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Illumination with a moving light source
Volume 31, number 2
OPTICS COMMUNICATIONS
November 1979
ILLUMINATION WITH A MOVING LIGHT SOURCE Kazuyoshi ITOH and Yoshihiro OHTSUKA
Department of Engineering Science, Hokkaido University, Sapporo 060, Japan Received 3 May 1979 Revised manuscript received 7 September 1979
Image formation is discussed under the condition that an object is illuminated with a moving light source and a timeaveraged image is observed. An optical system with such illumination works as if the object were illuminated with spatially, partially coherent light from a certain fixed source.
1. Introduction
Many attempts [1-8] have been made so far to produce quasi-incoherent secondary sources using coherent light beams. Among these attempts, the methods which involve deflection or shifting of the coherent light beams [2-8] are of interest. The principle of these methods was briefly mentioned by McKechnie [1 ]. It was pointed out that one [7] of the methods acts so as to transform the coherent light beam into a quasi-monochromatic incoherent ring source. Although this ring source is conceptual, illumination with such a moving light source might be equivalent to spatially, partially coherent illumination in a practical sense. As far as theoretical aspects are concerned, however, no exact explanation seems to be given yet. The purpose of this communication is to present the equivalent of spatial coherence condition of the moving source illumination.
2. Effective mutual intensity We shall be concerned with a time-averaged image of an object illuminated with a moving light source. It has been shown by Welford [9] that if a pupil of a coherently illuminated optical system varies in time, the time-averaged image is equivalent to the image of a system with a certain fixed pupil and partially coherent illumination. Although the practical situation to be considered here is different from such an optical system, his basic concept is virtually applicable to the present analysis. The purpose of this section is to more generalize the treatment in association with moving source illumination. The optical system under consideration is schematically shown in fig. 1. Consider first a fixed aperture A illuminated with a quasi-monochromatic primary source S. This aperture limits the area of a secondary light source. An object situated in a plane B is projected on to an image plane C. All thepoints in the aperture, object, and image planes are designated, respectively, by two-dimenisonal vectorial coordinates P, Q and R. Let Kn(P , Q) and Km(Q,R ) be the amplitude point spread functions or transmission functions between the respective points and let J(P1 ,P2) be the mutual intensity at two points P1 and P2 in the aperture plane. I f F ( Q ) stands for the complex transmittance of the object, the intensity in the image plane is given by [10]
I(R ) =
ffs(Q 1, Q2) F(Q 1) F *(Q2) Km (Q 1, R ) K* (Q2, R) dQ 1dQ2,
(1)
BB
119
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OPTICS COMMUNICATIONS
:~
~.,~
'i
~ "~'
~.::~1
A
k
November 1979
Image Plane
J,~l,-,
KI(P.O) ~ U 2 . V q~ffJ a KmCO.n)L.~f C
Fig. 1. Schematicillustration of the optical system.
