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A novel laser intensity function and its fitting method
Optics & Laser Technology 47 (2013) 183–188
Contents lists available at SciVerse ScienceDirect
Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
A novel laser intensity function and its fitting method Yuan Wen-quan a,n, Gong Yan a,nn, Qin Shuo b a b
State Key Laboratory of Applied Optics, Changchun Institute of Optics, Fine Mechanics and Physics, The Chinese Academy of Sciences, Changchun 130033, China The Air-force Aviation University, Changchun 130021, China
a r t i c l e i n f o
abstract
Article history: Received 17 April 2012 Received in revised form 5 September 2012 Accepted 7 September 2012 Available online 12 October 2012
A novel transcendental function is presented to describe laser intensity distribution instead of Gaussian functions, and its fitting method is given in this paper. The transcendental function based on the Gaussian functions is fitted with multivariate optimization method, which is carried out with direction search method and least square method in practice. Four kinds of distribution laser intensity are fitted with the transcendental function in this article, validity and effectiveness of the fitting method are verified by these numerical simulations. Crown Copyright & 2012 Published by Elsevier Ltd. All rights reserved.
Keywords: Intensity distribution Transcendental function Multivariate optimization
1. Introduction Thermal lensing effects often occur in modern lasers such as high-power lasers, lithography objectives and so on, lenses in these apparatus will have deformations, stress and strain due to laser heating [1,2], the refractive indices of optical materials will change [3,4], stress birefringence [5,6] and self-focusing [7] also occur. Since optical performance of these apparatus is seriously affected due to thermal lensing effects, the effects should be considered at early time of optical design. And in thermal analysis of thermal lensing effects, the raw discrete data of laser intensity distribution on surfaces of lenses should be analyzed by optical software at first. Then, the raw discrete data will be fitted as laser intensity functions. Finally, the functions will be multiplied with material absorption factor, and applied on surfaces of lenses as thermal load. Usually, laser intensity functions were supposed to be Gaussian forms which include Gaussian form [8–11], sub-Gaussian form [12], super-Gaussian form [13–15] and ellipticalGaussian form [16,17] in the past research. But laser intensity distribution on surfaces of lenses does not often obey Gaussian forms due to refraction of aspheric or thick lens [18–20], so it seems unsuitable to fit the raw discrete data of laser intensity distributed on surfaces of lenses with Gaussian functions. The other ways to describe laser intensity in nonlinear optics are Gaussian decomposition method (GD) [21–23] and Fresnel–Kirchhoff method (FK) [24–26], but their mathematical forms are complicated for thermal analysis.
n
Corresponding author. Tel.: þ86 431 86708163. Principal corresponding author. Tel.: þ 86 431 86178996. E-mail addresses: [email protected] (Y. Wen-quan), [email protected] (G. Yan). nn
In this article, a novel transcendental function is proposed to describe distribution of laser intensity on lenses, and its form includes Gaussian forms. Since the function is unable to be fitted directly, we solve parameters of the function with multivariate optimization method, and get the laser intensity function finally.
2. Raising of a transcendental laser intensity function In the research related with laser beams, laser intensity distribution on surfaces of optical components was usually described as Gaussian form [9–11], which is shown as follows: 2
Iðx,yÞ ¼ I0 e2ððx
þ y2 Þ=w2 Þ
,
ð1Þ
where I0 denotes energy density of the laser beam at the center (W/ m2), and w is the radius of laser beam. The geometric interpretation is depicted in Fig. 1. While, several other researchers thought that if add a factor ‘l’ to Gaussian function, the new function can be described laser intensity distribution better [12–15]. And its form is described as follows: 2
Iðx,yÞ ¼ I0 e2ððx
þ y2 Þ=w2 Þl
,
ð2Þ
where l is a positive real factor. And when l o 1, it is called subGaussian distribution; when l ¼ 1, it is called Gaussian distribution, which has the same form as Eq. (1); when l 41, it is called superGaussian distribution, and when l-1, Eq. (3) describes uniform distribution. In addition, laser intensity distribution described by Eq. (1) is circular, an elliptical-Gaussian function was raised to describe laser distribution in some articles [16,17], and its form is described as follows: 2
Iðx,yÞ ¼ I0 e2ððx
0030-3992/$ - see front matter Crown Copyright & 2012 Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.optlastec.2012.09.009
=u2 Þ þ ðy2 =v2 ÞÞ
,
ð3Þ
184
Y. Wen-quan et al. / Optics & Laser Technology 47 (2013) 183–188
where u and v are waist radius of laser beam. When u¼v, it has the same form as Eq. (1). Obviously Eq. (1) is a special kind of Eqs. (2) or (3), therefore Eqs. (2) and (3) can be more suitable for describing laser intensity distribution. And it also can be seen from structure of above equations that the indices which is in bracket of exponent are all quadratic, here we use two positive real factors instead of them, and construct a transcendental function as Eq. (4) to describe distribution of laser intensity: j
j
k
Iðx,yÞ ¼ I0 e2ðð9x9 =u Þ þ ð9y9
=vk ÞÞl
,
ð4Þ
where j, k and l are all positive real factors. When j¼k¼2 and l¼ 1, Eqs. (4) and (3) have the same form; when j¼k¼2 and u¼v¼w, Eqs. (4) and (2) have the same form. In other words, function described by Eq. (4) includes Eqs. (1)–(3), therefore it is more applicable.
