Abstract

This report shows how the pattern information in an object through which a coherent beam of light is passed can be transformed into information existing in the focal plane. The information in the focal plane can be interpreted as sidebands of a zero-frequency carrier. There is also a close connection between the sideband interpretation and the sampling theorem of information theory and this connection is derived and discussed.

The mathematical methods used in the first (one-dimensional) part of the following analysis are an application to optics of certain classical methods in modulation and communication theory. These are the method of symmetrical and antisymmetrical sideband distribution, the method of paired echoes and sampling theorem analysis.

In the two dimensional analysis, a more general method of separating functions into symmetrical and antisymmetrical parts is used. This leads to a new general analysis of the diffraction patterns of unsymmetrical pupil functions. In the course of the analysis a two-dimensional quadrature function is used. The quadrature function in one dimension is the Hilbert transform.

© 1962 Optical Society of America

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References

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  1. See, for example, M. Born and E. WolfPrinciples of Optics (Pergamon Press, New York, 1959), pp. 419–420.
  2. We note in passing that the use of the Fourier transforms (3) and (4) imply the assumption that the lens is large enough so that it does not act as an optical stop.
  3. S. Goldman, Information Theory (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1953), pp. 71–75.
  4. Application of the sampling theorems to optical-image transformations is not new. See, for example, E. L. O’Neill and T. Asakura, “Optical Image Formations in Terms of Entropy Transmission,” No. P-163 (Itek Corporation, Waltham, Massachusetts, April1960). They also refer to earlier work of Gabor and Gamo.
  5. S. Goldman, Frequency Analysis, Modulation and Noise (McGraw-Hill Book Company, Inc., New York, 1948), pp. 102–108, 167–181.
  6. Quadrature functions are discussed in Appendix II.
  7. We are here referring to distortion of the intensity pattern, which should not be confused with Seidel distortion.
  8. See reference 1, pp. 425–427.
  9. The name “schlieren method” has also been used for certain types of central darkground methods of observation which are symmetrical and do not give rise to quadrature distortion.
  10. It is worthy of note that if a large unmodulated signal (carrier) coherent with the incident radiation were added to the disturbance in the image plane, then it would effectively eliminate the component in V¯ with which it was in quadrature as far as intensity modulation is concerned. The addition of such components may become practical with the use of optical masers.
  11. See reference 5, pp. 90–98, 176–181.
  12. Courant-Hilbert, Methoden der Mathematischen Physik (Verlag Julius Springer, Berlin, 1931), Vol. I, p. 62.
  13. See Goldman, reference 3, Appendix V.
  14. In Sec. IV of the main paper, it is shown that the real part of U is symmetrical with respect to the origin and the imaginary part is antisymmetrical.
  15. Unsymmetrical with respect to the origin.
  16. For examples of one dimensional quadrature functions see Goldman, reference 3, pp. 338–339.

Asakura, T.

Application of the sampling theorems to optical-image transformations is not new. See, for example, E. L. O’Neill and T. Asakura, “Optical Image Formations in Terms of Entropy Transmission,” No. P-163 (Itek Corporation, Waltham, Massachusetts, April1960). They also refer to earlier work of Gabor and Gamo.

Born, M.

See, for example, M. Born and E. WolfPrinciples of Optics (Pergamon Press, New York, 1959), pp. 419–420.

Gabor,

Application of the sampling theorems to optical-image transformations is not new. See, for example, E. L. O’Neill and T. Asakura, “Optical Image Formations in Terms of Entropy Transmission,” No. P-163 (Itek Corporation, Waltham, Massachusetts, April1960). They also refer to earlier work of Gabor and Gamo.

Gamo,

Application of the sampling theorems to optical-image transformations is not new. See, for example, E. L. O’Neill and T. Asakura, “Optical Image Formations in Terms of Entropy Transmission,” No. P-163 (Itek Corporation, Waltham, Massachusetts, April1960). They also refer to earlier work of Gabor and Gamo.

Goldman,

For examples of one dimensional quadrature functions see Goldman, reference 3, pp. 338–339.

Goldman, S.

S. Goldman, Frequency Analysis, Modulation and Noise (McGraw-Hill Book Company, Inc., New York, 1948), pp. 102–108, 167–181.

S. Goldman, Information Theory (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1953), pp. 71–75.

O’Neill, E. L.