where J(Q1, Q2) is the mutual intensity at two points Q1 and Q2, and is also related to J(P1, P2) as
J(Q1, Q2) = f f J ( P 1 ,P2)Kn(P1, Q1)K*n(P2, Q2) dP1 dP2AA
(2)
Here, asterisks denote the complex conjugate. Referring to fig. 1, we next consider transversal translation of a combined system of the primary source and the aperture, while the illuminating and image-forming systems are fixed. Under these circumstances, the image quality possibly changes, since the coherence condition of illumination might be affected by the translation. On the other hand, the same effects in the image will also take place if the illuminating and image-forming systems are conversely translated against the fixed source-aperture system. For the sake of simple treatment, the analysis will be made for the latter case. It should be kept in mind that the former method of the moving source illumination is proposed for the practical usage. Let the image intensity be averaged over a time interval ( - T , T). This time interval is divided into a set o f N short intervals (ti, t i + r) (i = 1,2, ...,N), where r = ti+ 1 - t i ~ 2T. The optical disturbances across the aperture and the image planes can be expressed by V(P, t) = U(P, t) exp (-2rrj~t) and V(R, t)= U(R, t) exp (-21rj~t), respectively, where U denotes a slowly varying envelope function [11 ] and ~ is the mid-frequency of light from the quasi-monochromatic primary source. Here, U(P, t) is a random variable and accordingly U(R, t) is also a random variable. First of all, an optical arrangement where the source and aperture system is fixed but shifted transversally by a constant vector P0 is considered. Suppose the image be integrated over the only ith short interval of ~ime (t i, ti+ r). With the help of the slowly varying envelope functions [10,i2], the time-integrated image over the short interval is then given by
ti+r
t
,-ti+r
fti v(n,')u'(R,Odt=fflffLf BB (AA t i X K n (P1 + P0, Q 1) K*(P2 + P0, Q2 ) dP1 dP2 } F(Q 1) F*(Q2 ) Km (Q 1, R) K* (Q2, R ) dQ 1 OQ2 ,
(3)
where A 1 and A 2 denote the transit times of light from P1 to R and P2 to R, respectively, which depend on t h e variables P1, Q1 and P2, Q2, However, the time difference IA1 --A2[ is negligible if it is much shorter than the coherence time. Then we have A 1 = A 2 = A. A time-dependent translation can be discussed now by puttingP 0 =P(t). We shall confine our attention to a sufficiently slow motion of the source system. Then, K n [P1 + P(t), Q1] and K n [/°2 + P(t), Q2] become approximately time-independent for every short interval (ti, t i + T) ( i - 1,2, ..., N). By replacing Kn(P 1 + PO, Q1) and K h (P2 + P0, Q2) in eq. (3) with K n [P1 +P(ti), Q1] and K n [P2 +P(ti), Q2], we readily obtain the fractional image imegrated over the 120
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OPTICS COMMUNICATIONS
November 1979
ith short interval. Summation of all the fractional images then forms the image integrated over the whole observation interval ( - T , T). Thus, we have (2T) -1
f, u(R, t) U*(R, t) dt = (2T) -1 ~-
-T
ff
i=1
X gn [el +
/ss[y AA ti--A
U(P1, t) U*(P 2 , t) dt
e(ti), all K* [/'2 +P(ti), 02] de1 dP2 } F(Q1)F*(Q2)Km( a l ,
]
R) K* (Q2, R) dOl dO2 .
(4)
!
Since U(P,t) follows a random process, the recorded image given by eq. (4) is also a random variable. The most important quantity is the expectation or ensemble average of the recorded images. Under a reasonable assumption that the optical disturbances across the aperture is stationary and ergodic, the infinite time average is equal to the ensemble average. That is,
(5)
(U(P1, t)U*(P 2, t)) = J(el, P2),
where ( ) means the ensemble average. It should be remembered that J(P1, P2) has been defined as a quantity of an infinite time average [10]. Then the ensemble average of the recorded images can be obtained from eqs. (4) and
(5): I-T,T(R) =f f JE(Q1, Q2)F(Q1)F*(Q2)Km(Q1,R)K* (Q2,R)dQ1
dQ2 ,
(6)
BB
where
JE(Q1,02) = ffJ(el,e2) AA
{(2T) -1
f T gn [P1 + P(t), Q1]K*n[/°2+ P(t), Q 2 ] d t } dP 1 dP 2 .
