Abstract

The transform method of Bennett and Rice is used to obtain the output spectrum of a nonlinearly processed interferogram. The interferogram of a multi-element spectrum is considered as band-limited Gaussian noise. The theory is applied to nonlinear processing in holographic Fourier spectroscopy and inverse Fourier spectroscopy and to digital processing in conventional Fourier spectroscopy.

© 1978 Optical Society of America

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References

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  1. S. O. Rice, “Mathematical Analysis of Random Noise,” Bell Syst. Tech. J. 23, 282 (1944).
    [Crossref]
  2. W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958).
  3. W. R. Bennett and S. O. Rice, Philos. Mag. 18, 422 (1934).
  4. A. Kozma, J. Opt. Soc. Am. 56, 493 (1966).
    [Crossref]
  5. H. Dammann, J. Opt. Soc. Am. 60, 1640 (1970); C. H. F. Velzel, Opt. Commun. 3, 133 (1971); J. W. Goodman and G. R. Knight, J. Opt. Soc. Am. 58, 1276 (1968); J. N. Tokarski, Appl. Opt. 8, 151 (1969).
    [Crossref]
  6. Hon-ming Lai and Shih-yu Feng, J. Opt. Soc. Am. 66, 761 (1976).
    [Crossref]
  7. A. Aspect, Thèse (Orsay, 1971).
  8. J. C. Perrin, Thèse (Orsay, 1974).
  9. C. H. F. Velzel, Philips Res. Rep. 27, 297 (1972).
  10. C. H. F. Velzel, Philips Res. Rep. 31, 97 (1976).
  11. See Ref. 2, p. 158.
  12. Rayleigh, “Wave Theory of Light,” Encyclopaedia Brittannica XXIV, 1888, Sec. 4.
  13. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves by Rough Surfaces (Pergamon, Oxford, 1963), Chap. 7.
  14. See, e.g., E. C. Titchmarch, Introduction to the Theory of Fourier Integrals (Oxford U.P., New York, 1937).
  15. M. J. Lighthill, An Introduction to Fourier Analysis and Generalized Functions (Columbia U.P., New York, 1959).
  16. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, 1968).
  17. C. H. F. Velzel, Opt. Acta 20, 585 (1973).
    [Crossref]
  18. M. Forman, J. Phys. 28, C2-58 (1967); see also G. A. Vanasse and H. Sakai, “Fourier Spectroscopy,” in Progress in Optics VI, edited by E. Wolf (North-Holland, Amsterdam, 1967).
    [Crossref]
  19. M. de Haan and C. H. F. Veizel, “Nonlinear Frequency Mixing in Optical Video Recording,” Philips Res. Rep. 32, 436 (1977).

1977 (1)

M. de Haan and C. H. F. Veizel, “Nonlinear Frequency Mixing in Optical Video Recording,” Philips Res. Rep. 32, 436 (1977).

1976 (2)

Hon-ming Lai and Shih-yu Feng, J. Opt. Soc. Am. 66, 761 (1976).
[Crossref]

C. H. F. Velzel, Philips Res. Rep. 31, 97 (1976).

1973 (1)

C. H. F. Velzel, Opt. Acta 20, 585 (1973).
[Crossref]

1972 (1)

C. H. F. Velzel, Philips Res. Rep. 27, 297 (1972).

1970 (1)

1967 (1)

M. Forman, J. Phys. 28, C2-58 (1967); see also G. A. Vanasse and H. Sakai, “Fourier Spectroscopy,” in Progress in Optics VI, edited by E. Wolf (North-Holland, Amsterdam, 1967).
[Crossref]

1966 (1)

A. Kozma, J. Opt. Soc. Am. 56, 493 (1966).
[Crossref]

1944 (1)

S. O. Rice, “Mathematical Analysis of Random Noise,” Bell Syst. Tech. J. 23, 282 (1944).
[Crossref]

1934 (1)

W. R. Bennett and S. O. Rice, Philos. Mag. 18, 422 (1934).

1888 (1)

Rayleigh, “Wave Theory of Light,” Encyclopaedia Brittannica XXIV, 1888, Sec. 4.

Abramowitz, M.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, 1968).

Aspect, A.

A. Aspect, Thèse (Orsay, 1971).

Beckmann, P.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves by Rough Surfaces (Pergamon, Oxford, 1963), Chap. 7.

Bennett, W. R.

W. R. Bennett and S. O. Rice, Philos. Mag. 18, 422 (1934).

Dammann, H.

Davenport, W. B.

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958).

de Haan, M.

