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High-precision method for disorder in beam tracing in an optical system with elastic optics
Optics Communications 306 (2013) 35–41
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High-precision method for disorder in beam tracing in an optical system with elastic optics Shipeng Feng n, Dongxu Li College of Aerospace Science and Engineering, National University of Defense Technology, Changsha, Hunan Province 410073, China
art ic l e i nf o
a b s t r a c t
Article history: Received 15 January 2013 Received in revised form 26 April 2013 Accepted 20 May 2013 Available online 3 June 2013
Disorder in beam tracing should be calculated accurately when designing an optical system. In conventional numerical calculation methods for the disordered beam path, the optics were treated as rigid bodies and the misalignment caused by the deformation of the optics was neglected. In this work, the expression for the deformation of the surface normal was derived based on the elastic body assumption using differential geometry and the transient response finite-element method. Furthermore, with the aid of this expression, concrete steps for a numerical calculation method for disorder in beam tracing were given. Then, the validity of this improved method was tested with two examples and with an experimental result from another study. & 2013 Elsevier B.V. All rights reserved.
Keywords: Elastic optics Disorder in beam tracing Differential geometry Finite-element method
1. Introduction The application environments of optical systems are becoming more and more complex, and the fields in which optical systems are applied are continuously expanding. As a result, all types of disturbances in the environment, such as mechanical oscillations, temperature changes, atmosphere turbulence and so on, influence the stability of optical systems [1]. Among these types of disturbances, mechanical oscillations generally interfere with optical systems [2]. Mechanical oscillations cause beam path disorder, decrease the performance of optical systems and even result in optical system failure. Many academics have investigated this problem. Mahdieh et al. [3] and Gao et al. [4] focused on the quality of the beam affected by the misalignment of optical elements. To improve the stability of optical systems operating in a severe environment, the amount of beam disorder should be calculated accurately. Wang et al. [5] used a simple ray-tracing method to measure the vertex radius of curvature of an aspheric mirror. Cho et al. [6] investigated a method to calculate the coupling efficiency for various misalignment parameters. Chen et al. [7] used a ray-tracing method to design a solar concentrator. The accuracy of numerical methods to calculate a disordered beam path directly influences the design quality of optical systems. However, in conventional numerical calculation methods for the disordered beam path, the optical lenses were treated as rigid bodies. Only a few feature points were marked on the surfaces of
n
Corresponding author. Tel.: +86 731 845 73186. E-mail address: [email protected] (S. Feng).
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the lens. Then, the deformations of the optical stands because of external excitation were calculated by FEM whereas the displacements of the feature points were given based on the results of the deformation of the optical stands. Therefore, through coordinate transformation, which used the displacements of the feature points, the disordered beam path could be calculated. Rakich [8] used this method to solve the problem of third-order misalignment aberrations. It was reasonable that the optical lenses were assumed to be rigid bodies under slight external excitation, while it would be so rough that the optical lenses were treated as rigid bodies if the external excitation was severe or if accurate results were needed. Recently, Iparraguirre and Rio-Gaztelurrutia [9] completed a perfect investigation of the laser modes in misaligned resonators. However, Iparraguirre also ignored the elasticity of the lenses of the resonators. The conventional method (CM) to calculate beam path disorder is a method in which the deformation of the optical stand and mount is computed by FEM and the elastic deformation of the optical elements caused by vibration is neglected. Although the deformation might be very slight, small changes in the reflection or refraction surfaces could lead to large misalignments. Thus, conventional calculation methods should be improved. Based on the elastic body assumption, the expression for the deformation of the surface normal was derived using differential geometry and the transient response finite-element method. Furthermore, with the aid of this expression, concrete steps for a numerical calculation method for disorder in beam tracing were listed. Primarily, the contact point between the beam and the surface of a mirror or lens was solved. Finally, the validity of this improved method was tested with two examples. In the two
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S. Feng, D. Li / Optics Communications 306 (2013) 35–41
examples, the simulation results for the method considering the deformation of optics, the simulation results for the conventional method and the experimental results were compared with each other. The validity of this improved method was verified.
2. Calculation of the surface normal after deformation of the optics The conventional methods for disorder in beam tracing ignored the elastic deformation of the optics and derived the rigid displacement of the optics by calculating the deformation of the optical mounts. Based on the assumption of elastic optics, calculations of the surface deformation of the optical lens could be translated into calculations of the change of the surface normal. Based on differential geometry, the expression for the deformation of the surface normal was given. The displacement of any point on the surface could be expressed as functions of the coordinates x, y and z after surface deformation. 8 > < p ¼ pðx; y; zÞ q ¼ qðx; y; zÞ ð1Þ > : s ¼ sðx; y; zÞ In Eq. (1), p, q and s are the displacements of the point (x, y, z) in the x, y and z directions, respectively. Then, the deformation displacements in the directions of x, y and z should be added to the parametric equations for the changed surface. Thus, the changed parametric equations could be given by 8 > < x′ ¼ x þ pðx; y; zÞ y′ ¼ y þ qðx; y; zÞ ð2Þ > : z′ ¼ z þ sðx; y; zÞ Then, the vector form of the parametric equation for the changed surface could be expressed by ! ! ! ! ! r ′ ¼ r ′ðu; vÞ ¼ x′ðu; vÞ i þ y′ðu; vÞ j þ z′ðu; vÞ k ð3Þ where u and v are the independent variables of the parametric equation. The parameters x, y, and z are functions of u and v and from Eq. (2), the parameters x′, y′, and z′, which are expressed by the parameters x, y, and z, are functions of u and v as well. Therefore, the partial derivatives of the parametric vector equation of the surface with respect to the parameters u and v could be given as follows: 8 ! ! !