(7)
-T
Note that the summation in eq. (4) has been replaced by an integration from the assumption that r < 2T. On colaparison of eqs. (6) and (7) with eqs. (1) and (2), we see that JE(Q1, Q2) plays the same role as J(Q1, Q2) in the conventional fixed source system. We now imagine two identical objects placed in front of identical imaging systems: one is illuminated with a moving source, and the other is illuminated with a fixed source. If JE(Q 1 , Q2) is made equal to J(QI' Q2), it is evident from eqs. (1) and (6) that the same images are obtained from the two systems. In this sense, JE(Q1, Q2) may be called the effective mutual intensity. In practice, however, we can observe only the recorded images given by eq. (4). The variation of the recorded images would be closely related to the statistical characteristics of the optical disturbances across the aperture and also to the observation time 2T. It is worthwhile to describe briefly this dependence for a special case. When a primary source of coherence time r c satisfies the condition of r c ,~ r with r < 2T, the stationarity for the optical disturbances across the aperture allows us to put
ti+r-A r -1 f U(PI,t)U*(P2, t)dt~-J(P1,P2). ti- A
(8)
Since substitution of eq. (8) into eq. (4) does not produce any appreciable error, the recorded images will not differ appreciably from one another. For such a primary source, it is concluded that the variation mentioned above is almost negligible. In other words, one sample of the recorded images would give almost the same intensity distribution as the ensemble averages. As is evident, perfectly monochromatic light sources also yield the same results.
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3. Time-averaged secondary light source The expression for the effective mutual intensity defined in the preceding section essentially includes an integration of a finite time interval. According to McKechnie [1 ], however, the locus of the moving light source can be regarded as an incoherent extended source in a practical sense. His interpretation will be found to be valid for a certain type of moving source systems. We shall consider a point-like source, which can be approximated by Dirac 6 function, centered on a moving point P(t) in the aperture plane. Let non-zero velocity of the moving point be S(P). Then, coordinate transformation results in the alternative expression for the effective mutual intensity: JE(Q1, Q2)
=L-1SfloIS(P) I-1Kn(P, Q1) K*(P, Q2)ds
(9)
,
c where I 0 = J(P1, P2), L is defined as the length of the path C along which the point source moves within the interval the path element is given by ds = IS(P)I dt. On the other hand, the mutual intensity at Q1 and Q2 for a spatially incoherent source o situated in front of the same optical system is given by [10]
(-T, T), S =L/(2T), and
J(Q1, Q2) = f I(P)Kn(P, Q1)K* (P, Q2)OP,
(10)
o
where I(P) is the intensity distribution of o. From the comparison of eq. (9) with eq. (10), it is understood that the locus C delineated by the moving point source acts as if it were an incoherent curved line source whose intensity distribution is given by I(P) =IoL-1SIS(P)I-1. The locus, including the intensity distribution, may be called the timeaveraged secondary light source. K6hler's method would be recommended for uniform illumination with a simple locus such as a straight line. Until now, a point-like source has been considered, but the fact whether the source is point-like or not is not essential. Let us now suppose a special type of the transmission function which satisfies the condition,
Kn(P+PO, Q) = gn(e , a)kn(eo, a),
(11)
where P and P0 represent the points in the aperture plane. This assumption holds for some of optical configurations for Fourier transform where the vignetting effect is negligible [13]. When such an illuminating system is employed, eq. (7) becomes JE(Q1, Q2) = Jo(Q1,
Q2)JE,k(Q1, Q2),
(12)
where J0(Q1 ' Q2) =
ffJ(P1,
P2)
Kn(PI' Q1) K*n(P2,Q2) dP1 dP2
'
(13)
AA T
JE,k(Q1, Q2) = (2T) -1
f ~. [e(t), Q1] k*[P(t), Q2]d t ,
(14)
-T or
JE,k(Q1, Q2)
=L-IS
f ls(P)l-lg,(e, Q1)k*n(P, Q2) d s .