3. Solving of parameters in the transcendental function 3.1. Transformation of the transcendental function The nonlinear function should be transformed to a linear one before its fitting with least square method. Since Eq. (4) is nonlinear, we transform it to a new form as follows: 1=l j k 1 ln Iðx,yÞ ¼ 9x9 þ 9y9 , 2 I0 uj vk
ð5Þ
As can be seen from Eq. (5), it is unable to fit the equation with least square method directly for unknown terms on each side of it.
3.2. Solving method of the parameters
Fig. 1. Schematic diagram of laser intensity distribution on optical component.
The standard deviation between the raw discrete data of laser intensity and its fitting function Iðx,yÞ described by Eq. (4) can be
Fig. 2. Schematic diagram of computing process in Example 3.
Fig. 3. Laser intensity distribution of Example 1(a) grid of raw discrete data; (b) grid of fitting function; (c) contrast between raw discrete data and fitting function at x¼ 0; (d) contrast between raw discrete data and fitting function at y¼ 0.
Fig. 4. Laser intensity distribution of Example 2(a) grid of raw discrete data; (b) grid of fitting function; (c) contrast between raw discrete data and fitting function at x¼ 0; (d) contrast between raw discrete data and fitting function at y¼ 0.
186
Y. Wen-quan et al. / Optics & Laser Technology 47 (2013) 183–188
written as: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n u1 X j k j k l ½I0 e2ðð9xi 9 =u Þ þ ð9yi 9 =v ÞÞ Iðxi ,yi Þ2 , S¼t ni¼1
ð6Þ
where i is sequence number of raw discrete laser intensity datasets, and there are n groups of discrete data sets in all. Therefore, solving of parameters in Eq. (4) has been transformed to getting a group of I0 , j, k, l, u and v which make S to get a minimum value. Now we can suppose I0 , j, k and l as known items, and set 2 3 1 Iðx1 ,y1 Þ 1=l 2 j k3 2 ln½ I0 6 7 9x1 9 9y1 9 6 7 6 7 " # 6 1 Iðx2 ,y2 Þ 1=l 7 6 9x 9j 9y 9k 7 j 6 7 u ln½ 6 2 7 2 2 I 7, 0 A¼6 , b¼6 7, x ¼ 6 7 k 6 ^ 7 v ^ 5 6 7 ^ 4 6 j k 1=l 7 4 9xn 9 9yn 9 1 Iðxn ,yn Þ 5 2 ln½ I0 write them with Eq. (5) to a matrix form as follows: Ax ¼ b,
ð7Þ
Certainly we can get the normal equations as Eq. (8) which has the same solution x as the n 2 least square problem of Eq. (7): AT Ax ¼ AT b,
ð8Þ
To solve Eq. (8), we compute the Cholesky factorization [27] T
A A ¼ LLT ,
ð9Þ
where L is lower triangular, and then the solution x can be computed by solving the triangular systems Ly ¼ AT b and LT x ¼ y,
values of u and v can also be got. Additionally, if intervals of I0 , j, k and l can be estimated exactly, a series of u and v can be got by traversing I0 , j, k and l in their estimative intervals via enumeration. Thus, we can get a group of fitting laser intensity functions and associated standard deviations S by Eq. (6). The group of I0 , j, k, l, u and v which make the standard deviation (S) get a minimum value are the best solution for Eq. (4), and the associated function is the best solution for fitting raw discrete data of laser intensity distribution. If quantities of I0 , j, k and l are Ni ði ¼ 1,2,3,4Þ in their intervals, time complexity of the above method is product of the quantities Q ½Oð 4i ¼ 1 N i Þ. Obviously, solving efficiency of this method is very low when every Ni ði ¼ 1,2,3,4Þ is a large value, even that the method is unfeasible. However, Eq. (6) can be seen as a function which is S for I0 , j, k and l, and the function is continuously differentiable in intervals of I0 , j, k and l. If initial guesses of I0 , j, k and l are given, values of u and v can be got by solving Eq. (7). And we can get the next group of I0 , j, k and l with steepest descent method [28,29], which is defined by ðI0 ,j,k,lÞm þ 1 ¼ ðI0 ,j,k,lÞm lm rSm
ðm ¼ 0,1,2, . . .Þ,
ð10Þ
where rSm ¼ ½@Sm =@I0 ,@Sm =@j,@Sm =@k,@Sm =@l ðm ¼ 0,1,2, . . .Þ, m the sequence number of I0 , j, k and l, lm the small enough number. The above process can be repeated until S converges to a minimum, and the associated I0 , j, k, l, u and v is the best solution for Eq. (4). Therefore, the problem how to get the minimum of S in Eq. (6) has been changed into a problem of multivariate optimization. As a matter of fact, it is uneasy to calculate rSm in programming, here we use direction search method [30,31] to change values of I0 , j, k and l instead of steepest descent method, and the whole process is shown in Algorithm 1.