the conventional method for disorder in beam tracing ignores the effect of deformation of the optical surfaces because the direction of the normal could not be solved through analytical calculation methods based on transformation of the coordinate system. In the method described in this work, the optical surfaces could be divided into many small fields, based on the assumption of elastic optics. In every small field, the prediction and calculation of the surface deformation could be given by node displacements. The concrete steps of the numerical calculation method that considers the deformation of the optical surfaces for disorder in beam tracing are given as follows: First, screen the FEM nodes and determine the contact field. Screen nodes on the optical surface and calculate the distances between the nodes and the beam path. Then, pick three nodes, called P, Q and R, which have smaller distances than the other nodes. The element surface formed by Node P, Node Q and Node R is the field in which the beam makes contact with the optical surface. Second, calculate the contact point E between the beam path line and the surface PQR. Third, calculate the normal vectors at Node P, Node Q and Node R with Eq. (5). ! Fourth, calculate the normal vector n E at point E. From the normal vectors calculated in the third step, the ! normal n E could be calculated with the weighted mean method. Fifth, express the beam path after the beam interaction with ! the optics with the normal n E . Through the five steps above, the coordinates and normal vector of the contact point E between the beam path and the optical surface after deformation could be calculated. Then, the disorder beam could be precisely traced. 4. Comparison of numerical simulations and experimental results 4.1. Example 1 4.1.1. Description of the example To test the validity of this improved method, example 1 was given. Example 1 was shown by Shao et al. [10], they investigated the problem of beam path disorder with numerical simulations and experiment and neglected the deformation of the optics. In this work, the example was calculated with consideration of the optical deformation and the numerical result was compared with the simulation and experimental results in Shao's work [10]. A sketch of the one-mirror optical system in example 1 is shown in Fig. 1. A He–Ne laser was fixed on an optical table. The laser beam was reflected by a mirror and was transmitted to the CCD, which could evaluate the amount of beam disorder by detecting the spot position. The pixel resolution of the CCD was 9.84 μm in the horizontal direction and 10.22 μm in the vertical direction. The distance between the CCD and the reflector was 2 m. The mirror of the reflector was made of quartz glass, and the CCD
Laser Generator
Reflector
3-axis Accelerometer
2m Optical Table
3. High-precision method for disorder in beam tracing under the assumption of elastic optics The surface of optics can deform if the external excitation is severe or if the stiffness of the optical elements is low. The amount of beam path disorder is given by the combined effects of rigid displacement and deformation of the optical surfaces. However,
Vibrating Table
Fig. 1. A sketch of the one-mirror optical system for example 1.
S. Feng, D. Li / Optics Communications 306 (2013) 35–41
material of the mirror mount was aluminium alloy. In this example, the reflector was excited by the vibrating table and a misalignment was formed. The actual excitation of the reflector was tested with a 3-axis accelerometer fixed to the vibrating table near the reflector. 4.1.2. Simulation and calculation First, screen the FEM nodes of the nodes on the mirror surface and determine the contact field. Based on the conventional method used by Shao, the analytical model by FEM is illustrated in Fig. 2. The acceleration excitation, which was a type of random excitation with an amplitude between 7 4 m/s2 was loaded on the reflecting mirror. The mean value of the random excitation, which had a uniform distribution, was 0 m/s2. The displacement of any node on the mirror's surface at any moment could be calculated through transient dynamic analysis. Assume that P is a node on the mirror's surface P : ðx1 ; y1 ; z1 Þ
ð6Þ
The line equation of the spatial beam path could be expressed by L:
x−x0 y−y0 z−z0 ¼ ¼ p q r
ð7Þ
where (x0,y0,z0) is a point on the line and (p,q,r) is the direction vector of the line. Then, the distance between the node P and the beam path line L could be given by i j k p q r x1 −x0 y1 −y0 z1 −z0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d ¼ ð8Þ p2 þ q2 þ r 2
240mm
110mm
Φ 20mm
37
On the surface of the mirror, screen three FEM nodes with the shortest distances from the beam path, and call these nodes P, Q and R. Then, the contact field was determined from the nodes P, Q and R. Second, calculate the contact point E between the beam path line and the surface PQR. Assume that there are three points P(x1,y1,z1), Q(x2,y2,z2), R(x3, y3,z3). Thus, the equation of the plane PQR could be given by x−x y−y1 z−z1 1 x−x2 y−y2 z−z2 ¼ 0 ð9Þ x−x y−y3 z−z3 3 By substituting Eq. (7) into Eq. (9), the contact point E could be obtained. Third, calculate the normal vectors at Node P, Node Q and Node R with Eq. (5). From the method from differential geometry described in the ! ! second part of this paper, the parametric equations for r P0 , r Q0 , !0 and r R located at Node P, Node Q and Node R, respectively, after deformation could be calculated with Eq. (4). In addition, with the ! ! ! aid of Eq. (5), the normal vectors n P , n q , and n r at the nodes P, Q and R could be calculated. 8 !0 !0 ! r Pu r Pv > > nP¼ ! > ! > > j r 0Pu r 0Pv j > > > !0 !0 < ! r r n Q ¼ !Q u !Q v ð10Þ > j r 0Q u r 0Q v j > > > !0 !0 > > r Ru r Rv > >! nR¼ ! : ! j r 0Ru r 0Rv j ! Fourth, calculate the normal vector n E at point E. ! The normal vector n E at point E could be given by the weighted mean. −1 ! −1 ! −1 ! dPE ⋅ n P þ dQ E ⋅ n Q þ dRE ⋅ n R ! nE¼ ð11Þ −1 −1 −1 dPE þ dQ E þ dRE where dPE, dQE, and dRE were the distances between points P, Q and R and point E. The weight coefficients were the reciprocals of the distances between points P, Q and R and point E because the node that was closest to E had a greater weight than the other nodes. Fifth, obtain the beam disorder by calculating the contact point between the beam path and the surface of the CCD. The disordered beam expression could be given through the reflection law described by the spatial vector ! ! ! ! a ′ ¼ a þ 2ðj a j cos βÞ⋅ n E ð12Þ ! ! where a was the incident beam, a ′ was the disordered reflecting beam and β was the angle of reflection. Assume that the coordinates of E were (x4,y4,z4) and that the ! direction of the beam vector a ′was (p′,q′,r′). Then, the equation for the disordered beam could be expressed by L′ :
x−x4 y−y4 z−z4 ¼ ¼ p′ q′ r′
ð13Þ
50mm
30mm
Finally, the contact point F between the disordered beam and the photosensitive surface of the CCD could be calculated with simultaneous equations for the straight line L′ and the photosensitive surface of the CCD. The amount of beam disorder could be expressed by the coordinates of point F.
