω i .
A ( t ) = + A ( ω ) exp ( i ω t ) d ω
A out ( t ) = + d ω exp ( i ω t ) m = 0 ( β R ) m A ( β m ω ) × exp [ i ω L o c ( 2 m + 1 2 m 2 υ c ) ] ,
A ( ω ) = exp [ ( ω ω 0 σ ω ) 2 ] ,
+ d t
I n = ( β R ) 2 n + d ω [ m = 0 ( β R ) m × exp [ ( ω β n + m ω o ) 2 / σ ω 2 ] exp ( i ϕ ) ] 2 ,
ϕ = 2 ω L 0 c [ m ( m 2 + 2 n m ) υ / c ] ;
I n = R 2 n · k F k ( n ) ,
υ / c > 2 σ ω / ω 0 .
υ / c < ( 1 R ) 2 λ / 4 L 0 .

Metrics