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Acoustically modified spatial coherence in optical fresnel diffraction region
Volume 35, number 2
OPTICS COMMUNICATIONS
November 1980
ACOUSTICALLY MODIFIED SPATIAL COHERENCE IN OPTICAL FRESNEL DIFFRACTION REGION
Yoshihiro OHTSUKA, Yoshiteru NOZOE and Yoh IMAI Department of Engineering Science, Faculty of Engineering, Hokkaido University, Sapporo 060, Japan
Received 21 July 1980
Theoretical and experimental interpretations are given to the fact that in the optical Fresnel diffraction region acoustically modified spatial coherence is independent of the distance from the acoustic cell, but takes an identical expression across any planes parallel to the propagation direction of the acoustic wave.
1. Introduction Although the theoretical aspects of optical coherence have been studied extensively in the past [ 11, the experimental work seems to have been rather limited because of the difficulty of obtaining an intense, coherence-controllable light source. As is wellknown, spatially, partially coherent light has been produced so far from an incoherent extended thermal source by passing it through a small opening, the theoretical basis of which stems from the Van CittertZernike theorem [2]. In recent years, laser light scattered by one or two moving diffusers has frequently been used as a secondary quasi-incoherent light source [3,4]. This technique is very simple for practical use, but seems not promising for the quantitative control of optical coherence because the scattered light is influenced heavily in coherence by the roughness properties as well as the dynamics of the moving diffusers. Otherwise, it has also been known that coherence deterioration of a laser beam can be achieved if it is transmitted through a liquid crystal under an external d.c. field [S]. In contrast, it has been revealed very recently that progressing acoustic waves also act as a spatial coherence modulator [6-lo]. The most remarkable point of using acoustic waves is that an h.f. electrical potential applied to the acoustic transducer permits us to modify spatial coherence quantitatively. Those
studies have been made by paying our attention to the, exit plane of the acoustic cell through which a light beam passes. For this reason, if we want to illuminate an object with such an acoustically modulated laser beam, an imaging system has to be constructed in order to transfer the spatial coherence function from the exit plane of the acoustic cell to the object plane. This arrangement limits the practical use. According to the most recent researches made by the present authors, however, it has been proven that such an imaging system is not required due to the fact that the acoustically modified spatial coherence function is represented by the same expression in the optical Fresnel diffraction region as that across the exit plane of the acoustic cell. This communication is to report both the theory and experiment.
2. Theory The optical system to be illustrated is shown in fig. 1 For the sake of simple treatment, spatially coherent light normally illuminates an acoustic cell along which a progressing acoustic wave travels in the x-direction. As a result, an acoustically modulated optical wavefield moving together with the acoustic wave is created in the optical Fresnel diffraction region. Let the z-axis whose origin is at the exit plane of the acoustic cell indicate the propagation direction of the modulated 1.57
Volume 35, number 2
OPTICS COMMUNICATIONS
November 1980
frequency
Incident
(w t nfl) propagating at an angle 19~= In addition to this, k, is generally much larger than (nK), so that the square root term in the exponent of eq. (4) can be approximated to be sin-‘(nK/k,).
7
{k; -- (HK)~}~‘~ = k{l
- (1/2)(nK/k)2},
(5)
where k = o/c is the wavenumber of the incident light. In view of eqs. (4) and (5) we can write eq. (2) as Fig. 1. A schematic representation
V(z, x, t) = V. exp(iot
of the optical system.
light, and then the optical disturbance is represented in the near field by V(z, x, t) with z > 0 which satisfies the scalar wave equation v%=
(l/c2)(a+/at2),
(1)
where c is the velocity of light outside the cell. It should be reminded that the spatial coherence function [2] can be calculated from V(z, x, t) if eq. (1) is solved for V(z, x, t). A traditional method to solve eq. (1) is to write V(z, x, t) in a Fourier series, since V(z, x, t) has a periodic nature in the near field of interest. If the incident light of the angular frequency w has initially a uniform amplitude V. over the entrance plane of the acoustic cell, the optical disturbance at a point p(z, x) and at an instant t may thus be written in a Fourier representation as
5 U,(z) n=_-m
exp(in(fi2t
- Kx)} ,
(2)
where U,(z) is the amplitude factor for the nth component and a, K are the acoustic angular frequency and wavenumber, respectively. Substitution of eq. (2) into eq. (1) yields a differential equation with respect to the nth component U,(z): {U;(z)/dz2}
+ ((w + ns2)2/c2-
since the individual zero independently. given by U,(z)=
e
(~IK)~}U~(Z) = 0, (3)
terms of the summation must be The solution for eq. (3) is then
exp[-i{kz
- (HK)~}“~z],
(4)
where @ = U,(z = 0) and k, = (o t &2)/c. Note that {V,U,(z)) is the amplitude of a plane wave of the 158
L$ exp [i {n(Rt - Kx) + (nK)2z/2k}].