(15)
c Since eq. (13) is identical with eq. (2), Jo(Q1, Q2) is the mutual intensity under the fixed central illumination. It 122
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November 1979
is worth noting that the effective mutual intensity in the plane B is degraded by JE,k(Q1, Q2) even if the secondary source may be spatially coherent across the aperture plane. When a focussed spot of a spatially coherent beam is scanning the aperture plane, it is not necessary to consider the profile of the spot. Only the path delineated in the aperture plane should be taken into account. In an actual recording system, the expectation of the finite time average given by eq. (6) is usually obtained, so that the effective mutual intensity will determine the image quality. In this sense, the present method has the advantage that the image quality can be simply controlled by taking different paths of the moving source. From eqs. (12), (14) and (6), we have
I_T,T(R)=ffJo(Q1,Q2)F(Q1)F*(Q2)
[
f
(2T) -1
BB
T
Kw(Q1,R,t)K~(Q2,R,t)dt
] dQ1 dQ2 ,
(16)
-T
where Kw(Q, R, t) = kn [P(t), Q] Km (Q, R). It is interesting that the same expression as is derived by Welford [9] comes out ifJo(Q1, Q2) = const and Jo(Q1, Q2)F(Q1)F*(Q2) = U(Q1)U*(Q2) are substituted in this formula. In this case, the moving source is equivalent to the moving pupil.
4. Measurable effective mutual intensity From a practical point of view, it is desirable to represent the effective mutual intensity in terms of the optical disturbances across the object plane. To achieve this, let U(Q, t) denote the optical disturbances across the object plane. Then, the expectation of the time-averaged image intensity over the interval (-T, T) is obtained similarly to eq. (6) and given by
}
I-T.T(R)=f(2T)-aj,
_TO(Q1 , t)V*(Q2, t)dt F(Q1)F(Q1)Km(Q1,R)K*(Q2,R)dQ1
dQ2 ,
(17)
BB
where the transit times of light from Q1 and Q2 to R are neglected. This is another expression of eq. (6). Since eqs. (6) and (17) represent the identical image for any object or transmission function, the following identity must hold: JE(Q1, Q2) = {(\-2T)- 1
f
U(Q1, t) U*(Q2, t) d
.
(18)
-T Therefore, the effective mutual intensity can be estimated from the observed mutual intensity in the object plane within a finite interval of time. However, it is important to synchronize the observation with the motion of the source, since the optical disturbances across the object plane are no longer stationary due to the motion of the source. When the motion is periodic and the observation interval is much longer than the period, this requirement becomes less significant.
5. Summary
It has been shown that the concept of the effective mutual intensity can be used in order to specify the effect of the moving source illumination. Since the effective mutual intensity has the same role as the usual mutual intensity, the moving source system can be regarded as a conventional system illuminated with spatially, partially coherent light from a certain fixed source. Therefore, any method of image analysis or technique of image processing available for the conventional partially coherent systems may also be applicable to the moving source systems. Experimental verification is now under study and the results will be presented in the near future. 123
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Acknowledgements The authors are grateful to Mr. T. Saimi at Matsushita Electric Industrial Co. Ltd. and Dr. S. Yokozeki at Osaka University for their encouragement and significant suggestion to see refs. [ 4 - 6 ] at the beginning o f this work.
References [1 ] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11 ] [121 [131
124
T.S. McKechnie, in: Laser speckle and related phenomena, ed. J.C. Dainty (Springer-Verlag, New York, 1975) p. 123. C.E. Thomas, Appl. Optics 7 (1968) 517. M.J. Bowman, Appl. Optics 7 (1968) 2280. Y. Belvaux, S. Lowenthal and T. Saimi, Optics Comm. 5 (1972) 143. E. Mr6z, R. Pawluczyk and M. Pluta, Optica Applicata 1 (1971) 9. R. Pawluczyk and E. Mr6z, Optica Acta 20 (1973) 379. D.J. Cronin and A.E. Su ith, Opt. Engng 12 (1973) 50. B.W. Reuter and O. Wess, in: Proc. ICO Conf. (Madrid, 1978) p. 275. W.T. Welford, Optics Comm. 4 (1971) 275. M. Born and E. Wolf, Principles of optics, 4th Ed. (Pergamon Press, London, 1970) Ch. 10. L. Mandel, in: Progress in optics, Vol. 2, ed. E. Wolf (North-Holland Pub., Amsterdam, 1963) p. 187. F. Zernike, Physica 5 (1938) 785. J.W. Goodman, Introduction to Fourier optics (McGraw-Hill Book, New York, 1968) Ch. 5.