Application of the sampling theorems to optical-image transformations is not new. See, for example, E. L. O’Neill and T. Asakura, “Optical Image Formations in Terms of Entropy Transmission,” No. P-163 (Itek Corporation, Waltham, Massachusetts, April1960). They also refer to earlier work of Gabor and Gamo.

Wolf, E.

See, for example, M. Born and E. WolfPrinciples of Optics (Pergamon Press, New York, 1959), pp. 419–420.

Other (16)

See, for example, M. Born and E. WolfPrinciples of Optics (Pergamon Press, New York, 1959), pp. 419–420.

We note in passing that the use of the Fourier transforms (3) and (4) imply the assumption that the lens is large enough so that it does not act as an optical stop.

S. Goldman, Information Theory (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1953), pp. 71–75.

Application of the sampling theorems to optical-image transformations is not new. See, for example, E. L. O’Neill and T. Asakura, “Optical Image Formations in Terms of Entropy Transmission,” No. P-163 (Itek Corporation, Waltham, Massachusetts, April1960). They also refer to earlier work of Gabor and Gamo.

S. Goldman, Frequency Analysis, Modulation and Noise (McGraw-Hill Book Company, Inc., New York, 1948), pp. 102–108, 167–181.

Quadrature functions are discussed in Appendix II.

We are here referring to distortion of the intensity pattern, which should not be confused with Seidel distortion.

See reference 1, pp. 425–427.

The name “schlieren method” has also been used for certain types of central darkground methods of observation which are symmetrical and do not give rise to quadrature distortion.

It is worthy of note that if a large unmodulated signal (carrier) coherent with the incident radiation were added to the disturbance in the image plane, then it would effectively eliminate the component in V¯ with which it was in quadrature as far as intensity modulation is concerned. The addition of such components may become practical with the use of optical masers.

See reference 5, pp. 90–98, 176–181.

Courant-Hilbert, Methoden der Mathematischen Physik (Verlag Julius Springer, Berlin, 1931), Vol. I, p. 62.

See Goldman, reference 3, Appendix V.

In Sec. IV of the main paper, it is shown that the real part of U is symmetrical with respect to the origin and the imaginary part is antisymmetrical.

Unsymmetrical with respect to the origin.

For examples of one dimensional quadrature functions see Goldman, reference 3, pp. 338–339.

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Figures (8)

Fig. 1
Fig. 1

Basic optical image forming system.

Fig. 2
Fig. 2

Rectangle of expansion for the two-dimensional Fourier series analysis of an enclosed transilluminated object.

Fig. 3
Fig. 3

A half-plane stop in the focal plane.

Fig. 4
Fig. 4

Separation of an asymmetrical aperture into parts which are symmetrical and which are unsymmetrical with respect to a particular origin.

Fig. 5
Fig. 5

Diagram showing the asymmetrical nature of the pupil function of an off-axis image point.

Fig. 6
Fig. 6

Relations of index signs and quadrants in a two-dimensional Fourier series.

Fig. 7
Fig. 7

Positive and negative half-axes of the axis of ξ.

Fig. 8
Fig. 8

Division of the ξ, η plane into two half-planes.

Equations (97)