Fig. 5. Laser intensity distribution of Example 3(a) grid of raw discrete data; (b) grid of fitting function; (c) contrast between raw discrete data and fitting function at x¼ 0; (d) contrast between raw discrete data and fitting function at y¼ 0.
Y. Wen-quan et al. / Optics & Laser Technology 47 (2013) 183–188
187
P sum of their steps [Oð 4i ¼ 1 Ni Þ], which is much more efficient Q than that of enumeration [Oð 4i ¼ 1 N i Þ]. Here we use Example 3 in Section 4 to illustrate computing process of this algorithm, which is depicted in Fig. 2. And in Fig. 2, length of arrows donates I0 while radius of balls for S.
Algorithm 1. Direction searching for fitting transcendental function. set initial guesses for ðI0 ,j,k,lÞ0 solve ðu,vÞ0 with least square sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n j k 1 P j k l S0 ¼ ½I0 e2ðð9xi 9 =u Þ þ ð9yi 9 =v ÞÞ Iðxi ,yi Þ2 9ðI0 ,j,k,lÞ 0 ni¼1 for m ¼ 0,1,2, . . . ðI0 ,j,k,lÞm þ 1 ¼ one group of ðI0 ,j,k,lÞm 7ðlm ,0,0,0Þ,ðI0 ,j,k,lÞm 7 ð0, lm ,0,0Þ,ðI0 ,j,k,lÞm 7 ð0,0, lm ,0Þ,ðI0 ,j,k,lÞm 7 ð0,0,0, lm Þg which makes Sm þ 1 getting a minimum sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n j k 1 P j k l ½I0 e2ðð9xi 9 =u Þ þ ð9yi 9 =v ÞÞ Iðxi ,yi Þ2 9ðI0 ,j,k,lÞ Sm þ 1 ¼ mþ1 ni¼1
4. Numerical simulations Now, we fit laser intensity distribution on surfaces of lenses in a 193 nm lithography objective with Eq. (4) via Algorithm 1. Since the Eq. (4) can also describe circular-Gaussian and elliptical-Gaussian distribution, we do not fit them with Eq. (4) here, but fit several special kinds of laser intensity distribution on surfaces of lenses which are illustrated as Figs. 3–6. In addition, the raw discrete data and their fitting functions are drawn together for comparison, and they are drawn in grid form for depicting conveniently. Functions of the examples and their standard deviations (S) are shown in Table 1, and the units of all variables are SI.
if Sm þ 1 Z Sm then output Sm ,ðI0 ,j,k,l,u,vÞm & exit end As seen from Algorithm 1, the standard deviation (S) decreases nearly along the steepest descent direction when I0 , j, k and l vary. And if step sizes of the parameters are set, the time complexity is
Fig. 6. Laser intensity distribution of Example 4(a) grid of raw discrete data; (b) grid of fitting function; (c) contrast between raw discrete data and fitting function at x¼ 0; (d) contrast between raw discrete data and fitting function at y¼ 0.
Table 1 Functions of the examples. Items
Functions
S
Example 1
Iðx,yÞ ¼ 6237 e2ðð9x9
Example 2
Iðx,yÞ ¼ 830:1 e
Example 3
Iðx,yÞ ¼ 1160 e2ðð9x9
Example 4
1:96
2ðð9x9
2:86
Iðx,yÞ ¼ 6274 e2ðð9x9
2:36
=0:01232Þ1:96 Þ þ ð9y9
2:27
3:32
=0:12572:27 Þ þ ð9y9
1:89
=0:012322:86 Þ þ ð9y9 =0:02708
3:32
Þ þ ð9y9
=0:018862:36 ÞÞ1:39
=0:12551:89 ÞÞ7:31
1:38
=0:087701:38 ÞÞ2:99
7:49
=0:059047:49 ÞÞ1:29
Figurations
122.0
Shown in Fig. 3
69.58
Shown in Fig. 4
42.10
Shown in Fig. 5
201.4
Shown in Fig. 6
188
Y. Wen-quan et al. / Optics & Laser Technology 47 (2013) 183–188
As seen from the above computing, it is proper to describe the special kinds of laser intensity distribution with Eq. (4). Since Eq. (4) includes Gaussian functions, it is much more suitable than Gaussian functions in describing distribution of laser intensity on surfaces of optical components, and its fitting method depicted in Algorithm 1 is feasible.