OPTICS COMMUNICATIONS
November 1979
ILLUMINATION WITH A MOVING LIGHT SOURCE Kazuyoshi ITOH and Yoshihiro OHTSUKA
Department of Engineering Science, Hokkaido University, Sapporo 060, Japan Received 3 May 1979 Revised manuscript received 7 September 1979
Image formation is discussed under the condition that an object is illuminated with a moving light source and a timeaveraged image is observed. An optical system with such illumination works as if the object were illuminated with spatially, partially coherent light from a certain fixed source.
1. Introduction
Many attempts [1-8] have been made so far to produce quasi-incoherent secondary sources using coherent light beams. Among these attempts, the methods which involve deflection or shifting of the coherent light beams [2-8] are of interest. The principle of these methods was briefly mentioned by McKechnie [1 ]. It was pointed out that one [7] of the methods acts so as to transform the coherent light beam into a quasi-monochromatic incoherent ring source. Although this ring source is conceptual, illumination with such a moving light source might be equivalent to spatially, partially coherent illumination in a practical sense. As far as theoretical aspects are concerned, however, no exact explanation seems to be given yet. The purpose of this communication is to present the equivalent of spatial coherence condition of the moving source illumination.
2. Effective mutual intensity We shall be concerned with a time-averaged image of an object illuminated with a moving light source. It has been shown by Welford [9] that if a pupil of a coherently illuminated optical system varies in time, the time-averaged image is equivalent to the image of a system with a certain fixed pupil and partially coherent illumination. Although the practical situation to be considered here is different from such an optical system, his basic concept is virtually applicable to the present analysis. The purpose of this section is to more generalize the treatment in association with moving source illumination. The optical system under consideration is schematically shown in fig. 1. Consider first a fixed aperture A illuminated with a quasi-monochromatic primary source S. This aperture limits the area of a secondary light source. An object situated in a plane B is projected on to an image plane C. All thepoints in the aperture, object, and image planes are designated, respectively, by two-dimenisonal vectorial coordinates P, Q and R. Let Kn(P , Q) and Km(Q,R ) be the amplitude point spread functions or transmission functions between the respective points and let J(P1 ,P2) be the mutual intensity at two points P1 and P2 in the aperture plane. I f F ( Q ) stands for the complex transmittance of the object, the intensity in the image plane is given by [10]
I(R ) =
ffs(Q 1, Q2) F(Q 1) F *(Q2) Km (Q 1, R ) K* (Q2, R) dQ 1dQ2,
(1)
BB
119
Volume 31, number 2 s'
OPTICS COMMUNICATIONS
:~
~.,~
'i
~ "~'
~.::~1
A
k
November 1979
Image Plane
J,~l,-,
KI(P.O) ~ U 2 . V q~ffJ a KmCO.n)L.~f C
Fig. 1. Schematicillustration of the optical system.