(6)
It is now possible to formulate the mutual intensity [2] which represents the optical spatial coherence at two points Pl(xl, z) and P2(x2, z) in a plane parallel to the exit plane of the acoustic cell. From the definition J12(z) = tV(z, x 1, t)V*(z, x2, t)>, we obtain
X (eXp{i(n - m)a2t)) exp{i(mx2
- nxl)K}
(7)
where ( . .. ) stands for the time average and * denotes the complex conjugate. Since Owhennfm, (exp (i(n - m) f22t))=
(8)
i 1 whenn=m, eq. (7) is reduced to
V(z, x, t) = V. exp(iwt) X
X ,,c_
- kz)
l@12 w{inK(q J12(z) = I Vo12 c n=_-m
- xl)).
(9)
This formula evidently shows that the mutual intensity is dependent on only the two-point separation (x2 - x1) but independent of z in space. Consequently, we can say that the same mutual intensity as eq. (9) is always established across any near-field planes parallel to the exit plane of the acoustic cell. It should be recalled that, according to the knowledge of a phase grating, its self-imaging planes [ 11,121 exist in the Fresnel diffraction region when the condition z = (4rqk)/K2 = (2qA2/X), (q = 1,2,3, ...). is satisfied, A and h being the sound and light wavelengths, respectively. From this fact, it is understood intuitively, that the mutual intensity becomes the same across such imaging planes, but a point to be emphasized here is that we have always the same mutual intensity as eq.
.
OPTICS COMMUNICATIONS
Volume 35, number 2
(9) across any planes regardless of the self-imaging planes. To the extent where the acoustic wave can be treated as a pure phase grating in the so-called RamanNath regime [ 131, the optical wavefronts undergo only phase-modulation. In this case, the optical disturbance across the exit plane of the acoustic cell is given by V(0, x, t) = V. exp [i {wt t u sin(L% - Kx)}],
0
A
0
November 1980
Z=Omm
2=27mm z=36mm
Q$,4a
2=45mm
a 0,
(10) 1
where IJ is called the Raman-Nath parameter dependent on the acoustic power level. With the help of the Bessel function Jn(u) of the first kind and nth order, eq. (10) is developed in a Fourier series
0
I
0.5
I
I
-y
1.0
Fig. 2. The degree of coherence as a function of the two-point separation divided by the ultrasonic wavelength.
V(0, x, t) = V. exp(iwt) 3. Experimental X C J,(u) exp (in(fl2t - Kx)}. n=--m
(11)
Comparison of this formula with eq. (6) at z = 0 allows us to have @ = J,(u). Thus eq. (9) becomes m J12(z) = lVo12 C Ji(u)exp{inK(x2 n=_-m
- ~1))
= IVo12Jo[2ul sin{K(x2 -x1)/2)1].
(12)
This is the same expression as has been derived previously across the exit plane of the acoustic cell [6,7]. For spatial coherence measurements it is convenient to derive the degree of coherence at P,(x,, z) and P2(r2, z) (see fig. 1). Its definition [2] leads to
Ir(Ax)l = 1312(~)l/~~11(z)J22(z) = IJ, [2u I sin (KAx/2)1] I,
(13)
with Ax = (x2 - x1). The value of this function can be measured experimentally from Young’s double-slit interference experiments using the formula [2] IT(A
[(Imax - Imin)/(l,,
x Wl +4>/24&1,
+Imin)I
4. Conclusions (14)
where Imax, Imin are the maximum and minimum intensities of the interference fringes and I,, I2 are each intensity of the light from the double slits independently.
The acoustic cell used was full of pure water, along which a progressing acoustic wave of 10 MHz was excited by an X-cut quartz transducer. A well-collimated He-Ne laser beam was employed for spatially coherent illumination upon the entrance plane of the acoustic cell. For the Young’s double-slit experiments, a specially designed double-slit apparatus [7] whose separation is variable was set at a required distance from the acoustic cell (see fig. 1). In order to verify the theoretical result experimentally, it was examined whether or not lr(Ax)l is independent of the distance z under the constant acoustic power level giving u = 1.0. The results measured for four planes from z = 0 to z = 45 mm are plotted in fig. 2. As has been expected, all the plotted curves take almost the same curve, ignoring some amount of experimental errors. Additionally, the averaged plotted-curve was confirmed to fit well with the theoretical curve [7] calculated from eq. (13).