5. Conclusions In this article, we have presented a novel transcendental function to describe laser intensity distribution on surfaces of lenses, and fitted it with multivariate optimization method. First, we set initial guesses of several parameters, solve the other parameters with least square method, and get standard deviation (S) of the function. Then, the above predefined parameters are set as variables, and we use direction search method instead of steepest descent method to change every parameter which makes the standard deviation (S) descend nearly along the steepest direction. Finally, a transcendental function whose standard deviation S gets a minimum can be solved, and the function is the best description for distribution of laser intensity. Four kinds of distribution laser intensity are taken as examples to verify validity and effectiveness of this numerical method. Contrast between the discrete data and their fitting functions shows that the transcendental function is more suitable than Gaussian functions for describing distribution of laser intensity; and time P complexity of the algorithm in this article [Oð 4i ¼ 1 Ni Þ] is much Q more efficient than that of traditional enumeration [Oð 4i ¼ 1 N i Þ] when steps of every parameter are N i ði ¼ 1,2,3,4Þ. Acknowledgment This work is supported by the National 02 Project of China (Grant No. 2009ZX02005). References [1] Yang Huomu, Feng Guoying, Zhou Shouhuan. Thermal effects in high-power Nd: YAG disk-type solid state laserOptics & Laser Technology 2011;43:1006–15. [2] Khizhnyak A, Lopiitchouk M, Peshko I. Lens transformations of high power solid-state laser beams. Optics & Laser Technology 1998;30:341–8. [3] Haitao Chen, Huajun Yang, Zou Xuefang, Liu Xueqiong. The refractive index changes derived from thermal effects in Bragg fiber laser. Optik 2011;122:1478–80. [4] Mathey Pierre. Thermal strain distributions and change of refractive indices in 4mm crystals illuminated with Gaussian CW laser beams. Optics communications 2001;193:217–26. [5] SuKim Hyun, Lee Sungman. Dependence of the beam characteristics of the thermal-birefringence compensated symmetric resonator with two Nd YAG laser rods on the curvature of laser-rod end surfacesOptics & Laser Technology 2007;39:116–22. [6] Shoji Ichiro, Taira Takunori, Ikesue Akio. Thermally-induced-birefringence 3þ effects of highly Nd -doped Y3 Al5 O12 ceramic lasers. Optical Materials 2007;29:1271–6.
[7] Patil SD, Navare ST, Takale MV, Dongare MB. Self-focusing of cosh-Gaussian laser beams in a parabolic medium with linear absorption. Optics and Lasers in Engineering 2009;47:604–6. [8] Karim MA, Awwal AAS, Nasiruddin AM, Basit A, Vedak DS, Smith CC, et al. Gaussian laser-beam-diameter measurement using sinusoidal and triangular rulings. Optics Letters 1987;12:93–5. [9] Sowdari Dharani, Majumdar Pradip. Finite element analysis of laser irradiated metal heating and melting processes. Optics & Laser Technology 2010;42:855–65. [10] Xiaolu Song, Bingbin Li, Zhen Guo, Shiyu Wang, Defang Cai, Jianguo Wen. Influences of pump beam distribution on thermal lensing spherical aberration in an LD end-pumped Nd: YAG laserOptics Communications 2009;282:4779–83. [11] Niu Ronglian, Liu Jie, Liu Chengcheng, Liu Ying, Qin Lianjie. Theoretical investigation on thermal lensing effects of Yb:KY(WO4)2 in diode-pumped lasers. Optik 2011;122:1931–4. [12] Koushki E, Farzaneh A, Mousavi SH. Closed aperture z-scan technique using the Fresnel–Kirchhoff diffraction theory for materials with high nonlinear refractions. Applied Physics B 2010;99:565–70. [13] Santarsiero Massimo, Borghi Riccardo. Correspondence between superGaussian and flattened Gaussian beams. Optical Society of America 1999;16:188–90. [14] Grow Taylor D, Ishaaya Amiel A, Vuong Luat T, Gaeta-Alexander L, Gavish Nir, Fibich Gadi. Collapse dynamics of super-Gaussian beams. Optics Express 2006;14:5468–75. [15] Baida Lu, Shirong Luo. Beam propagation factor of apertured super-Gaussian beams. Optik 2001;112:503–6. [16] Gill TS, Saini NS, Kaul SS. Dynamics of self-focusing and self-phase modulation of elliptic Gaussian laser beam in a Kerr-medium. Pramana-Journal of Physics 2000;55:423–31. [17] Xingpeng Yan, Mali Gong, Fahong He, Qiang Liu, Xing Fu, Dong shengWang. Numerical modeling of the thermal lensing effect in a grazing-incidence laser. Optics Communications 2009;282:1851–7. [18] ftekharuddin KM, Awwal AAS, Karim MA. Gaussian-to-Bessel beam transformation using a split refracting system. Applied Optics 1993;32:2252–6. [19] Karim MA, Cherri AK, Awwal AAS, Basit A. Refracting system for annular laser beam transformation. Applied Optics 1987;26:2446–9. [20] Jahan SR, Karim MA. Refraction systems for Gaussian-to-uniform beam transformations. Optics & Laser Technology 1989;21:27–30. [21] Weaire D, Wherrett BS, Miller DAB, Smith SD. Effect of low-power nonlinear refraction on laser-beam propagation in InSb. Optics Letters 1979;4:331–4. [22] Tsigaridas G, Fakis M, Polyzos I, Persephonis P, Giannetas V. Z-scan analysis for high-order nonlinearities through Gaussian decomposition. Optics Communications 2003;225:253–68. [23] Tsigaridas G, Fakis M, Polyzos I, Persephonis P, Giannetas V. Z-scan technique for elliptic Gaussian beams. Applied Physics B 2003;77:71–5. [24] Deng L, He K, Zhou T, Li C. Formation and evolution of far-field diffraction patterns of divergent and convergent Gaussian beams passing through self-focusing and self-defocusing media. Journal of Optics APure and Applied Optics. 