Abstract

A general formulation is given, including the nonlinearity of a film’s t-E curve, on the intensity distribution of the reconstructed spectrum in holographic Fourier-transform spectroscopy, and a discussion on resolution follows. It is shown that Rayleigh’s criterion never applies except for a doublet under a special situation. The amplitude cancellation mechanism, which is responsible for a recently reported very good resolution, is not of much advantage when the spectrum is not a simple doublet. A third-order approximation is used to study the nonlinear effect, and a doublet case is analyzed in detail for positions and intensities of the higher-order lines.

© 1976 Optical Society of America

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References

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  1. G. W. Stroke and A. T. Funkhouser, Phys. Lett. 16, 272 (1965).
    [Crossref]
  2. G. W. Stroke, Physica (Utr.) 33, 253 (1967).
    [Crossref]
  3. K. Yoshihara and A. Kitade, Jpn. J. Appl. Phys. 6, 116 (1967).
    [Crossref]
  4. K. Kamiya, K. Yoshihara, and K. Okada, Jpn. J. Appl. Phys. 7, 1129 (1968).
    [Crossref]
  5. W. T. Plummer, Jpn. J. Appl. Phys. 6, 1250 (1967).
    [Crossref]
  6. H. Abbott, F. T. Johnson, R. Licata, and P. Oliver, Am. J. Phys. 39, 412 (1971).
    [Crossref]
  7. P. F. Parshin and A. A. Chumachenko, Usp. Fiz. Nauk. 103, 553 (1971) [Sov. Phys.-Usp. 14, 219 (1971)].
    [Crossref]
  8. L. K. Su, S. T. Hsue, and S. Y. Feng, Phys. Lett. A 53, 177 (1975).
    [Crossref]
  9. G. W. Stroke, An Introduction to Coherent Optics and Holography (Academic, New York1969), 2nd ed., p. 157.
  10. H. M. Lai and S. Y. Feng, Phys. Lett. A 54, 88 (1975).
    [Crossref]
  11. A simpler case of Fraunhofer diffraction pattern due to one sinusoidal grating can be found in J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 65–68.
  12. J. W. Goodman, in Ref. 11, p. 130–131.
  13. A. Kozma, Introduction to Optical Data Processing (McGraw-Hill, New York, 1967), Vol. 1, Chap. 9.
  14. J. W. Goodman and G. R. Knight, J. Opt. Soc. Am. 58, 1276 (1968).
    [Crossref]
  15. O. Bzyngdahl and A. Lohmann, J. Opt. Soc. Am. 58, 1325 (1968).
    [Crossref]
  16. R. J. Collier, B. Burckhardt, and L. H. Lin, Optical Holography (Academic, New York, 1971), p. 338, Eq. (12.1).
  17. R. J. Collier, in Ref. 16, p. 341–343.
  18. A. Kozma, J. Opt. Soc. Am. 56, 428 (1966).
    [Crossref]

1975 (2)

L. K. Su, S. T. Hsue, and S. Y. Feng, Phys. Lett. A 53, 177 (1975).
[Crossref]

H. M. Lai and S. Y. Feng, Phys. Lett. A 54, 88 (1975).
[Crossref]

1971 (2)

H. Abbott, F. T. Johnson, R. Licata, and P. Oliver, Am. J. Phys. 39, 412 (1971).
[Crossref]

P. F. Parshin and A. A. Chumachenko, Usp. Fiz. Nauk. 103, 553 (1971) [Sov. Phys.-Usp. 14, 219 (1971)].
[Crossref]

1968 (3)

1967 (3)

W. T. Plummer, Jpn. J. Appl. Phys. 6, 1250 (1967).
[Crossref]

G. W. Stroke, Physica (Utr.) 33, 253 (1967).
[Crossref]

K. Yoshihara and A. Kitade, Jpn. J. Appl. Phys. 6, 116 (1967).
[Crossref]

1966 (1)

1965 (1)

G. W. Stroke and A. T. Funkhouser, Phys. Lett. 16, 272 (1965).
[Crossref]

Abbott, H.

H. Abbott, F. T. Johnson, R. Licata, and P. Oliver, Am. J. Phys. 39, 412 (1971).
[Crossref]

Burckhardt, B.

R. J. Collier, B. Burckhardt, and L. H. Lin, Optical Holography (Academic, New York, 1971), p. 338, Eq. (12.1).

Bzyngdahl, O.

Chumachenko, A. A.