where J(Q1, Q2) is the mutual intensity at two points Q1 and Q2, and is also related to J(P1, P2) as
J(Q1, Q2) = f f J ( P 1 ,P2)Kn(P1, Q1)K*n(P2, Q2) dP1 dP2AA
(2)
Here, asterisks denote the complex conjugate. Referring to fig. 1, we next consider transversal translation of a combined system of the primary source and the aperture, while the illuminating and image-forming systems are fixed. Under these circumstances, the image quality possibly changes, since the coherence condition of illumination might be affected by the translation. On the other hand, the same effects in the image will also take place if the illuminating and image-forming systems are conversely translated against the fixed source-aperture system. For the sake of simple treatment, the analysis will be made for the latter case. It should be kept in mind that the former method of the moving source illumination is proposed for the practical usage. Let the image intensity be averaged over a time interval ( - T , T). This time interval is divided into a set o f N short intervals (ti, t i + r) (i = 1,2, ...,N), where r = ti+ 1 - t i ~ 2T. The optical disturbances across the aperture and the image planes can be expressed by V(P, t) = U(P, t) exp (-2rrj~t) and V(R, t)= U(R, t) exp (-21rj~t), respectively, where U denotes a slowly varying envelope function [11 ] and ~ is the mid-frequency of light from the quasi-monochromatic primary source. Here, U(P, t) is a random variable and accordingly U(R, t) is also a random variable. First of all, an optical arrangement where the source and aperture system is fixed but shifted transversally by a constant vector P0 is considered. Suppose the image be integrated over the only ith short interval of ~ime (t i, ti+ r). With the help of the slowly varying envelope functions [10,i2], the time-integrated image over the short interval is then given by
ti+r
t
,-ti+r
fti v(n,')u'(R,Odt=fflffLf BB (AA t i X K n (P1 + P0, Q 1) K*(P2 + P0, Q2 ) dP1 dP2 } F(Q 1) F*(Q2 ) Km (Q 1, R) K* (Q2, R ) dQ 1 OQ2 ,
(3)
where A 1 and A 2 denote the transit times of light from P1 to R and P2 to R, respectively, which depend on t h e variables P1, Q1 and P2, Q2, However, the time difference IA1 --A2[ is negligible if it is much shorter than the coherence time. Then we have A 1 = A 2 = A. A time-dependent translation can be discussed now by puttingP 0 =P(t). We shall confine our attention to a sufficiently slow motion of the source system. Then, K n [P1 + P(t), Q1] and K n [/°2 + P(t), Q2] become approximately time-independent for every short interval (ti, t i + T) ( i - 1,2, ..., N). By replacing Kn(P 1 + PO, Q1) and K h (P2 + P0, Q2) in eq. (3) with K n [P1 +P(ti), Q1] and K n [P2 +P(ti), Q2], we readily obtain the fractional image imegrated over the 120
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November 1979
ith short interval. Summation of all the fractional images then forms the image integrated over the whole observation interval ( - T , T). Thus, we have (2T) -1
f, u(R, t) U*(R, t) dt = (2T) -1 ~-
-T
ff
i=1
X gn [el +
/ss[y AA ti--A
U(P1, t) U*(P 2 , t) dt
e(ti), all K* [/'2 +P(ti), 02] de1 dP2 } F(Q1)F*(Q2)Km( a l ,
]
R) K* (Q2, R) dOl dO2 .
(4)
!
Since U(P,t) follows a random process, the recorded image given by eq. (4) is also a random variable. The most important quantity is the expectation or ensemble average of the recorded images. Under a reasonable assumption that the optical disturbances across the aperture is stationary and ergodic, the infinite time average is equal to the ensemble average. That is,
(5)
(U(P1, t)U*(P 2, t)) = J(el, P2),
where ( ) means the ensemble average. It should be remembered that J(P1, P2) has been defined as a quantity of an infinite time average [10]. Then the ensemble average of the recorded images can be obtained from eqs. (4) and
(5): I-T,T(R) =f f JE(Q1, Q2)F(Q1)F*(Q2)Km(Q1,R)K* (Q2,R)dQ1
dQ2 ,
(6)
BB
where
JE(Q1,02) = ffJ(el,e2) AA
{(2T) -1
f T gn [P1 + P(t), Q1]K*n[/°2+ P(t), Q 2 ] d t } dP 1 dP 2 .
