Abstract

Combined effects of amplitude and phase variations on the irradiance in the image of a sinusoidal complex object are investigated for a partially coherent optical system. For this purpose, the general expression for the irradiance in the image of a sinusoidal complex object that has amplitude transmittance (C + A cos2πx0u) and phase distribution of the form exp(B cos2πx0u) is formulated for a partially coherent optical system. The image irradiance obtained in such an optical system consists of the fundamental frequency and a number of harmonics produced by the nonlinear modulation of the object. The characteristics of the image are assessed in terms of the image contrast, which is defined as the ratio of the coefficient of a harmonic component to the background. By use of the general formula, the image contrast of the fundamental and harmonic components up to sixth order can be calculated for the partially coherent optical system illuminated with bounded and annular illuminations. The typical nonlinear effect occurs in the low-spatial-frequency region. Especially, maxima of the contrast curves occur in the low-frequency region; they may be caused by nonlinearity of the system and effects of phase variations. Analysis and computation showed that, in image formation in a partially coherent optical system, phase variation in the object is extremely influential for the appearance of the image and its contrast.

© 1975 Optical Society of America

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References

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  1. T. Suzuki, Tech. Rep. Osaka Univ. 12, 61 (1962).
  2. R. J. Becherer and G. B. Parrent, J. Opt. Soc. Am. 57, 1479 (1967).
    [Crossref]
  3. R. E. Swing and J. R. Clay, J. Opt. Soc. Am. 57, 1180 (1967).
    [Crossref]
  4. M. De and S. C. Som, J. Opt. Soc. Am. 53, 779 (1963).
    [Crossref]
  5. H. H. Hopkins, Proc. R. Soc. A217, 408 (1953).
  6. G. N. Watson, Theory of Bessel Functions (Cambridge U.P., London, 1922).
  7. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 530.
  8. E. H. Linfoot, Recent Advances in Optics (Oxford U. P., London, 1955), p. 120.

1967 (2)

1963 (1)

1962 (1)

T. Suzuki, Tech. Rep. Osaka Univ. 12, 61 (1962).

1953 (1)

H. H. Hopkins, Proc. R. Soc. A217, 408 (1953).

Becherer, R. J.

Born, M.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 530.

Clay, J. R.

De, M.

Hopkins, H. H.

H. H. Hopkins, Proc. R. Soc. A217, 408 (1953).

Linfoot, E. H.

E. H. Linfoot, Recent Advances in Optics (Oxford U. P., London, 1955), p. 120.

Parrent, G. B.

Som, S. C.

Suzuki, T.

T. Suzuki, Tech. Rep. Osaka Univ. 12, 61 (1962).

Swing, R. E.

Watson, G. N.

G. N. Watson, Theory of Bessel Functions (Cambridge U.P., London, 1922).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 530.

J. Opt. Soc. Am. (3)

Proc. R. Soc. (1)

H. H. Hopkins, Proc. R. Soc. A217, 408 (1953).

Tech. Rep. Osaka Univ. (1)

T. Suzuki, Tech. Rep. Osaka Univ. 12, 61 (1962).

Other (3)

G. N. Watson, Theory of Bessel Functions (Cambridge U.P., London, 1922).

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), p. 530.

E. H. Linfoot, Recent Advances in Optics (Oxford U. P., London, 1955), p. 120.

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Figures (14)

FIG. 1
FIG. 1

Arrangement of the image-forming system.

FIG. 2
FIG. 2

Object model. Amplitude transmittance is C + A cos2πx0u, and exp[iB cos2πx0u] is the phase distribution.

FIG. 3
FIG. 3

Curves of Bessel function Jn(B) of the first kind and order n.

FIG. 4
FIG. 4

Image contrast for the fundamental and harmonic components in the image of the sinusoidal complex object illuminated with the bounded effective source. Object parameters: A = C = 1 and B =2π.

FIG. 5
FIG. 5

Same as Fig. 4. Object parameters: A = 1/2, C = 1, and B = 2π.

FIG. 6
FIG. 6

Same as Fig. 4. Object parameters: A = C = 1 and B = 4π.

FIG. 7
FIG. 7

Same as Fig. 4. Object parameters: A =1/2, C = 1, and B =3.6π.

