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Maximum diameter of the read-out laser beam in double-exposure speckle photography
Optics Communications 221 (2003) 279–288 www.elsevier.com/locate/optcom
Maximum diameter of the read-out laser beam in double-exposure speckle photography F. Salazar a,*, F. Gasc on b a
Escuela Superior de Ingenieros de Minas, U.P.M. Rıos Rosas, 21, 28003 Madrid, Spain b Escuela Superior de Arquitectura, U.S. Reina Mercedes, 2, 41012 Sevilla, Spain
Received 5 August 2002; received in revised form 4 February 2003; accepted 17 March 2003
Abstract In this work, a previous proposed simple speckle model is applied to obtain a criterion to determine the upper limit of the read-out laser beam diameter for having information about the in-plane displacements of a material from a double-exposure speckle photography. The supposed rough plane surface of the sample is subjected to a general movement: translation, rotation, and strain. For that general case, an expression for the maximum diameter of the scanning beam is calculated. The theory is verified experimentally in the particular cases of a translation, a rotation and an anisotropic thermal strain. Ó 2003 Elsevier Science B.V. All rights reserved. Keywords: Speckle model; Double-exposure photography; In-plane motion; Interrogation beam; Maximum diameter; Young fringes
1. Introduction Problems related with industry and physics, such as mechanical and thermal deformations, have motivated the use of different techniques for their solutions. Traditionally the experiments to determine properties of materials have been carried out with machines or devices that interacted with samples. However, since the 1970s optical techniques have more importance due to various advantages: time, cost, accuracy, etc. Furthermore, speckle techniques play an important role because the speckle phenomenon does not interact with the object during measurement and the results are more precise. Speckle interferometry and speckle photography have been extensively researched and applied in distinct fields as deformations [1–4], temperature measurement [5], or roughness measurement [6]. In the present article we are interested in the speckle photography due to the possibilities of this technique to determine in-plane displacements.
*
Corresponding author. Fax: +34-1336-6952. E-mail addresses: [email protected] (F. Salazar), [email protected] (F. Gasco´n).
0030-4018/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0030-4018(03)01399-3
280
F. Salazar, F. Gascon / Optics Communications 221 (2003) 279–288
In this context, the theory about the double-exposure speckle photography and the displacement of the speckle in a general deformation has been very well explained by Yamaguchi [7,8], even though the goal of that investigation was not to research the limits of the beam to be used in the point-wise filtering technique in any deformation experiment. However, other authors have addressed the investigation to that issue. In this way, Li et al. [9] and Chiang and Li [10], established the upper limit of measurement of speckle displacement based on a fringe-discernibility criterion which combines the effect of the amplitude modulation of Young fringes and the secondary speckle size. More recently, Sj€ odahl [11] has investigated in detail the effect of the finite window used in electronic speckle photography on the cross-correlation algorithm. From another point of view, in a previous paper [3], we demonstrated theoretically and experimentally the maximum value of the scanning beam diameter for measuring displacements in the case of an isotropic thermal deformation. This value is a possible criterion for obtaining valid information of the displacements through the diffraction pattern in the point-wise filtering technique. Now, given the importance of this read-out laser beam technique, our objective is to develop a general theory that allows us to find the upper limit for the interrogation laser beam in any plane general movement rather than restricted to thermal expansion of isotropic samples. This motion consists of a translation, a rotation around an axis perpendicular to the plane of the sample, and a strain. The developed theory is founded partly on a simple speckle model, which we have previously proposed [12].
2. A model of speckle In a previous article [12] we proposed a simple speckle model to explain the minimum diameter of the read-out laser beam in the point-wise filtering method. Taken into consideration that the characteristics of the intensity associated with this speckle pattern may be understood through the image-plane intensity autocorrelation function [13], this model assumes that the intensity of each spot on the photographic plate has the form !2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi J1 ðpD x2 þ y 2 =ðkz0 ÞÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Is ðx; yÞ ¼ 4I0 ; ð1Þ pD x2 þ y 2 =ðkz0 Þ where I0 is the intensity in the center of the spot, J1 is the first-order Bessel function of the first kind, D is the diameter of the lens used to take the photograph, z0 is the distance from the lens to the photographic plate, x and y are the coordinates of a point on the plate, and k is the wavelength of the radiation used. Let N be the number of spots recorded on the photographic plate for the first exposure. If we suppose all spots on the plate are identical to each other except that their centers are randomly distributed, the intensity Is ðx; yÞ on the plane of the photographic plate is I1 ¼
N X
Is ðx; yÞ dðx xi Þdðy yi Þ;
ð2Þ
i¼1
where ðxi ; yi Þ represent random coordinates of points on the plate, d is the DiracÕs delta distribution, and represents the convolution operator. When the sample is deformed, it is assumed that the internal structure of each speckle is maintained as Is ðx; yÞ. Hence the intensity corresponding to the second exposure is given by the function I2 ¼
N X
½Is ðx; yÞ dð x xi Þdð y yi Þ ½dð x fi ðx; yÞÞdð y gi ðx; yÞÞ;
ð3Þ
i¼1
where fi and gi are functions of coordinates x and y which express the in-plane displacement components of the spot centered at the point x ¼ xi , y ¼ yi .
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281
The amplitude transmission function for the developed plate is tðx; yÞ ¼ a bðI1 þ I2 Þs;
ð4Þ
where s is the time exposure, in our case equal for both, and a and b are two constants of proportionality. To obtain information on movements brought about on the surface of the object, we use the point-wise filtering technique. According to this technique, a collimated circular laser beam with diameter d is projected perpendicularly onto the developed plate, centered on the origin (0, 0) in the OXY system of the photograph, Fig. 1. Let us suppose that the read-out laser beam has homogeneous intensity and zero intensity outside it. This means that the beam function may be expressed by a bidimensional pulse distribution, that is, pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! x2 þ y 2 Bðx; yÞ ¼ P ; ð5Þ d where P is the circle-function. The amplitude U 0 ða; bÞ of the diffracted wave by the developed photographic plate is in the Fraunhofer approximation and outside the incident direction [12] U 0 ða; bÞ ¼ 2CbsH ða; bÞ
Nd X
f cos ½pðafi þ bgi Þ exp ½ 2pjðaxi þ byi þ afi =2 þ bgi =2Þg Qða; bÞ;
ð6Þ
i¼1
Fig. 1. An incident laser beam of diameter d is diffracted by a subjective speckle field (double-exposure) recorded on a photographic plate. The pattern is analyzed on the X 0 , Y 0 plane.