50mm Fig. 2. The FEM model of the reflecting mirror.
4.1.3. Analysis of simulations In the calculation method for the beam disorder under the assumption of elastic optics, the direction of a normal vector was calculated by the displacement of a node. As a result, the accuracy of the normal direction would be influenced by the scale of the mesh. Additionally, the accuracy of the calculation of the disordered beam
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S. Feng, D. Li / Optics Communications 306 (2013) 35–41
Table 1 Simulation results for different scales for the mesh. Global edge length
Direction
5 mm
Horizontal 20.8 Vertical 190.6 Horizontal 19.6 Vertical 184.2 Horizontal 19.6 Vertical 184.8
1 mm 0.1 mm
Result 1 (μm)
Result 2 (μm)
Result 3 (μm)
Standard deviation (μm)
18.9 187.3 19.7 185.7 19.7 184.0
19.0 180.6 19.6 185.0 19.6 185.3
1.5 7.2 0.1 1.0 0.1 0.9
path would also be affected. To investigate the influence of the scale of the mesh on the accuracy of the calculation, a few simulation results with different scales for the mesh are compared with each other in Table 1. In Table 1, the global edge length is the maximal edge length for all of the elements. For any scale for the mesh, the simulation was performed three times, and the maximal horizontal and vertical misalignments are listed. Additionally, the standard deviations for every scale of the mesh were calculated. The standard deviation is a parameter that expresses the amount of variation in the results and could be calculated by
Horizontal Vertical
200 150
ð14Þ
where s is the standard deviation, n is the number of results, fi is the value of the ith result, and E(f) is the mean value of the results. In Table 1, if the global edge length was 5 mm, the standard deviations of the results for the misalignment calculation were quite large. The result in the horizontal direction was 1.5 μm whereas the result in the vertical direction was 7.2 μm. A large standard deviation meant that the calculation results for the amount of beam disorder were not stable and that the calculation errors were large. However, if the global edge length was 1 mm, the standard deviations of the results of the misalignment calculations clearly decreased. The result in the horizontal direction was 0.1 μm whereas the result in the vertical direction was 1.0 μm. If the global edge length was 0.1 mm, the standard deviations of the results of the misalignment calculations decreased slightly. Therefore, if the global edge length was 1 mm, the mesh was fine enough to solve this problem. This method was used to determine how fine the meshing should be. The traces of the contact point between the beam line and the photosensitive surface of the CCD are illustrated in Fig. 3, for the simulation calculated with a mesh scale of 0.1 mm. In Fig. 3, the contact points for every sampling time and the connecting lines connecting consecutive points are shown. In Fig. 3, the positions of the points on the surface of the CCD were accurately calculated with the theory. However, because the sampling time was discrete, the connecting lines between the points approximately fit the actual track of the contact point on the surface of the CCD. In Fig. 4, the horizontal and vertical coordinates of the contact point F at each sampling time are illustrated. To investigate the statistical properties of the contact point F of the disordered beam, the horizontal and vertical coordinates of point F were treated with a statistical test. The results of the statistical test are shown in Table 2. Because the optical element was excited by a random acceleration excitation, which had a uniform distribution with a mean value of 0 m/s2, the amount of beam disorder should not deviate far from a uniform distribution with a mean value of 0 m/s2. In Table 2, the mean values and the deviations of the amount of beam disorder are given. Because the mean values were much less than
100 X and Y/10-6m
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 n s¼ ∑ ðf −Eðf ÞÞ2 ni¼1 i
Fig. 3. Simulation of the amount of beam end disorder.
50 0 -50 -100 -150 -200
0
1
2
3
4
5 t/s
6
7
8
9
10
Fig. 4. The horizontal and vertical coordinates of the contact point F at each sampling time.
Table 2 The statistical properties of the simulation results for example 1. Direction
Mean value E (μm)
Horizontal −0.9 Vertical −15.9
Standard deviation s (μm)
Results of χ2 test
Maximum (μm)
Minimum (μm)
10.8 103.2
3.7 8.6
19.6 184.8
−19.6 −183.7
the standard deviations in the horizontal and vertical directions, the calculation results for the beam disorder fit the distribution with a mean value of 0 μm. To test whether the calculated results for the beam disorder fit the uniform distribution, the χ2 test was used. The equation for the χ2 test was given by ðM i −T i Þ2 Ti i¼1 10
χ2 ¼ ∑
ð15Þ
where Mi was the number of points in the ith interval, and Ti was the number of points in the ith interval in the theoretical
S. Feng, D. Li / Optics Communications 306 (2013) 35–41
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treatment. The intervals were defined to be one tenth of the field between the maximal result and the minimal result. In the horizontal direction, the interval was 3.92 μm whereas in the vertical direction, the interval was 36.9 μm. In Table 2, the result of the χ2 test in the horizontal direction was 3.7 and the result of the χ2 test in the vertical direction was 8.6. Based on the theory of the χ2 test, the test in this example had 9 degrees of freedom, and the critical value χ 20:05 ð9Þ at a 5% significance level was 16.9. ( 2 χ Horizontal ¼ 3:7 oχ 20:05 ð9Þ ð16Þ χ 2Vertical ¼ 8:6 o χ 20:05 ð9Þ
experimental data was 0.5%. In the vertical direction, the deviation between the result calculated with the conventional method and the experimental data was 0.8% whereas the deviation between the result calculated with the new method and the experimental data was 0.4%. Based on Table 3, the results of the numerical method for the beam path disorder calculation that considers the deformation of the optics more accurately reflected the experimental results than did the results from the conventional method.
