Abstract

The line width of a well-stabilized laser operating far above threshold is determined by random fluctuations of the phase. This paper discusses several types of experiments which can give information about the details of this phase random process. In order to study the laser phase noise experimentally the laser signal (containing phase noise only) must be passed through some type of interferometer which will convert the phase noise to intensity noise. The various properties of this derived intensity noise which may then be determined are its probability density, first and second moments, autocorrelation function, and spectrum. These measurable quantities depend on two factors; the first and more fundamental is the joint probability distribution for the change in phase in a given time. The second factor is the manner of operation of the interferometer in changing phase to intensity noise. We discuss both two-beam and multiple-beam interferometers and derive theoretical expressions for the above-mentioned properties of the output intensity fluctuations. It is interesting that although in both cases the output intensity fluctuations are nongaussian random processes, it is nevertheless possible to derive a number of useful theoretical results.

© 1966 Optical Society of America

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References

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  1. J. A. Armstrong and A. W. Smith, Phys. Rev. Letters 14, 68 and 208 (1965);also A. W. Smith and J. A. Armstrong, Phys. Letters 16, 38 (1965).
    [Crossref]
  2. J. A. Armstrong and A. W. Smith, Phys. Rev. 140, A155 (1965).
    [Crossref]
  3. C. Freed and H. A. Haus, Appl. Phys. Letters 6, 85 (1965);also C. Freed and H. A. Haus, Phys. Rev. 141, A287 (1966).
    [Crossref]
  4. F. T. Arecchi, A. Berne, and P. Bulamacchi, Phys. Rev. Letters 16, 32 (1966).
    [Crossref]
  5. D. E. McCumber, Phys. Rev. 141, A306 (1966).
    [Crossref]
  6. H. Risken, Z. Physik 186, 85 (1965).
    [Crossref]
  7. J. A. Armstrong and A. W. Smith, Appl. Phys. Letters 4, 196 (1964);see also Ref. 2 above.
    [Crossref]
  8. L. Mandel, Phys. Rev. 138, B753 (1965).
    [Crossref]
  9. H. Hodara, Proc. IEEE 53, 696 (1965).
    [Crossref]
  10. A. Blaquiere, Ann. Radioelec. 8, 36, 153 (1953).
  11. R. J. Glauber, in Quantum Optics and Electronics, C. DeWitt, A. Blandin, and C. Cohen-Tannoudji, Eds. (Gordon and Breach, New York, 1965), p. 169.
  12. W. E. Lamb, in Quantum Optics and Electronics, C. DeWitt, A. Blandin, and C. Cohen-Tannoudji, Eds. (Gordon and Breach, New York, 1965), p. 379.
  13. W. Magnus and F. Oberhettinger, Special Functions of Mathematical Physics (Chelsea Publishing Co., New York, 1949), p. 98.
  14. P. Connes, J. Phys. Radium 19, 262 (1958).
    [Crossref]

1966 (2)

F. T. Arecchi, A. Berne, and P. Bulamacchi, Phys. Rev. Letters 16, 32 (1966).
[Crossref]

D. E. McCumber, Phys. Rev. 141, A306 (1966).
[Crossref]

1965 (6)

H. Risken, Z. Physik 186, 85 (1965).
[Crossref]

J. A. Armstrong and A. W. Smith, Phys. Rev. Letters 14, 68 and 208 (1965);also A. W. Smith and J. A. Armstrong, Phys. Letters 16, 38 (1965).
[Crossref]

J. A. Armstrong and A. W. Smith, Phys. Rev. 140, A155 (1965).
[Crossref]

C. Freed and H. A. Haus, Appl. Phys. Letters 6, 85 (1965);also C. Freed and H. A. Haus, Phys. Rev. 141, A287 (1966).
[Crossref]

L. Mandel, Phys. Rev. 138, B753 (1965).
[Crossref]

H. Hodara, Proc. IEEE 53, 696 (1965).
[Crossref]

1964 (1)

J. A. Armstrong and A. W. Smith, Appl. Phys. Letters 4, 196 (1964);see also Ref. 2 above.
[Crossref]

1958 (1)

P. Connes, J. Phys. Radium 19, 262 (1958).
[Crossref]

1953 (1)

A. Blaquiere, Ann. Radioelec. 8, 36, 153 (1953).

Arecchi, F. T.

F. T. Arecchi, A. Berne, and P. Bulamacchi, Phys. Rev. Letters 16, 32 (1966).
[Crossref]

Armstrong, J. A.

