V. P. Ryabukho, A. A. Chausski , and A. E. Grinevich
Institute of Problems in Precision Mechanics and Control, Russian Academy of Sciences, Saratov Saratov State University
共Submitted April 16, 1999兲
Pis’ma Zh. Tekh. Fiz. 25, 5–10共December 26, 1999兲
A telescope system incorporating an illuminating spatially modulated laser beam, a diffuser in the entry plane, and a random phase screen in the spatial frequency plane was used to
analyze the formation of average-intensity interference fringes in the image plane of the diffuser. It is shown that the system can operate as a shift interferometer where the contrast of the fringes is independent of the diffuser characteristics. Analytic expressions are obtained for the contrast of the fringes as a function of the parameters of the screen and the illuminating beam and it is established that the statistical anisotropy of the screen influences the contrast of the fringes. © 1999 American Institute of Physics.关S1063-7850共99兲01712-7兴
In Refs. 1–3 Ryabukho and Chausski established that the parameters of the phase inhomogeneities of an object satisfying the ‘‘random phase screen’’ model4,5can be deter- mined using a spatially modulated laser probe beam focused onto the surface of the screen. In order to observe average- intensity interference fringes which carry information on the inhomogeneity parameters, these authors1–3 suggested mov- ing the object or the inhomogeneities relative to the probe beam. Equivalent averaging is performed by scanning the laser beam over the object. In the present paper we consider an alternative method of obtaining average-intensity fringes with both the object and the probe beam fixed. This method involves simultaneously probing the object with numerous identical focused spatially modulated laser beams 共SMLBs兲 obtained using a primary auxiliary diffuser which functions as an irregular diffraction grating.
The optical system is shown schematically in Fig. 1. In the absence of the diffuser S1the object S2 is illuminated by a focused SMLB and the object must be displaced in the transverse direction to observe the interference fringes.1–3 The diffuser S1 multiplies the SMLB, i.e., the diffraction field behind it is a set of SMLBs propagating in different directions and simultaneously probing the object S2. As a result of uncorrelated addition in the image of the diffuser S1, the diffraction patterns from these beams form average- intensity fringes.
The formation of the interference fringes in the image S1
⬘
can have another interpretation which is more convenient for a formal analysis. Since the diffuser S1is illuminated by two waves whose directions of propagation differ by the angle, two identical speckle fields form behind it, propagating at the angle relative to each other.6In the rear focal plane of the lens L1 the fields acquire the transverse shift 0⫽f⫽ f /⌳, where ⌳ is the period of the fringes in the SMLB.
As a result of this shift the speckle fields beyond the object S2also become partially decorrelated, and in the image plane where the shift of the fields again becomes zero, the contrast of the average-intensity fringes is reduced. Hence, the con-
trast of the fringes should be determined by the modulus of the normalized correlation function B12(,0) of the interfer- ing fields U1() and U2() in the image plane:
V⫽V0
冏
B12共,0兲 B12共,0⫽0兲
冏
, B12共,0兲⫽
具
U1共兲U2*共兲典
,共1兲
where V0 is the contrast of the fringes in the illuminating diffuser S1 of the SMLB; the angular brackets denote a sta- tistical averaging operation.
Let us assume that the diffuser S1 and the object S2 are random phase screens having the transmission functions t1(r) and t2(). Then, using two successive Fourier transfor- mations and assuming that the entire scattered field falls within the aperture of the lens, U1() may be written as
U1共兲⫽
冕
⫺⬁ ⬁冕
⫺⬁ ⬁ U0共r兲exp共ik1r兲t1共1兲 ⫻exp冉
ik fr冊
t2共兲exp冉
i k f冊
d 2rd2, 共2兲where U0(r) is the complex amplitude of one of the waves in the SMLB and k1 is the wave vector of this wave. The expression for U2() is similar, with the vector k1 replaced by k2, where兩⌬k12兩⫽兩k1⫺k2兩⫽k2sin(/2).
Substituting Eq. 共2兲 for U1() and U2() into Eq. 共1兲, changing the order in which integration and averaging are carried out, and taking into account the independence of the random functions t1(r) and t2() yields the following ex- pression for the correlation function of the complex ampli- tudes of the fields in the image plane:
TECHNICAL PHYSICS LETTERS VOLUME 25, NUMBER 12 DECEMBER 1999
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B12共,⌬k12兲⫽
冕
⫺⬁ ⬁具
If共兲典
d2冕
⫺⬁ ⬁ Bf ⫻冉
⌬⫹f k⌬k12冊
t2共⌬兲 ⫻exp冉
⫺ikf ⌬冊
d2⌬, 共3兲 where具
If()典
⫽兰⫺⬁⬁ 兰⫺⬁⬁ t1(⌬r)exp(i (k/f) ⌬r)d2⌬r is theaverage intensity in the rear focal plane of the lens L1 共the spatial spectrum of the diffuser S1), the function Bf(⌬)
⫽兰⫺⬁⬁ 兰⫺⬁⬁ I0(r)exp(i (k/f)⌬r)d2r is the autocorrelation function of the field illuminating the object S2, I0(r)
⫽兩U0(r)兩2 is the average intensity in the SMLB, and
t1(⌬r) and t2(⌬) are the normalized autocorrelation
functions of the transmission coefficients t1(r) and t2() of the diffuser S1 and the object S2.
