Abstract

A calculation of the fringe visibility in a laser-illuminated two-beam interferometer is presented and discussed. The laser is assumed to be oscillating in several adjacent axial modes whose intensities are determined by the Doppler-broadened emission line. An exact expression is given for the visibility as a function of path-length difference, and it is shown to be the product of an exponential envelope and a periodic function whose period is very nearly twice the length of the laser. An important special case is considered to illustrate the values of path-length difference for which the visibility is minimum. Finally, the visibility is plotted in a way which permits estimation of depth of field in holography or acceptable path mismatch in interferometry. For any laser operating in the assumed manner, calculation is simplified by the form of the equations. An example is given.

© 1967 Optical Society of America

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References

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  1. T. S. Jaseja, A. Javan, and C. H. Townes, Phys. Rev. Letters 10, 165 (1963).
    [Crossref]
  2. T. Morokuma, K. F. Nefflen, T. R. Lawrence, and T. M. Klucher, J. Opt. Soc. Am. 53, 394 (1963).
    [Crossref]
  3. D. R. Herriott (private communication). See also Herriott’s article in J. Opt. Soc. Am. 52, 37 (1962).
  4. R. Stark, Spectra-Physics, Inc. (private communication).
  5. A. D. Jacobson and F. J. McClung, Appl. Opt. 4, 1509 (1965).
    [Crossref]
  6. M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), 2nd ed., pp. 320–323, 506, 507.
  7. A. Javan, E. A. Ballik, and W. L. Bond, J. Opt. Soc. Am. 52, 96 (1962).
    [Crossref]
  8. K. A. Stetson and R. L. Powell, J. Opt. Soc. Am. 56, 1161 (1966).
    [Crossref]
  9. Handbook of Chemistry and Physics (Chemical Rubber Publishing Co., Cleveland, 1962), 43rd ed., p. 283.

1966 (1)

1965 (1)

1963 (2)

T. S. Jaseja, A. Javan, and C. H. Townes, Phys. Rev. Letters 10, 165 (1963).
[Crossref]

T. Morokuma, K. F. Nefflen, T. R. Lawrence, and T. M. Klucher, J. Opt. Soc. Am. 53, 394 (1963).
[Crossref]

1962 (1)

Ballik, E. A.

Bond, W. L.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), 2nd ed., pp. 320–323, 506, 507.

Herriott, D. R.

D. R. Herriott (private communication). See also Herriott’s article in J. Opt. Soc. Am. 52, 37 (1962).

Jacobson, A. D.

Jaseja, T. S.

T. S. Jaseja, A. Javan, and C. H. Townes, Phys. Rev. Letters 10, 165 (1963).
[Crossref]

Javan, A.

T. S. Jaseja, A. Javan, and C. H. Townes, Phys. Rev. Letters 10, 165 (1963).
[Crossref]

A. Javan, E. A. Ballik, and W. L. Bond, J. Opt. Soc. Am. 52, 96 (1962).
[Crossref]

Klucher, T. M.

Lawrence, T. R.

McClung, F. J.

Morokuma, T.

Nefflen, K. F.

Powell, R. L.

Stark, R.

R. Stark, Spectra-Physics, Inc. (private communication).

Stetson, K. A.

Townes, C. H.

T. S. Jaseja, A. Javan, and C. H. Townes, Phys. Rev. Letters 10, 165 (1963).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), 2nd ed., pp. 320–323, 506, 507.

Appl. Opt. (1)

J. Opt. Soc. Am. (3)

Phys. Rev. Letters (1)

T. S. Jaseja, A. Javan, and C. H. Townes, Phys. Rev. Letters 10, 165 (1963).
[Crossref]

Other (4)

Handbook of Chemistry and Physics (Chemical Rubber Publishing Co., Cleveland, 1962), 43rd ed., p. 283.

D. R. Herriott (private communication). See also Herriott’s article in J. Opt. Soc. Am. 52, 37 (1962).

R. Stark, Spectra-Physics, Inc. (private communication).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), 2nd ed., pp. 320–323, 506, 507.

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Figures (3)

Fig. 1
Fig. 1

Illustration of assumed spectral intensity distribution for mode configuration (3, −1, 1; .4). The relative widths of the oscillating cavity modes are greatly exaggerated.

Fig. 2
Fig. 2

Calculated fringe visibility as a function of path-length difference for R=0.63. Curve (3, −1, 1; 0.4) corresponds to the mode configuration depicted in Fig. 1.

Fig. 3
Fig. 3

Calculated fringe visibility as a function of R with e−2 mode threshold and f=0.5. Numbers affixed to the curves are values of γΔℒ/L. Significant discontinuities occur in vicinity of dashed lines.