FIG. 8
FIG. 8

Image contrast for the fundamental and second harmonics in the image of the amplitude object whose parameters are A = C = 1 and B = 0.

FIG. 9
FIG. 9

Effects of the phase variations on the bias term and the contrast of the fundamental-frequency component in the image of the sinusoidal complex object illuminated with the bounded effective source. (a) Bias term and (b) fundamental component.

FIG. 10
FIG. 10

Same as Fig. 9 but showing the effects on the contrasts of harmonics. (a) Second harmonics, (b) third harmonics, (c) fourth harmonics, (d) fifth harmonics, (e) sixth harmonics. In (c), (d), and (e), the scale factors of the vertical axis are doubled.

FIG. 11
FIG. 11

Effects of the variation of the source extension on the image contrast for fundamental and harmonic components. R designates the ratio of the extension of the bounded effective source to that of the aperture of the objective.

FIG. 12
FIG. 12

Effects of the phase variations on the contrast of harmonics in the image of the sinusoidal complex object illuminated with the annular effective source. Contrast for (a) fundamental component, (b) second harmonics, and (c) third harmonics.

FIG. 13
FIG. 13

Same as Fig. 12 but for (a) fourth harmonics, (b) fifth harmonics, and (c) sixth harmonics.

FIG. 14
FIG. 14

Contrast curves for the fundamental-frequency component with phase variations in a transparent object with the form exp[iB cos2πx0u] through a phase-contrast microscope. The phase parameter is changed from π/8 to π.

Equations (52)