(7)
-T
Note that the summation in eq. (4) has been replaced by an integration from the assumption that r < 2T. On colaparison of eqs. (6) and (7) with eqs. (1) and (2), we see that JE(Q1, Q2) plays the same role as J(Q1, Q2) in the conventional fixed source system. We now imagine two identical objects placed in front of identical imaging systems: one is illuminated with a moving source, and the other is illuminated with a fixed source. If JE(Q 1 , Q2) is made equal to J(QI' Q2), it is evident from eqs. (1) and (6) that the same images are obtained from the two systems. In this sense, JE(Q1, Q2) may be called the effective mutual intensity. In practice, however, we can observe only the recorded images given by eq. (4). The variation of the recorded images would be closely related to the statistical characteristics of the optical disturbances across the aperture and also to the observation time 2T. It is worthwhile to describe briefly this dependence for a special case. When a primary source of coherence time r c satisfies the condition of r c ,~ r with r < 2T, the stationarity for the optical disturbances across the aperture allows us to put
ti+r-A r -1 f U(PI,t)U*(P2, t)dt~-J(P1,P2). ti- A
(8)
Since substitution of eq. (8) into eq. (4) does not produce any appreciable error, the recorded images will not differ appreciably from one another. For such a primary source, it is concluded that the variation mentioned above is almost negligible. In other words, one sample of the recorded images would give almost the same intensity distribution as the ensemble averages. As is evident, perfectly monochromatic light sources also yield the same results.
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November 1979
3. Time-averaged secondary light source The expression for the effective mutual intensity defined in the preceding section essentially includes an integration of a finite time interval. According to McKechnie [1 ], however, the locus of the moving light source can be regarded as an incoherent extended source in a practical sense. His interpretation will be found to be valid for a certain type of moving source systems. We shall consider a point-like source, which can be approximated by Dirac 6 function, centered on a moving point P(t) in the aperture plane. Let non-zero velocity of the moving point be S(P). Then, coordinate transformation results in the alternative expression for the effective mutual intensity: JE(Q1, Q2)
=L-1SfloIS(P) I-1Kn(P, Q1) K*(P, Q2)ds
(9)
,
c where I 0 = J(P1, P2), L is defined as the length of the path C along which the point source moves within the interval the path element is given by ds = IS(P)I dt. On the other hand, the mutual intensity at Q1 and Q2 for a spatially incoherent source o situated in front of the same optical system is given by [10]
(-T, T), S =L/(2T), and
J(Q1, Q2) = f I(P)Kn(P, Q1)K* (P, Q2)OP,
(10)
o
where I(P) is the intensity distribution of o. From the comparison of eq. (9) with eq. (10), it is understood that the locus C delineated by the moving point source acts as if it were an incoherent curved line source whose intensity distribution is given by I(P) =IoL-1SIS(P)I-1. The locus, including the intensity distribution, may be called the timeaveraged secondary light source. K6hler's method would be recommended for uniform illumination with a simple locus such as a straight line. Until now, a point-like source has been considered, but the fact whether the source is point-like or not is not essential. Let us now suppose a special type of the transmission function which satisfies the condition,
Kn(P+PO, Q) = gn(e , a)kn(eo, a),
(11)
where P and P0 represent the points in the aperture plane. This assumption holds for some of optical configurations for Fourier transform where the vignetting effect is negligible [13]. When such an illuminating system is employed, eq. (7) becomes JE(Q1, Q2) = Jo(Q1,
Q2)JE,k(Q1, Q2),
(12)
where J0(Q1 ' Q2) =
ffJ(P1,
P2)
Kn(PI' Q1) K*n(P2,Q2) dP1 dP2
'
(13)
AA T
JE,k(Q1, Q2) = (2T) -1
f ~. [e(t), Q1] k*[P(t), Q2]d t ,
(14)
-T or
JE,k(Q1, Q2)
=L-IS
f ls(P)l-lg,(e, Q1)k*n(P, Q2) d s .
