Fast Convergent Conditions of an Iterative Method for Retrieving the

CHINESE JOURNAL OF PHYSICS

VOL. 48, NO. 3

June 2010

Fast Convergent Conditions of an Iterative Method for Retrieving the Kerr-Effect-Aberrated Phase Profile of a Laser Beam Pao-Keng Yang and Wen-Tse Shih Department of Opto-Electronic System Engineering, Ming-Hsin University of Science and Technology, Hsinchu, Taiwan 304, R.O.C. (Received September 22, 2009) An iterative method based on the Huygens-Fresnel principle was used to retrieve the Kerr-effect-aberrated phase profile from the measured beam intensity distributions in two longitudinally separated cross sections. The time of iteration was found to depend on the sampling cross sections of the light beam. We first analyzed the unaberrated TEM00 Gaussian beam with a parabolic phase profile by the q-parameter transformation and found that the convergence will be accelerated if two cross sections are located in regions with small radii of curvature in the wavefront. We also show that the fast convergent conditions are applicable to retrieving the Kerr-effect-aberrated phase profile of a laser beam. If the nonlinear medium is placed at the waist’s position (z = 0), the choice with one cross section located at z = 21 z0 and the other at z = 32 z0 , leading to fast convergence, is recommended for retrieving the Kerr-effect-aberrated phase profile, which can be used to give information on the optical nonlinearities of materials. Using the suggested cross sections can greatly save time in the iterative calculation. PACS numbers: 41.85.Ew, 42.25.Kb, 42.30.Rx

I. INTRODUCTION

The Kerr effect is characterized by an intensity-dependent index of refraction when the medium is exposed to an intense optical field. As light propagates through a nonlinear medium, the Kerr effect leads to a phase delay which is largest on the beam axis and smaller outside the axis. Consequently, the wavefront is deformed and the laser light is focused. This effect is called self-focusing, which has important implications for passive mode-locking of lasers [1–3] and for the optical damage of media. The Z-scan [4, 5] is a popular and sensitive technique for measuring the strength of the Kerr nonlinearity of an optical material. Instead of directly measuring the deformed phase profile, Z-scan measures the on-axis intensity at some point behind the focus as a function of the position when a plate of the sample under test is translated through the beam waist. The fitting curve is derived under some assumptions on the incident beam. In the original formulation, the Z-scan technique assumes that the incident beam is Gaussian. However, the lasers used in laboratories rarely generate perfect Gaussian beams. The most natural way to characterize the Kerr effect is to measure the induced aberration in the phase profile after the laser beam passes through a sample. Practically, it is more complicated to measure the phase profile than the intensity distribution. To directly measure the phase profile of a laser beam will involve sophisticated 2-dimensional (2-D) interferometric techniques [6, 7]. Owing to the rapid progress in the technology of CCD and CMOS cameras, measuring the 2-D light

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intensity distribution becomes an easy task. A number of different iterative algorithms have been developed to retrieve the phase profile from pure intensity measurements [8–15]. Most of these methods retrieve the phase profile from the information involving the Fourier intensity measurement, which needs a lens or the propagation of the optical field to the farfield to conduct the Fourier transformation. Matsuoka and Yamakawa presented an iterative method for calculating the phase profile of a laser beam if the intensity distribution had been measured at two longitudinally separated cross sections 16]. Their idea works without additional optics and therefore looks more attractive than other methods which require imaging systems. However, their results showed that the procedure converged slowly after more than 500 times of iteration. In this paper we study how the locations of two cross sections affect the converging speed and look for the conditions to accelerate the convergence for retrieving the Kerr-effect-aberrated phase profile.

y1

y2 x1

x2 z CS2

CS1

d FIG. 1:

II. PHASE RETRIEVAL FROM AN ITERATIVE ALGORITHM USING THE HUYGENS-FRESNEL INTEGRAL

Suppose the optical field distribution function is u1 (x1 , y1 ) in the cross-section 1 (CS1) and u2 (x2 , y2 ) in the cross-section 2 (CS2), as shown in Fig. 1. The two cross sections are separated by a distance of d. The optical field is a complex function which can be rewritten in a form using the magnitude and phase, i.e. u1 (x1 , y1 ) = |u1 (x1 , y1 )| eiφ1 (x1 ,y1 ) in CS 1 and u2 (x2 , y2 ) = |u2 (x2 , y2 )| eiφ2 (x2 ,y2 ) in CS 2. The magnitude functions, |u1 (x1 , y1 )| and |u2 (x2 , y2 )|, can be determined by 2-D intensity measurements; however, the phase functions, φ1 (x1 , y1 ) and φ2 (x2 , y2 ), remain undetermined. According to the Huygens-Fresnel principle, the optical field in CS2 can be obtained by propagating the field in CS1 to CS2. This can be written in an integral form [17] of 2π Z Z ∞ u1 (x1 , y1 )ei λ r i ∞ χ(θ), (1) dx1 dy1 u2 (x2 , y2 ) = λ −∞ r −∞ p where r ≡ (x2 − x1 )2 + (y2 − y1 )2 + d2 and χ(θ) denotes the obliquity factor. On the other hand, the optical field in CS1 can be obtained by back-propagating the field in CS2

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measure |u1(x1,y1)| and |u2(x2,y2)| guess f1(x1,y1)=0 evaluate u1(x1,y1) calculate u2(x2,y2) using the new u1(x1,y1) from Eq. (1) to get new |u2(x2,y2)| and f2(x2,y2)

use calculated f1(x1,y1) and measured |u1(x1,y1)| to construct a new u1(x1,y1) N

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use calculated f2(x2,y2) and measured |u2(x2,y2)| to construct a new u2(x2,y2)

calculate u1(x1,y1) using the new u2(x2,y2) from Eq. (2) to get new |u1(x1,y1)| and f1(x1,y1)

FIG. 2: Flow chart of the iterative procedure using the Huygens-Fresnel integral.

to CS1. This is given by i u1 (x1 , y1 ) = λ

Z

∞

dx2 −∞

Z

2π

u2 (x2 , y2 )e−i λ r dy2 χ(θ). r −∞ ∞

(2)

The iteration procedure is detailed by the flow chart in Fig. 2. Given the magnitude functions of |u1 (x1 , y1 )| and |u2 (x2 , y2 )| obtained by the 2-D intensity measurements, we first guess φ1 (x1 , y1 ) = 0 (corresponding to a flat wavefront) and then propagate u1 (x1 , y1 ) to CS2 to get a new u2 (x2 , y2 ). Keeping the new φ2 (x2 , y2 ), replacing |u2 (x2 , y2 )| by the measured |u2 (x2 , y2 )|, and back-propagating u2 (x2 , y2 ) to CS1, a new u1 (x1 , y1 ) is obtained. Substituting the φ1 (x1 , y1 ) calculated from the new u1 (x1 , y1 ) for the initially guessed φ1 (x1 , y1 ) and repeating the above procedure, we can get the phase functions in the two cross sections as long as the procedure is convergent (i.e., both calculated |u1 (x1 , y1 )| and |u2 (x2 , y2 )| approach the measured results).

III. RESULTS AND DISCUSSION

III-1. TEM00 Gaussian beam Given arbitrary intensity distributions in two cross sections, it is time-consuming to carry out the iteration procedure in Fig. 2. To investigate the guiding rules on how to choose the locations of the two cross-sections for fast convergence, we first analyzed a TEM00 Gaussian beam systematically. The optical field in a TEM00 Gaussian, propagating


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measure w1,w2 guess R1=infinity calculate q1

calculate q2=q1+d to get new w2 and R2

using calculated R1 and measured w1 to construct a new q1 N

Y

end

calculated w2 =measured w2

calculated w1 =measured w1

Y

end

N

using calculated R2 and measured w2 to construct a new q2

calculate q1=q2-d to get new w1 and R1

FIG. 3: Flow chart of the iterative procedure using the q-parameter transformation for a TEM00 Gaussian beam.

along the z-direction with a waist spot-size of w0 , has the form u(r, z) =

r2 r2 A1 −ik 2q(z) w0 r2 e = iA1 eiφ0 e− w2 · e−ik 2R , q(z) w

(3)

where φ0 = tan−1 (λz/πw02 ), w, and R are the constant phase shift, the radius of the spot size, and the radius of curvature of the wavefront. The parameters R and w constitute the λ complex parameter q by 1q = R1 − i πw 2 . In some cross section in a TEM00 Gaussian beam, the phase profile is characterized by R and the intensity distribution by w. By the ABCD law of the q-parameter transformation [18], Eq. (1) and Eq. (2) can be simply replaced by Eq. (4) and Eq. (5), respectively: q2 = q1 + d,

(4)

q1 = q2 − d.

