Abstract

Described is a reflectometer capable of making reflectivity measurements of low-loss highly reflecting mirror coatings and transmission measurements of low-loss antireflection coatings. The technique directly measures the intensity decay time of an optical cavity comprised of low-loss elements. We develop the theoretical framework for the device and discuss in what conditions and to what extent the decay time represents a true measure of mirror reflectivity. Current apparatus provides a decay time resolution of 10 nsec and has demonstrated a cavity total loss resolution of 5 ppm.

© 1984 Optical Society of America

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  1. V. E. Sanders, Appl. Opt. 16, 19 (1977); V. E. Sanders, Rockwell International; private communication.
    [CrossRef] [PubMed]
  2. J. M. Herbelin, J. A. McKay, M. A. Kwok, R. H. Ueunten, D. S. Urevig, D. J. Spencer, D. J. Benard, Appl. Opt. 19, 144 (1980).
    [CrossRef] [PubMed]
  3. The measurement of decay time to determine cavity losses is by no means original to us. The technique has its roots in passive RLC circuit theory and in microwave networks. Although cavity decay time is discussed in the literature (compare Ref. 9), we are not aware of any published work describing apparatus or experiments making use of it. We are, however, aware that techniques measuring cavity decay have indeed been used by some individuals to measure optical cavity characteristics. A rotating mirror has been used to gate laser light into an optical cavity (M. Ford, Ph.D. Thesis, U. Glasgow, Scotland, 1979, unpublished). Madey uses a similar technique to measure the decay time of a free-electron laser cavity (J. Madey, Stanford U.; private communication).
  4. M. Born, E. Wolf, Principles of Modern Optics (Pergamon, New York, 1980). Equation (2) of the text is a generalization of Eq. (12), p. 325, in Born and Wolf.
  5. Compare H. W. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966); S. A. Collins, Appl. Opt. 3, 1263 (1964).
    [CrossRef] [PubMed]
  6. We have not rigorously justified the use of the Laplace transform method for optical cavities. Replacement of the laser and cavity by equivalent electrical circuits lends credence to our procedure. Such equivalent circuits have been studied, for example, by E. I. Gordon, Bell Syst. Tech. J. 43, 507 (1963).
  7. On Laplace transform calculus see, for example, B. P. Lathi, Signals Systems and Communication (Wiley, New York, 1965).
  8. The scrutinizing reader will have noted that the boundary condition that E(t = 0−) = E(t = 0+) is not met in Eq. (10) because of the omission of the phase factor in Eq. (2). The condition can be regained by replacing t on the right-hand sides of Eqs. (10) by t − t0 where t0 is the light travel time from the input to the output mirror.
  9. Ref. 4, p. 328.
  10. We have derived formulas assuming pointlike losses. The problem of distributed losses is discussed by A. Yariv, Quantum Electronics (Wiley, New York, 1975), p. 141.
  11. A. Yariv, Introduction to Optical Electronics (Holt, Rinehart, & Winston, New York, 1971), p.79.
  12. Ref. 4, p. 42.

1980 (1)

1977 (1)

1966 (1)

1963 (1)

We have not rigorously justified the use of the Laplace transform method for optical cavities. Replacement of the laser and cavity by equivalent electrical circuits lends credence to our procedure. Such equivalent circuits have been studied, for example, by E. I. Gordon, Bell Syst. Tech. J. 43, 507 (1963).

Benard, D. J.

Born, M.

M. Born, E. Wolf, Principles of Modern Optics (Pergamon, New York, 1980). Equation (2) of the text is a generalization of Eq. (12), p. 325, in Born and Wolf.

Ford, M.

The measurement of decay time to determine cavity losses is by no means original to us. The technique has its roots in passive RLC circuit theory and in microwave networks. Although cavity decay time is discussed in the literature (compare Ref. 9), we are not aware of any published work describing apparatus or experiments making use of it. We are, however, aware that techniques measuring cavity decay have indeed been used by some individuals to measure optical cavity characteristics. A rotating mirror has been used to gate laser light into an optical cavity (M. Ford, Ph.D. Thesis, U. Glasgow, Scotland, 1979, unpublished). Madey uses a similar technique to measure the decay time of a free-electron laser cavity (J. Madey, Stanford U.; private communication).