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where C is a constant for a given incident laser beam, Nd is the number of spots illuminated by the read-out laser beam, and the spatial frequencies are a ¼ cos hx =k and b ¼ cos hy =k. Moreover, in Eq. (6) appear the functions H ða; bÞ and Qða; bÞ which are the Fourier transforms of Is ðx; yÞ and Bðx; yÞ, respectively. The corresponding mathematical expressions for H and Q are the following: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4k4 z40 kz0 kz0 2 2 H ða; bÞ ¼ 4I0 2 4 P a þb P a 2 þ b2 ð7Þ pD D D which is commonly referred as diffraction halo, and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dJ1 pd a2 þ b2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : Qða; bÞ ¼ 2 a2 þ b2
ð8Þ
Eq. (6) is a complicated convolution due to the finite extent of the laser beam diameter. In this way, the result of convoluting Qða; bÞ with the first term of the second member in the equality (6) is the smoothing of the geometrical shape of the halo H ða; bÞ and of the functions in the summation. However, that convolution may be carried out if we consider that the equivalent width Wf of a function f ðx; yÞ is equal to the reciprocal of the equivalent width WF of its transform F ða; bÞ, that is, Wf WF ¼ 1 [14]. Applying that last relation to functions Bðx; yÞ and Qða; bÞ we can deduce, that if the diameter d of B is wide the equivalent width of Q is small and high. In that case it means that Qða; bÞ is similar to a DiracÕs delta generalized function, and therefore, the aforementioned convolution does not modifiy significantly the form of the first term in Eq. (6). Hence, the expression for the amplitude disturbance is [12]: U 0 ða; bÞ ¼ 2CbsH ða; bÞ
Nd X
f cos½pðafi þ bgi Þ exp ½ 2pjðaxi þ byi þ afi =2 þ bgi =2Þg:
ð9Þ
i¼1
This means the halo function H ða; bÞ is practically zero for ða2 þ b2 Þ1=2 P D=ðkz0 Þ. On the other hand the intensity is shown to be of the following form [12] 2
I KjU 0 j ; where K is a constant.
3. Criterion for the maximun diameter of the scanning beam We are interested in the analysis of the in-plane displacement suffered by different parts of a sample in an experiment. Let us suppose that each point of the plane object is subjected to any small displacement. To analyze the plane movement, we chose the center of beam (x ¼ 0, y ¼ 0) as the kinematic reduction center. Hence, the neighboring points of the object under study will undergo a rigid displacement formed by a translation t, equal to the displacement of the reduction center, plus a rotation Xz around an axis parallel to OZ and passing through that center and a strain eij . As we have chosen the point (x ¼ 0, y ¼ 0) as the reduction center, said point is subject only to a displacement t. If the displacements are linear functions of the coordinates, the displacement of a generic spot i will have as components on OX and OY, respectively: fi ¼ tx Xz yi þ e11 xi þ e12 yi tx þ fi0 ; gi ¼ ty þ Xz xi þ e12 xi þ e22 yi ty þ gi0 ;
ð10Þ
where fi0 and gi0 represent the components of the displacement due to the rotation and strain. Therefore, the modulus of the amplitude (9) is
F. Salazar, F. Gascon / Optics Communications 221 (2003) 279–288
Nd X 0
U ¼ 2jCjbsH ða; bÞ cos p atx þ bty þ afi0 þ bgi0 exp 2pj axi þ byi þ atx =2 þ bty =2 i¼1
0 0 þ afi =2 þ bgi =2 :
283
ð11Þ
The fringes that correspond to the translation will appear if pðatx þ bty Þ is at least p=2 for the cosine to get the first minimum, therefore pðatx þ bty Þ is not small. The development of the cosine of the addition pðatx þ bty Þ þ pðafi0 þ bgi0 Þ is cosðpðatx þ bty ÞÞ, if cosðpðafi0 þ bgi0 ÞÞ 1 and sinðpðafi0 þ bgi0 ÞÞ 0. Then in order to reach the cosine as common factor and fringes to appear, the following inequality must be verified jafi0 þ bgi0 j 1: Taking into account this inequality, we obtain for the intensity Nd X
2 2 exp ½ 2pjðaxi þ byi Þ exp 2pj atx =2 þ bty =2 þ afi0 =2 I 0 ¼ 4KjCj b2 s2 ½ H ða; bÞ i¼1 2
2
þ bgi0 =2 cos p atx þ bty :
ð12Þ
ð13Þ
However, taking into account the translations tx and ty are constants because they correspond to the reduction center, the term exp½2pjðatx =2 þ bty =2Þ, which absolute value is the unity, may be placed as common factor of the summatory, and then expression (13) reduces to 2 Nd X
2
0
2 2 2 2 0 0 I ¼ 4KjCj b s ½ H ða; bÞ exp ½ 2pjðaxi þ byi Þ exp 2pj afi =2 þ bgi =2 cos p atx þ bty : i¼1 ð14Þ Condition (12) indicates some circumstances under which translation t is detectable in the area illuminated by the read-out laser beam, therefore that condition may be considered as a possible criterion to determine the maximum diameter of the read-out beam to obtain information about displacements. With this result we study different cases of interest. 3.1. Pure translation In the case of plane motion of the object its displacements have a constant value t for all the points of the sample, fi0 ¼ gi0 ¼ 0, and Eq. (14) gives Nd 0
X U ¼ 2jCjbsH ða; bÞ cos p atx þ bty ð15Þ exp ½ 2pjðaxi þ byi Þ i¼1 and the intensity formula is simplified to 2 Nd X 2 2 I 0 ¼ 4KjCj b2 s2 ½ H ða; bÞ exp ½ 2pjðaxi þ byi Þ cos2 pðatx þ bty Þ : i¼1
ð16Þ
Therefore the intensity observed in the conical corona of interest in the space of the frequencies is the product of a function that decreases with ða2 þ b2 Þ because of ½H ða; bÞ2 , by the summation of random phase complexes and by a harmonic function of ða2 þ b2 Þ1=2 with frequency jtj=2. That harmonic function represents the familiar Young fringes which are modulated by the secondary speckle noise (random