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F ( x ) = A ( x ) exp [ j ϕ ( x ) ] .
{ 1 - [ A ( x ) ] 2 } .
U ( ξ ) = C S 1 F ( x ) exp [ - j k ( ξ f x ) ] d x
V ( x ) = C 1 S 2 U ( ξ ) exp ( - j k x D ξ ) d ξ = ? C 2 F ( x M ) .
k = 2 π / λ
M = - ( D / f )
( - 1 2 x 0 < x < 1 2 x 0 ) ,
F ( x ) = A ( x ) = a 0 2 + n = 1 ( a n cos n c x + b n sin n c x ) ,
c = 2 π / x 0 .
U ( ξ ) = C - x 0 / 2 + x 0 / 2 ( a 0 2 + n = 1 { a n 2 [ exp ( j n c x ) + exp ( - j n c x ) ] - j b n 2 [ exp ( j n c x ) - exp ( - j n c x ) ] } ) exp ( - j k f ξ x ) d x = C - x 0 / 2 x 0 / 2 a 0 2 exp ( - j k f ξ x ) d x + n = 1 C a n 2 - x 0 / 2 x 0 / 2 exp [ - j ( k f ξ - n c ) x ] d x + n = 1 C a n 2 - x 0 / 2 + x 0 / 2 exp [ - j ( k f ξ + n c ) ] d x + n = 1 j C b n 2 - x 0 / 2 + x 0 / 2 exp [ - j ( k f ξ + n c ) x ] d x + n = 1 - j C b n 2 - x 0 / 2 + x 0 / 2 exp [ - j ( k f ξ - n c ) x ] d x .
U 0 ( ξ ) = C - x 0 / 2 + x 0 / 2 exp ( - j k f ξ x ) d x = C exp ( - j k f ξ x ) - j ( k / f ) ξ | - x 0 / 2 + x 0 / 2 = C exp ( - j k f ξ x 0 2 ) - exp ( + j k f ξ x 0 2 ) - j ( k / f ) ξ = C x 0 { sin [ ( k ξ / f ) ( 1 2 x 0 ) ] ( k ξ / f ) ( 1 2 x 0 ) } = C x 0 { sin [ π ( ξ / f ) ( x 0 / λ ) ] π ξ x 0 / f λ } .
U ( ξ ) = a 0 2 U 0 ( ξ ) + n = 1 a n 2 U 0 ( ξ - n c f k ) + n = 1 a n 2 U 0 ( ξ + n c f k ) - n = 1 j b n 2 U 0 ( ξ - n c f k ) + n = 1 j b n 2 U 0 ( ξ + n c f k ) ,
n c f k = 2 π n f x 0 k = n λ f x 0
U 0 ( ξ - n c f k ) = C x 0 ( sin { ( k x 0 / 2 f ) [ ξ - ( 2 π n / x 0 ) ( f / k ) ] ( k x 0 / 2 f ) [ ξ - ( 2 π n / x 0 ) ( f / k ) ] ) = C x 0 { sin [ ( k x 0 ξ / 2 f ) - n π ] ( k x 0 ξ / 2 f ) - n π } .
U ( ξ ) = a 0 2 U 0 ( ξ ) + n = 1 ( a n 2 + b n 2 ) 1 2 2 exp [ - j tan - 1 ( b n a n ) ] U 0 ( ξ - n λ f x 0 ) + n = 1 ( a n 2 + b n 2 ) 1 2 2 exp [ + j tan - 1 ( b n a n ) ] U 0 ( ξ + n λ f x 0 ) .
ξ x 0 / f λ = 1 ;             i . e . , when             ξ = λ f / x 0 .
U ( ξ ) = C x 0 m = - + ( a m - j b m 2 ) sin [ ( k x 0 ξ / 2 f ) - n π ] [ ( k x 0 ξ / 2 f ) - n π ]
a - m = a m ,
b - m = - b m .
k x 0 / f = 2 π x 0 / λ f .
F ( x ) = exp [ j ϕ ( x ) ] .
B = ( 2 / x 0 ) [ ϕ ( 1 2 x 0 ) - ϕ ( - 1 2 x 0 ) ]
ϕ ( x ) - B x = α 0 2 + n = 1 ( α n cos n c x + β n sin n c x ) .
U ( ξ ) = C - x 0 / 2 + x 0 / 2 F ( x ) exp ( - j k f ξ x ) d x = C - x 0 / 2 + x 0 / 2 exp j [ ϕ ( x ) - k f ξ x ] d x = C - x 0 / 2 + x 0 / 2 exp { j [ - k f ξ x + B x + α 0 2 + n = 1 ( α n cos n c x + β n sin n c x ) ] } d x .