Gordon, E. I.

We have not rigorously justified the use of the Laplace transform method for optical cavities. Replacement of the laser and cavity by equivalent electrical circuits lends credence to our procedure. Such equivalent circuits have been studied, for example, by E. I. Gordon, Bell Syst. Tech. J. 43, 507 (1963).

Herbelin, J. M.

Kogelnik, H. W.

Kwok, M. A.

Lathi, B. P.

On Laplace transform calculus see, for example, B. P. Lathi, Signals Systems and Communication (Wiley, New York, 1965).

Li, T.

McKay, J. A.

Sanders, V. E.

Spencer, D. J.

Ueunten, R. H.

Urevig, D. S.

Wolf, E.

M. Born, E. Wolf, Principles of Modern Optics (Pergamon, New York, 1980). Equation (2) of the text is a generalization of Eq. (12), p. 325, in Born and Wolf.

Yariv, A.

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart, & Winston, New York, 1971), p.79.

We have derived formulas assuming pointlike losses. The problem of distributed losses is discussed by A. Yariv, Quantum Electronics (Wiley, New York, 1975), p. 141.

Appl. Opt. (3)

Bell Syst. Tech. J. (1)

We have not rigorously justified the use of the Laplace transform method for optical cavities. Replacement of the laser and cavity by equivalent electrical circuits lends credence to our procedure. Such equivalent circuits have been studied, for example, by E. I. Gordon, Bell Syst. Tech. J. 43, 507 (1963).

Other (8)

On Laplace transform calculus see, for example, B. P. Lathi, Signals Systems and Communication (Wiley, New York, 1965).

The scrutinizing reader will have noted that the boundary condition that E(t = 0−) = E(t = 0+) is not met in Eq. (10) because of the omission of the phase factor in Eq. (2). The condition can be regained by replacing t on the right-hand sides of Eqs. (10) by t − t0 where t0 is the light travel time from the input to the output mirror.

Ref. 4, p. 328.

We have derived formulas assuming pointlike losses. The problem of distributed losses is discussed by A. Yariv, Quantum Electronics (Wiley, New York, 1975), p. 141.

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart, & Winston, New York, 1971), p.79.

Ref. 4, p. 42.

The measurement of decay time to determine cavity losses is by no means original to us. The technique has its roots in passive RLC circuit theory and in microwave networks. Although cavity decay time is discussed in the literature (compare Ref. 9), we are not aware of any published work describing apparatus or experiments making use of it. We are, however, aware that techniques measuring cavity decay have indeed been used by some individuals to measure optical cavity characteristics. A rotating mirror has been used to gate laser light into an optical cavity (M. Ford, Ph.D. Thesis, U. Glasgow, Scotland, 1979, unpublished). Madey uses a similar technique to measure the decay time of a free-electron laser cavity (J. Madey, Stanford U.; private communication).

M. Born, E. Wolf, Principles of Modern Optics (Pergamon, New York, 1980). Equation (2) of the text is a generalization of Eq. (12), p. 325, in Born and Wolf.

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Figures (5)

Fig. 1
Fig. 1

Conceptual schematic of decay measuring device. Optical switch shuts off light when detector output reaches a preset threshold level.

Fig. 2
Fig. 2

Schematic of optical and electrical circuit for cavity decay time measurement.

Fig. 3
Fig. 3

Comparator timing. Comparator 1 turns the Pockels device off when detector output reaches Voff, comparator 2 turns the clock on at Vclock, and comparator 3 turns the clock off when the detector output falls to 1/e Vclock.

Fig. 4
Fig. 4

Typical cavity decay curve from a two-mirrored 10-m cavity having a fall time of 23.2 μsec starts on the left-hand side as the lower of the two curves. The second curve is the decay from a RC network having the same decay constant. The decays from the two events are perfectly merged showing that the cavity decay is indeed exponential.