2005;7:409–15. [25] Nascimento CM, Alencar MARK, Chavez-Cerda S, Silva MGA, Meneghetti MR, Hickmann JM. Experimental demonstration of novel effects on the far-field diffraction patterns of a Gaussian beam in a Kerr medium. Journal of Optics APure and Applied Optics. 2006;8:947–51. [26] Yao BL, Ren LY, Hou X. Z-scan theory based on a diffraction model. Journal of the Optical Society of America B 2003;20:1290–4. [27] Heath MT. Scientific computing: an introductory survey2nd ed New York: McGraw-Hill; 1997. [28] Anjidani M, Effati S. Steepest descent method for solving zero-one nonlinear programming problems. Applied Mathematics and Computation 2007;193: 197–202. [29] Walter B, Richardson Jr. Steepest descent using smoothed gradients. Applied Mathematics and Computation 2000;112:241–54. [30] Schonauer W. Experiments with search directions for a generalized CG method. Applied Numerical Mathematics 1999;30:241–56. [31] Luh Hsing, Tsaih Ray. An efficient search direction for linear programming problems. Computers & Operations Research 2002;29:195–203.
Contents lists available at SciVerse ScienceDirect
Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
A novel laser intensity function and its fitting method Yuan Wen-quan a,n, Gong Yan a,nn, Qin Shuo b a b
State Key Laboratory of Applied Optics, Changchun Institute of Optics, Fine Mechanics and Physics, The Chinese Academy of Sciences, Changchun 130033, China The Air-force Aviation University, Changchun 130021, China
a r t i c l e i n f o
abstract
Article history: Received 17 April 2012 Received in revised form 5 September 2012 Accepted 7 September 2012 Available online 12 October 2012
A novel transcendental function is presented to describe laser intensity distribution instead of Gaussian functions, and its fitting method is given in this paper. The transcendental function based on the Gaussian functions is fitted with multivariate optimization method, which is carried out with direction search method and least square method in practice. Four kinds of distribution laser intensity are fitted with the transcendental function in this article, validity and effectiveness of the fitting method are verified by these numerical simulations. Crown Copyright & 2012 Published by Elsevier Ltd. All rights reserved.
Keywords: Intensity distribution Transcendental function Multivariate optimization
1. Introduction Thermal lensing effects often occur in modern lasers such as high-power lasers, lithography objectives and so on, lenses in these apparatus will have deformations, stress and strain due to laser heating [1,2], the refractive indices of optical materials will change [3,4], stress birefringence [5,6] and self-focusing [7] also occur. Since optical performance of these apparatus is seriously affected due to thermal lensing effects, the effects should be considered at early time of optical design. And in thermal analysis of thermal lensing effects, the raw discrete data of laser intensity distribution on surfaces of lenses should be analyzed by optical software at first. Then, the raw discrete data will be fitted as laser intensity functions. Finally, the functions will be multiplied with material absorption factor, and applied on surfaces of lenses as thermal load. Usually, laser intensity functions were supposed to be Gaussian forms which include Gaussian form [8–11], sub-Gaussian form [12], super-Gaussian form [13–15] and ellipticalGaussian form [16,17] in the past research. But laser intensity distribution on surfaces of lenses does not often obey Gaussian forms due to refraction of aspheric or thick lens [18–20], so it seems unsuitable to fit the raw discrete data of laser intensity distributed on surfaces of lenses with Gaussian functions. The other ways to describe laser intensity in nonlinear optics are Gaussian decomposition method (GD) [21–23] and Fresnel–Kirchhoff method (FK) [24–26], but their mathematical forms are complicated for thermal analysis.
n
Corresponding author. Tel.: þ86 431 86708163. Principal corresponding author. Tel.: þ 86 431 86178996. E-mail addresses: [email protected] (Y. Wen-quan), [email protected] (G. Yan). nn
In this article, a novel transcendental function is proposed to describe distribution of laser intensity on lenses, and its form includes Gaussian forms. Since the function is unable to be fitted directly, we solve parameters of the function with multivariate optimization method, and get the laser intensity function finally.