Equations (44)

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V = ( I max - I min ) / ( I max + I min ) ,
V ( Δ ) = [ C 2 ( Δ ) + S 2 ( Δ ) ] 1 2 / C ( 0 ) ,
C ( Δ ) = 2 j ( x ) cos ( x Δ ) d x ,
S ( Δ ) = 2 j ( x ) sin ( x Δ ) d x ,
x = k - k ,
j ( x ) = j 0 e - α 2 x 2 n = N M exp [ - β 2 ( x - x n ) 2 ] ,
x n = ( f + n ) Δ x ,
- 1 2 f 1 2 ,
Δ x = π / L ,
α = c ( ln 2 ) 1 2 / π Δ ν D ,
β = c ( ln 2 ) 1 2 / π Δ ν M .
q λ = 2 L ,
β 10 5 α .
V ( Δ ) = exp [ - γ ( Δ / 2 β ) 2 ] × { N M exp [ - γ R 2 ( f + n ) 2 ] } - 1 S M , N ,
S M , N 2 = { n = N M exp [ - γ R 2 ( f + n ) 2 ] cos ( γ n π Δ / L ) } 2 + { n = N M exp [ - γ R 2 ( f + n ) 2 ] sin ( γ n π Δ / L ) } 2 ,
γ = β 2 / ( α 2 + β 2 ) ,
R = α Δ x .
Δ = 2 L m / γ ,
R = ( ln 2 ) 1 2 c / L Δ ν D .
V 0 ( Δ ) = exp [ - ( Δ / 2 β ) 2 ] | sin ( p π Δ / 2 L ) p sin ( π Δ / 2 L ) | ,
Δ / 2 L = l / p ,             1 l < p ,
R 2 / 2 - R 4 ( f + N ) 2
- α 2 x 2 - β 2 ( x - x n ) 2 = - ( α 2 + β 2 ) ( x - γ x n ) 2 - α 2 γ x n 2 ,
j ( x ) = j 0 n = N M exp [ - ( α 2 + β 2 ) ( x - γ x n ) 2 - α 2 γ x n 2 ] .
( Δ ) = 2 j 0 n = N M exp [ - α 2 γ x n 2 ] I n I n = - k ¯ exp [ - ( α 2 + β 2 ) ( x - γ x n ) 2 ] × cos ( x Δ ) d x . }
( α 2 + β 2 ) ( - k ¯ - γ x n ) 2 1 ,
u x - γ x n
cos [ ( u + γ x n ) Δ ] = cos ( u Δ ) cos ( γ x n Δ ) - sin ( u Δ ) sin ( γ x n Δ ) .
- e - a 2 u 2 sin ( b u ) d u = 0
- e - a 2 u 2 cos ( b u ) d u = [ ( π ) 1 2 / a ] e - b 2 / 4 a 2 ,
I n = ( π α 2 + β 2 ) 1 2 exp [ - γ ( Δ / 2 β ) 2 ] cos ( γ x n Δ ) .
C ( Δ ) = 2 j 0 ( π α 2 + β 2 ) 1 2 exp [ - γ ( Δ / 2 β ) 2 ] × N M exp [ - γ α 2 x n 2 ] cos ( γ x n Δ ) .
S ( Δ ) = 2 j 0 ( π α 2 + β 2 ) 1 2 exp [ - γ ( Δ / 2 β ) 2 ] × N M exp [ - γ α 2 x n 2 ] sin ( γ x n Δ ) .
( f + n ) R = α x n ,
V ( Δ ) = exp [ - γ ( Δ / 2 β ) 2 ] × { N M exp [ - γ R 2 ( f + n ) 2 ] } - 1 S M , N ,
S M , N 2 = { N M exp [ - γ R 2 ( f + n ) 2 ] cos [ γ ( f + n ) π Δ / L ] } 2 + { N M exp [ - γ R 2 ( f + n ) 2 ] sin [ γ ( f + n ) π Δ / L ] } 2 .
a n exp [ - γ R 2 ( f + n ) 2 ] y γ π Δ / L ,
S M , N 2 = n = N M a n cos [ ( f + n ) y ] m = N M a m cos [ ( f + m ) y ] + n = N M a n sin [ ( f + n ) y ] m = N M a m sin [ ( f + m ) y ] , = n = N M a n m = N M a m cos [ ( f + n ) y ] cos [ ( f + m ) y ] + sin [ ( f + n ) y ] sin [ ( f + m ) y ] , = n = N M a n m = N M a m cos [ ( n - m ) y ] .
V 0 ( Δ ) lim α 0 V ( Δ ) .
V 0 ( Δ ) = exp [ - ( Δ / 2 β ) 2 ] p { [ N M cos ( n π Δ / L ) ] 2 + [ N M sin ( n π Δ / L ) ] 2 } 1 2 ,
y = π Δ / L ,
cos ( n y ) = sin { [ n + ( 1 2 ) ] y } - sin { [ n - ( 1 2 ) ] y } 2 sin ( y / 2 ) sin ( n y ) = cos { [ n - ( 1 2 ) ] y } - cos { [ n + ( 1 2 ) ] y } 2 sin ( y / 2 ) } ,
N M cos ( n y ) = sin { [ M + ( 1 2 ) ] y } - sin { [ N - ( 1 2 ) ] y } 2 sin ( y / 2 ) , N M sin ( n y ) = cos { [ N - ( 1 2 ) ] y } - cos { [ M + ( 1 2 ) ] y } 2 sin ( y / 2 ) .
V 0 ( Δ ) = exp [ - ( Δ / 2 β ) 2 ] p { 1 - cos [ ( M + 1 - N ) y ] 2 sin 2 ( y / 2 ) } 1 2 = exp [ - ( Δ / 2 β ) 2 ] | sin ( p y / 2 ) p sin ( y / 2 ) |