(5)

The iteration procedure using the q-parameter transformation is shown in Fig. 3. Fig. 4 depicts results obtained by choosing two cross sections to be around the beam waist. The results show that the proposed procedures converge after a few times of iteration. Fig. 5 shows the results when the two cross sections are chosen to be far away from the beam waist and on the same side of the waist. The two cross sections in Fig. 5 are kept at the same separation as those in Fig. 4. The results in Fig. 5 show that the proposed procedures do not converge efficiently. Comparing Fig. 4 with Fig. 5, the locations of the two selected cross-sections will affect the convergence of iteration. Fig. 6 shows the results

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FIG. 4: Iterative results for a Gaussian beam with parameter conditions chosen to be λ = 1.06 µm, w0 = 0.1 mm, w1 = 0.1967 mm, w2 = 0.1426 mm, z0 = 29.5263 mm, and d = 80 mm. The two cross sections are chosen to be around the waist and the waist is between the two cross sections.

for the beam with a large waist spot-size, w0 (i.e., a low-divergence beam). The conditions for Fig. 6 are the same as those for Fig. 4 except for the difference in the waist spot-size. Comparing Fig. 4 with Fig. 6, it seems that the focusing of the laser beam will affect the convergence of the iteration, but later analyses will show that the choices for cross-sections’ locations for fast convergence is independent of the waist spot-size. To specify why some choices on the cross sections exhibit faster convergence than others, we first look at the variation in the radius of curvature of the wavefront along the propagating direction for a focused Gaussian beam in Fig. 7, where the behavior of 1/R versus z was drawn. In the regions close to the waist (z = 0) or far away from the waist (z −z0 or z z0 where z0 = πw02 /λ is referred to as the confocal parameter), 1/R approaches zero, and the radius of curvature of the wavefront goes to infinity. Thus, we can approximate the beam by a plane wave whose phase profile can be represented by a flat plane in these regions. A plane wave tends to retain the phase profile in the propagation from one cross-section to the other, that is, φ1 (x, y) ∼ = φ2 (x, y). To accelerate the convergence,


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FIG. 5: Iterative results for a Gaussian beam with parameter conditions chosen to be λ = 1.06 µm, w0 = 0.1 mm, w1 = 0.6847 mm, w2 = 0.9536 mm, z0 = 29.5263 mm, and d = 80 mm. The two cross sections are chosen to be far away from and on the same side of the beam waist.

we shall avoid choosing two cross sections between which the light propagates like a plane wave. This will occur in two cases: [i] the two cross sections are far away from the waist and the separation is smaller than the radius of curvature; [ii] the two cross sections are around the waist and the separation is much smaller than the depth of focus (d 2z0 ). The beam parameters used in Fig. 5 satisfy case [i] and those in Fig. 6 satisfy case [ii]. This explains why the converging speed in Fig. 5 and Fig. 6 is slower than that in Fig. 4. In the regions around z = ±z0 (marked by region 1 and region 2 in Fig. 7), the wavefront has a small radius of curvature, and the off-axis light-rays will deviate from the z-direction to some extent. The large ray deviation implies that the phase profile will change significantly during propagation, and this will be helpful to convergence. In order to avoid plane-wavelike propagation, one can choose the two cross sections to be both in region 1 or region 2 or one in region 1 and the other in region 2. For quantitative comparison, we first define the criterion for the number of iteration steps needed to achieve convergence, NC . The iteration is said to be converged and can stop when both δw1 /w1 and δw2 /w2 become

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FIG. 6: Iterative results for a Gaussian beam with parameter conditions chosen to be λ = 1.06 µm, w0 = 0.4 mm, w1 = 0.4022 mm, w2 = 0.4008 mm, z0 = 472.42 mm, and d = 80 mm. The two cross sections are chosen to be around the waist but for a low-divergence beam, and the waist is between the two cross sections.