M. de Haan and C. H. F. Veizel, “Nonlinear Frequency Mixing in Optical Video Recording,” Philips Res. Rep. 32, 436 (1977).

Feng, Shih-yu

Forman, M.

M. Forman, J. Phys. 28, C2-58 (1967); see also G. A. Vanasse and H. Sakai, “Fourier Spectroscopy,” in Progress in Optics VI, edited by E. Wolf (North-Holland, Amsterdam, 1967).
[Crossref]

Kozma, A.

A. Kozma, J. Opt. Soc. Am. 56, 493 (1966).
[Crossref]

Lai, Hon-ming

Lighthill, M. J.

M. J. Lighthill, An Introduction to Fourier Analysis and Generalized Functions (Columbia U.P., New York, 1959).

Perrin, J. C.

J. C. Perrin, Thèse (Orsay, 1974).

Rayleigh,

Rayleigh, “Wave Theory of Light,” Encyclopaedia Brittannica XXIV, 1888, Sec. 4.

Rice, S. O.

S. O. Rice, “Mathematical Analysis of Random Noise,” Bell Syst. Tech. J. 23, 282 (1944).
[Crossref]

W. R. Bennett and S. O. Rice, Philos. Mag. 18, 422 (1934).

Root, W. L.

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958).

Spizzichino, A.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves by Rough Surfaces (Pergamon, Oxford, 1963), Chap. 7.

Stegun, I.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, 1968).

Titchmarch, E. C.

See, e.g., E. C. Titchmarch, Introduction to the Theory of Fourier Integrals (Oxford U.P., New York, 1937).

Veizel, C. H. F.

M. de Haan and C. H. F. Veizel, “Nonlinear Frequency Mixing in Optical Video Recording,” Philips Res. Rep. 32, 436 (1977).

Velzel, C. H. F.

C. H. F. Velzel, Philips Res. Rep. 31, 97 (1976).

C. H. F. Velzel, Opt. Acta 20, 585 (1973).
[Crossref]

C. H. F. Velzel, Philips Res. Rep. 27, 297 (1972).

Bell Syst. Tech. J. (1)

S. O. Rice, “Mathematical Analysis of Random Noise,” Bell Syst. Tech. J. 23, 282 (1944).
[Crossref]

Encyclopaedia Brittannica XXIV (1)

Rayleigh, “Wave Theory of Light,” Encyclopaedia Brittannica XXIV, 1888, Sec. 4.

J. Opt. Soc. Am. (3)

J. Phys. (1)

M. Forman, J. Phys. 28, C2-58 (1967); see also G. A. Vanasse and H. Sakai, “Fourier Spectroscopy,” in Progress in Optics VI, edited by E. Wolf (North-Holland, Amsterdam, 1967).
[Crossref]

Opt. Acta (1)

C. H. F. Velzel, Opt. Acta 20, 585 (1973).
[Crossref]

Philips Res. Rep. (3)

M. de Haan and C. H. F. Veizel, “Nonlinear Frequency Mixing in Optical Video Recording,” Philips Res. Rep. 32, 436 (1977).

C. H. F. Velzel, Philips Res. Rep. 27, 297 (1972).

C. H. F. Velzel, Philips Res. Rep. 31, 97 (1976).

Philos. Mag. (1)

W. R. Bennett and S. O. Rice, Philos. Mag. 18, 422 (1934).

Other (8)

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves by Rough Surfaces (Pergamon, Oxford, 1963), Chap. 7.

See, e.g., E. C. Titchmarch, Introduction to the Theory of Fourier Integrals (Oxford U.P., New York, 1937).

M. J. Lighthill, An Introduction to Fourier Analysis and Generalized Functions (Columbia U.P., New York, 1959).

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, 1968).

See Ref. 2, p. 158.

W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958).

A. Aspect, Thèse (Orsay, 1971).

J. C. Perrin, Thèse (Orsay, 1974).

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Figures (4)

FIG. 1
FIG. 1

Functions Qm(k) for m = 1, 2 and 3.

FIG. 2
FIG. 2

(a) Transmission characteristic of Eq. (24) for T0 = ½, α = 1/Eav, β = 1/6Eav. (b) Power ratio at k = k0 between the first- and third-order spectra.

FIG. 3
FIG. 3

Characteristic of a 7-step digitizer (L = 3). The broken line shows the characteristic for L → ∞.

FIG. 4
FIG. 4

(a) Power of the first-order spectrum for a piecewise linear limiter (L → ∞) as a function of a/σ. (b) Power at k = k0 of the third-order spectrum.

Tables (1)

Tables Icon

TABLE I Power ratio 9|h,|2δ4/16|h1|2 as a function of L for several values of a/σ.