In the optical Fresnel diffraction region, acoustically modified spatial coherence is represented by an identical formula across any plane parallel to the exit plane of the acoustic cell, as long as the acoustic wave can be treated as a pure phase grating. For the experimental studies of a certain object illuminated with an acoustically modulated laser beam, the results obtained 159
Volume 35, number 2
OPTICS COMMUNICATIONS
here allow us to put the object at any place desired in the Fresnel diffraction region with no imaging system to transfer spatial coherence from the acoustic cell to the object plane. This arrangement would be conveniently available for partical uses.
Acknowledgement The authors wish to thank Dr. M. Imai for his valuable comments.
References [l] See, for example, L. Mandel and E. Wolf, eds., Coherence and fluctuation of light (Dover, New York, 1970), Vols. land2.
160
November 1980
[ 21 M. Born and E. Wolf, Principles of optics, 4 th ed. (Pergamon, New York, 1970) Ch. 10. [ 31 H. Arsenalt and S. Lowenthal, Optics Comm. 1 (1970) 451. [4] E. Schroder, OpticsComm. 3 (1971) 68. [S] F. Scudieri, M. Bertolotti and R. Bartolino, Appl. Optics 13 (1974) 181. [6] Y. Ohtsuka, Optics Comm. 17 (1976) 234. [7] Y. Ohtsuka and Y. Imai, J. Opt. Sot. Am. 69 (1978) 684. [8] Y. Ohtsuka, Jpn. J. Appl. Phys. 17 (1978) 1775. [9] Y. Ohtsuka, Can. J. Phys. 57 (1979) 1420. [lo] Y. Imai and Y. Ohtsuka, Appl. Optics 19 (1980) 542. [ll] J.T. Winthrop and C.R. Worthington, J. Opt. Sot. Am. 55 (1965) 373. 1121 E.A. Hiedemann and M.A. Brezeale, J. Opt. Sot. Am. 49 . (1959) 372. [13] C.V. Raman and N.S.N. Nath, Proc. Ind. Acad. Sci. 2A (1935) 406,413; 3A (1936) 75, 119.
OPTICS COMMUNICATIONS
November 1980
ACOUSTICALLY MODIFIED SPATIAL COHERENCE IN OPTICAL FRESNEL DIFFRACTION REGION
Yoshihiro OHTSUKA, Yoshiteru NOZOE and Yoh IMAI Department of Engineering Science, Faculty of Engineering, Hokkaido University, Sapporo 060, Japan
Received 21 July 1980
Theoretical and experimental interpretations are given to the fact that in the optical Fresnel diffraction region acoustically modified spatial coherence is independent of the distance from the acoustic cell, but takes an identical expression across any planes parallel to the propagation direction of the acoustic wave.
1. Introduction Although the theoretical aspects of optical coherence have been studied extensively in the past [ 11, the experimental work seems to have been rather limited because of the difficulty of obtaining an intense, coherence-controllable light source. As is wellknown, spatially, partially coherent light has been produced so far from an incoherent extended thermal source by passing it through a small opening, the theoretical basis of which stems from the Van CittertZernike theorem [2]. In recent years, laser light scattered by one or two moving diffusers has frequently been used as a secondary quasi-incoherent light source [3,4]. This technique is very simple for practical use, but seems not promising for the quantitative control of optical coherence because the scattered light is influenced heavily in coherence by the roughness properties as well as the dynamics of the moving diffusers. Otherwise, it has also been known that coherence deterioration of a laser beam can be achieved if it is transmitted through a liquid crystal under an external d.c. field [S]. In contrast, it has been revealed very recently that progressing acoustic waves also act as a spatial coherence modulator [6-lo]. The most remarkable point of using acoustic waves is that an h.f. electrical potential applied to the acoustic transducer permits us to modify spatial coherence quantitatively. Those
studies have been made by paying our attention to the, exit plane of the acoustic cell through which a light beam passes. For this reason, if we want to illuminate an object with such an acoustically modulated laser beam, an imaging system has to be constructed in order to transfer the spatial coherence function from the exit plane of the acoustic cell to the object plane. This arrangement limits the practical use. According to the most recent researches made by the present authors, however, it has been proven that such an imaging system is not required due to the fact that the acoustically modified spatial coherence function is represented by the same expression in the optical Fresnel diffraction region as that across the exit plane of the acoustic cell. This communication is to report both the theory and experiment.