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I ( u , v ) = γ ( x , y ) ϕ ( x , y ) 2 d x d y ,
ϕ ( x , y ) = o ( s - x , t - y ) f ( s , t ) e 2 π i ( u s + v t ) d s d t ,
u = ( n sin α ) ξ λ , u = ( n sin α ) ξ λ , v = ( n sin α ) η λ , v = ( n sin α ) η λ and             x = ( n sin α ) μ λ y = ( n sin α ) ν λ
O ( u ) = ( C + A cos 2 π x 0 u ) e i B cos 2 π x 0 u ,
I ( u ) = ( C 2 + 1 2 A 2 ) + 2 A C cos 2 π x 0 u + 1 2 A 2 cos 4 π x 0 u .
o ( x ) = O ( u ) e - 2 π i x u d u .
e i B cos 2 π x 0 u = n = - i n J n ( B ) e 2 π i n x 0 u ,
o ( x ) = n = - i n J n ( B ) [ C δ ( x - n x 0 ) + ( A / 2 ) δ { x - ( n + 1 ) x 0 } + ( 1 2 A ) δ { x - ( n - 1 ) x 0 } ] ,
ϕ ( x ) = n = - i n J n ( B ) e 2 π i u ( x + n x 0 ) × [ C f ( x + x 0 ) + ( A / 2 ) f { x + ( n + 1 ) x 0 } e 2 π i x 0 u + ( A / 2 ) f { x + ( n - 1 ) x 0 } e - 2 π i x 0 u ] .
ϕ * ( x ) = m = - i - m J m ( B ) e - 2 π i u ( x + m x 0 ) × [ C f * ( x + x 0 ) + ( A / 2 ) f * { x + ( m + 1 ) x 0 } e - 2 π i x 0 u + ( A / 2 ) f * { x + ( m - 1 ) x 0 } e 2 π i x 0 u ] .
I ( u ) = n = - m = - i n - m J n ( B ) J m ( B ) e 2 π i ( n - m ) x 0 u × [ C 2 γ ( x ) f ( x + n x 0 ) f * ( x + m x 0 ) d x + ( C A / 2 ) γ ( x ) f { x + ( n + 1 ) x 0 } f * ( x + m x 0 ) e 2 π i x 0 u d x + ( C A / 2 ) γ ( x ) f { x + ( n - 1 ) x 0 } f * ( x + x m 0 ) e - 2 π i x 0 u d x + ( C A / 2 ) γ ( x ) f ( x + n x 0 ) f * { x + ( m - 1 ) x 0 } e 2 π i x 0 u d x + ( C A / 2 ) γ ( x ) f ( x + n x 0 ) f * { x + ( m + 1 ) x 0 } e - 2 π i x 0 u d x + ( A / 2 ) 2 γ ( x ) f ( x + ( n + 1 ) x 0 } f * { x + ( m + 1 ) x 0 } d x + ( A / 2 ) 2 γ ( x ) f { x + ( n - 1 ) x 0 } f * { x + ( m - 1 ) x 0 } d x + ( A / 2 ) 2 γ ( x ) f { x + ( n - 1 ) x 0 } f * { x + ( m + 1 ) x 0 } e - 2 π i ( 2 x 0 ) u d x + ( A / 2 ) 2 γ ( x ) f { x + ( n + 1 ) x 0 } f * { x + ( m - 1 ) x 0 } e 2 π i ( 2 x 0 ) u d x ] ,
T ( n , n - k ) = r ( x ) f ( x + n x 0 ) f * { x + ( n - k ) x 0 } d x = T r ( n , n - k ) + i T i ( n , n - k ) ,
I ( u ) = k = - I k ( u ) = I 0 ( u ) + { I 1 ( u ) + I - 1 ( u ) } + { I 2 ( u ) + I - 2 ( u ) } + ,
I k ( u ) = i k e 2 π i k x 0 u n = - J n ( B ) J n - k ( B ) [ C 2 T ( n , n - k ) + ( A / 2 ) 2 T ( n + 1 , n - k + 1 ) + ( A / 2 ) 2 T ( n - 1 , n - k - 1 ) + ( C A / 2 ) e 2 π i x 0 u { T ( n + 1 , n - k ) + T ( n , n - k - 1 ) } + ( C A / 2 ) e - 2 π i x 0 u { T ( n - 1 , n - k ) + T ( n , n - k + 1 ) } × ( A / 2 ) 2 e - 2 π i ( 2 x 0 ) u T ( n - 1 , n - k + 1 ) + e 2 π i ( 2 x 0 ) u T ( n + 1 , n - k - 1 ) ] .
I 0 ( u ) = n = - J n 2 ( B ) [ C 2 T ( n , n ) + ( A / 2 ) 2 T ( n + 1 , n + 1 ) + ( A / 2 ) 2 T ( n - 1 , n - 1 ) + C A cos 2 π x 0 u { T r ( n + 1 , n ) + T r ( n , n - 1 ) } + ( A 2 / 2 ) cos 2 π ( 2 x 0 ) u T r ( n + 1 , n - 1 ) - C A sin 2 π x 0 u { T i ( n + 1 , n ) + T i ( n , n - 1 ) } - ( A 2 / 2 ) sin 2 π ( 2 x 0 ) u T i ( n + 1 , n - 1 ) ] ,