Equations (39)

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f ( z ) = n = 1 N ( 1 + cos k n z ) ,
f ¯ = N k 1 k 2 ( 1 + cos k z ) P ( k ) d k
σ 2 = N [ k 1 k 2 cos 2 k z P ( k ) d k - ( k 1 k 2 cos k z P ( k ) d k ) 2 ] ,
f ¯ = N [ 1 + cos k 0 z s ( Δ k z ) ] ,
σ 2 = ½ N [ 1 + cos ( 2 k 0 z ) s ( 2 Δ k z ) - 2 cos 2 ( k 0 z ) s 2 ( Δ k z ) ] ,
k 0 = ( k 1 + k 2 ) / 2 ,             Δ k = ( k 2 - k 1 ) / 2 ,             s ( x ) ( sin x ) / x .
R ( z , z ) = σ 2 k 1 k 2 P ( k ) cos k z cos k z d k .
R ( z , z ) = ( σ 2 / 2 ) { cos ( k 0 z + k 0 z ) s ( Δ k z + Δ k z ) + cos ( k 0 z - k 0 z ) s ( Δ k z - Δ k z ) } .
y = y ( x ) ,
G ( z , z ) = y ( z ) y ( z ) ,
y = 1 2 π - Y ( s ) e i x s d s .
G ( z , z ) = ( 1 2 π ) 2 - - d s d s Y ( s ) Y ( s ) M ( s , s ) ,
M ( s , t ) = exp { - ½ [ σ 2 s 2 + 2 R ( z , z ) s t + σ 2 t 2 ] } .
G ( z , z ) = m h m 2 R m ( z , z )
h m = 1 2 π 1 m ! - Y ( s ) e - σ 2 s 2 / 2 s m d s .
S ( k ) = 2 z 1 z 2 z 1 z 2 G ( z , z ) cos k z cos k z d z d z .
S ( k ) = 2 m h m 2 Q m ( k ) σ 2 m ,
Q 0 = ( z 1 z 2 cos k z d z ) 2
P ˜ ( k ) = ½ P ( k ) ,             k > 0 P ˜ ( k ) = ½ P ( - k ) ,             k < 0
Q m ( k ) = - P ˜ ( k - k ) Q m - 1 ( k ) d k .
x = f ( z ) - N + a i cos k i z .
h 10 = 1 2 π - Y ( s ) e - σ 2 s 2 / 2 J 1 ( a i s ) d s ,
h 210 = 1 2 π - Y ( s ) e - σ 2 s 2 / 2 J 2 ( a i s ) J 1 ( a j s ) d s .
T - T 0 = - α ( E - E av ) + β ( E - E av ) 3 ,
Y ( s ) = - 2 π i [ α δ ( 1 ) ( s ) + β δ ( 3 ) ( s ) ] ,
h 1 = - i ( α - 3 σ x 2 β ) , h 3 = - i β .
E ( k ) = F ( k ) z 1 z 2 ( 1 + cos k z ) d z v ,
z = z + Z ( f ( z ) - N )
E m ( k ) = 2 z 1 z 2 cos k z cos [ k Z ( f ( z ) - N ) ] d z F ( k ) .
y = 2 cos s 0 x
Y ( s ) = δ ( s - s 0 ) + δ ( s + s 0 ) .
Y ( s ) = i [ δ ( s - s 0 ) - δ ( s + s 0 ) ] .
h 1 = ( 1 / π ) s 0 e - σ 2 s 0 2 / 2 , h 3 = ( 1 / 6 π ) s 0 3 e - σ 2 s 0 2 / 2 .
y = H ( x ) ,
Δ E j ( k ) = ( 1 / k ) e i k z j ( - 1 ) j .
Y ( s ) = ½ δ ( s ) + 1 / i s .
h 1 = 1 / i σ ( 2 π ) 1 / 2 , h 3 = 1 / 12 i σ 3 ( 2 π ) 1 / 2 .
h 1 = 1 i σ ( 2 π ) 1 / 2 1 2 L + 1 l = - L L e - l 2 a 2 / L 2 σ 2 h 3 = 1 12 i σ 3 ( 2 π ) 1 / 2 1 2 L + 1 × l = - L L ( 1 - 2 l 2 a 2 L 2 σ 2 ) e - l 2 a 2 / L 2 σ 2 ,
h 1 = [ 1 / i σ ( 2 π ) 1 / 2 ] ( σ / a ) erf ( a / σ ) , h 3 = [ 1 / 12 i σ 3 ( 2 π ) 1 / 2 ] e - ( a / σ ) 2 .