P. F. Parshin and A. A. Chumachenko, Usp. Fiz. Nauk. 103, 553 (1971) [Sov. Phys.-Usp. 14, 219 (1971)].
[Crossref]

Collier, R. J.

R. J. Collier, B. Burckhardt, and L. H. Lin, Optical Holography (Academic, New York, 1971), p. 338, Eq. (12.1).

R. J. Collier, in Ref. 16, p. 341–343.

Feng, S. Y.

H. M. Lai and S. Y. Feng, Phys. Lett. A 54, 88 (1975).
[Crossref]

L. K. Su, S. T. Hsue, and S. Y. Feng, Phys. Lett. A 53, 177 (1975).
[Crossref]

Funkhouser, A. T.

G. W. Stroke and A. T. Funkhouser, Phys. Lett. 16, 272 (1965).
[Crossref]

Goodman, J. W.

J. W. Goodman and G. R. Knight, J. Opt. Soc. Am. 58, 1276 (1968).
[Crossref]

A simpler case of Fraunhofer diffraction pattern due to one sinusoidal grating can be found in J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 65–68.

J. W. Goodman, in Ref. 11, p. 130–131.

Hsue, S. T.

L. K. Su, S. T. Hsue, and S. Y. Feng, Phys. Lett. A 53, 177 (1975).
[Crossref]

Johnson, F. T.

H. Abbott, F. T. Johnson, R. Licata, and P. Oliver, Am. J. Phys. 39, 412 (1971).
[Crossref]

Kamiya, K.

K. Kamiya, K. Yoshihara, and K. Okada, Jpn. J. Appl. Phys. 7, 1129 (1968).
[Crossref]

Kitade, A.

K. Yoshihara and A. Kitade, Jpn. J. Appl. Phys. 6, 116 (1967).
[Crossref]

Knight, G. R.

Kozma, A.

A. Kozma, J. Opt. Soc. Am. 56, 428 (1966).
[Crossref]

A. Kozma, Introduction to Optical Data Processing (McGraw-Hill, New York, 1967), Vol. 1, Chap. 9.

Lai, H. M.

H. M. Lai and S. Y. Feng, Phys. Lett. A 54, 88 (1975).
[Crossref]

Licata, R.

H. Abbott, F. T. Johnson, R. Licata, and P. Oliver, Am. J. Phys. 39, 412 (1971).
[Crossref]

Lin, L. H.

R. J. Collier, B. Burckhardt, and L. H. Lin, Optical Holography (Academic, New York, 1971), p. 338, Eq. (12.1).

Lohmann, A.

Okada, K.

K. Kamiya, K. Yoshihara, and K. Okada, Jpn. J. Appl. Phys. 7, 1129 (1968).
[Crossref]

Oliver, P.

H. Abbott, F. T. Johnson, R. Licata, and P. Oliver, Am. J. Phys. 39, 412 (1971).
[Crossref]

Parshin, P. F.

P. F. Parshin and A. A. Chumachenko, Usp. Fiz. Nauk. 103, 553 (1971) [Sov. Phys.-Usp. 14, 219 (1971)].
[Crossref]

Plummer, W. T.

W. T. Plummer, Jpn. J. Appl. Phys. 6, 1250 (1967).
[Crossref]

Stroke, G. W.

G. W. Stroke, Physica (Utr.) 33, 253 (1967).
[Crossref]

G. W. Stroke and A. T. Funkhouser, Phys. Lett. 16, 272 (1965).
[Crossref]

G. W. Stroke, An Introduction to Coherent Optics and Holography (Academic, New York1969), 2nd ed., p. 157.

Su, L. K.

L. K. Su, S. T. Hsue, and S. Y. Feng, Phys. Lett. A 53, 177 (1975).
[Crossref]

Yoshihara, K.

K. Kamiya, K. Yoshihara, and K. Okada, Jpn. J. Appl. Phys. 7, 1129 (1968).
[Crossref]

K. Yoshihara and A. Kitade, Jpn. J. Appl. Phys. 6, 116 (1967).
[Crossref]

Am. J. Phys. (1)

H. Abbott, F. T. Johnson, R. Licata, and P. Oliver, Am. J. Phys. 39, 412 (1971).
[Crossref]

J. Opt. Soc. Am. (3)

Jpn. J. Appl. Phys. (3)

K. Yoshihara and A. Kitade, Jpn. J. Appl. Phys. 6, 116 (1967).
[Crossref]

K. Kamiya, K. Yoshihara, and K. Okada, Jpn. J. Appl. Phys. 7, 1129 (1968).
[Crossref]

W. T. Plummer, Jpn. J. Appl. Phys. 6, 1250 (1967).
[Crossref]

Phys. Lett. (1)