exp ( j z sin θ ) = k = - + J k ( z ) exp ( j k θ ) = J 0 ( z ) + [ J 1 ( z ) exp ( j θ ) + J - 1 ( z ) exp ( - j θ ) ] + [ J 2 ( z ) exp ( j 2 θ ) + J - 2 ( z ) exp ( - j 2 θ ) ] + ,
exp ( j z cos θ ) = exp [ j z sin ( π 2 - θ ) ] = k = - + J k ( z ) exp [ j k ( π 2 - θ ) ] = k = - + J k ( z ) exp ( j k π 2 ) × exp ( - j k θ ) ,
J - n ( z ) = ( - 1 ) n J n ( z ) ,
J n ( z ) = [ z n / 2 n ( n ! ) ] { 1 - [ z 2 / 2 2 ( n + 1 ) ] + } .
U ( ξ ) = C - x 0 / 2 + x 0 / 2 exp [ - j ( k f ξ - B ) x ] × exp ( j α 0 2 ) × n = 1 exp ( j α n cos n c x ) n = 1 exp ( j β n sin n c x ) d x .
exp ( j α n cos n c x ) = k = - + J k ( α n ) exp ( j k π 2 ) exp ( - j k n c x ) = J 0 ( α n ) + j J 1 ( α n ) [ exp ( j n c x ) + exp ( - j n c x ) ] - J 2 ( α n ) [ exp ( j 2 n c x ) + exp ( - j 2 n c x ) ] - .
exp ( j β n sin n c x ) = k = - J k ( β n ) exp ( j k n c x ) = J 0 ( β n ) + J 1 ( β n ) [ exp ( j n c x ) - exp ( - j n c x ) ] + J 2 ( β n ) [ exp ( j 2 n c x ) + exp ( - j 2 n c x ) ] + .
ϕ ( x ) = B x + 1 2 α 0 + α n cos n c x .
U ( ξ ) = C - x 0 / 2 + x 0 / 2 exp [ - j ( k f ξ - B ) x ] exp ( j α 0 2 ) exp ( j α n cos n c x ) d x = C - x 0 / 2 + x 0 / 2 exp [ - j ( k f ξ - B ) x ] exp ( j α 0 2 ) × { J 0 ( α n ) + j J 1 ( α n ) [ exp ( j n c x ) + exp ( - j n c x ) ] - J 2 ( α n ) [ exp ( j 2 n c x ) + exp ( - j 2 n x ) ] } d x = C - x 0 / 2 + x 0 / 2 exp [ - j ( k f ξ - B ) x ] exp ( j α 0 2 ) J 0 ( α n ) d x + j C - x 0 / 2 + x 0 / 2 exp [ - j ( k f ξ - B - n c ) x ] exp ( j α 0 2 ) J 1 ( α n ) d x + j C - x 0 / 2 + x 0 / 2 exp [ - j ( k f ξ - B + n c ) x ] exp ( j α 0 2 ) J 1 ( α n ) d x - C - x 0 / 2 + x 0 / 2 exp [ - j ( k f ξ - B - 2 n c ) x ] × exp ( j α 0 2 ) J 2 ( α n ) d x - C - x 0 / 2 + x 0 / 2 exp [ - j ( k f ξ - B + 2 n c ) x ] exp ( j α 0 2 ) J 2 ( α n ) d x + = exp ( j α 0 2 ) J 0 ( α n ) U 0 ( ξ - B f k ) + j exp ( j α 0 2 ) J 1 ( α n ) U 0 ( ξ - B f k - n c f k ) + j exp ( j α 0 2 ) J 1 ( α n ) U 0 ( ξ - B f k + n c f k ) - exp ( j α 0 2 ) J 2 ( α n ) U 0 ( ξ - B f k - 2 n c f k ) - exp ( j α 0 2 ) J 2 ( α n ) U 0 ( ξ - B f k + 2 n c f k ) + .
ϕ ( x ) = B x + 1 2 α 0 + β n sin n c x ,
U ( ξ ) = exp ( j α 0 2 ) J 0 ( β n ) U 0 ( ξ - B f k ) + exp ( j α 0 2 ) J 1 ( β n ) U 0 ( ξ - B f k - n c f k ) - exp ( j α 0 2 ) J 1 ( β n ) U 0 ( ξ - B f k + n c f k ) + exp ( j α 0 2 ) J 2 ( β n ) U 0 ( ξ - B f k - 2 n c f k ) + exp ( j α 0 2 ) J 2 ( β n ) U 0 ( ξ - B f k + 2 n c f k ) + .