smaller than 0.1% after NC times of iteration. δw1 and δw2 are defined as the difference between the calculated and measured spot sizes in CS1 and CS2, respectively. Table 1 shows the values of NC in several cases. After futher analyses, the iteration with CS1 located at z = −z0 and CS2 at z = z0 (corresponding to case 5 in Table 1) does not lead to the fastest convergence. This can be ascribed to the large difference in the radius of curvature between the initially guessed value and that at z = z0 . Since the wavefront at z = z0 is more curved than that in other positions, an initially flat wavefront will take more times of iteration to converge to a strongly curved wavefront. Four choices on the sampling cross sections for faster convergence are found to be (CS1,CS2)=(L1,L2), (L1,L4), (L2,L3), or (L3,L4) where L1, L2, L3, and L4 are four locations marked by dotted circles in Fig. 7. To be more precise, in the following calculation, we use z = −3z0 /2 for L1, z = −z0 /2 for L2, z = z0 /2 for L3, and z = 3z0 /2 for L4. Case 2, using the parameters as those in Fig. 5, converges very slowly and has a large NC (= 837). The outcome from the suggested four

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region 2

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FIG. 7: Behavior of 1/R versus z for a Gaussian beam. The parameters are the same as those used in Fig. 4. TABLE I: Number of the iteration steps for convergence in various cases with different choices of the two cross sections and beam parameters. A Nd:YAG laser with λ = 1.064µm was chosen for simulation. Cases 5–8 represent the four suggested choices: Case 5: (CS1,CS2)=(L1,L2), Case 6: (CS1,CS2)=(L1,L4), Case 7: (CS1,CS2)=(L2,L3), Case 8: (CS1,CS2)=(L3,L4). R1,0 denotes the initial guess of R1 in the iterative procedure. Case w0 (mm) 1 (FIG.4) 0.1 2 (FIG.5) 0.1 3 0.1 4 (FIG.6) 0.4 5 0.1 6 (L1,L2) 0.1 7 (L1,L4) 0.1 8 (L2,L3) 0.1 9 (L3,L4) 0.1 10 (L1,L3) 0.1 11 (L3,L4) 0.4

z0 (mm) 29.5263 29.5263 29.5263 472.42 29.5263 29.5263 29.5263 29.5263 29.5263 29.5263 472.42

w1 (mm) 0.1967 0.6847 0.6847 0.4022 0.1414 0.1803 0.1803 0.1180 0.1118 0.1118 0.4472

w2 (mm) 0.1426 0.9536 0.9536 0.4008 0.1414 0.1118 0.1803 0.1180 0.1803 0.1803 0.7211

d (mm) R1,0 (mm) 80 ∞ 80 ∞ 80 -10 80 ∞ 59.0525 ∞ 29.5263 ∞ 88.5787 ∞ 29.5263 ∞ 29.5263 ∞ 59.0525 ∞ 472.42 ∞

R1 (mm) -98.6844 205.0485 -33.4322 -4760.0323 -61.3324 -63.9765 -144.2826 -73.8068 73.8435 -74.9738 1185.7363

R2 (mm) 125.1529 283.8034 46.6014 6843.0635 61.5571 -73.8519 144.3732 73.8145 64.0073 64.3134 1024.2258

NC 13 837 991 26 149 6 7 3 5 35 5

choices on the sampling cross sections (cases 6–9) converges very fast after only 3–7 times of iteration. Case 8, corresponding to the choice of (CS1,CS2)=(L2,L3) where CS1 and CS2 are symmetric around the waist and at a distance of d = z0 , has the minimum iterative times (NC = 3). It is interesting that the choice of (CS1,CS2)=(L1,L3) with NC = 35 (see case 10 in Table 1) does not converge as fast as the above four choices. Case 9 and case 11 correspond to the same choice as (CS1,CS2)=(L3,L4) but with different values in the waist spot-size. The results show that the converging speed is independent of the waist spot-size,