2. Raising of a transcendental laser intensity function In the research related with laser beams, laser intensity distribution on surfaces of optical components was usually described as Gaussian form [9–11], which is shown as follows: 2
Iðx,yÞ ¼ I0 e2ððx
þ y2 Þ=w2 Þ
,
ð1Þ
where I0 denotes energy density of the laser beam at the center (W/ m2), and w is the radius of laser beam. The geometric interpretation is depicted in Fig. 1. While, several other researchers thought that if add a factor ‘l’ to Gaussian function, the new function can be described laser intensity distribution better [12–15]. And its form is described as follows: 2
Iðx,yÞ ¼ I0 e2ððx
þ y2 Þ=w2 Þl
,
ð2Þ
where l is a positive real factor. And when l o 1, it is called subGaussian distribution; when l ¼ 1, it is called Gaussian distribution, which has the same form as Eq. (1); when l 41, it is called superGaussian distribution, and when l-1, Eq. (3) describes uniform distribution. In addition, laser intensity distribution described by Eq. (1) is circular, an elliptical-Gaussian function was raised to describe laser distribution in some articles [16,17], and its form is described as follows: 2
Iðx,yÞ ¼ I0 e2ððx
0030-3992/$ - see front matter Crown Copyright & 2012 Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.optlastec.2012.09.009
=u2 Þ þ ðy2 =v2 ÞÞ
,
ð3Þ
184
Y. Wen-quan et al. / Optics & Laser Technology 47 (2013) 183–188
where u and v are waist radius of laser beam. When u¼v, it has the same form as Eq. (1). Obviously Eq. (1) is a special kind of Eqs. (2) or (3), therefore Eqs. (2) and (3) can be more suitable for describing laser intensity distribution. And it also can be seen from structure of above equations that the indices which is in bracket of exponent are all quadratic, here we use two positive real factors instead of them, and construct a transcendental function as Eq. (4) to describe distribution of laser intensity: j
j
k
Iðx,yÞ ¼ I0 e2ðð9x9 =u Þ þ ð9y9
=vk ÞÞl
,
ð4Þ
where j, k and l are all positive real factors. When j¼k¼2 and l¼ 1, Eqs. (4) and (3) have the same form; when j¼k¼2 and u¼v¼w, Eqs. (4) and (2) have the same form. In other words, function described by Eq. (4) includes Eqs. (1)–(3), therefore it is more applicable.
3. Solving of parameters in the transcendental function 3.1. Transformation of the transcendental function The nonlinear function should be transformed to a linear one before its fitting with least square method. Since Eq. (4) is nonlinear, we transform it to a new form as follows: 1=l j k 1 ln Iðx,yÞ ¼ 9x9 þ 9y9 , 2 I0 uj vk
ð5Þ
As can be seen from Eq. (5), it is unable to fit the equation with least square method directly for unknown terms on each side of it.
3.2. Solving method of the parameters
Fig. 1. Schematic diagram of laser intensity distribution on optical component.
The standard deviation between the raw discrete data of laser intensity and its fitting function Iðx,yÞ described by Eq. (4) can be
Fig. 2. Schematic diagram of computing process in Example 3.
Fig. 3. Laser intensity distribution of Example 1(a) grid of raw discrete data; (b) grid of fitting function; (c) contrast between raw discrete data and fitting function at x¼ 0; (d) contrast between raw discrete data and fitting function at y¼ 0.
Fig. 4. Laser intensity distribution of Example 2(a) grid of raw discrete data; (b) grid of fitting function; (c) contrast between raw discrete data and fitting function at x¼ 0; (d) contrast between raw discrete data and fitting function at y¼ 0.