G. W. Stroke and A. T. Funkhouser, Phys. Lett. 16, 272 (1965).
[Crossref]

Phys. Lett. A (2)

L. K. Su, S. T. Hsue, and S. Y. Feng, Phys. Lett. A 53, 177 (1975).
[Crossref]

H. M. Lai and S. Y. Feng, Phys. Lett. A 54, 88 (1975).
[Crossref]

Physica (Utr.) (1)

G. W. Stroke, Physica (Utr.) 33, 253 (1967).
[Crossref]

Usp. Fiz. Nauk. (1)

P. F. Parshin and A. A. Chumachenko, Usp. Fiz. Nauk. 103, 553 (1971) [Sov. Phys.-Usp. 14, 219 (1971)].
[Crossref]

Other (6)

G. W. Stroke, An Introduction to Coherent Optics and Holography (Academic, New York1969), 2nd ed., p. 157.

A simpler case of Fraunhofer diffraction pattern due to one sinusoidal grating can be found in J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 65–68.

J. W. Goodman, in Ref. 11, p. 130–131.

A. Kozma, Introduction to Optical Data Processing (McGraw-Hill, New York, 1967), Vol. 1, Chap. 9.

R. J. Collier, B. Burckhardt, and L. H. Lin, Optical Holography (Academic, New York, 1971), p. 338, Eq. (12.1).

R. J. Collier, in Ref. 16, p. 341–343.

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Figures (5)

FIG. 1
FIG. 1

Beat pattern formed by two sinusoidal intensity gratings of equal amplitude. Also shown is a reconstruction beam of width D and centered at 1 2 ( q 2 - q 1 ) - 1. D is smaller than the beat period (q2q1)−1. The positions of “node” and “antinode” are defined as shown.

FIG. 2
FIG. 2

Reconstructed doublet spectrum as a function of the position of the reconstruction beam center. Curve 1: center at the middle between a node and neighboring antinode. Curve 2: center at node. Curve 3: center at antinode. (a) for D(q2q1) = 1, (b) for D ( q 2 - q 1 ) = 1 2. Dotted lines in (a) are the intensity distribution of the two overlapping doublet lines taken individually. The doublet lines are of equal intensity.

FIG. 3
FIG. 3

Reconstructed doublet spectrum as a function of different relative intensities T1:T2 of the two doublet lines. The reconstruction beam center is at a node. (a) for D ( q 2 - q 1 ) = 1 2, (b) for D(q2q1) = 0.8.

FIG. 4
FIG. 4

Reconstructed triplet spectrum as a function of the product value D(q2q1). The three lines of the triplet are of equal intensity. Curve 1: D(q2q1) = 1, curve 2: D ( q 2 - q 1 ) = 1 2, curve 3: D ( q 2 - q 1 ) = 1 4. (a) under the condition x 0 = 1 2 ( q 2 - q 1 ) - 1, (b) under the condition x 0 = 1 4 ( q 2 - q 1 ) - 1.

FIG. 5
FIG. 5

Reconstructed doublet spectrum with nonlinear effect, assuming all lines are well resolved. Theoretical intensity ratios are indicated for second-order triplet and third-order quartet.

Equations (18)