ϕ ( x ) = B x + 1 2 α 0 + α n cos n c x + β m sin m c x ,
U ( ξ ) = C - x 0 / 2 + x 0 / 2 exp [ - j ( k f ξ - B ) x ] exp ( j α 0 2 ) × { J 0 ( α n ) + j J 1 ( α n ) [ exp ( j n c x ) + exp ( - j n c x ) ] - J 2 ( α n ) [ exp ( j 2 n c x ) + exp ( - j 2 n c x ) ] + } × { J 0 ( β m ) + J 1 ( β m ) [ exp ( j m c x ) - exp ( - j m c x ) ] + J 2 ( β m ) [ exp ( j 2 m c x ) + exp ( - j 2 m c x ) ] } d x = exp ( j α 0 2 ) J 0 ( α n ) J 0 ( β m ) U 0 ( ξ - B f k ) + exp ( j α 0 2 ) J 0 ( α n ) J 1 ( β m ) U 0 ( ξ - B f k - m c f k ) - exp ( j α 0 2 ) J 0 ( α n ) J 1 ( β m ) U 0 ( ξ - B f k + m c f k ) + j exp ( j α 0 2 ) J 0 ( β m ) J 1 ( α n ) U 0 ( ξ - B f k - n c f k ) + j exp ( j α 0 2 ) J 0 ( β m ) J 1 ( α n ) U 0 ( ξ - B f k + n c f k ) + j exp ( j α 0 2 ) J 1 ( α n ) J 1 ( β m ) × U 0 [ ξ - B f k + ( m + n ) c f k ] + all other sum and difference terms .
J 0 ( α n ) = 1 = J 0 ( β n ) , J 1 ( α n ) = 1 2 α n ,             J 1 ( β n ) = 1 2 β n , J 2 ( α n ) = 0 = J 2 ( β n ) = J 3 = J 4 , etc .
U ( ξ ) = exp ( j α 0 2 ) U 0 ( ξ - B f k ) + j exp ( j α 0 2 ) n = 1 α n 2 U 0 ( ξ - B f k - n c f k ) + j exp ( j α 0 2 ) n = 1 α n 2 U 0 ( ξ - B f k + n c f k ) + exp ( j α 0 2 ) n = 1 β n 2 U 0 ( ξ - B f k - n c f k ) - exp ( j α 0 2 ) n = 1 β n 2 U 0 ( ξ - B f k + n c f k ) ,
n c f / k = n λ f / x 0 .
F ( x ) = A ( x ) exp [ j ϕ ( x ) ]
U ( ξ , η ) = C S 1 F ( x , y ) exp [ - j k ( ξ f x + η f y ) ] d x d y
V ( x , y ) = C 1 S 2 U ( ξ , η ) exp [ - j k ( x D ξ + y D η ) ] d ξ d η = ? C 2 F ( x / M , y / M ) .
x = r cos θ ,
y = r sin θ ,
ξ = ρ cos ϕ ,
η = ρ sin ϕ .
F ¯ ( r , θ ) = F ( x , y )
Ū ( ρ , ϕ ) = U ( ξ , η )
F ¯ ( r , θ ) = F ¯ s ( r , θ ) + F ¯ a ( r , θ ) ,
F ¯ s ( r , θ ) = 1 2 [ F ¯ ( r , θ ) + F ¯ ( r , θ + π ) ]
F ¯ a ( r , θ ) = 1 2 [ F ¯ ( r , θ ) - F ¯ ( r , θ + π ) ]
Ū ( ρ , ϕ ) = C F ¯ ( r , θ ) exp [ - j k r ρ f cos ( θ - ϕ ) ] r d r d θ .
Ū ( ρ , ϕ + π ) = C F ¯ ( r , θ ) exp [ - j k r ρ f cos ( θ - ϕ - π ) ] × r d r d θ = Ū * ( ρ , ϕ ) ,
Ū ( ρ , ϕ ) = Ū s ( ρ , ϕ ) + j Ū a ( ρ , ϕ ) ,
V ¯ ( r , θ ) = C 1 ϕ = 0 2 π ρ = 0 Ū ( ρ , ϕ ) exp [ - j k D ρ r cos ( θ - ϕ ) ] × ρ d ρ d ϕ = C 2 F ¯ ( r M , θ ) .