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once the two cross sections are located at one of the four choices. We also found that the exchange of CS1 for CS2 in an asymmetric configuration will not affect the converging speed. Case 3 in Table 1 with the same parameters as case 2 but with a different guess on the initial value of R1 will lead to different converging values of R1 and R2 . In case 3, the beam is tightly focused and the waist is located at some position in the region between CS1 and CS2. In case 2, the waist is located outside the region between CS1 and CS2. Therefore, in addition to the two intensity profiles, the information about the waist’s position or waist spot-size is necessary to uniquely determine the values of R1 and R2 . III-2. Non-Gaussian beam with nonlinear-Kerr-effect-induced phase distortion The above discussions are only restricted to a Gaussian beam. Note that the procedure based on Eq. (1) and Eq. (2) is not restricted to the TEM00 Gaussian beam and can be applied to a laser beam with arbitrary intensity and phase distributions. To show that the fast convergent conditions for a TEM00 Gaussian beam can also be applied to a non-Gaussian beam with phase distortion due to the nonlinear Kerr effect, we use the iteration procedure proposed in Fig. 2 to calculate the distorted phase profile resulting from the nonlinear Kerr effect. The configuration is shown in Fig. 8 where a Kerr medium (sample) is located at the waist’s position. Among the four suggested choices on the two cross sections, only the condition of (CS1,CS2)=(L3,L4) can be chosen to retrieve the phase profile because the two cross sections should be at the same side of the beam waist. We assumed that the incident laser beam is Gaussian and has the same beam parameters as those used in case 9 of Table 1. After the laser beam passes through the sample, an intensity-dependent phase shift of ∆φ(r) = ∆φ0 exp(−2r 2 /w02 ) will be introduced owing to the third-order nonlinearity. We added the nonlinear phase shift at the waist’s position and then propagated the optical field from the waist to CS1 and CS2 to get the two intensity profiles for iteration. The results with ∆φ0 = 0.4π are shown in Fig. 9. Due to the cylindrical symmetry, we represent u(x, y) by u(r) and φ(x, y) by φ(r). The function of phase is restricted in the range of (−π, π). In Fig. 9, |u1 (r1 )|, the square root of the intensity distribution in CS1, deviates from a Gaussian distribution and has a shallow dip at r = 0. The results in Fig. 9 show that the phase profile and intensity distributions converge rapidly after 6 times of iteration. This suggests that the fast convergent conditions are also applicable to retrieving the Kerr-aberrated phase profile. The initial guess on the phase profile in CS1 is assumed to be a flat plane, which will lead to a solution with the beam waist located outside the region between CS1 and CS2 for a Gaussian beam (referring to the case 9 with (CS1,CS2)=(L3,L4) in Table 1 where R1 and R2 have the same sign). This is consistent with the configuration shown in Fig. 8. With the retrieved phase in Fig. 9, the optical field in CS1 in Fig. 8 can be backpropagated to the waist’s position to directly obtain the nonlinear phase shift, ∆φ(r) = ∆φ0 exp(−2r 2 /w02 ), which can be used to determine the nonlinear optical properties of materials. In the technique of the Z-scan method, a small nonlinear phase change ∆φ0 1 and small thickness of the sample are assumed. The results in Fig. 9 with ∆φ0 = 0.4π > 1 show that the iterative phase-retrieval procedure will converge even when the assumption of


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Sample

CS1 CS2 FIG. 8: Configuration for analyzing the phase distortion resulting from a Kerr medium located at the waist.

small nonlinear phase change is not valid. The 2-D intensity measurements in two separated planes can be taken at one time if an aberration-free beam-splitter is inserted in between the sample and CS1 in Fig. 8. However, the data in the Z-scan technique is taken point by point as the test sample is translated along the z-direction. Although we assumed an infinitesimal thickness for the sample and a Gaussian profile for the incident intensity in calculating the results in Fig. 9, the proposed phase-retrieval procedure is still applicable to the incident beam with a non-Gaussian intensity profile and to the sample with a finite thickness. In conclusion, we present a choice on the locations of two intensity cross sections for fast convergence when an iterative algorithm based on the Huygens-Fresnel principle is used to retrieve the Kerr-effect-aberrated phase profile, which can be used to determine the nonlinear optical properties of materials. If the two cross sections are chosen to be located at some positions in the regions with small radii of curvature, the convergence will be greatly sped up. This can save time in the iterative calculation. If the nonlinear medium is placed at the waist’s position (z = 0), the choice with one cross section located at z = 21 z0 and the other at z = 32 z0 is recommended for retrieving the Kerr-effect-aberrated phase profile. With this choice, the calculation can converge after only a few times of iteration.

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FIG. 9: Results obtained by using the iterative procedure in Fig. 2, which directly applies the Huygens-Fresnel integral to analyzing the configuration shown in Fig. 8. The condition of (CS1,CS2)=(L3,L4) was chosen to retrieve the phase profile.

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