J. A. Armstrong and A. W. Smith, Phys. Rev. Letters 14, 68 and 208 (1965);also A. W. Smith and J. A. Armstrong, Phys. Letters 16, 38 (1965).
[Crossref]

J. A. Armstrong and A. W. Smith, Phys. Rev. 140, A155 (1965).
[Crossref]

J. A. Armstrong and A. W. Smith, Appl. Phys. Letters 4, 196 (1964);see also Ref. 2 above.
[Crossref]

Berne, A.

F. T. Arecchi, A. Berne, and P. Bulamacchi, Phys. Rev. Letters 16, 32 (1966).
[Crossref]

Blaquiere, A.

A. Blaquiere, Ann. Radioelec. 8, 36, 153 (1953).

Bulamacchi, P.

F. T. Arecchi, A. Berne, and P. Bulamacchi, Phys. Rev. Letters 16, 32 (1966).
[Crossref]

Connes, P.

P. Connes, J. Phys. Radium 19, 262 (1958).
[Crossref]

Freed, C.

C. Freed and H. A. Haus, Appl. Phys. Letters 6, 85 (1965);also C. Freed and H. A. Haus, Phys. Rev. 141, A287 (1966).
[Crossref]

Glauber, R. J.

R. J. Glauber, in Quantum Optics and Electronics, C. DeWitt, A. Blandin, and C. Cohen-Tannoudji, Eds. (Gordon and Breach, New York, 1965), p. 169.

Haus, H. A.

C. Freed and H. A. Haus, Appl. Phys. Letters 6, 85 (1965);also C. Freed and H. A. Haus, Phys. Rev. 141, A287 (1966).
[Crossref]

Hodara, H.

H. Hodara, Proc. IEEE 53, 696 (1965).
[Crossref]

Lamb, W. E.

W. E. Lamb, in Quantum Optics and Electronics, C. DeWitt, A. Blandin, and C. Cohen-Tannoudji, Eds. (Gordon and Breach, New York, 1965), p. 379.

Magnus, W.

W. Magnus and F. Oberhettinger, Special Functions of Mathematical Physics (Chelsea Publishing Co., New York, 1949), p. 98.

Mandel, L.

L. Mandel, Phys. Rev. 138, B753 (1965).
[Crossref]

McCumber, D. E.

D. E. McCumber, Phys. Rev. 141, A306 (1966).
[Crossref]

Oberhettinger, F.

W. Magnus and F. Oberhettinger, Special Functions of Mathematical Physics (Chelsea Publishing Co., New York, 1949), p. 98.

Risken, H.

H. Risken, Z. Physik 186, 85 (1965).
[Crossref]

Smith, A. W.

J. A. Armstrong and A. W. Smith, Phys. Rev. 140, A155 (1965).
[Crossref]

J. A. Armstrong and A. W. Smith, Phys. Rev. Letters 14, 68 and 208 (1965);also A. W. Smith and J. A. Armstrong, Phys. Letters 16, 38 (1965).
[Crossref]

J. A. Armstrong and A. W. Smith, Appl. Phys. Letters 4, 196 (1964);see also Ref. 2 above.
[Crossref]

Ann. Radioelec. (1)

A. Blaquiere, Ann. Radioelec. 8, 36, 153 (1953).

Appl. Phys. Letters (2)

C. Freed and H. A. Haus, Appl. Phys. Letters 6, 85 (1965);also C. Freed and H. A. Haus, Phys. Rev. 141, A287 (1966).
[Crossref]

J. A. Armstrong and A. W. Smith, Appl. Phys. Letters 4, 196 (1964);see also Ref. 2 above.
[Crossref]

J. Phys. Radium (1)

P. Connes, J. Phys. Radium 19, 262 (1958).
[Crossref]

Phys. Rev. (3)

L. Mandel, Phys. Rev. 138, B753 (1965).
[Crossref]

D. E. McCumber, Phys. Rev. 141, A306 (1966).
[Crossref]

J. A. Armstrong and A. W. Smith, Phys. Rev. 140, A155 (1965).
[Crossref]

Phys. Rev. Letters (2)

J. A. Armstrong and A. W. Smith, Phys. Rev. Letters 14, 68 and 208 (1965);also A. W. Smith and J. A. Armstrong, Phys. Letters 16, 38 (1965).
[Crossref]

F. T. Arecchi, A. Berne, and P. Bulamacchi, Phys. Rev. Letters 16, 32 (1966).
[Crossref]

Proc. IEEE (1)

H. Hodara, Proc. IEEE 53, 696 (1965).
[Crossref]

Z. Physik (1)

H. Risken, Z. Physik 186, 85 (1965).
[Crossref]

Other (3)

R. J. Glauber, in Quantum Optics and Electronics, C. DeWitt, A. Blandin, and C. Cohen-Tannoudji, Eds. (Gordon and Breach, New York, 1965), p. 169.