C 2 F ¯ ( r M , θ ) = C 1 ϕ = 0 2 π ρ = 0 Ū ( ρ , ϕ ) exp [ - j k D ρ r cos ( θ - ϕ ) ] ρ d ρ d ϕ = C 1 ϕ 0 ϕ 0 + π 0 ( { Ū s cos [ k ρ r D cos ( θ - ϕ ) ] + Ū a sin [ k ρ r D cos ( θ - ϕ ) ] } + j { Ū a cos [ k ρ r D cos ( θ - ϕ ) ] - Ū s sin [ k ρ r D cos ( θ - ϕ ) ] } ) ρ d ρ d ϕ + ( C 1 ϕ = 0 ϕ 0 ρ = 0 + C 1 ϕ = ϕ 0 + π 2 π ρ = 0 ) same integrand = 2C 1 ϕ 0 ϕ 0 + π 0 { Ū s cos [ k ρ r D cos ( θ - ϕ ) ] + Ū a sin [ k ρ r D cos ( θ - ϕ ) ] } ρ d ρ d ϕ = C 2 F ¯ ( r M , θ ) .
V ¯ ( r , θ ) = C 1 ϕ 0 ϕ 0 + π 0 ( { Ū s cos [ k ρ r D cos ( θ - ϕ ) ] + Ū a sin [ k ρ r D cos ( θ - ϕ ) ] } + j { Ū a cos [ k ρ r D cos ( θ - ϕ ) ] - Ū s sin [ k ρ r D cos ( θ - ϕ ) ] } ) ρ d ρ d ϕ = C 2 2 F ¯ ( r M , θ ) + j C 2 2 F ¯ Q ( r M , θ ) ,
F ¯ = j G ¯ ,
V ¯ ( r , θ ) = j C 2 2 [ G ¯ ( r M , θ ) + j G ¯ Q ( r M , θ ) ] .
F ¯ = A ¯ + j B ¯ ,
V ¯ ( r , θ ) = 1 2 C 2 [ ( A ¯ + j A ¯ Q ) + j ( B ¯ + j B ¯ Q ) ] .
I = V ¯ · V ¯ * = ( 1 2 C 2 ) 2 { ( A ¯ - B ¯ Q ) 2 + ( A ¯ Q + B ¯ 2 ) } .
F ¯ = K exp [ j β ( r , θ ) ] = K ( cos β + j sin β ) = K ( 1 + j β ) , approximately ,
I = 1 4 K 2 C 2 2 { ( 1 - β Q ) 2 + β 2 } = 1 4 K 2 C 2 2 ( 1 - 2 β Q ) , approximately .
F ¯ ( r , θ ) = A ( r , θ ) + j B ( r , θ ) .
V ¯ ( r , θ ) = [ A S + 1 2 A N S + 1 2 j A Q ( N S ) ] + j [ B S + 1 2 B N S + 1 2 j B Q ( N S ) ] = [ A S + 1 2 A N S - 1 2 B Q ( N S ) ] + j [ B S + 1 2 B N S + 1 2 A Q ( N S ) ]
A S + j B S = S Ū exp [ - j k D ρ r cos ( θ - ϕ ) ] ρ d ρ d ϕ
[ 1 2 A N S + 1 2 j A Q ( N S ) ] + j [ 1 2 B N S + 1 2 j B Q ( N S ) ] = N S Ū exp [ - j k D ρ r cos ( θ - ϕ ) ] ρ d ρ d ϕ
[ A - ( A S + 1 2 A N S ) ] + j [ B - ( B S + 1 2 B N S ) ] .
- 1 2 B Q ( N S ) + 1 2 j A Q ( N S )
F ¯ = F ¯ exp ( j β ) = A + j B = [ C + a ( r , θ ) + j b ( r , θ ) ] exp ( j α ) ,
a C             and             b C
I = V ¯ · V ¯ * = [ C + a s + 1 2 a N S - 1 2 b Q ( N S ) ] 2 + [ b s + 1 2 b N S + 1 2 a Q ( N S ) ] 2 = C 2 + 2 C [ a s + 1 2 a N S - 1 2 b Q ( N S ) ] , approximately ,
F ( s , t ) = μ = - + v = - + a μ v exp [ j ( μ s + v t ) ] ,
a μ v = 1 4 π 2 - π + π d s - π + π d t F ( s , t ) exp [ - j ( μ s + v t ) ] .
F ( s , t ) = μ = - + v = - + a μ v exp [ j ( π a μ s + π b v t ) ] ,
a μ v = 1 4 a b - a + a d s - b + b d t F ( s , t ) × exp [ - j ( π a μ s + π b v t ) ] .
a - μ , - v = a μ , v *
a + μ , - v = a - μ , + v * ,