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summation) that appears within the halo. This means that in pure translation the laser beam diameter can be very large and still produce high visibility fringes. However, it cannot be too small. The read-out beam needs to be large in comparison with the displacement. Otherwise the number of correlated speckles within the illuminated area will be small in relation to the uncorrelated speckles and the contrast will be low. 3.2. Translation, rotation, and strain In the present case, the problem to find the upper limit for the explorer laser beam is more difficult due to the fact that fi0 and gi0 are not zero. Hence, in Eq. (11) we cannot reach the cosine as a common factor and consequently the Young fringes are not noticeable. However, we can give a solution if we consider the inequality (12). 2 The inequality (12) must be verified for all possible values of a and b allowed by ½H ða; bÞ . It means that a and b can become so large that the direction they represent points to the border of the halo, that is, jaj 6 D=ðz0 kÞ and jbj 6 D=ðz0 kÞ. Let us study the inequalities 0
af þ bg0 6 af 0 þ bg0 ¼ jaj f 0 þ jbj g0 6 D f 0 þ g0 : ð17Þ i i i i i i i kz0 i Let us suppose that the terms of the displacements corresponding to deformations and rotations are small enough to verify 0 0 f þ g z0 k : ð18Þ i i D Then (17) gets jafi0 þ bgi0 j 1, and (12) is verified. Hence, the inequality (18) becomes a sufficient condition. Eq. (18) may be written as kz0 : D If that addition is much less than kz0 =D, it must be also true: je11 xi þ e12 yi Xz yi j þ je12 xi þ e22 yi þ Xz yi j
je11 xi þ e12 yi Xz yi j
kz0 ; D
ð19Þ
ð20Þ
kz0 : ð21Þ D These inequations have as solution a set of points inside a region bounded by four straight lines. To give a quantitative criterion for the maximum diameter of the scanning beam, we replace the inequalities (20) and (21) by theirs respective equalities, obtaining as limit the zone represented in Fig. 2. The interior of the region and more precisely its central zone, represents the set of spots that verify the inequality (19), they become all the set adequate enough for the measurement of t. Upon illuminating the double-exposure speckle photograph with the read-out laser beam, all the recorded speckles whose xi , yi are within the zone will contribute appropriately to the Young fringes. Then the diameter of the read-out should be less than two times the minimum distance from the center (0,0) to each straight lines. Therefore, the diameter d must verify the inequality 9 8 > > = < 2z0 k=D 2z0 k=D ð22Þ d min qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : > 2> : e2 þ ðe12 Xz Þ2 e2 þ ðe12 þ Xz Þ ; je12 xi þ e22 yi þ Xz yi j
11
22
F. Salazar, F. Gascon / Optics Communications 221 (2003) 279–288
285
Fig. 2. Four straight lines bound the region where the translation can be detected. The interrogation laser beam must be inside them.
In short, the spot recorded at ðxi ; yi Þ during the first exposure and the same spot but shifted and recorded during the second exposure form a pair. Since in general there are many pairs within the area where the scanning beam intersects with the photographic plate, we could obtain different maximum and minimum directions for each of these pairs. This is not so, however, for the displacements that the spots undergo within the limits of the read-out laser beam, since these displacements are approximately equal. This means that any pair within the area that the scanning beam intersects, fulfilling the inequality (22), will produce almost identical maximum and minimum directions. Once the upper limit is established for the read-out laser beam, we illuminate the photographic plate with a laser beam narrow enough to fulfill the condition of Eq. (22). Next, we measure the directions of the diffraction maxima (Young fringes) and calculate displacement suffered by the sample point. Effectively, this is sufficient enough to analyze the phase of the cosine of the expression (14) for a point within the area illuminated by the scanning beam. The directions of the intensity maxima will be given by the classical equation pðatx þ bty Þ ¼ np, or ð23Þ jDrj sin hn cos n ¼ nk; qffiffiffiffiffiffiffiffiffiffiffiffiffi where jDrj ¼ tx2 þ ty2 represents the displacement module, hn is the angle formed by the OZ axis and the direction of observation of the nth order maximum, and n is the angle between the displacement direction and the direction defined by the center of the diffraction halo O0 and the observation point P (Fig. 1). When the observation point P is taken on the straight line through O0 and perpendicular to the nth intensity fringe, Eq. (23) is transformed in the well known jDrj sin hn ¼ nk. 3.3. Isotropic deformation As inequality (22) gives a possible criterion to know the maximum value of the interrogation beam diameter to obtain information of the displacements, it should be applicable to the particular case of an isotropic deformation. In an isotropic deformation e11 ¼ e22 ¼ e, and e12 ¼ 0, hence (22) gives 2z0 k=D d qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : e2 þ X2z
ð24Þ
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In the case that no rotation is present or it is small in relation to e, Eq. (24) becomes
2 zD0 k d jej
ð25Þ
which agrees, approximately, with the first criterion done in [3] for isotropic bodies ! 1:2ðzD0 kÞ rs ¼ d< ; jej jej where rs is the speckle diameter.