Substituting expression 共3兲 into expression 共1兲 shows that the contrast of the fringes does not depend on
具
If()典
andt1(⌬r), i.e., it does not depend on the properties of thediffuser S1.
For a fairly large-aperture illuminating SMLB for which the width of the function Bf(⌬) is substantially less than
the width of the function t2(⌬) so that Bf(⌬) may be
replaced by a ␦-function, the expression for the contrast of the fringes has the extremely simple form
V⫽V0t2
冉
0⫽ fk⌬k12
冊
. 共4兲The contrast is determined by the normalized correlation function of the boundary field beyond the object as a func- tion of the magnitude and direction of the relative shift 0, i.e., the period ⌳ and the orientation of the fringes in the SMLB. Thus, this system operates as a shift interferometer where the contrast of the fringes depends on the statistical anisotropy of the object.
For an arbitrary aperture 2W of the illuminating SMLB, an analytic expression for the fringe contrast can be obtained under the following assumptions: the intensity distribution I0(r) is Gaussian, I0(r)⫽I0exp(⫺2r2/W2); the inhomogene- ities of the object obey normal statistics, and their correlation coefficient K(⌬) is Gaussian, K(⌬)⫽exp(⫺⌬2/l2), where l is the correlation length of the inhomogeneities. For t2 (⌬) we can then use the approximation2 t2(⌬)
⬇(1⫺exp(⫺2))exp(⫺⌬2/⬜2)⫹exp(⫺2), where 2 is the dispersion of the phase fluctuations, and
⬜⫽l关⫺ln兵⫺2ln关exp(⫺1)(exp(2)⫺1)⫹1兴其兴1/2 is the cor- relation length of the field beyond the object S2(⬜⬇l, for
⭐1,⬜⬇l/ for⬎1).
Using these approximations in expression 共3兲 gives the following expression for the fringe contrast 共1兲 in the paraxial region of the image⫽0:
V⫽V0 0⫹共1⫺0兲⬜ 2共 ⬜2⫹f 2兲⫺1exp关⫺ 0 2/共 ⬜2⫹f 2兲兴 0⫹共1⫺0兲⬜2共⬜2⫹f 2兲⫺1 , 共5兲
where 0⫽exp(⫺2) andf⫽
冑
2 f /W is the correlation length of the field illuminating the object S2. Note that a similar expression for the fringe contrast is obtained using a single SMLB focused onto the surface of a moving object subject to the conditionf⫽冑
2w0, where w0is the radius of the constriction of the focused Gaussian beam. If the period FIG. 1. Telescope measuring system with illuminating spa- tially modulated laser beam, diffuser in entry plane, and object being monitored in spatial frequency plane: SMLB — illuminating spatially modulated laser beam with paral- lel interference fringes, L1 and L2 — collimating lenses,S1 — diffuser in front focal plane of the lens L1, S2 —
object being monitored in rear focal plane of lens L1, and
S1⬘— image of diffuser S1.
FIG. 2. Contrast of average-intensity interference fringes in image of dif- fuser: a — as a function of the fringe period⌳ in the illuminating beam for an object with ⫽1.15 and l⫽17m for various values of the beam aperture 2W and thus various correlation lengths of the object-probing field
f for f⫽110 mm. 1 — 2W⫽3 mm, f⫽20.8m; 2 — 2W⫽5 mm,
f⫽12.5m; 3 — 2W⫽12 mm, f⫽5.2m; b — as a function of the
beam aperture for various fringe periods: 1 —⌳⫽8 mm, 2 — ⌳⫽5.5 mm, and 3 —⌳⫽3 mm.
⌳ of the fringes is fairly small, when0 2⬎
⬜2⫹f 2
the depen- dence of the fringe contrast on the statistical anisotropy of the object disappears.
The theoretical results agree fairly accurately with the experimental ones. Figure 2 gives the experimental points and theoretical curves obtained using expression 共5兲 for the fringe contrast V/V0as a function of the fringe period⌳ and the aperture 2W of the illuminating SMLB.
The formation of an image of the interference fringes in the system shown in Fig. 1 can also be considered from the point of view of the classical analysis of linear optical systems.5 However, the approach used here is clearer from the physical point of view and allows us to establish an anal- ogy with the processes of formation of interference patterns in systems using a single probe SMLB.1–3It should also be noted that these results can be applied to optical imaging systems of a more general nature.
This work was supported by RFBR Grant No. 96-15- 96389, under the Program ‘‘Leading Scientific Schools in the Russian Federation.’’
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2
V. P. Ryabukho and A. A. Chausski, Pis’ma Zh. Tekh. Fiz. 23共19兲, 47
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6
V. P. Ryabukho, Yu. A. Avetisyan, and A. B. Sumanova, Opt. Spektrosk. 79, 299共1995兲 关Opt. Spectrosc. 79, 275 共1995兲兴.
Translated by R. M. Durham
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