Based on Eq. (16), the test was accepted. In other words, the calculated results for the beam disorder fit the uniform distribution. In summary, based on the assumption of elastic optics, the calculation results for the beam disorder depended on the properties of the excitation. Therefore, the position of the contact point F on the photosensitive surface of the CCD fit the statistical properties of the uniform distribution with a mean value of 0 μm. In other words, the results calculated by the method in this work fit the distribution of the excitation well. Shao et al. [10] developed the experimental system for this example and tested the maximal disorder amount of the beam. What follows in this work is a comparison among the simulation results from the method that considers the deformation of the mirror, the simulation results from the conventional method and the experimental results obtained by Shao; this comparison is given in Table 3. In Table 3, the results calculated from the method based on the assumption of elastic optics and the method based on the assumption of rigid optics were compared with Shao's experimental data. The maximal misalignments in the horizontal and vertical directions are given in Table 3. In the horizontal direction, the deviation between the result calculated with the conventional method and the experimental data was 4.1% whereas the deviation between the result calculated with the new method and the
4.2.1. Description of the example To compare the method given in this work and the conventional method, example 2 was used. In example 2, the optical system contains two optical reflection elements and one refraction element. The optical reflection elements are the same as the element in example 1. The mount of the refraction element is also the same as the mount in example 1. The three optical elements were excited by a vibrating table. If the excitation was not present, the centres of the three optical elements were at the same height, and the spacings of the optical elements were 0.1 m. Additionally, the distance between the second mirror and the photosensitive surface of the CCD was 2 m. The results calculated by different methods for the misalignment were compared with each other as the amplitude of the excitation changed. The optical system for example 2 is illustrated in Fig. 5. The optical elements were excited with the same type of random excitation at different amplitudes. In example 2, three types of different excitations were added to the optical system. The amplitudes of the three types of excitation were 74 m/s2, 78 m/s2 and 7 12 m/s2, and the mean value of the three types of excitation was 0 m/s2.
Table 3 Comparison of the simulated and experimental results.
Experiment New method Conventional method
Horizontal direction
Vertical direction
Results (μm)
Error compared with experimental result
Results (μm)
Error compared with experimental result
19.7 19.6 18.9
0 0.5% 4.1%
184.0 184.8 185.5
0 0.4% 0.8%
CCD
4.2. Example 2
4.2.2. Simulation and calculation The numerical method given in this work and the conventional method were used to solve example 2. The calculation steps were the same as those for example 1 although the process was more complex. The amount of beam path disorder after the simulation is shown in Fig. 6. The amount of beam path disorder under three different excitations was calculated using two different methods. The simulation results are shown in Table 4. As shown in Table 4, the amount of beam path disorder increased as the excitation amplitude increased. In the horizontal direction, the amounts of beam disorder calculated by the two methods increased from 24.9 μm and 23.7 μm to 94.9 μm and 79.4 μm, respectively. Additionally, in
0.1m
Laser Generator Reflector
0.1m
Refractor
Reflector
Optical Table 2m
Vibrating Table
Fig. 5. A sketch of the three-mirror optical system for example 2.
3 Axis Accelerometer
40
S. Feng, D. Li / Optics Communications 306 (2013) 35–41
Fig. 6. Comparison of the simulation results for example 2. (a), (c), and (e) are the results for the simulation with deformation of the optics under excitation amplitudes of 7 4, 7 8, and 7 12 m/s2, respectively. (b), (d), and (f) are the results for the simulation without deformation of the optics under excitation amplitudes of 7 4, 7 8, and 7 12 m/s2, respectively.
Table 4 Comparison of simulated results for different methods and excitations. Direction
Horizontal misalignment Vertical misalignment
Method
New method Conventional method New method Conventional method
Excitation 1
Excitation 2
Excitation 3
Result (μm)
Deviation (%)
Result (μm)
Deviation (%)
Result (μm)
Deviation (%)
24.9 23.8 229.7 216.9
4.7
62.9 55.3 577.9 521.2
12.1
94.9 79.4 872.6 735.5
16.4
5.6
the vertical direction, the amounts of beam disorder calculated by the two methods increased from 229.7 μm and 216.9 μm to 872.6 μm and 735.5 μm, respectively. For the same excitation, the amount of beam disorder calculated with the method described in this work was larger than the amount of beam disorder calculated with the conventional method. Additionally, as the excitation amplitude increases, the deviations of results calculated with the method based on the assumption of elastic optics and with the conventional method
9.8
15.7
increased. In Table 4, the deviations calculated with the two methods increased from 4.7% for excitation 1 to 16.4% for excitation 3 in the horizontal direction. Additionally, the deviations calculated with the two methods increased from 5.6% for excitation 1 to 15.7% for excitation 3 in the vertical direction. The reason for this phenomenon is that the method in this work includes the amount of beam disorder caused by optical deformation whereas the conventional method does not. If amplitude of the excitation increases, the deformation of the optics becomes greater and
S. Feng, D. Li / Optics Communications 306 (2013) 35–41
influences the amount of beam disorder much more. As a result, the deviations between the results calculated by the two methods increased as the excitation increased. 4.3. Discussion It is necessary to be attentive to the optical mesh because the precision of the method given in this work relies on the scale of the mesh. The finer the mesh is, the more accurate the calculation results are. Additionally, if the amount of bending of the lens is great, the influence of the scale of the mesh is obvious. In this work, an example of a one-reflector optical system under external excitation was given. In example 1, by comparing the calculated results at different mesh scales, how fine the meshing should be was pointed out. Then, to investigate the correlation of the calculation results and the excitation, the simulation results were analysed statistically. The statistical properties of simulation results and the excitation were similar. The reliability of the method in this work was tested. Because of limitations on experimental conditions, we did not perform experimental research. However, to validate the numerical method given in this work, the calculation results were compared with the simulation results and experimental data from Shao. The deviation between the simulation results of the method described in this work and the experimental data was less than that between the simulation results of the conventional method and the experimental data. The results showed that the amount of beam disorder was caused by both the rigid displacement and the elastic deformation of the optics. Hence, the calculation method could be improved by introducing the deformation of optics to beam path tracing. The deviation between the method described in this work and the conventional method under three different excitation conditions was shown with example 2. As the external excitations became larger and larger, the deviation of the calculated results for the two methods became larger and larger. This result arose because the deformation of the optical lens became greater as the amplitude of the external excitation increased. The calculated results for the conventional method that did not consider the deformation of the optics had greater errors under the increased excitation conditions. The results showed that the deformation of optics must be introduced to methods to calculate the beam path disorder to obtain more accurate results when the external excitation is violent. It was also shown that the method described in this work could be applied to a complex optical system. Compared with the conventional method, the method in this work requires the same workload to calculate the deformation of the optical mount and stand but requires more computational
41
time to calculate the deformation of the optics. As a result, the computational cost for the improved accuracy is the computational time for the deformation of the optics. Because the method described in this work is based on a mesh method and does not use an analytical equation for the surface of the optic, this method could be used in optical systems with optics of arbitrary shape, such as optics with aspheric surfaces.