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Y. Wen-quan et al. / Optics & Laser Technology 47 (2013) 183–188
written as: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n u1 X j k j k l ½I0 e2ðð9xi 9 =u Þ þ ð9yi 9 =v ÞÞ Iðxi ,yi Þ2 , S¼t ni¼1
ð6Þ
where i is sequence number of raw discrete laser intensity datasets, and there are n groups of discrete data sets in all. Therefore, solving of parameters in Eq. (4) has been transformed to getting a group of I0 , j, k, l, u and v which make S to get a minimum value. Now we can suppose I0 , j, k and l as known items, and set 2 3 1 Iðx1 ,y1 Þ 1=l 2 j k3 2 ln½ I0 6 7 9x1 9 9y1 9 6 7 6 7 " # 6 1 Iðx2 ,y2 Þ 1=l 7 6 9x 9j 9y 9k 7 j 6 7 u ln½ 6 2 7 2 2 I 7, 0 A¼6 , b¼6 7, x ¼ 6 7 k 6 ^ 7 v ^ 5 6 7 ^ 4 6 j k 1=l 7 4 9xn 9 9yn 9 1 Iðxn ,yn Þ 5 2 ln½ I0 write them with Eq. (5) to a matrix form as follows: Ax ¼ b,
ð7Þ
Certainly we can get the normal equations as Eq. (8) which has the same solution x as the n 2 least square problem of Eq. (7): AT Ax ¼ AT b,
ð8Þ
To solve Eq. (8), we compute the Cholesky factorization [27] T
A A ¼ LLT ,
ð9Þ
where L is lower triangular, and then the solution x can be computed by solving the triangular systems Ly ¼ AT b and LT x ¼ y,
values of u and v can also be got. Additionally, if intervals of I0 , j, k and l can be estimated exactly, a series of u and v can be got by traversing I0 , j, k and l in their estimative intervals via enumeration. Thus, we can get a group of fitting laser intensity functions and associated standard deviations S by Eq. (6). The group of I0 , j, k, l, u and v which make the standard deviation (S) get a minimum value are the best solution for Eq. (4), and the associated function is the best solution for fitting raw discrete data of laser intensity distribution. If quantities of I0 , j, k and l are Ni ði ¼ 1,2,3,4Þ in their intervals, time complexity of the above method is product of the quantities Q ½Oð 4i ¼ 1 N i Þ. Obviously, solving efficiency of this method is very low when every Ni ði ¼ 1,2,3,4Þ is a large value, even that the method is unfeasible. However, Eq. (6) can be seen as a function which is S for I0 , j, k and l, and the function is continuously differentiable in intervals of I0 , j, k and l. If initial guesses of I0 , j, k and l are given, values of u and v can be got by solving Eq. (7). And we can get the next group of I0 , j, k and l with steepest descent method [28,29], which is defined by ðI0 ,j,k,lÞm þ 1 ¼ ðI0 ,j,k,lÞm lm rSm
ðm ¼ 0,1,2, . . .Þ,
ð10Þ
where rSm ¼ ½@Sm =@I0 ,@Sm =@j,@Sm =@k,@Sm =@l ðm ¼ 0,1,2, . . .Þ, m the sequence number of I0 , j, k and l, lm the small enough number. The above process can be repeated until S converges to a minimum, and the associated I0 , j, k, l, u and v is the best solution for Eq. (4). Therefore, the problem how to get the minimum of S in Eq. (6) has been changed into a problem of multivariate optimization. As a matter of fact, it is uneasy to calculate rSm in programming, here we use direction search method [30,31] to change values of I0 , j, k and l instead of steepest descent method, and the whole process is shown in Algorithm 1.
Fig. 5. Laser intensity distribution of Example 3(a) grid of raw discrete data; (b) grid of fitting function; (c) contrast between raw discrete data and fitting function at x¼ 0; (d) contrast between raw discrete data and fitting function at y¼ 0.
Y. Wen-quan et al. / Optics & Laser Technology 47 (2013) 183–188
187
P sum of their steps [Oð 4i ¼ 1 Ni Þ], which is much more efficient Q than that of enumeration [Oð 4i ¼ 1 N i Þ]. Here we use Example 3 in Section 4 to illustrate computing process of this algorithm, which is depicted in Fig. 2. And in Fig. 2, length of arrows donates I0 while radius of balls for S.
Algorithm 1. Direction searching for fitting transcendental function. set initial guesses for ðI0 ,j,k,lÞ0 solve ðu,vÞ0 with least square sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n j k 1 P j k l S0 ¼ ½I0 e2ðð9xi 9 =u Þ þ ð9yi 9 =v ÞÞ Iðxi ,yi Þ2 9ðI0 ,j,k,lÞ 0 ni¼1 for m ¼ 0,1,2, . . . ðI0 ,j,k,lÞm þ 1 ¼ one group of ðI0 ,j,k,lÞm 7ðlm ,0,0,0Þ,ðI0 ,j,k,lÞm 7 ð0, lm ,0,0Þ,ðI0 ,j,k,lÞm 7 ð0,0, lm ,0Þ,ðI0 ,j,k,lÞm 7 ð0,0,0, lm Þg which makes Sm þ 1 getting a minimum sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n j k 1 P j k l ½I0 e2ðð9xi 9 =u Þ þ ð9yi 9 =v ÞÞ Iðxi ,yi Þ2 9ðI0 ,j,k,lÞ Sm þ 1 ¼ mþ1 ni¼1
4. Numerical simulations Now, we fit laser intensity distribution on surfaces of lenses in a 193 nm lithography objective with Eq. (4) via Algorithm 1. Since the Eq. (4) can also describe circular-Gaussian and elliptical-Gaussian distribution, we do not fit them with Eq. (4) here, but fit several special kinds of laser intensity distribution on surfaces of lenses which are illustrated as Figs. 3–6. In addition, the raw discrete data and their fitting functions are drawn together for comparison, and they are drawn in grid form for depicting conveniently. Functions of the examples and their standard deviations (S) are shown in Table 1, and the units of all variables are SI.