4. Experimental results In order to illustrate and verify the developed theory, we will apply it to some particular cases. We used typical experimental speckle photography devices (sample, lens, and photographic plate), with magnification factor M ¼ 1, k ¼ 632:8 nm, and F number ¼ 3:5, thus z0 =D ¼ 7:0. 4.1. Pure translation We will first carry out a pure in-plane translation experiment for which Xz ¼ eij ¼ 0. The translation was brought in the sample with the value jDrj ¼ tx ¼ 20 lm. As fi0 ¼ gi0 ¼ 0, hence the inequality (22) becomes d 1;
ð26Þ
and we see that this inequality is always fulfilled, independent of t value. An analogous result may be obtained through Eq. (12), that is, 0 1 (it is always true). This means that in a pure translation, we are going to have sharp Young fringes no matter how great the diameter of the read-out laser beam is. This is logical because, as it has been stated, inequalities do not depend on the translation. To illustrate these results Fig. 3 shows the diffraction patterns of scanning beam diameters corresponding to 1.0, 2.0, and 3.3 cm, respectively (Figs. 3(a)–(c)). In all cases the Young fringes may be measured due to the high contrast of the pattern. 4.2. Rotation A slab of a material was subjected to a rotation of Xz ¼ 350 lrad around an instantaneous rotation axis orthogonal to the slab. An area of the photographic record was illuminated by the circular read-out beam. The
Fig. 3. Diffraction patterns for diameters of 1.0 cm (a), 2.0 cm (b), and 3.3 cm (c) in a pure translation. Note the high contrast of the fringes.
F. Salazar, F. Gascon / Optics Communications 221 (2003) 279–288
287
Fig. 4. YoungÕs fringes in case of rotation. The diameters used were 1.0 cm (a), 1.8 cm (b), and 2.4 cm (c), respectively. It can be observed the lost of visibility of YoungÕs fringes when the diameter of the laser beam is increased.
center O of the illuminated area is r0 ¼ 5 cm apart from the instantaneous rotation axis. If we take the point O as the reduction center, the displacement of that point O can be regarded as a translation of value 17.5 lm. Condition (22) yields d 2:6 cm: Figs. 4(a)–(c) show the diffraction patterns formed for values of d equal to 1.0, 1.8, and 2.4 cm, respectively. For the value of the beam near the limit a blurred pattern is obtained, not being possible to assure the correct measurement of the displacement. 4.3. Anisotropic thermal strain In this section, a case involving a homogeneous thermal strain of an anisotropic material is studied to demonstrate the developed theory. Three mutually orthogonal slabs are cut from a block of anisotropic material (slate) [15]. The slabs are kept at a homogeneous temperature T1 during the first exposition and at a temperature T2 , also homogeneous, during the second exposition. One side of each slab will rest on a flat support. Let aij be the thermal linear expansion tensor of the sample. The strain originated by the increase in temperature (T2 T1 ) is homogeneous and is defined by the strain tensor eij ¼ aij ðT2 T1 Þ. Strain Table 1 a11 a22 a33 a12 a13 a23
6:9 ðlKÞ1 5.7 5.1 0.9 1.2 )0.3
t31 t32 t21 t23 t12 t13
)267.6 (lm) 114.7 169.9 )1181.6 325.5 )184.0
Xx Xy Xz
47.6 (lrad) 90.4 0.0
aI aII aIII
4.2 ðlKÞ1 5.7 7.7
288
F. Salazar, F. Gascon / Optics Communications 221 (2003) 279–288
Fig. 5. The pattern corresponds to diameters of 1.0 cm (a), 2.8 cm (b), and 3.3 cm (c), respectively, for the case of a homogeneous thermal deformation of an anisotropic plate of slate. In (a) the maxima and minima are visible and its distance is measurable. However, the correct location of the maxima of intensity is difficult in (b) due to the poor visibility of the fringes. In (c) measurement of the displacement is not possible .
analysis gives aij , its principal eigenvalues aI , aII , aIII , the translation components ti , and the rotation components Xi . These values are shown in Table 1. The optical arrangement was similar to the aforementioned in paragraph 4.1., but with a lens of F number F ¼ 3:6. Applying Table 1 to the slab corresponding to a11 , a22 , and a12 , for an increase of temperature of 30.8 °C we obtain e11 ¼ 213 106 ; e22 ¼ 176 106 ; e12 ¼ 28 106 and Eq. (22) gives d 4:2 cm: The distance measurement between fringes for the value d ¼ 1:0 cm is possible (Fig. 5(a)). In the case of d ¼ 2:8 cm (Fig. 5(b)), it begins to be difficult to locate the maxima of intensity, and when the diameter d reaches the value 3.3 cm (Fig. 5(c)), the visibility decreases and it is not possible to assure the location of maxima and minima. Hence, the measure of the displacement is not possible. Acknowledgements We are in debt to an anonymous referee for useful suggestions and comments on the paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
R.P. Khetan, F.P. Chiang, Appl. Opt. 15 (1976) 2205. F.P. Chiang, J. Adachi, R. Anastasi, J. Beaty, Opt. Eng. 21 (1982) 379. F. Salazar, F. Gasc on, Opt. Commun. 103 (1993) 235. M. Sj€ odahl, Appl. Opt. 34 (1995) 5799, and references therein. C. Shakher, A.K. Nirala, Appl. Opt. 33 (1994) 2125. C.J. Tay, S.L. Toh, H.M. Shang, J. Zhang, Appl. Opt. 34 (1995) 2324. I. Yamaguchi, Opt. Acta 28 (1981) 1359. I. Yamaguchi, J. Opt. Soc. Am. A 1 (1984) 81. D.W. Li, T.B. Chen, F.P. Chiang, J. Opt. Soc. Am. A 2 (1985) 657. F.P. Chiang, D.W. Li, Opt. Eng. 24 (1985) 937. M. Sj€ odahl, Appl. Opt. 36 (1997) 2875. F. Gasc on, F. Salazar, Opt. Commun. 172 (1999) 77. J.W. Goodman, Statistical properties of laser speckle patterns, in: J.C. Dainty (Ed.), Laser Speckle and Related Phenomena, Topics in Applied Physics of Topic in Applied Physics, vol. 9, Springer, Berlin, 1984, pp. 40–42, Chapter 2. [14] R.N. Bracewell, in: The Fourier Transform and its Applications, McGraw-Hill International Editions, New York, 1986, p. 151. [15] F. Salazar, F. Gasc on, Opt. Commun. 119 (1995) 361.