5. Conclusions To increase the calculation accuracy for disordered beam path tracing, this work improved upon the conventional method. The calculation method in this work was based on the assumption of elastic optics and included the optic, the rigid displacement and elastic deformation, which influenced the beam disorder. If the external excitation was violent, the deformation of the optic could not be neglected. Thus, the improved method for disorder in beam tracing in this work has a greater precision than the conventional method. Additionally, compared with the conventional method, the method described in this work does not obviously increase the complexity or the workload and is suitable for calculations of the beam path in precise optical instruments. However, it is necessary to be attentive to the optical mesh because the precision of the method given in this work relies on the scale of the mesh. The finer the mesh is, the more accurate the calculation results are. Additionally, if the amount of bending of the optic is great, the influence of the scale of the mesh is clear. The next step is to use the method described in this work to investigate how the different amounts of bending of the optics influence the precision of the disorder in beam tracing. References [1] X.L. Ji, H.T. Eyyuboglu, Y. Baykal, Optics Express 18 (2010) 6604. [2] Q. Wang, L.Y. Tan, J. Ma, S.Y. Yu, Y.J. Jiang, Optics Express 20 (2012) 1033. [3] M.H. Mahdieh, M. Shirmahi, M. Alavi-Nejad, Beam quality in unstable optical resonators and the effects of misalignment—art. in: W. Gries, T.P. Pearsall (Eds.), Workshop on Laser Applications in Europe, 2006, 61570X. [4] Y.Q. Gao, B.Q. Zhu, D.Z. Liu, Z.Q. Lin, Journal of Optics 12 (2010) 095704. [5] H. Wang, Y. Li, L.J. Zeng, C.Y. Yin, Z.J. Feng, Optics Communications 232 (2004) 61. [6] H.S. Cho, W.H. Kim, H.H. Park, Optics and Lasers in Engineering 42 (2004) 503. [7] C.F. Chen, C.H. Lin, H.T. Jan, Y.L. Yang, Optics Communications 282 (2009) 360. [8] A. Rakich, Calculation of third-order misalignment aberrations with the optical plate diagram, in: J. Bentley, A. Gupta, R.N. Youngworth, (Eds.), International Optical Design Conference 2010. [9] I. Iparraguirre, T. del Rio-Gaztelurrutia, Optics Communications 284 (2011) 418. [10] J. Shao, J. Ye, Z. Hu, Z. Zhang, M. Huang, Optics and Precision Engineering 19 (2011) 7.
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
High-precision method for disorder in beam tracing in an optical system with elastic optics Shipeng Feng n, Dongxu Li College of Aerospace Science and Engineering, National University of Defense Technology, Changsha, Hunan Province 410073, China
art ic l e i nf o
a b s t r a c t
Article history: Received 15 January 2013 Received in revised form 26 April 2013 Accepted 20 May 2013 Available online 3 June 2013
Disorder in beam tracing should be calculated accurately when designing an optical system. In conventional numerical calculation methods for the disordered beam path, the optics were treated as rigid bodies and the misalignment caused by the deformation of the optics was neglected. In this work, the expression for the deformation of the surface normal was derived based on the elastic body assumption using differential geometry and the transient response finite-element method. Furthermore, with the aid of this expression, concrete steps for a numerical calculation method for disorder in beam tracing were given. Then, the validity of this improved method was tested with two examples and with an experimental result from another study. & 2013 Elsevier B.V. All rights reserved.