2. Theory The optical system to be illustrated is shown in fig. 1 For the sake of simple treatment, spatially coherent light normally illuminates an acoustic cell along which a progressing acoustic wave travels in the x-direction. As a result, an acoustically modulated optical wavefield moving together with the acoustic wave is created in the optical Fresnel diffraction region. Let the z-axis whose origin is at the exit plane of the acoustic cell indicate the propagation direction of the modulated 1.57
Volume 35, number 2
OPTICS COMMUNICATIONS
November 1980
frequency
Incident
(w t nfl) propagating at an angle 19~= In addition to this, k, is generally much larger than (nK), so that the square root term in the exponent of eq. (4) can be approximated to be sin-‘(nK/k,).
7
{k; -- (HK)~}~‘~ = k{l
- (1/2)(nK/k)2},
(5)
where k = o/c is the wavenumber of the incident light. In view of eqs. (4) and (5) we can write eq. (2) as Fig. 1. A schematic representation
V(z, x, t) = V. exp(iot
of the optical system.
light, and then the optical disturbance is represented in the near field by V(z, x, t) with z > 0 which satisfies the scalar wave equation v%=
(l/c2)(a+/at2),
(1)
where c is the velocity of light outside the cell. It should be reminded that the spatial coherence function [2] can be calculated from V(z, x, t) if eq. (1) is solved for V(z, x, t). A traditional method to solve eq. (1) is to write V(z, x, t) in a Fourier series, since V(z, x, t) has a periodic nature in the near field of interest. If the incident light of the angular frequency w has initially a uniform amplitude V. over the entrance plane of the acoustic cell, the optical disturbance at a point p(z, x) and at an instant t may thus be written in a Fourier representation as
5 U,(z) n=_-m
exp(in(fi2t
- Kx)} ,
(2)
where U,(z) is the amplitude factor for the nth component and a, K are the acoustic angular frequency and wavenumber, respectively. Substitution of eq. (2) into eq. (1) yields a differential equation with respect to the nth component U,(z): {U;(z)/dz2}
+ ((w + ns2)2/c2-
since the individual zero independently. given by U,(z)=
e
(~IK)~}U~(Z) = 0, (3)
terms of the summation must be The solution for eq. (3) is then
exp[-i{kz
- (HK)~}“~z],
(4)
where @ = U,(z = 0) and k, = (o t &2)/c. Note that {V,U,(z)) is the amplitude of a plane wave of the 158
L$ exp [i {n(Rt - Kx) + (nK)2z/2k}].
(6)
It is now possible to formulate the mutual intensity [2] which represents the optical spatial coherence at two points Pl(xl, z) and P2(x2, z) in a plane parallel to the exit plane of the acoustic cell. From the definition J12(z) = tV(z, x 1, t)V*(z, x2, t)>, we obtain
X (eXp{i(n - m)a2t)) exp{i(mx2
- nxl)K}
(7)
where ( . .. ) stands for the time average and * denotes the complex conjugate. Since Owhennfm, (exp (i(n - m) f22t))=
(8)
i 1 whenn=m, eq. (7) is reduced to
V(z, x, t) = V. exp(iwt) X
X ,,c_
- kz)
l@12 w{inK(q J12(z) = I Vo12 c n=_-m
- xl)).
(9)
This formula evidently shows that the mutual intensity is dependent on only the two-point separation (x2 - x1) but independent of z in space. Consequently, we can say that the same mutual intensity as eq. (9) is always established across any near-field planes parallel to the exit plane of the acoustic cell. It should be recalled that, according to the knowledge of a phase grating, its self-imaging planes [ 11,121 exist in the Fresnel diffraction region when the condition z = (4rqk)/K2 = (2qA2/X), (q = 1,2,3, ...). is satisfied, A and h being the sound and light wavelengths, respectively. From this fact, it is understood intuitively, that the mutual intensity becomes the same across such imaging planes, but a point to be emphasized here is that we have always the same mutual intensity as eq.
.
OPTICS COMMUNICATIONS
Volume 35, number 2
(9) across any planes regardless of the self-imaging planes. To the extent where the acoustic wave can be treated as a pure phase grating in the so-called RamanNath regime [ 131, the optical wavefronts undergo only phase-modulation. In this case, the optical disturbance across the exit plane of the acoustic cell is given by V(0, x, t) = V. exp [i {wt t u sin(L% - Kx)}],
0
A
0
November 1980
Z=Omm
2=27mm z=36mm
Q$,4a
2=45mm
a 0,
(10) 1
where IJ is called the Raman-Nath parameter dependent on the acoustic power level. With the help of the Bessel function Jn(u) of the first kind and nth order, eq. (10) is developed in a Fourier series
0
I
0.5
I
I
-y
1.0
Fig. 2. The degree of coherence as a function of the two-point separation divided by the ultrasonic wavelength.