Maximum diameter of the read-out laser beam in double-exposure speckle photography F. Salazar a,*, F. Gasc on b a
Escuela Superior de Ingenieros de Minas, U.P.M. Rıos Rosas, 21, 28003 Madrid, Spain b Escuela Superior de Arquitectura, U.S. Reina Mercedes, 2, 41012 Sevilla, Spain
Received 5 August 2002; received in revised form 4 February 2003; accepted 17 March 2003
Abstract In this work, a previous proposed simple speckle model is applied to obtain a criterion to determine the upper limit of the read-out laser beam diameter for having information about the in-plane displacements of a material from a double-exposure speckle photography. The supposed rough plane surface of the sample is subjected to a general movement: translation, rotation, and strain. For that general case, an expression for the maximum diameter of the scanning beam is calculated. The theory is verified experimentally in the particular cases of a translation, a rotation and an anisotropic thermal strain. Ó 2003 Elsevier Science B.V. All rights reserved. Keywords: Speckle model; Double-exposure photography; In-plane motion; Interrogation beam; Maximum diameter; Young fringes
1. Introduction Problems related with industry and physics, such as mechanical and thermal deformations, have motivated the use of different techniques for their solutions. Traditionally the experiments to determine properties of materials have been carried out with machines or devices that interacted with samples. However, since the 1970s optical techniques have more importance due to various advantages: time, cost, accuracy, etc. Furthermore, speckle techniques play an important role because the speckle phenomenon does not interact with the object during measurement and the results are more precise. Speckle interferometry and speckle photography have been extensively researched and applied in distinct fields as deformations [1–4], temperature measurement [5], or roughness measurement [6]. In the present article we are interested in the speckle photography due to the possibilities of this technique to determine in-plane displacements.
*
Corresponding author. Fax: +34-1336-6952. E-mail addresses: [email protected] (F. Salazar), [email protected] (F. Gasco´n).
0030-4018/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0030-4018(03)01399-3
280
F. Salazar, F. Gascon / Optics Communications 221 (2003) 279–288
In this context, the theory about the double-exposure speckle photography and the displacement of the speckle in a general deformation has been very well explained by Yamaguchi [7,8], even though the goal of that investigation was not to research the limits of the beam to be used in the point-wise filtering technique in any deformation experiment. However, other authors have addressed the investigation to that issue. In this way, Li et al. [9] and Chiang and Li [10], established the upper limit of measurement of speckle displacement based on a fringe-discernibility criterion which combines the effect of the amplitude modulation of Young fringes and the secondary speckle size. More recently, Sj€ odahl [11] has investigated in detail the effect of the finite window used in electronic speckle photography on the cross-correlation algorithm. From another point of view, in a previous paper [3], we demonstrated theoretically and experimentally the maximum value of the scanning beam diameter for measuring displacements in the case of an isotropic thermal deformation. This value is a possible criterion for obtaining valid information of the displacements through the diffraction pattern in the point-wise filtering technique. Now, given the importance of this read-out laser beam technique, our objective is to develop a general theory that allows us to find the upper limit for the interrogation laser beam in any plane general movement rather than restricted to thermal expansion of isotropic samples. This motion consists of a translation, a rotation around an axis perpendicular to the plane of the sample, and a strain. The developed theory is founded partly on a simple speckle model, which we have previously proposed [12].
2. A model of speckle In a previous article [12] we proposed a simple speckle model to explain the minimum diameter of the read-out laser beam in the point-wise filtering method. Taken into consideration that the characteristics of the intensity associated with this speckle pattern may be understood through the image-plane intensity autocorrelation function [13], this model assumes that the intensity of each spot on the photographic plate has the form !2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi J1 ðpD x2 þ y 2 =ðkz0 ÞÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Is ðx; yÞ ¼ 4I0 ; ð1Þ pD x2 þ y 2 =ðkz0 Þ where I0 is the intensity in the center of the spot, J1 is the first-order Bessel function of the first kind, D is the diameter of the lens used to take the photograph, z0 is the distance from the lens to the photographic plate, x and y are the coordinates of a point on the plate, and k is the wavelength of the radiation used. Let N be the number of spots recorded on the photographic plate for the first exposure. If we suppose all spots on the plate are identical to each other except that their centers are randomly distributed, the intensity Is ðx; yÞ on the plane of the photographic plate is I1 ¼
N X
Is ðx; yÞ dðx xi Þdðy yi Þ;
ð2Þ
i¼1
where ðxi ; yi Þ represent random coordinates of points on the plate, d is the DiracÕs delta distribution, and represents the convolution operator. When the sample is deformed, it is assumed that the internal structure of each speckle is maintained as Is ðx; yÞ. Hence the intensity corresponding to the second exposure is given by the function I2 ¼
N X
½Is ðx; yÞ dð x xi Þdð y yi Þ ½dð x fi ðx; yÞÞdð y gi ðx; yÞÞ;
ð3Þ
i¼1
where fi and gi are functions of coordinates x and y which express the in-plane displacement components of the spot centered at the point x ¼ xi , y ¼ yi .
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281
The amplitude transmission function for the developed plate is tðx; yÞ ¼ a bðI1 þ I2 Þs;
ð4Þ
where s is the time exposure, in our case equal for both, and a and b are two constants of proportionality. To obtain information on movements brought about on the surface of the object, we use the point-wise filtering technique. According to this technique, a collimated circular laser beam with diameter d is projected perpendicularly onto the developed plate, centered on the origin (0, 0) in the OXY system of the photograph, Fig. 1. Let us suppose that the read-out laser beam has homogeneous intensity and zero intensity outside it. This means that the beam function may be expressed by a bidimensional pulse distribution, that is, pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! x2 þ y 2 Bðx; yÞ ¼ P ; ð5Þ d where P is the circle-function. The amplitude U 0 ða; bÞ of the diffracted wave by the developed photographic plate is in the Fraunhofer approximation and outside the incident direction [12] U 0 ða; bÞ ¼ 2CbsH ða; bÞ
Nd X
f cos ½pðafi þ bgi Þ exp ½ 2pjðaxi þ byi þ afi =2 þ bgi =2Þg Qða; bÞ;
ð6Þ
i¼1
Fig. 1. An incident laser beam of diameter d is diffracted by a subjective speckle field (double-exposure) recorded on a photographic plate. The pattern is analyzed on the X 0 , Y 0 plane.