n = - i - k J n ( B ) J n + k ( B ) T ( n + a , n + b ) = n = - i - k J n ( B ) J n - k ( B ) T ( n - k + a , n - k + b ) ,
I k ( u ) + I - k ( u ) = n = - J n ( B ) J n - k ( B ) × [ cos ( 2 π k x 0 u + k π / 2 ) { 2 C 2 T r ( n , n - k ) + ( A 2 / 2 ) T r ( n + 1 , n - k + 1 ) + ( A 2 / 2 ) T r ( n - 1 , n - k - 1 ) } + C A cos { 2 π ( k + 1 ) x 0 u + k π / 2 } { T r ( n + 1 , n - k ) + T r ( n , n - k - 1 ) } + C A cos { 2 π ( k - 1 ) x 0 u + k π / 2 } × { T r ( n - 1 , n - k ) + T r ( n , n - k + 1 ) } + ( A 2 / 2 ) cos { 2 π ( k - 2 ) x 0 u + k π / 2 } T r ( n - 1 , n - k + 1 ) + ( A 2 / 2 ) cos { 2 π ( k + 2 ) x 0 u + k π / 2 } T r ( n + 1 , n - k - 1 ) - sin ( 2 π k x 0 u + k π / 2 ) × { 2 C 2 T i ( n , n - k ) + ( A 2 / 2 ) T i ( n + 1 , n - k + 1 ) + ( A 2 / 2 ) T i ( n - 1 , n - k - 1 ) } - C A sin { 2 π ( k + 1 ) x 0 u + k π / 2 } { T i ( n + 1 , n - k ) + T i ( n , n - k - 1 ) } - C A sin { 2 π ( k - 1 ) x 0 u + k π / 2 } × { T i ( n - 1 , n - k ) + T i ( n , n - k + 1 ) } - ( A 2 / 2 ) sin { 2 π ( k - 2 ) x 0 u + k π / 2 } T i ( n - 1 , n - k + 1 ) - ( A 2 / 2 ) sin { 2 π ( k + 2 ) x 0 u + k π / 2 } T i ( n + 1 , n - k - 1 ) ] .
I ( u ) = k = 0 [ D k cos 2 π k x 0 u + D k sin 2 π k x 0 u ] = k = 0 E k cos ( 2 π k x 0 u - θ k ) ,
E k = ( D k 2 + D k 2 ) 1 / 2 , θ k = tan - 1 ( D k / D k ) ,             k = 1 , 2 , 3 ,
n = - J n ( B ) J n - k ( B ) T ( n , n - k ) = n = - J n + p ( B ) J n + p - k ( B ) T ( n + p , n + p - k ) ,
D 0 = n = - P 0 n T ( n , n ) , D 2 k - 1 D 2 k - 1 = n = - { P ( 2 k - 1 ) n Q ( 2 k - 1 ) n T r ( n + k - 1 , n - k ) ± Q ( 2 k - 1 ) n P ( 2 k - 1 ) n T i ( n + k - 1 , n - k ) } , ( for odd order ) D 2 k D 2 k = n = - { P 2 k n Q 2 k n T r ( n + k , n - k ) ± Q 2 k n P 2 k n T i ( n + k , n - k ) } , ( for even order ) k = 1 , 2 , 3 , in which P 0 n = C 2 J n 2 ( B ) + ( A 2 / 4 ) { J n - 1 2 ( B ) + J n + 1 2 ( B ) - 2 J n + 1 ( B ) J n - 1 ( B ) } , P 1 n = C A { J n - 1 2 ( B ) + J n 2 ( B ) - J n + 1 ( B ) J n - 1 ( B ) - J n ( B ) J n - 2 ( B ) } ,
P 2 n = - 2 C 2 J n + 1 ( B ) J n - 1 ( B ) + ( A 2 / 2 ) { J n 2 ( B ) - J n ( B ) J n - 2 ( B ) - J n + 2 ( B ) J n ( B ) + J n + 2 ( B ) J n - 2 ( B ) } , P 3 n = C A { - J n ( B ) J n - 2 ( B ) - J n + 1 ( B ) J n - 1 ( B ) + J n + 2 ( B ) J n - 2 ( B ) + J n + 1 ( B ) J n - 3 ( B ) } , P 4 n = 2 C 2 J n + 2 ( B ) J n - 2 ( B ) + ( A 2 / 2 ) { - J n + 1 ( B ) J n - 1 ( B ) + J n + 1 ( B ) J n - 3 ( B ) + J n + 3 ( B ) J n - 1 ( B ) - J n + 3 ( B ) J n - 3 ( B ) } , P 5 n = C A { J n + 1 ( B ) J n - 3 ( B ) + J n + 2 ( B ) J n - 2 ( B ) - J n + 3 ( B ) J n - 3 ( B ) - J n + 2 ( B ) J n - 4 ( B ) } , P 6 n = - 2 C 2 J n + 3 ( B ) J n - 3 ( B ) + ( A 2 / 2 ) { J n + 2 ( B ) J n - 2 ( B ) - J n + 2 ( B ) J