W. E. Lamb, in Quantum Optics and Electronics, C. DeWitt, A. Blandin, and C. Cohen-Tannoudji, Eds. (Gordon and Breach, New York, 1965), p. 379.

W. Magnus and F. Oberhettinger, Special Functions of Mathematical Physics (Chelsea Publishing Co., New York, 1949), p. 98.

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Figures (6)

F. 1
F. 1

(a) Schematic representation of a two-beam interferometer. (b) Schematic representation of a multiple-beam interferometer.

F. 2
F. 2

(a) Diagram of Eq. (1), indicating the different values of Δϕ which correspond to a given value of i. (b) Sample function of the phase random process. At some instant of time a multiple-beam interferometer would be sampling the values of ϕ(t) indicated by the points τ, 3τ, 5τ, ⋯. Note that changes in ϕ occurring during overlapping time intervals are not independent.

F. 3
F. 3

Probability density P(i) for two-beam interferometer. The parameter δτ is the ratio of the interferometer length to the laser-phase coherence length.

F. 4
F. 4

The spectrum of the output intensity fluctuations from a two-beam interferometer exposed to pure phase noise of band width δ/2π.

F. 5
F. 5

Monte-Carlo histograms showing the intensity probability densities for a 10-beam interferometer exposed to pure phase noise. R = 0.772. The histograms for each δτ were obtained by evaluating Eq. (15) 5000 times, using a different random sequence of the 100ϕmn each time.

F. 6
F. 6

The relative mean squared intensity fluctuation ρ of the output of various interferometers exposed to pure phase noise. The lowest curve is for a two-beam interferometer; the others are for multiple-beam interferometers whose mirrors have the indicated reflectances R.

Equations (37)