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where C is a constant for a given incident laser beam, Nd is the number of spots illuminated by the read-out laser beam, and the spatial frequencies are a ¼ cos hx =k and b ¼ cos hy =k. Moreover, in Eq. (6) appear the functions H ða; bÞ and Qða; bÞ which are the Fourier transforms of Is ðx; yÞ and Bðx; yÞ, respectively. The corresponding mathematical expressions for H and Q are the following: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4k4 z40 kz0 kz0 2 2 H ða; bÞ ¼ 4I0 2 4 P a þb P a 2 þ b2 ð7Þ pD D D which is commonly referred as diffraction halo, and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dJ1 pd a2 þ b2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : Qða; bÞ ¼ 2 a2 þ b2
ð8Þ
Eq. (6) is a complicated convolution due to the finite extent of the laser beam diameter. In this way, the result of convoluting Qða; bÞ with the first term of the second member in the equality (6) is the smoothing of the geometrical shape of the halo H ða; bÞ and of the functions in the summation. However, that convolution may be carried out if we consider that the equivalent width Wf of a function f ðx; yÞ is equal to the reciprocal of the equivalent width WF of its transform F ða; bÞ, that is, Wf WF ¼ 1 [14]. Applying that last relation to functions Bðx; yÞ and Qða; bÞ we can deduce, that if the diameter d of B is wide the equivalent width of Q is small and high. In that case it means that Qða; bÞ is similar to a DiracÕs delta generalized function, and therefore, the aforementioned convolution does not modifiy significantly the form of the first term in Eq. (6). Hence, the expression for the amplitude disturbance is [12]: U 0 ða; bÞ ¼ 2CbsH ða; bÞ
Nd X
f cos½pðafi þ bgi Þ exp ½ 2pjðaxi þ byi þ afi =2 þ bgi =2Þg:
ð9Þ
i¼1
This means the halo function H ða; bÞ is practically zero for ða2 þ b2 Þ1=2 P D=ðkz0 Þ. On the other hand the intensity is shown to be of the following form [12] 2
I KjU 0 j ; where K is a constant.
3. Criterion for the maximun diameter of the scanning beam We are interested in the analysis of the in-plane displacement suffered by different parts of a sample in an experiment. Let us suppose that each point of the plane object is subjected to any small displacement. To analyze the plane movement, we chose the center of beam (x ¼ 0, y ¼ 0) as the kinematic reduction center. Hence, the neighboring points of the object under study will undergo a rigid displacement formed by a translation t, equal to the displacement of the reduction center, plus a rotation Xz around an axis parallel to OZ and passing through that center and a strain eij . As we have chosen the point (x ¼ 0, y ¼ 0) as the reduction center, said point is subject only to a displacement t. If the displacements are linear functions of the coordinates, the displacement of a generic spot i will have as components on OX and OY, respectively: fi ¼ tx Xz yi þ e11 xi þ e12 yi tx þ fi0 ; gi ¼ ty þ Xz xi þ e12 xi þ e22 yi ty þ gi0 ;
ð10Þ
where fi0 and gi0 represent the components of the displacement due to the rotation and strain. Therefore, the modulus of the amplitude (9) is
F. Salazar, F. Gascon / Optics Communications 221 (2003) 279–288
Nd X 0
U ¼ 2jCjbsH ða; bÞ cos p atx þ bty þ afi0 þ bgi0 exp 2pj axi þ byi þ atx =2 þ bty =2 i¼1
0 0 þ afi =2 þ bgi =2 :
283
ð11Þ
The fringes that correspond to the translation will appear if pðatx þ bty Þ is at least p=2 for the cosine to get the first minimum, therefore pðatx þ bty Þ is not small. The development of the cosine of the addition pðatx þ bty Þ þ pðafi0 þ bgi0 Þ is cosðpðatx þ bty ÞÞ, if cosðpðafi0 þ bgi0 ÞÞ 1 and sinðpðafi0 þ bgi0 ÞÞ 0. Then in order to reach the cosine as common factor and fringes to appear, the following inequality must be verified jafi0 þ bgi0 j 1: Taking into account this inequality, we obtain for the intensity Nd X
2 2 exp ½ 2pjðaxi þ byi Þ exp 2pj atx =2 þ bty =2 þ afi0 =2 I 0 ¼ 4KjCj b2 s2 ½ H ða; bÞ i¼1 2
2
þ bgi0 =2 cos p atx þ bty :
ð12Þ
ð13Þ
However, taking into account the translations tx and ty are constants because they correspond to the reduction center, the term exp½2pjðatx =2 þ bty =2Þ, which absolute value is the unity, may be placed as common factor of the summatory, and then expression (13) reduces to 2 Nd X
2
0
2 2 2 2 0 0 I ¼ 4KjCj b s ½ H ða; bÞ exp ½ 2pjðaxi þ byi Þ exp 2pj afi =2 þ bgi =2 cos p atx þ bty : i¼1 ð14Þ Condition (12) indicates some circumstances under which translation t is detectable in the area illuminated by the read-out laser beam, therefore that condition may be considered as a possible criterion to determine the maximum diameter of the read-out beam to obtain information about displacements. With this result we study different cases of interest. 3.1. Pure translation In the case of plane motion of the object its displacements have a constant value t for all the points of the sample, fi0 ¼ gi0 ¼ 0, and Eq. (14) gives Nd 0
X U ¼ 2jCjbsH ða; bÞ cos p atx þ bty ð15Þ exp ½ 2pjðaxi þ byi Þ i¼1 and the intensity formula is simplified to 2 Nd X 2 2 I 0 ¼ 4KjCj b2 s2 ½ H ða; bÞ exp ½ 2pjðaxi þ byi Þ cos2 pðatx þ bty Þ : i¼1
ð16Þ
Therefore the intensity observed in the conical corona of interest in the space of the frequencies is the product of a function that decreases with ða2 þ b2 Þ because of ½H ða; bÞ2 , by the summation of random phase complexes and by a harmonic function of ða2 þ b2 Þ1=2 with frequency jtj=2. That harmonic function represents the familiar Young fringes which are modulated by the secondary speckle noise (random