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E ( x ) = α = 1 N I α [ 1 + m cos ( 2 π q α x ) ] ,
t ( x ) = H 0 + μ = 1 α = 1 N H μ ( α ) cos 2 π μ q α x + μ = 1 ν = 1 α < β N N H μ ν ( α β ) { cos [ 2 π ( μ q α + ν q β ) x ] + cos [ 2 π ( μ q α - ν q β ) x ] } + μ = 1 ν = 1 τ = 1 α < β < γ N N N H μ ν τ ( α β γ ) { cos [ 2 π ( μ q α + ν q β + τ q γ ) x ] + cos [ 2 π ( μ q α + ν q β - τ q γ ) x ] + cos [ 2 π ( μ q α - ν q β + τ q γ ) x ] + cos [ 2 π ( μ q α - ν q β - τ q γ ) x ] } + = T 0 + a T a cos ( 2 π Q a x ) ,
A ( θ ) A 0 ( T 0 sinc ( u ) + 1 2 a T a [ exp ( i 2 π Q a x 0 ) sinc ( u - D Q a ) + exp ( - i 2 π Q a x 0 ) sinc ( u + D Q a ) ] ) ,
A ( θ ) A 0 ( T 0 sinc ( u ) + 1 2 α = 1 N T α [ exp ( i 2 π q α x 0 ) sinc ( u - D q α ) + exp ( - i 2 π q α x 0 ) sinc ( u + D q α ) ] ) ,
I ( θ ) T 0 2 sinc 2 ( u ) + 1 4 α = 1 N T α 2 [ sinc 2 ( u - D q α ) + sinc 2 ( u + D q α ) ] ,
T 1 sinc ( u - D q 1 ) + exp [ i 2 π x 0 ( q 2 - q 1 ) ] T 2 sinc ( u - D q 2 ) 2 ,
T 1 sinc ( u - D q 1 ) + exp [ i 2 π x 0 ( q 2 - q 1 ) ] T 2 sinc ( u - D q 2 ) + exp [ i 2 π x 0 ( q 3 - q 1 ) ] T 3 sinc ( u - D q 3 ) 2 ,
sinc ( u - D q 1 ) - sinc ( u - D q 2 ) + sinc ( u - D q 3 ) 2 .
[ sinc ( u - D q 1 ) - sinc ( u - D q 3 ) ] 2 + sinc 2 ( u - D q 2 ) .
( 2 n + 1 ) 1 2 ( q α 2 - q α 1 ) = ( 2 m + 1 ) 1 2 ( q β 2 - q β 1 ) ,             n , m = 0 , 1 , 2 , .
t ( E ) = j = 0 n k j E j ,
t ( x ) = T ( 0 ) + α = 1 N T α ( 1 ) cos 2 π q α x + α = 1 N T α ( 2 ) cos 4 π q α x + α < β N N T α β ( 2 ) [ cos 2 π ( q α + q β ) x + cos 2 π ( q α - q β ) x ] + α = 1 N T α ( 3 ) cos 6 π q α x + α β N N T α β ( 3 ) [ cos 2 π ( 2 q α + q β ) x + cos 2 π ( 2 q α - q β ) x ] + α < β < γ N N N T α β γ ( 3 ) [ cos 2 π ( q α + q β + q γ ) x + cos 2 π ( q α + q β - q γ ) x + cos 2 π ( q α - q β + q γ ) x + cos 2 π ( q α - q β - q γ ) x ] ,
T ( 0 ) = k 0 + k 1 I 0 + k 2 I 0 2 + k 3 I 0 3 + m 2 ( k 2 + 3 k 3 I 0 ) α = 1 N I α 2 2 , T α ( 1 ) = m ( k 1 + 2 k 2 I 0 + 3 k 3 I 0 2 ) I α + ( 3 4 m 3 k 3 I α 3 ) ( 1 + 2 β α N I β I α ) , T α ( 2 ) = 1 2 m 2 ( k 2 + 3 k 3 I 0 ) I α 2 ,             T α β ( 2 ) = m 2 ( k 2 + 3 k 3 I 0 ) I α I β ,             T α ( 3 ) = 1 4 m 3 k 3 I α 3 ,             T α β ( 3 ) = 3 4 m 3 k 3 I α 2 I β ,             T α β γ ( 3 ) = 3 2 m 3 k 3 I α I β I γ ;
I 0 = α = 1 N I α .
T ( 0 ) sinc ( u ) + 1 2 T 12 ( 2 ) { exp [ i 2 π ( q 2 - q 1 ) x 0 ] sinc [ u - D ( q 2 - q 1 ) ] + exp [ - i 2 π ( q 2 - q 1 ) x 0 ] sinc [ u + D ( q 2 - q 1 ) ] } 2 ,
L ( ξ ) = 1 2 π 0 e i ξ E t ( E ) d E ,
t ( E ) = - e - i ξ E L ( ξ ) d ξ             ( E > 0 ) .
H 0 = - d ξ L ( ξ ) e - i ξ I 0 α = 1 N J 0 ( ξ I α m ) , H μ ( α ) = 2 ( - i ) μ - d ξ L ( ξ ) e - i ξ I 0 J μ ( ξ I α m ) β α N J 0 ( ξ I β m ) , H μ ν ( α β ) = 2 ( - i ) μ + ν - d ξ L ( ξ ) e - i ξ I 0 × J μ ( ξ I α m ) J ν ( ξ I β m ) γ α , β N J 0 ( ξ I γ m ) , H μ ν τ ) ( α β γ ) = 2 ( - i ) μ + ν + τ - d ξ L ( ξ ) e - i ξ I 0 × J μ ( ξ I α m ) J ν ( ξ I β m ) J τ ( ξ I γ m ) δ α , β , γ N J 0 ( ξ I δ m ) ,