Abstract

Characteristic fringes of a spectrometer are produced by interference of a large number of wavefronts of regularly increasing phase difference. This phase difference implies a temporal delay between the wavefronts. So, the response of such an instrument to a light pulse of very short duration may not be given by the conventional formula, which generally corresponds to a steady-state situation. In this paper, the temporal response of Fabry–Perot interferometers to pulses of light of various lengths and kinds is considered and inferences are drawn regarding how to estimate the pulse lengths and how to carry out the spectral analysis of such pulses.

© 1975 Optical Society of America

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References

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  1. W. H. Steel, Interferometry (Cambridge U.P., Cambridge, 1967), p. 111.
  2. C. Roychoudhuri, Multipass Fabry–Perot Interferometer for Brillouin Scattering Measurements, Ph. D. Thesis (University of Rochester, New York, 1973).
  3. A. Kastler, Nouv. Rev. Opt. 5, 133 (1974).
    [Crossref]
  4. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), pp. 323–347.
  5. L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 260 (1965).
    [Crossref]
  6. D. J. Bradley, M. S. Engwell, A. W. McCullough, G. Magyar, and M. C. Richardson, Philos. Trans. R. Soc. 263A, 225 (1968); see, especially, p. 232.

1974 (1)

A. Kastler, Nouv. Rev. Opt. 5, 133 (1974).
[Crossref]

1968 (1)

D. J. Bradley, M. S. Engwell, A. W. McCullough, G. Magyar, and M. C. Richardson, Philos. Trans. R. Soc. 263A, 225 (1968); see, especially, p. 232.

1965 (1)

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 260 (1965).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), pp. 323–347.

Bradley, D. J.

D. J. Bradley, M. S. Engwell, A. W. McCullough, G. Magyar, and M. C. Richardson, Philos. Trans. R. Soc. 263A, 225 (1968); see, especially, p. 232.

Engwell, M. S.

D. J. Bradley, M. S. Engwell, A. W. McCullough, G. Magyar, and M. C. Richardson, Philos. Trans. R. Soc. 263A, 225 (1968); see, especially, p. 232.

Kastler, A.

A. Kastler, Nouv. Rev. Opt. 5, 133 (1974).
[Crossref]

Magyar, G.

D. J. Bradley, M. S. Engwell, A. W. McCullough, G. Magyar, and M. C. Richardson, Philos. Trans. R. Soc. 263A, 225 (1968); see, especially, p. 232.

Mandel, L.

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 260 (1965).
[Crossref]

McCullough, A. W.

D. J. Bradley, M. S. Engwell, A. W. McCullough, G. Magyar, and M. C. Richardson, Philos. Trans. R. Soc. 263A, 225 (1968); see, especially, p. 232.

Richardson, M. C.

D. J. Bradley, M. S. Engwell, A. W. McCullough, G. Magyar, and M. C. Richardson, Philos. Trans. R. Soc. 263A, 225 (1968); see, especially, p. 232.

Roychoudhuri, C.

C. Roychoudhuri, Multipass Fabry–Perot Interferometer for Brillouin Scattering Measurements, Ph. D. Thesis (University of Rochester, New York, 1973).

Steel, W. H.

W. H. Steel, Interferometry (Cambridge U.P., Cambridge, 1967), p. 111.

Wolf, E.

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 260 (1965).
[Crossref]

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), pp. 323–347.

Nouv. Rev. Opt. (1)

A. Kastler, Nouv. Rev. Opt. 5, 133 (1974).
[Crossref]

Philos. Trans. R. Soc. (1)

D. J. Bradley, M. S. Engwell, A. W. McCullough, G. Magyar, and M. C. Richardson, Philos. Trans. R. Soc. 263A, 225 (1968); see, especially, p. 232.

Rev. Mod. Phys. (1)

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 260 (1965).
[Crossref]

Other (3)

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), pp. 323–347.

W. H. Steel, Interferometry (Cambridge U.P., Cambridge, 1967), p. 111.

C. Roychoudhuri, Multipass Fabry–Perot Interferometer for Brillouin Scattering Measurements, Ph. D. Thesis (University of Rochester, New York, 1973).

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Figures (10)

FIG. 1
FIG. 1

(a) Multiple reflections in a Fabry–Perot interferometer (FP) formed by two plane parallel and partially transmitting mirrors of identical nature. (b) The transmission characteristics of a FP, known as the Airy curve.