if Sm þ 1 Z Sm then output Sm ,ðI0 ,j,k,l,u,vÞm & exit end As seen from Algorithm 1, the standard deviation (S) decreases nearly along the steepest descent direction when I0 , j, k and l vary. And if step sizes of the parameters are set, the time complexity is
Fig. 6. Laser intensity distribution of Example 4(a) grid of raw discrete data; (b) grid of fitting function; (c) contrast between raw discrete data and fitting function at x¼ 0; (d) contrast between raw discrete data and fitting function at y¼ 0.
Table 1 Functions of the examples. Items
Functions
S
Example 1
Iðx,yÞ ¼ 6237 e2ðð9x9
Example 2
Iðx,yÞ ¼ 830:1 e
Example 3
Iðx,yÞ ¼ 1160 e2ðð9x9
Example 4
1:96
2ðð9x9
2:86
Iðx,yÞ ¼ 6274 e2ðð9x9
2:36
=0:01232Þ1:96 Þ þ ð9y9
2:27
3:32
=0:12572:27 Þ þ ð9y9
1:89
=0:012322:86 Þ þ ð9y9 =0:02708
3:32
Þ þ ð9y9
=0:018862:36 ÞÞ1:39
=0:12551:89 ÞÞ7:31
1:38
=0:087701:38 ÞÞ2:99
7:49
=0:059047:49 ÞÞ1:29
Figurations
122.0
Shown in Fig. 3
69.58
Shown in Fig. 4
42.10
Shown in Fig. 5
201.4
Shown in Fig. 6
188
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As seen from the above computing, it is proper to describe the special kinds of laser intensity distribution with Eq. (4). Since Eq. (4) includes Gaussian functions, it is much more suitable than Gaussian functions in describing distribution of laser intensity on surfaces of optical components, and its fitting method depicted in Algorithm 1 is feasible.
5. Conclusions In this article, we have presented a novel transcendental function to describe laser intensity distribution on surfaces of lenses, and fitted it with multivariate optimization method. First, we set initial guesses of several parameters, solve the other parameters with least square method, and get standard deviation (S) of the function. Then, the above predefined parameters are set as variables, and we use direction search method instead of steepest descent method to change every parameter which makes the standard deviation (S) descend nearly along the steepest direction. Finally, a transcendental function whose standard deviation S gets a minimum can be solved, and the function is the best description for distribution of laser intensity. Four kinds of distribution laser intensity are taken as examples to verify validity and effectiveness of this numerical method. Contrast between the discrete data and their fitting functions shows that the transcendental function is more suitable than Gaussian functions for describing distribution of laser intensity; and time P complexity of the algorithm in this article [Oð 4i ¼ 1 Ni Þ] is much Q more efficient than that of traditional enumeration [Oð 4i ¼ 1 N i Þ] when steps of every parameter are N i ði ¼ 1,2,3,4Þ. Acknowledgment This work is supported by the National 02 Project of China (Grant No. 2009ZX02005). References [1] Yang Huomu, Feng Guoying, Zhou Shouhuan. Thermal effects in high-power Nd: YAG disk-type solid state laserOptics & Laser Technology 2011;43:1006–15. [2] Khizhnyak A, Lopiitchouk M, Peshko I. Lens transformations of high power solid-state laser beams. Optics & Laser Technology 1998;30:341–8. [3] Haitao Chen, Huajun Yang, Zou Xuefang, Liu Xueqiong. The refractive index changes derived from thermal effects in Bragg fiber laser. Optik 2011;122:1478–80. [4] Mathey Pierre. Thermal strain distributions and change of refractive indices in 4mm crystals illuminated with Gaussian CW laser beams. Optics communications 2001;193:217–26. [5] SuKim Hyun, Lee Sungman. Dependence of the beam characteristics of the thermal-birefringence compensated symmetric resonator with two Nd YAG laser rods on the curvature of laser-rod end surfacesOptics & Laser Technology 2007;39:116–22. [6] Shoji Ichiro, Taira Takunori, Ikesue Akio. Thermally-induced-birefringence 3þ effects of highly Nd -doped Y3 Al5 O12 ceramic lasers. Optical Materials 2007;29:1271–6.
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