Keywords: Elastic optics Disorder in beam tracing Differential geometry Finite-element method
1. Introduction The application environments of optical systems are becoming more and more complex, and the fields in which optical systems are applied are continuously expanding. As a result, all types of disturbances in the environment, such as mechanical oscillations, temperature changes, atmosphere turbulence and so on, influence the stability of optical systems [1]. Among these types of disturbances, mechanical oscillations generally interfere with optical systems [2]. Mechanical oscillations cause beam path disorder, decrease the performance of optical systems and even result in optical system failure. Many academics have investigated this problem. Mahdieh et al. [3] and Gao et al. [4] focused on the quality of the beam affected by the misalignment of optical elements. To improve the stability of optical systems operating in a severe environment, the amount of beam disorder should be calculated accurately. Wang et al. [5] used a simple ray-tracing method to measure the vertex radius of curvature of an aspheric mirror. Cho et al. [6] investigated a method to calculate the coupling efficiency for various misalignment parameters. Chen et al. [7] used a ray-tracing method to design a solar concentrator. The accuracy of numerical methods to calculate a disordered beam path directly influences the design quality of optical systems. However, in conventional numerical calculation methods for the disordered beam path, the optical lenses were treated as rigid bodies. Only a few feature points were marked on the surfaces of
n
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the lens. Then, the deformations of the optical stands because of external excitation were calculated by FEM whereas the displacements of the feature points were given based on the results of the deformation of the optical stands. Therefore, through coordinate transformation, which used the displacements of the feature points, the disordered beam path could be calculated. Rakich [8] used this method to solve the problem of third-order misalignment aberrations. It was reasonable that the optical lenses were assumed to be rigid bodies under slight external excitation, while it would be so rough that the optical lenses were treated as rigid bodies if the external excitation was severe or if accurate results were needed. Recently, Iparraguirre and Rio-Gaztelurrutia [9] completed a perfect investigation of the laser modes in misaligned resonators. However, Iparraguirre also ignored the elasticity of the lenses of the resonators. The conventional method (CM) to calculate beam path disorder is a method in which the deformation of the optical stand and mount is computed by FEM and the elastic deformation of the optical elements caused by vibration is neglected. Although the deformation might be very slight, small changes in the reflection or refraction surfaces could lead to large misalignments. Thus, conventional calculation methods should be improved. Based on the elastic body assumption, the expression for the deformation of the surface normal was derived using differential geometry and the transient response finite-element method. Furthermore, with the aid of this expression, concrete steps for a numerical calculation method for disorder in beam tracing were listed. Primarily, the contact point between the beam and the surface of a mirror or lens was solved. Finally, the validity of this improved method was tested with two examples. In the two
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examples, the simulation results for the method considering the deformation of optics, the simulation results for the conventional method and the experimental results were compared with each other. The validity of this improved method was verified.
2. Calculation of the surface normal after deformation of the optics The conventional methods for disorder in beam tracing ignored the elastic deformation of the optics and derived the rigid displacement of the optics by calculating the deformation of the optical mounts. Based on the assumption of elastic optics, calculations of the surface deformation of the optical lens could be translated into calculations of the change of the surface normal. Based on differential geometry, the expression for the deformation of the surface normal was given. The displacement of any point on the surface could be expressed as functions of the coordinates x, y and z after surface deformation. 8 > < p ¼ pðx; y; zÞ q ¼ qðx; y; zÞ ð1Þ > : s ¼ sðx; y; zÞ In Eq. (1), p, q and s are the displacements of the point (x, y, z) in the x, y and z directions, respectively. Then, the deformation displacements in the directions of x, y and z should be added to the parametric equations for the changed surface. Thus, the changed parametric equations could be given by 8 > < x′ ¼ x þ pðx; y; zÞ y′ ¼ y þ qðx; y; zÞ ð2Þ > : z′ ¼ z þ sðx; y; zÞ Then, the vector form of the parametric equation for the changed surface could be expressed by ! ! ! ! ! r ′ ¼ r ′ðu; vÞ ¼ x′ðu; vÞ i þ y′ðu; vÞ j þ z′ðu; vÞ k ð3Þ where u and v are the independent variables of the parametric equation. The parameters x, y, and z are functions of u and v and from Eq. (2), the parameters x′, y′, and z′, which are expressed by the parameters x, y, and z, are functions of u and v as well. Therefore, the partial derivatives of the parametric vector equation of the surface with respect to the parameters u and v could be given as follows: 8 ! ! !
the conventional method for disorder in beam tracing ignores the effect of deformation of the optical surfaces because the direction of the normal could not be solved through analytical calculation methods based on transformation of the coordinate system. In the method described in this work, the optical surfaces could be divided into many small fields, based on the assumption of elastic optics. In every small field, the prediction and calculation of the surface deformation could be given by node displacements. The concrete steps of the numerical calculation method that considers the deformation of the optical surfaces for disorder in beam tracing are given as follows: First, screen the FEM nodes and determine the contact field. Screen nodes on the optical surface and calculate the distances between the nodes and the beam path. Then, pick three nodes, called P, Q and R, which have smaller distances than the other nodes. The element surface formed by Node P, Node Q and Node R is the field in which the beam makes contact with the optical surface. Second, calculate the contact point E between the beam path line and the surface PQR. Third, calculate the normal vectors at Node P, Node Q and Node R with Eq. (5). ! Fourth, calculate the normal vector n E at point E. From the normal vectors calculated in the third step, the ! normal n E could be calculated with the weighted mean method. Fifth, express the beam path after the beam interaction with ! the optics with the normal n E . Through the five steps above, the coordinates and normal vector of the contact point E between the beam path and the optical surface after deformation could be calculated. Then, the disorder beam could be precisely traced. 4. Comparison of numerical simulations and experimental results 4.1. Example 1 4.1.1. Description of the example To test the validity of this improved method, example 1 was given. Example 1 was shown by Shao et al. [10], they investigated the problem of beam path disorder with numerical simulations and experiment and neglected the deformation of the optics. In this work, the example was calculated with consideration of the optical deformation and the numerical result was compared with the simulation and experimental results in Shao's work [10]. A sketch of the one-mirror optical system in example 1 is shown in Fig. 1. A He–Ne laser was fixed on an optical table. The laser beam was reflected by a mirror and was transmitted to the CCD, which could evaluate the amount of beam disorder by detecting the spot position. The pixel resolution of the CCD was 9.84 μm in the horizontal direction and 10.22 μm in the vertical direction. The distance between the CCD and the reflector was 2 m. The mirror of the reflector was made of quartz glass, and the CCD
Laser Generator
Reflector
3-axis Accelerometer
2m Optical Table
3. High-precision method for disorder in beam tracing under the assumption of elastic optics The surface of optics can deform if the external excitation is severe or if the stiffness of the optical elements is low. The amount of beam path disorder is given by the combined effects of rigid displacement and deformation of the optical surfaces. However,
Vibrating Table
Fig. 1. A sketch of the one-mirror optical system for example 1.