(15)
c Since eq. (13) is identical with eq. (2), Jo(Q1, Q2) is the mutual intensity under the fixed central illumination. It 122
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is worth noting that the effective mutual intensity in the plane B is degraded by JE,k(Q1, Q2) even if the secondary source may be spatially coherent across the aperture plane. When a focussed spot of a spatially coherent beam is scanning the aperture plane, it is not necessary to consider the profile of the spot. Only the path delineated in the aperture plane should be taken into account. In an actual recording system, the expectation of the finite time average given by eq. (6) is usually obtained, so that the effective mutual intensity will determine the image quality. In this sense, the present method has the advantage that the image quality can be simply controlled by taking different paths of the moving source. From eqs. (12), (14) and (6), we have
I_T,T(R)=ffJo(Q1,Q2)F(Q1)F*(Q2)
[
f
(2T) -1
BB
T
Kw(Q1,R,t)K~(Q2,R,t)dt
] dQ1 dQ2 ,
(16)
-T
where Kw(Q, R, t) = kn [P(t), Q] Km (Q, R). It is interesting that the same expression as is derived by Welford [9] comes out ifJo(Q1, Q2) = const and Jo(Q1, Q2)F(Q1)F*(Q2) = U(Q1)U*(Q2) are substituted in this formula. In this case, the moving source is equivalent to the moving pupil.
4. Measurable effective mutual intensity From a practical point of view, it is desirable to represent the effective mutual intensity in terms of the optical disturbances across the object plane. To achieve this, let U(Q, t) denote the optical disturbances across the object plane. Then, the expectation of the time-averaged image intensity over the interval (-T, T) is obtained similarly to eq. (6) and given by
}
I-T.T(R)=f(2T)-aj,
_TO(Q1 , t)V*(Q2, t)dt F(Q1)F(Q1)Km(Q1,R)K*(Q2,R)dQ1
dQ2 ,
(17)
BB
where the transit times of light from Q1 and Q2 to R are neglected. This is another expression of eq. (6). Since eqs. (6) and (17) represent the identical image for any object or transmission function, the following identity must hold: JE(Q1, Q2) = {(\-2T)- 1
f
U(Q1, t) U*(Q2, t) d
.
(18)
-T Therefore, the effective mutual intensity can be estimated from the observed mutual intensity in the object plane within a finite interval of time. However, it is important to synchronize the observation with the motion of the source, since the optical disturbances across the object plane are no longer stationary due to the motion of the source. When the motion is periodic and the observation interval is much longer than the period, this requirement becomes less significant.
5. Summary
It has been shown that the concept of the effective mutual intensity can be used in order to specify the effect of the moving source illumination. Since the effective mutual intensity has the same role as the usual mutual intensity, the moving source system can be regarded as a conventional system illuminated with spatially, partially coherent light from a certain fixed source. Therefore, any method of image analysis or technique of image processing available for the conventional partially coherent systems may also be applicable to the moving source systems. Experimental verification is now under study and the results will be presented in the near future. 123
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Acknowledgements The authors are grateful to Mr. T. Saimi at Matsushita Electric Industrial Co. Ltd. and Dr. S. Yokozeki at Osaka University for their encouragement and significant suggestion to see refs. [ 4 - 6 ] at the beginning o f this work.
References [1 ] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11 ] [121 [131
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T.S. McKechnie, in: Laser speckle and related phenomena, ed. J.C. Dainty (Springer-Verlag, New York, 1975) p. 123. C.E. Thomas, Appl. Optics 7 (1968) 517. M.J. Bowman, Appl. Optics 7 (1968) 2280. Y. Belvaux, S. Lowenthal and T. Saimi, Optics Comm. 5 (1972) 143. E. Mr6z, R. Pawluczyk and M. Pluta, Optica Applicata 1 (1971) 9. R. Pawluczyk and E. Mr6z, Optica Acta 20 (1973) 379. D.J. Cronin and A.E. Su ith, Opt. Engng 12 (1973) 50. B.W. Reuter and O. Wess, in: Proc. ICO Conf. (Madrid, 1978) p. 275. W.T. Welford, Optics Comm. 4 (1971) 275. M. Born and E. Wolf, Principles of optics, 4th Ed. (Pergamon Press, London, 1970) Ch. 10. L. Mandel, in: Progress in optics, Vol. 2, ed. E. Wolf (North-Holland Pub., Amsterdam, 1963) p. 187. F. Zernike, Physica 5 (1938) 785. J.W. Goodman, Introduction to Fourier optics (McGraw-Hill Book, New York, 1968) Ch. 5.