n - 4 ( B ) - J n + 4 ( B ) J n - 2 ( B ) + J n + 4 ( B ) J n - 4 ( B ) } , Q 1 n = - 2 C 2 J n ( B ) J n - 1 ( B ) + ( A 2 / 2 ) { J ( B ) J n - 1 ( B ) - J n - 1 ( B ) J n - 2 ( B ) - J n + 1 ( B ) J n ( B ) + J n + 1 ( B ) J n - 2 ( B ) } , Q 2 n = C A { - J n ( B ) J n - 1 ( B ) - J n + 1 ( B ) J n ( B ) + J n + 2 ( B ) J n - 1 ( B ) + J n + 1 ( B ) J n - 2 ( B ) } , Q 3 n = 2 C 2 J n + 1 ( B ) J n - 2 ( B ) + ( A 2 / 2 ) { - J n ( B ) J n - 1 ( B ) + J n ( B ) J n - 3 ( B ) + J n + 2 ( B ) J n - 1 ( B ) - J n + 2 ( B ) J n - 3 ( B ) } , Q 4 n = C A { J n + 1 ( B ) J n - 2 ( B ) + J n + 2 ( B ) J n - 1 ( B ) - J n + 3 ( B ) J n - 2 ( B ) - J n + 2 ( B ) J n - 3 ( B ) } , Q 5 n = - 2 C 2 J n + 2 ( B ) J n - 3 ( B ) + ( A 2 / 2 ) { J n + 1 ( B ) J n - 2 ( B ) - J n + 1 ( B ) J n - 4 ( B ) - J n + 3 ( B ) J n - 2 ( B ) + J n + 3 ( B ) J n - 4 ( B ) } , Q 6 n = C A { - J n + 2 ( B ) J n - 3 ( B ) - J n + 3 ( B ) J n - 2 ( B ) + J n + 4 ( B ) J n - 3 ( B ) + J n + 3 ( B ) J n - 4 ( B ) } .
γ ( x ) = γ ( - x ) ,             f ( x ) = f ( - x ) ;
T r ( n , n - k ) = T r ( k - n , - n ) , T i ( n , n - k ) = - T i ( k - n , - n ) .
I ( u ) = k = 0 D k cos 2 π k x 0 u .
C T k = D k / D 0 .
A T 1 = C 2 + A 2 / 2 2 A C C T 1 ,             A T 2 = 2 ( C 2 + A 2 / 2 ) A 2 C T 2 .
T ( n , m ) = f ( x + n x 0 ) f * ( x + m x 0 ) d x ,
T ( n , n ) = T ( 0 , 0 ) .
n = - J n ( B ) J n - k ( B ) = J k ( 0 ) = 0 , for k 0 , n = - J n 2 ( B ) = 1 , for k = 0 ,
D 0 = ( C 2 + 1 2 A 2 ) T ( 0 , 0 ) ,
D 1 = 2 C A T r ( 0 , - 1 ) ,
D 2 = A 2 T r ( 1 , - 1 ) / 2 ,
D k = 0 ,             for k 3.
A T 1 = T r ( 0 , - 1 ) T ( 0 , 0 ) ,             A T 2 = T r ( 1 , - 1 ) T ( 0 , 0 ) .
J n 2 ( 0 ) = { 1 for n = 0 , 0 for n 0 , J n ( 0 ) J n - k ( 0 ) = 0             for all n ( k 0 ) ,
D 0 = C 2 T ( 0 , 0 ) + A 2 T ( 1 , 1 ) / 2 , D 1 = 2 C A T r ( 1 , 0 ) , D 2 = A 2 T r ( 1 , - 1 ) / 2 , D k = 0 ,             for k 3.
C T 1 = 2 A C T r ( 1 , 0 ) C 2 T ( 0 , 0 ) + A 2 T ( 1 , 1 ) / 2 ,
C T 2 = A 2 T r ( 1 , - 1 ) / 2 C 2 T ( 0 , 0 ) + A 2 T ( 1 , 1 ) / 2 .
0 ( u ) = cos 2 π x 0 u .
I ( u ) = O ( u ) 2 = ( 1 + cos 4 π x 0 u ) / 2.
D 0 = T ( 1 , 1 ) / 2 , D 1 = 0 , D 2 = T r ( 1 , - 1 ) / 2 , D k = 0 ,             for k 3.
A T 2 = C T 2 = T r ( 1 - 1 ) T ( 1 , 1 ) .
O ( u ) = e i B cos 2 π x 0 u .
C T 2 k - 1 = ( - 1 ) k 2 n = - T i ( n + k - 1 , n - k ) J n + k - 1 ( B ) J n - k ( B ) n = - T ( n , n ) J n 2 ( B ) ,
C T 2 k = ( - 1 ) k k n = - T r ( n + k , n - k ) J n + k ( B ) J n - k ( B ) n = - T ( n , n ) J n 2 ( B ) ,
C T 0 = 1 ,             C T k = 0             for k 0.
f ( x ) = 1 , if x 1 , = 0 , otherwise ,
γ ( x ) = 1 , if x R , = 0 , otherwise ,
γ ( x ) = 1 , s 1 x s 2 , = 0 , elsewhere ,
f ( x ) = t e i π / 2 ,             s 1 x s 2 ,
= 1 ,             elsewhere ,