FIG. 2
FIG. 2

Multitudes of wave trains (A, B, C, …) of decreasing amplitudes but of the same delay time, τ0, between each other are formed by a pair of parallel and partially transmitting mirrors. τ0, called the delay time (see text), is the round trip time for light between the two mirrors. The resultant irradiance at a moment 0 is T2 |∑Rneinϕ|2.

FIG. 3
FIG. 3

(a) Temporal development of the transmission characteristics of a Fabry–Perot (FP): the up-going staircase indicates the development when the FP is in transmission mode and the down-going staircase indicates the development when the FP is in reflection mode. τ0 is the FP delay time. (b) For the sake of comparison, the steady-state transmission characteristics of an FP (the Airy curve) is shown alongside.

FIG. 4
FIG. 4

A Fabry–Perot interferometer in fringe mode (formation of the fringes with an incident divergent beam).

FIG. 5
FIG. 5

Of the multitudes of pulses formed by a Fabry–Perot, only a limited number of them can interfere when the incident pulse is of finite length. The upper rectangular curve indicates the length of the incident pulse. τ0 is the Fabry–Perot delay time.

FIG. 6
FIG. 6

A pulse of length shorter than the Fabry–Perot delay time, τ0, cannot produce normal interference effects, since the generated pulses are separated from each other. The upper rectangular curve indicates the length of the incident pulse.

FIG. 7
FIG. 7

A series of coherent short pulses of separation equal to the Fabry–Perot delay time, τ0, can produce regular interference effects. The top row indicates the incident series of pulses. The following rows indicate the train of pulses produced by the 0th, 1st, 2nd, … incident pulses on matching time scales. The overlapping pulses at a particular moment are indicated by a heavy arrow at the bottom. The amplitudes and the phases of the pulses are indicated above each one.

FIG. 8
FIG. 8

A situation that is very similar to that of Fig. 7 but with pulse separation equal to an integral multiple of the Fabry–Perot delay time, τ0. The broken lines on the bottom two rows indicate the overlapping of the adjacent pulses when the width is larger than τ0.

FIG. 9
FIG. 9

The possibility of interference with a single but very short pulse using two identical Fabry–Perots. Top row: A single short pulse produces a series of non-overlapping pulses through the first Fabry–Perot (FP). These pulses, in turn, pass through the second FP and produce many trains of pulses, which are indicated in the following rows on matched time scales. The heavy arrow at the bottom indicates the overlapping pulses at a particular moment. The amplitude and phase of any pulse are obtained by multiplying the expression shown above it by the factor that is indicated on the extreme right of the corresponding row.

FIG. 10
FIG. 10

A situation that is very similar to that of Fig. 9, but with two Fabry–Perots whose delay times are integral multiples of each other (τ01 = 4τ02). The broken lines at the bottom two rows indicate the overlapping of the adjacent pulses when the width is larger than τ02 but narrower than τ01.

Equations (75)