Fig. 5
Fig. 5

Cavity loss vs tilt of an etalon near the Brewster angle. Dots are experimental data. The theoretical (solid line) curve was obtained assuming a nominal cavity loss of 2 × 10−3 and a Brewster angle of 55.25°.

Tables (1)

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Table I Summary of Formulas Relating Various Cavity Parameters to the Cavity Decay Time in the Low-Total Low-Loss Limit a

Equations (26)

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R + S + T = 1.
E 0 exp ( - i ω t ) = E i exp ( - i ω t ) [ C m n q ( T i T o R p ) 1 / 2 1 1 - R exp ( i δ ) ] ,
δ = ω L c + φ x + φ y ,
H m n q = C m n q ( T i T o R p ) 1 / 2 { 1 - R [ 1 - i Δ ω L c - 1 2 ( Δ ω L c ) 2 + ( terms higher order in i Δ ω L c ) ] } - 1 ,
H m n q C m n q ( T i T o R p ) [ 1 - R ( 1 - i Δ ω L c ) ] - 1 A m n q c L [ γ c + i Δ ω m n q ] - 1 ,
Δ ω m n q = ω m n q - ω , L c ω m n q = 2 π q - m φ x - n φ y , γ c = ( c / L ) 1 - R R , A m n q = C m n q [ T i T o R p R ] 1 / 2 .
E i ( t ) = E s exp ( - i ω s t ) t < 0 , E i ( t ) = E s exp [ - ( γ s + i ω s ) t ] t > 0.
E o ( s ) = E i ( s ) H ( s ) ,
H ( s ) = c L m n q A m n q ( s + Ω m n q ) - 1 ,
E i ( s ) = 0 E s exp ( i ω s t ) exp ( s t ) d t + 0 E s × exp [ - ( γ s + i ω s ) t ] exp ( - s t ) d t = - E s ( Ω a + Ω b ) ( s + Ω a ) - 1 ( s - Ω b ) - 1 ,
Ω a = γ s + i ω s , Ω b = - i ω s ,
E o ( s ) = E s A c ( c L ) - ( Ω a + Ω b ) ( s + Ω a ) ( s - Ω b ) ( s + Ω c ) .
E o ( t ) = E s A c c / L Ω b + Ω c exp ( Ω b t )             t < 0 , E o ( t ) = E s A c [ c / L Ω c - Ω a exp ( - Ω a t ) + c / L ( Ω a + Ω b ) ( Ω a - Ω c ) ( Ω c + Ω b ) exp ( - Ω c t ) ]             t > 0.
E o ( t ) = E s A c c / L γ c + i Δ ω exp ( - i ω s t )             t < 0 ,
E o ( t ) = E s A c { c / L γ c - γ s + i Δ ω exp [ - ( γ s + i ω s ) t ] + c / L ( 1 - γ c / γ s - i Δ ω / γ s ) ( γ c + i Δ ω ) exp [ - ( γ c + i ω c ) t ] }             t > 0.
E o ( t ) = E s A c c / L γ c + i Δ ω exp [ - ( γ c + i ω c ) t ] .
I ( t ) = I o exp ( - t / τ c ) ,
τ c = 1 2 γ c , I o = E o ( t < 0 ) 2 ,
R 1 - R = 2 ( c L ) τ c .
F FSR Δ ν = c L ( Δ ν ) - 1 .
I ( ω ) H ( ω ) 2 = A c 2 ( { 1 - R [ 1 - 1 2 ( Δ ω L c ) 2 ] } 2 + R ( Δ ω L c ) 2 ) - 1 = ( A c L ) 2 { [ ( c / L ) ( 1 - R ) / R 1 / 4 ] 2 + ( Δ ω ) 2 } - 1 ,
F = π ( R ) 1 / 4 1 - R .
F = 2 π c L τ c .
F = π ( R ) 1 / 4 1 - R π { 1 - [ ( 1 - L 1 ) ( 1 - L 2 ) ] 1 / 2 } - 1
F 2 π i L i ,
Q = 2 π ν τ c ,

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