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I ( t ) = 2 E 0 2 { 1 + cos [ ϕ ( t ) ϕ ( t 2 τ ) ] } ,
E ( t ) = E 0 cos [ ω 0 t + ϕ ( t ) ] ,
P 2 τ ( Δ ϕ ) = ( 4 π δ τ ) 1 2 exp [ ( Δ ϕ ) 2 / 4 δ τ ] .
p ( i ) d i = d ( Δ ϕ ) ( 4 π δ τ ) 1 2 n = { exp [ ( Δ ϕ + 2 π n ) 2 4 δ τ ] + exp [ ( Δ ϕ 2 π n ) 2 4 δ τ ] }
= d ( Δ ϕ ) ( π δ τ ) 1 2 exp [ ( Δ ϕ ) 2 4 δ τ ] × { 1 + 2 n = 1 exp ( π 2 n 2 δ τ ) cosh ( n π Δ ϕ δ τ ) }
= d ( Δ ϕ ) ( π δ τ ) 1 2 exp ( ( Δ ϕ ) 2 4 δ τ ) × θ 3 [ i π Δ ϕ δ τ , exp ( π 2 δ τ ) ] .
p ( i ) = ( π δ τ ) 1 2 ( 2 i i 2 ) 1 2 exp { [ cos 1 ( i 1 ) ] 2 / 4 δ τ } × θ 3 { [ i π cos 1 ( i 1 ) / δ τ ] , exp ( π 2 / δ τ ) } , if 0 < i < 2 ;
ρ = 1 2 1 + exp ( 4 δ τ ) 2 exp ( 2 δ τ ) 1 + 2 exp ( δ τ ) + exp ( 2 δ τ ) .
R i ( t 1 , t 2 ) = i ( t 1 ) i ( t 2 ) = 1 + cos Δ ϕ 1 + cos Δ ϕ 2 + cos Δ ϕ 1 cos Δ ϕ 2 ,
R i ( T ) = [ 1 + exp ( δ τ ) ] 2 ; T 2 τ .
ϕ a ϕ ( t 2 ) ϕ ( t 1 2 τ ) ϕ b ϕ ( t 1 ) ϕ ( t 2 2 τ ) ϕ c ϕ ( t 2 2 τ ) ϕ ( t 1 2 τ ) ϕ d ϕ ( t 2 ) ϕ ( t 1 ) .
R i ( t 1 , t 2 ) = 1 + cos Δ ϕ 1 + cos Δ ϕ 2 + 1 2 cos ϕ c cos ϕ d + 1 2 cos ϕ a cos ϕ b 1 2 sin ϕ a sin ϕ b .
p ( ϕ a , ϕ b ) = 1 2 π δ [ 2 T ( 2 τ T ) ] 1 2 exp ( ( ϕ a ϕ b ) 2 4 δ T ) × exp ( ϕ b 2 2 δ ( 2 τ T ) ) ; T < 2 τ .
R i ( T ) = 1 + 2 exp ( δ τ ) + exp ( 2 δ τ ) cosh [ δ ( 2 τ T ) ] , T < 2 τ
= [ 1 + exp ( δ τ ) ] 2 , T 2 τ .
S i ( f ) = 4 0 R i ( T ) cos 2 π fTdT .
R i ( T ) = [ 1 + exp ( δ τ ) ] 2 + exp ( 2 δ τ ) × { cosh [ δ ( 2 τ T ) ] 1 } .
S i ( f ) = 4 exp ( 2 δ τ ) δ × [ ( 2 π f / δ ) + sin 4 π f τ + sinh ( 2 δ τ ) 1 + ( 2 π f / δ ) 2 sin 4 π f τ 2 π f / δ ] .
i ( t ) = I ( t ) / I 0 = m , n = 1 R ( m + n ) × cos { ϕ [ t ( 2 n 1 ) τ ] ϕ [ t ( 2 m 1 ) τ ] } .
i = n = 1 R 2 n + 2 m > n = 1 R m + n × cos ϕ m n p ( ϕ m n ) d ϕ m n .
p ( ϕ m n ) = [ 4 π ( m n ) δ τ ] 1 2 exp [ ϕ m n 2 / 4 δ ( m n ) τ ] .
i = R 2 1 R 2 [ 1 + 2 R exp ( δ τ ) 1 R exp ( δ τ ) ] .
i f m > n R m + n cos ϕ m n .
ρ = i 2 i 2 i 2 = 4 [ i f 2 i f 2 ] i 2 ,
i f 2 = m > n μ > ν R m + n + μ + ν cos ϕ m n cos ϕ μ ν = 1 2 m > n R 2 ( m + n ) [ 1 + cos 2 ϕ m n ] + ( m > n ) ( μ > ν ) R ( m + n + μ + ν ) cos ϕ m n cos ϕ μ ν .
ρ = 2 R 2 ( 1 + R K ) 2 ( ( 1 R K ) ( 1 + R 2 ) ( 1 R 3 K ) { [ 1 + K 4 ( 1 R 2 ) 1 R 2 K 4 ] × [ ( 1 R K ) ( 1 R 3 K ) + 2 R K ( 1 + R 2 ) ] + 4 R 2 K 2 } 2 K 2 ) .
ρ = 0 , if δ τ = 0 = 2 R 2 / ( 1 + R 2 ) , if δ τ .
ρ [ ( 1 + R 2 ) / 3 ( 1 R ) 2 ] ( δ τ ) , δ τ 1 .
( m > n ) ( μ > ν ) R ( m + n + μ + ν ) cos ϕ m n cos ϕ μ ν × p ( ϕ m n , ϕ μ ν ) d ϕ m n d ϕ μ ν .
p z ( z ) = p n ν ( ϕ n ν ) p μ m ( z ϕ n ν ) d ϕ n ν .
p z ( ϕ μ ν ϕ m n ) = p ( ϕ μ ν | ϕ m n ) = [ 4 π δ τ ( μ ν m + n ) ] 1 2 × exp [ ( ϕ μ ν ϕ m n ) 2 4 δ τ ( μ ν m + n ) ] .
p ( ϕ μ ν , ϕ m n ) = p ( ϕ μ ν | ϕ m n ) p ( ϕ m n ) = { 4 π δ τ [ ( μ ν m + n ) ( m n ) ] 1 2 } 1 2 × exp ( ( ϕ μ ν ϕ m n ) 2 4 δ τ ( μ ν m + n ) ) exp ( ϕ m n 2 4 δ τ ( m n ) ) .
cos ϕ m n cos ϕ μ ν = 1 2 exp [ ( μ ν m + n ) δ τ ] × { 1 + exp [ 4 ( m n ) δ τ ] } .
2 2 ( μ > ν ) ( m > n ) R ( m + n + μ + ν ) exp [ ( μ ν m + n ) δ τ ] × { 1 + exp [ 4 ( m n ) δ τ ] } .
m n = k = 1 , 2 , μ m = s = 1 , 2 , n ν = l = 1 , 2 , .
s = 1 k = 1 l = 1 ν = 1 R s + 2 k + 3 l + 4 ν × { exp [ ( s + l ) δ τ ] + exp [ ( s + l + 4 k ) δ τ ] } .
R 10 exp ( 2 δ τ ) ( 1 R 4 ) [ 1 R exp ( δ τ ) ] [ 1 R 3 exp ( δ τ ) ] × ( 1 1 R 2 + exp ( 4 δ τ ) 1 R 2 exp ( 4 δ τ ) ) .