S. Feng, D. Li / Optics Communications 306 (2013) 35–41
material of the mirror mount was aluminium alloy. In this example, the reflector was excited by the vibrating table and a misalignment was formed. The actual excitation of the reflector was tested with a 3-axis accelerometer fixed to the vibrating table near the reflector. 4.1.2. Simulation and calculation First, screen the FEM nodes of the nodes on the mirror surface and determine the contact field. Based on the conventional method used by Shao, the analytical model by FEM is illustrated in Fig. 2. The acceleration excitation, which was a type of random excitation with an amplitude between 7 4 m/s2 was loaded on the reflecting mirror. The mean value of the random excitation, which had a uniform distribution, was 0 m/s2. The displacement of any node on the mirror's surface at any moment could be calculated through transient dynamic analysis. Assume that P is a node on the mirror's surface P : ðx1 ; y1 ; z1 Þ
ð6Þ
The line equation of the spatial beam path could be expressed by L:
x−x0 y−y0 z−z0 ¼ ¼ p q r
ð7Þ
where (x0,y0,z0) is a point on the line and (p,q,r) is the direction vector of the line. Then, the distance between the node P and the beam path line L could be given by i j k p q r x1 −x0 y1 −y0 z1 −z0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d ¼ ð8Þ p2 þ q2 þ r 2
240mm
110mm
Φ 20mm
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On the surface of the mirror, screen three FEM nodes with the shortest distances from the beam path, and call these nodes P, Q and R. Then, the contact field was determined from the nodes P, Q and R. Second, calculate the contact point E between the beam path line and the surface PQR. Assume that there are three points P(x1,y1,z1), Q(x2,y2,z2), R(x3, y3,z3). Thus, the equation of the plane PQR could be given by x−x y−y1 z−z1 1 x−x2 y−y2 z−z2 ¼ 0 ð9Þ x−x y−y3 z−z3 3 By substituting Eq. (7) into Eq. (9), the contact point E could be obtained. Third, calculate the normal vectors at Node P, Node Q and Node R with Eq. (5). From the method from differential geometry described in the ! ! second part of this paper, the parametric equations for r P0 , r Q0 , !0 and r R located at Node P, Node Q and Node R, respectively, after deformation could be calculated with Eq. (4). In addition, with the ! ! ! aid of Eq. (5), the normal vectors n P , n q , and n r at the nodes P, Q and R could be calculated. 8 !0 !0 ! r Pu r Pv > > nP¼ ! > ! > > j r 0Pu r 0Pv j > > > !0 !0 < ! r r n Q ¼ !Q u !Q v ð10Þ > j r 0Q u r 0Q v j > > > !0 !0 > > r Ru r Rv > >! nR¼ ! : ! j r 0Ru r 0Rv j ! Fourth, calculate the normal vector n E at point E. ! The normal vector n E at point E could be given by the weighted mean. −1 ! −1 ! −1 ! dPE ⋅ n P þ dQ E ⋅ n Q þ dRE ⋅ n R ! nE¼ ð11Þ −1 −1 −1 dPE þ dQ E þ dRE where dPE, dQE, and dRE were the distances between points P, Q and R and point E. The weight coefficients were the reciprocals of the distances between points P, Q and R and point E because the node that was closest to E had a greater weight than the other nodes. Fifth, obtain the beam disorder by calculating the contact point between the beam path and the surface of the CCD. The disordered beam expression could be given through the reflection law described by the spatial vector ! ! ! ! a ′ ¼ a þ 2ðj a j cos βÞ⋅ n E ð12Þ ! ! where a was the incident beam, a ′ was the disordered reflecting beam and β was the angle of reflection. Assume that the coordinates of E were (x4,y4,z4) and that the ! direction of the beam vector a ′was (p′,q′,r′). Then, the equation for the disordered beam could be expressed by L′ :
x−x4 y−y4 z−z4 ¼ ¼ p′ q′ r′
ð13Þ
50mm
30mm
Finally, the contact point F between the disordered beam and the photosensitive surface of the CCD could be calculated with simultaneous equations for the straight line L′ and the photosensitive surface of the CCD. The amount of beam disorder could be expressed by the coordinates of point F.
50mm Fig. 2. The FEM model of the reflecting mirror.