F ( s , t ) = A 00 + n = 1 m = 1 [ 2 A m n cos ( π a m s + π b n t ) + 2 B m n sin ( π a m s + π b n t ) ] + n = - 1 - m + 1 + [ 2 A m n cos ( π a m s + π b n t ) + 2 B m n sin ( π a m s + π b n t ) ] = A 00 + n = 1 m = 1 [ 2 A m n cos ( π a m s + π b n t ) + 2 B m n sin ( π a m s + π b n t ) ] + n = 1 m = - 1 - [ 2 A m n cos ( π a m s + π b n t ) + 2 B m n sin ( π a m s + π b n t ) ] .
A 00 = 1 4 a b - a + a - b + b F ( s , t ) d s d t ,
A m n = 1 4 a b - a + a - b + b F ( s , t ) cos ( π a m s + π b n t ) d s d t ,
B m n = 1 4 a b - a + a - b + b F ( s , t ) sin ( π a m s + π b n t ) d s d t .
U ( ξ ) = - + f ( x ) exp ( - j ξ x d x ) = - + [ f e ( x ) + f 0 ( x ) ] [ cos ξ x - j sin ξ x ] d x = 2 0 f e ( x ) cos ξ x d x - j 2 0 f 0 ( x ) sin ξ x d x = U e ( ξ ) + j U 0 ( ξ ) .
f ( x ) = 1 2 π - + U ( ξ ) exp ( + j ξ x ) d ξ = 1 π 0 U c ( ξ ) cos ξ x d ξ - 1 π 0 U 0 ( ξ ) sin ξ x d ξ .
f Q ( x ) = 1 π 0 U e ( ξ ) cos ( ξ x + π 2 ) d ξ - 1 π 0 U 0 ( ξ ) sin ( ξ x + π 2 ) d ξ = - 1 π 0 U e ( ξ ) sin ξ x d ξ - 1 π 0 U 0 ( ξ ) cos ξ x d ξ .
f Q ( x ) 1 2 π - + U ( ξ ) exp [ + j ( ξ x + π 2 ) ] d ξ = 1 2 π - + [ U e ( ξ ) + j U 0 ] [ - sin ξ x + j cos ξ x ] d ξ = j π 0 [ U e ( ξ ) cos ξ x - U 0 ( ξ ) sin ξ x ] d ξ = j f ( x ) .
1 2 π 0 U ( ξ ) exp ( j ξ x ) d ξ = 1 2 π 0 [ U e ( ξ ) + j U 0 ( ξ ) ] [ cos ξ x + j sin ξ x ] d ξ = 1 2 π 0 { [ U e ( ξ ) cos ξ x - U 0 ( ξ ) sin ξ x ] + j [ U e ( ξ ) sin ξ x + U 0 ( ξ ) cos ξ x ] } d ξ = 1 2 f ( x ) - j 2 f Q ( x ) .
1 2 π U ( ξ ) j ξ x d ξ
1 2 π - ξ 1 + ξ 2 U ( ξ ) exp ( j ξ x ) d ξ = 1 2 π - ξ 1 + ξ 1 U ( ξ ) exp ( j ξ x ) d ξ + 1 2 π + ξ 1 + ξ 2 U ( ξ ) exp ( j ξ x ) d ξ = 1 π 0 ξ 1 [ U e ( ξ ) cos ξ x - U 0 ( ξ ) sin ξ x ] d ξ + 1 2 π ξ 1 ξ 2 [ U e ( ξ ) cos ξ x - U 0 ( ξ ) sin ξ x ] d ξ + j 2 π ξ 1 ξ 2 [ U e ( ξ ) sin ξ x + U 0 ( ξ ) cos ξ x ] d ξ .
f Q ( x ) = - 1 π - + f ( t ) x - t d t
f ( x ) = 1 π - + f Q ( t ) x - t d t ,
U ( ξ , η ) = - + f ( x , y ) exp [ - j ( ξ x + η y ) ] d x d y = U s + j U a .
f ( x , y ) = 1 4 π 2 - + U ( ξ , η ) exp [ j ( ξ x + η y ) ] d ξ d η = 1 4 π 2 - + ( U s + j U a ) [ cos ( ξ x + η y ) + j sin ( ξ x + η y ) ] d ξ d η = 1 2 π 2 half-plane [ U s cos ( ξ x + η y ) - U a sin ( ξ x + η y ) ] d ξ d η .
f Q ( x , y ) = 1 2 π 2 half-plane { U s cos ( ξ x + η y + π 2 ) - U a sin ( ξ x + η y + π 2 ) } d ξ d η = - 1 2 π 2 half-plane [ U s sin ( ξ x + η y ) + U a cos ( ξ x + η y ) ] d ξ d η .
- + [ f Q ( x , y ) ] 2 d x d y = - + [ f ( x , y ) ] 2 d x d y = 1 4 π 2 - + ( U s 2 + U a 2 ) d ξ d η .