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A = n = 0 T R n e i n ϕ = T / ( 1 - R e i ϕ ) ,
ϕ = ( 2 π ν / c ) 2 d cos θ ,
I = T / [ 1 + F sin 2 ϕ / 2 ] ,
T [ T / ( 1 - R ) ] 2 and F 4 R / ( 1 - R ) 2 .
I max = T / ( 1 - R ) 2 ,             I min = T / ( 1 + R ) 2 .
N = Δ ϕ / δ ϕ = ( π / 2 ) F = π R / ( 1 - R ) ,
δ ϕ = 2 ( 1 - R ) / R .
Δ ν fsr = c / 2 d cos θ .
I = T / [ 1 + F sin 2 ϕ / 2 ] I M = | n = 0 M - 1 T R n e i n ϕ | 2 .
M 2 N ,
τ 0 = 2 d / c = 1 / Δ ν fsr .
τ r = N τ 0 .
delay time ,             τ 0 θ = 2 d cos θ / c = τ 0 cos θ ,
response time ,             τ r θ = N τ 0 cos θ .
R λ / δ λ = n ( 0.97 N ) ,
n = 2 d cos θ / λ
R n M = ( M · 2 d cos θ ) / λ .
R ν / δ ν = ν M τ 0 cos θ ,
δ t M τ 0 cos θ = 1 / δ ν
δ ν δ t 1.
M δ t / τ 0 cos θ .
I m = | n = 0 m - 1 T R n e i n ϕ | 2 = T m · 1 + F m sin 2 ( m ϕ / 2 ) 1 + F sin 2 ( ϕ / 2 ) ,
T m [ T ( 1 - R m ) / ( 1 - R ) ] 2 < T
F m 4 R m / ( 1 - R m ) 2 < F .
sin 2 ( m δ ϕ m / 4 ) - ( F / 2 F m ) sin 2 ( δ ϕ m / 4 ) + ( 1 / 2 F m ) = 0.
N m = Δ ϕ m / δ ϕ m
N m m .
| I n - I n + 1 | = T 2 | | q = 0 n - 1 R q e i q ϕ | 2 - | q = 0 n R q e i q ϕ | 2 | .
M τ 0 cos θ < M τ 0
δ t > τ 0 .
δ ν δ t ~ 1
Δ ν fsr > δ ν ,
δ t = m τ 0
m τ 0 < M τ 0 ,
δ t = m τ 0 > M τ 0 ,
M τ 0 > δ t > τ 0
Δ ν fsr / M < δ ν < Δ ν fsr .
δ t < τ 0
δ ν δ t ~ 1.
Δ t = τ 0 .
I m + 1 = | n = 0 m T R n e i n ϕ | 2 .
Δ t = τ 0 cos θ n
2 d cos θ n = n λ = n c / ν
τ 0 cos θ n = n / ν
Δ t = τ 0 cos θ n = n / ν .
τ 0 cos θ n ± p = ( n ± p ) / ν = ( n / ν ) ± ( p / ν ) .
m = δ t / ( p / ν ) .
δ t = 5 × 10 - 13 s             at ν = 6 × 10 14 c / s and p ~ 10.
m = δ t ν / p = 30.
Δ t = q τ 0 .
I = T + T R 4 e i 4 ϕ + + T R 20 e i 20 ϕ 2 ,
I ( m / q + 1 ) = | n = 0 m / q T R n q e i n q ϕ | 2 ,
I = T q / [ 1 + F q sin 2 q ϕ / 2 ] ,
T q T 2 / ( 1 - R q ) 2 < T ,
F q 4 R / ( 1 - R q ) 2 < F ,
N q = 1 q · π 2 F q .
Δ ν fsr , q = Δ ν fsr / q = 1 / q τ 0 .
Δ ν ~ 1 / Δ t = 1 / q τ 0 ,
I = T + T R 4 e i 4 ϕ ( 1 + R 4 e i 4 ϕ + + R 16 e i 16 ϕ ) + T R 3 e i 3 ϕ ( 1 + R 4 e i 4 ϕ + + R 16 e i 16 ϕ ) + T R 2 e i 2 ϕ ( 1 + R 4 e i 4 ϕ + + R 16 e i 16 ϕ ) 2 .
Δ t = q τ 0             and             t = s τ 0
I = T + T ( R q e i q ϕ + R ( q - 1 ) e i ( q - 1 ) ϕ + + R ( q - s ) e i ( q - s ) ϕ ) × ( 1 + R q e i q ϕ + + R q ( m - 1 ) e i q ( m - 1 ) ϕ ) 2
I = | T + T R q - s e i ( q - s ) ϕ · 1 - R ( s + 1 ) e i ( s + 1 ) ϕ 1 - R e i ϕ . 1 - R q m e i q m ϕ 1 - R q e i q ϕ | 2 .
( q - s ) = 1.
I = | T · 1 - R ( q m + 1 ) e i ( q m + 1 ) ϕ 1 - R e i | 2 .
I = T / [ 1 + F sin 2 ϕ / 2 ] .
I = T 2 R m e i m ϕ + T 2 R m e i m ϕ + , ( m + 1 ) terms 2 = ( m + 1 ) 2 T 2 T R m e i m ϕ 2 .
τ 01 = q τ 02 .
Δ t τ 01 = q τ 02 .
ϕ = 2 π ν τ 01             and ϕ = 2 π ν τ 02 .
τ 01 = 4 τ 02 ,
I = T 2 R 5 e i 5 ϕ [ 1 + R 3 e i Φ + R 6 e i 2 Φ + + R i 15 e i 15 Φ ] 2 ,
I = | T 2 R m e i m ϕ n = 0 m R ( q - 1 ) n e i n Φ | 2 ,
Φ q ϕ - ϕ ,
Φ = 2 π ν ( q τ 02 - τ 01 ) = 0.
I = T 4 R 2 m | 1 - R ( q - 1 ) ( m + 1 ) 1 - R ( q - 1 ) | 2 .