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summation) that appears within the halo. This means that in pure translation the laser beam diameter can be very large and still produce high visibility fringes. However, it cannot be too small. The read-out beam needs to be large in comparison with the displacement. Otherwise the number of correlated speckles within the illuminated area will be small in relation to the uncorrelated speckles and the contrast will be low. 3.2. Translation, rotation, and strain In the present case, the problem to find the upper limit for the explorer laser beam is more difficult due to the fact that fi0 and gi0 are not zero. Hence, in Eq. (11) we cannot reach the cosine as a common factor and consequently the Young fringes are not noticeable. However, we can give a solution if we consider the inequality (12). 2 The inequality (12) must be verified for all possible values of a and b allowed by ½H ða; bÞ . It means that a and b can become so large that the direction they represent points to the border of the halo, that is, jaj 6 D=ðz0 kÞ and jbj 6 D=ðz0 kÞ. Let us study the inequalities 0
af þ bg0 6 af 0 þ bg0 ¼ jaj f 0 þ jbj g0 6 D f 0 þ g0 : ð17Þ i i i i i i i kz0 i Let us suppose that the terms of the displacements corresponding to deformations and rotations are small enough to verify 0 0 f þ g z0 k : ð18Þ i i D Then (17) gets jafi0 þ bgi0 j 1, and (12) is verified. Hence, the inequality (18) becomes a sufficient condition. Eq. (18) may be written as kz0 : D If that addition is much less than kz0 =D, it must be also true: je11 xi þ e12 yi Xz yi j þ je12 xi þ e22 yi þ Xz yi j
je11 xi þ e12 yi Xz yi j
kz0 ; D
ð19Þ
ð20Þ
kz0 : ð21Þ D These inequations have as solution a set of points inside a region bounded by four straight lines. To give a quantitative criterion for the maximum diameter of the scanning beam, we replace the inequalities (20) and (21) by theirs respective equalities, obtaining as limit the zone represented in Fig. 2. The interior of the region and more precisely its central zone, represents the set of spots that verify the inequality (19), they become all the set adequate enough for the measurement of t. Upon illuminating the double-exposure speckle photograph with the read-out laser beam, all the recorded speckles whose xi , yi are within the zone will contribute appropriately to the Young fringes. Then the diameter of the read-out should be less than two times the minimum distance from the center (0,0) to each straight lines. Therefore, the diameter d must verify the inequality 9 8 > > = < 2z0 k=D 2z0 k=D ð22Þ d min qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : > 2> : e2 þ ðe12 Xz Þ2 e2 þ ðe12 þ Xz Þ ; je12 xi þ e22 yi þ Xz yi j
11
22
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285
Fig. 2. Four straight lines bound the region where the translation can be detected. The interrogation laser beam must be inside them.
In short, the spot recorded at ðxi ; yi Þ during the first exposure and the same spot but shifted and recorded during the second exposure form a pair. Since in general there are many pairs within the area where the scanning beam intersects with the photographic plate, we could obtain different maximum and minimum directions for each of these pairs. This is not so, however, for the displacements that the spots undergo within the limits of the read-out laser beam, since these displacements are approximately equal. This means that any pair within the area that the scanning beam intersects, fulfilling the inequality (22), will produce almost identical maximum and minimum directions. Once the upper limit is established for the read-out laser beam, we illuminate the photographic plate with a laser beam narrow enough to fulfill the condition of Eq. (22). Next, we measure the directions of the diffraction maxima (Young fringes) and calculate displacement suffered by the sample point. Effectively, this is sufficient enough to analyze the phase of the cosine of the expression (14) for a point within the area illuminated by the scanning beam. The directions of the intensity maxima will be given by the classical equation pðatx þ bty Þ ¼ np, or ð23Þ jDrj sin hn cos n ¼ nk; qffiffiffiffiffiffiffiffiffiffiffiffiffi where jDrj ¼ tx2 þ ty2 represents the displacement module, hn is the angle formed by the OZ axis and the direction of observation of the nth order maximum, and n is the angle between the displacement direction and the direction defined by the center of the diffraction halo O0 and the observation point P (Fig. 1). When the observation point P is taken on the straight line through O0 and perpendicular to the nth intensity fringe, Eq. (23) is transformed in the well known jDrj sin hn ¼ nk. 3.3. Isotropic deformation As inequality (22) gives a possible criterion to know the maximum value of the interrogation beam diameter to obtain information of the displacements, it should be applicable to the particular case of an isotropic deformation. In an isotropic deformation e11 ¼ e22 ¼ e, and e12 ¼ 0, hence (22) gives 2z0 k=D d qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : e2 þ X2z
ð24Þ
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In the case that no rotation is present or it is small in relation to e, Eq. (24) becomes
2 zD0 k d jej
ð25Þ
which agrees, approximately, with the first criterion done in [3] for isotropic bodies ! 1:2ðzD0 kÞ rs ¼ d< ; jej jej where rs is the speckle diameter.