V(0, x, t) = V. exp(iwt) 3. Experimental X C J,(u) exp (in(fl2t - Kx)}. n=--m
(11)
Comparison of this formula with eq. (6) at z = 0 allows us to have @ = J,(u). Thus eq. (9) becomes m J12(z) = lVo12 C Ji(u)exp{inK(x2 n=_-m
- ~1))
= IVo12Jo[2ul sin{K(x2 -x1)/2)1].
(12)
This is the same expression as has been derived previously across the exit plane of the acoustic cell [6,7]. For spatial coherence measurements it is convenient to derive the degree of coherence at P,(x,, z) and P2(r2, z) (see fig. 1). Its definition [2] leads to
Ir(Ax)l = 1312(~)l/~~11(z)J22(z) = IJ, [2u I sin (KAx/2)1] I,
(13)
with Ax = (x2 - x1). The value of this function can be measured experimentally from Young’s double-slit interference experiments using the formula [2] IT(A
[(Imax - Imin)/(l,,
x Wl +4>/24&1,
+Imin)I
4. Conclusions (14)
where Imax, Imin are the maximum and minimum intensities of the interference fringes and I,, I2 are each intensity of the light from the double slits independently.
The acoustic cell used was full of pure water, along which a progressing acoustic wave of 10 MHz was excited by an X-cut quartz transducer. A well-collimated He-Ne laser beam was employed for spatially coherent illumination upon the entrance plane of the acoustic cell. For the Young’s double-slit experiments, a specially designed double-slit apparatus [7] whose separation is variable was set at a required distance from the acoustic cell (see fig. 1). In order to verify the theoretical result experimentally, it was examined whether or not lr(Ax)l is independent of the distance z under the constant acoustic power level giving u = 1.0. The results measured for four planes from z = 0 to z = 45 mm are plotted in fig. 2. As has been expected, all the plotted curves take almost the same curve, ignoring some amount of experimental errors. Additionally, the averaged plotted-curve was confirmed to fit well with the theoretical curve [7] calculated from eq. (13).
In the optical Fresnel diffraction region, acoustically modified spatial coherence is represented by an identical formula across any plane parallel to the exit plane of the acoustic cell, as long as the acoustic wave can be treated as a pure phase grating. For the experimental studies of a certain object illuminated with an acoustically modulated laser beam, the results obtained 159
Volume 35, number 2
OPTICS COMMUNICATIONS
here allow us to put the object at any place desired in the Fresnel diffraction region with no imaging system to transfer spatial coherence from the acoustic cell to the object plane. This arrangement would be conveniently available for partical uses.
Acknowledgement The authors wish to thank Dr. M. Imai for his valuable comments.
References [l] See, for example, L. Mandel and E. Wolf, eds., Coherence and fluctuation of light (Dover, New York, 1970), Vols. land2.
160
November 1980
[ 21 M. Born and E. Wolf, Principles of optics, 4 th ed. (Pergamon, New York, 1970) Ch. 10. [ 31 H. Arsenalt and S. Lowenthal, Optics Comm. 1 (1970) 451. [4] E. Schroder, OpticsComm. 3 (1971) 68. [S] F. Scudieri, M. Bertolotti and R. Bartolino, Appl. Optics 13 (1974) 181. [6] Y. Ohtsuka, Optics Comm. 17 (1976) 234. [7] Y. Ohtsuka and Y. Imai, J. Opt. Sot. Am. 69 (1978) 684. [8] Y. Ohtsuka, Jpn. J. Appl. Phys. 17 (1978) 1775. [9] Y. Ohtsuka, Can. J. Phys. 57 (1979) 1420. [lo] Y. Imai and Y. Ohtsuka, Appl. Optics 19 (1980) 542. [ll] J.T. Winthrop and C.R. Worthington, J. Opt. Sot. Am. 55 (1965) 373. 1121 E.A. Hiedemann and M.A. Brezeale, J. Opt. Sot. Am. 49 . (1959) 372. [13] C.V. Raman and N.S.N. Nath, Proc. Ind. Acad. Sci. 2A (1935) 406,413; 3A (1936) 75, 119.