4. Experimental results In order to illustrate and verify the developed theory, we will apply it to some particular cases. We used typical experimental speckle photography devices (sample, lens, and photographic plate), with magnification factor M ¼ 1, k ¼ 632:8 nm, and F number ¼ 3:5, thus z0 =D ¼ 7:0. 4.1. Pure translation We will first carry out a pure in-plane translation experiment for which Xz ¼ eij ¼ 0. The translation was brought in the sample with the value jDrj ¼ tx ¼ 20 lm. As fi0 ¼ gi0 ¼ 0, hence the inequality (22) becomes d 1;
ð26Þ
and we see that this inequality is always fulfilled, independent of t value. An analogous result may be obtained through Eq. (12), that is, 0 1 (it is always true). This means that in a pure translation, we are going to have sharp Young fringes no matter how great the diameter of the read-out laser beam is. This is logical because, as it has been stated, inequalities do not depend on the translation. To illustrate these results Fig. 3 shows the diffraction patterns of scanning beam diameters corresponding to 1.0, 2.0, and 3.3 cm, respectively (Figs. 3(a)–(c)). In all cases the Young fringes may be measured due to the high contrast of the pattern. 4.2. Rotation A slab of a material was subjected to a rotation of Xz ¼ 350 lrad around an instantaneous rotation axis orthogonal to the slab. An area of the photographic record was illuminated by the circular read-out beam. The
Fig. 3. Diffraction patterns for diameters of 1.0 cm (a), 2.0 cm (b), and 3.3 cm (c) in a pure translation. Note the high contrast of the fringes.
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287
Fig. 4. YoungÕs fringes in case of rotation. The diameters used were 1.0 cm (a), 1.8 cm (b), and 2.4 cm (c), respectively. It can be observed the lost of visibility of YoungÕs fringes when the diameter of the laser beam is increased.
center O of the illuminated area is r0 ¼ 5 cm apart from the instantaneous rotation axis. If we take the point O as the reduction center, the displacement of that point O can be regarded as a translation of value 17.5 lm. Condition (22) yields d 2:6 cm: Figs. 4(a)–(c) show the diffraction patterns formed for values of d equal to 1.0, 1.8, and 2.4 cm, respectively. For the value of the beam near the limit a blurred pattern is obtained, not being possible to assure the correct measurement of the displacement. 4.3. Anisotropic thermal strain In this section, a case involving a homogeneous thermal strain of an anisotropic material is studied to demonstrate the developed theory. Three mutually orthogonal slabs are cut from a block of anisotropic material (slate) [15]. The slabs are kept at a homogeneous temperature T1 during the first exposition and at a temperature T2 , also homogeneous, during the second exposition. One side of each slab will rest on a flat support. Let aij be the thermal linear expansion tensor of the sample. The strain originated by the increase in temperature (T2 T1 ) is homogeneous and is defined by the strain tensor eij ¼ aij ðT2 T1 Þ. Strain Table 1 a11 a22 a33 a12 a13 a23
6:9 ðlKÞ1 5.7 5.1 0.9 1.2 )0.3
t31 t32 t21 t23 t12 t13
)267.6 (lm) 114.7 169.9 )1181.6 325.5 )184.0
Xx Xy Xz
47.6 (lrad) 90.4 0.0
aI aII aIII
4.2 ðlKÞ1 5.7 7.7
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Fig. 5. The pattern corresponds to diameters of 1.0 cm (a), 2.8 cm (b), and 3.3 cm (c), respectively, for the case of a homogeneous thermal deformation of an anisotropic plate of slate. In (a) the maxima and minima are visible and its distance is measurable. However, the correct location of the maxima of intensity is difficult in (b) due to the poor visibility of the fringes. In (c) measurement of the displacement is not possible .
analysis gives aij , its principal eigenvalues aI , aII , aIII , the translation components ti , and the rotation components Xi . These values are shown in Table 1. The optical arrangement was similar to the aforementioned in paragraph 4.1., but with a lens of F number F ¼ 3:6. Applying Table 1 to the slab corresponding to a11 , a22 , and a12 , for an increase of temperature of 30.8 °C we obtain e11 ¼ 213 106 ; e22 ¼ 176 106 ; e12 ¼ 28 106 and Eq. (22) gives d 4:2 cm: The distance measurement between fringes for the value d ¼ 1:0 cm is possible (Fig. 5(a)). In the case of d ¼ 2:8 cm (Fig. 5(b)), it begins to be difficult to locate the maxima of intensity, and when the diameter d reaches the value 3.3 cm (Fig. 5(c)), the visibility decreases and it is not possible to assure the location of maxima and minima. Hence, the measure of the displacement is not possible. Acknowledgements We are in debt to an anonymous referee for useful suggestions and comments on the paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
R.P. Khetan, F.P. Chiang, Appl. Opt. 15 (1976) 2205. F.P. Chiang, J. Adachi, R. Anastasi, J. Beaty, Opt. Eng. 21 (1982) 379. F. Salazar, F. Gasc on, Opt. Commun. 103 (1993) 235. M. Sj€ odahl, Appl. Opt. 34 (1995) 5799, and references therein. C. Shakher, A.K. Nirala, Appl. Opt. 33 (1994) 2125. C.J. Tay, S.L. Toh, H.M. Shang, J. Zhang, Appl. Opt. 34 (1995) 2324. I. Yamaguchi, Opt. Acta 28 (1981) 1359. I. Yamaguchi, J. Opt. Soc. Am. A 1 (1984) 81. D.W. Li, T.B. Chen, F.P. Chiang, J. Opt. Soc. Am. A 2 (1985) 657. F.P. Chiang, D.W. Li, Opt. Eng. 24 (1985) 937. M. Sj€ odahl, Appl. Opt. 36 (1997) 2875. F. Gasc on, F. Salazar, Opt. Commun. 172 (1999) 77. J.W. Goodman, Statistical properties of laser speckle patterns, in: J.C. Dainty (Ed.), Laser Speckle and Related Phenomena, Topics in Applied Physics of Topic in Applied Physics, vol. 9, Springer, Berlin, 1984, pp. 40–42, Chapter 2. [14] R.N. Bracewell, in: The Fourier Transform and its Applications, McGraw-Hill International Editions, New York, 1986, p. 151. [15] F. Salazar, F. Gasc on, Opt. Commun. 119 (1995) 361.