4.1.3. Analysis of simulations In the calculation method for the beam disorder under the assumption of elastic optics, the direction of a normal vector was calculated by the displacement of a node. As a result, the accuracy of the normal direction would be influenced by the scale of the mesh. Additionally, the accuracy of the calculation of the disordered beam
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S. Feng, D. Li / Optics Communications 306 (2013) 35–41
Table 1 Simulation results for different scales for the mesh. Global edge length
Direction
5 mm
Horizontal 20.8 Vertical 190.6 Horizontal 19.6 Vertical 184.2 Horizontal 19.6 Vertical 184.8
1 mm 0.1 mm
Result 1 (μm)
Result 2 (μm)
Result 3 (μm)
Standard deviation (μm)
18.9 187.3 19.7 185.7 19.7 184.0
19.0 180.6 19.6 185.0 19.6 185.3
1.5 7.2 0.1 1.0 0.1 0.9
path would also be affected. To investigate the influence of the scale of the mesh on the accuracy of the calculation, a few simulation results with different scales for the mesh are compared with each other in Table 1. In Table 1, the global edge length is the maximal edge length for all of the elements. For any scale for the mesh, the simulation was performed three times, and the maximal horizontal and vertical misalignments are listed. Additionally, the standard deviations for every scale of the mesh were calculated. The standard deviation is a parameter that expresses the amount of variation in the results and could be calculated by
Horizontal Vertical
200 150
ð14Þ
where s is the standard deviation, n is the number of results, fi is the value of the ith result, and E(f) is the mean value of the results. In Table 1, if the global edge length was 5 mm, the standard deviations of the results for the misalignment calculation were quite large. The result in the horizontal direction was 1.5 μm whereas the result in the vertical direction was 7.2 μm. A large standard deviation meant that the calculation results for the amount of beam disorder were not stable and that the calculation errors were large. However, if the global edge length was 1 mm, the standard deviations of the results of the misalignment calculations clearly decreased. The result in the horizontal direction was 0.1 μm whereas the result in the vertical direction was 1.0 μm. If the global edge length was 0.1 mm, the standard deviations of the results of the misalignment calculations decreased slightly. Therefore, if the global edge length was 1 mm, the mesh was fine enough to solve this problem. This method was used to determine how fine the meshing should be. The traces of the contact point between the beam line and the photosensitive surface of the CCD are illustrated in Fig. 3, for the simulation calculated with a mesh scale of 0.1 mm. In Fig. 3, the contact points for every sampling time and the connecting lines connecting consecutive points are shown. In Fig. 3, the positions of the points on the surface of the CCD were accurately calculated with the theory. However, because the sampling time was discrete, the connecting lines between the points approximately fit the actual track of the contact point on the surface of the CCD. In Fig. 4, the horizontal and vertical coordinates of the contact point F at each sampling time are illustrated. To investigate the statistical properties of the contact point F of the disordered beam, the horizontal and vertical coordinates of point F were treated with a statistical test. The results of the statistical test are shown in Table 2. Because the optical element was excited by a random acceleration excitation, which had a uniform distribution with a mean value of 0 m/s2, the amount of beam disorder should not deviate far from a uniform distribution with a mean value of 0 m/s2. In Table 2, the mean values and the deviations of the amount of beam disorder are given. Because the mean values were much less than
100 X and Y/10-6m
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 n s¼ ∑ ðf −Eðf ÞÞ2 ni¼1 i
Fig. 3. Simulation of the amount of beam end disorder.
50 0 -50 -100 -150 -200
0
1
2
3
4
5 t/s
6
7
8
9
10
Fig. 4. The horizontal and vertical coordinates of the contact point F at each sampling time.
Table 2 The statistical properties of the simulation results for example 1. Direction
Mean value E (μm)
Horizontal −0.9 Vertical −15.9
Standard deviation s (μm)
Results of χ2 test
Maximum (μm)
Minimum (μm)
10.8 103.2
3.7 8.6
19.6 184.8
−19.6 −183.7
the standard deviations in the horizontal and vertical directions, the calculation results for the beam disorder fit the distribution with a mean value of 0 μm. To test whether the calculated results for the beam disorder fit the uniform distribution, the χ2 test was used. The equation for the χ2 test was given by ðM i −T i Þ2 Ti i¼1 10
χ2 ¼ ∑
ð15Þ
where Mi was the number of points in the ith interval, and Ti was the number of points in the ith interval in the theoretical
S. Feng, D. Li / Optics Communications 306 (2013) 35–41
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treatment. The intervals were defined to be one tenth of the field between the maximal result and the minimal result. In the horizontal direction, the interval was 3.92 μm whereas in the vertical direction, the interval was 36.9 μm. In Table 2, the result of the χ2 test in the horizontal direction was 3.7 and the result of the χ2 test in the vertical direction was 8.6. Based on the theory of the χ2 test, the test in this example had 9 degrees of freedom, and the critical value χ 20:05 ð9Þ at a 5% significance level was 16.9. ( 2 χ Horizontal ¼ 3:7 oχ 20:05 ð9Þ ð16Þ χ 2Vertical ¼ 8:6 o χ 20:05 ð9Þ
experimental data was 0.5%. In the vertical direction, the deviation between the result calculated with the conventional method and the experimental data was 0.8% whereas the deviation between the result calculated with the new method and the experimental data was 0.4%. Based on Table 3, the results of the numerical method for the beam path disorder calculation that considers the deformation of the optics more accurately reflected the experimental results than did the results from the conventional method.
Based on Eq. (16), the test was accepted. In other words, the calculated results for the beam disorder fit the uniform distribution. In summary, based on the assumption of elastic optics, the calculation results for the beam disorder depended on the properties of the excitation. Therefore, the position of the contact point F on the photosensitive surface of the CCD fit the statistical properties of the uniform distribution with a mean value of 0 μm. In other words, the results calculated by the method in this work fit the distribution of the excitation well. Shao et al. [10] developed the experimental system for this example and tested the maximal disorder amount of the beam. What follows in this work is a comparison among the simulation results from the method that considers the deformation of the mirror, the simulation results from the conventional method and the experimental results obtained by Shao; this comparison is given in Table 3. In Table 3, the results calculated from the method based on the assumption of elastic optics and the method based on the assumption of rigid optics were compared with Shao's experimental data. The maximal misalignments in the horizontal and vertical directions are given in Table 3. In the horizontal direction, the deviation between the result calculated with the conventional method and the experimental data was 4.1% whereas the deviation between the result calculated with the new method and the
4.2.1. Description of the example To compare the method given in this work and the conventional method, example 2 was used. In example 2, the optical system contains two optical reflection elements and one refraction element. The optical reflection elements are the same as the element in example 1. The mount of the refraction element is also the same as the mount in example 1. The three optical elements were excited by a vibrating table. If the excitation was not present, the centres of the three optical elements were at the same height, and the spacings of the optical elements were 0.1 m. Additionally, the distance between the second mirror and the photosensitive surface of the CCD was 2 m. The results calculated by different methods for the misalignment were compared with each other as the amplitude of the excitation changed. The optical system for example 2 is illustrated in Fig. 5. The optical elements were excited with the same type of random excitation at different amplitudes. In example 2, three types of different excitations were added to the optical system. The amplitudes of the three types of excitation were 74 m/s2, 78 m/s2 and 7 12 m/s2, and the mean value of the three types of excitation was 0 m/s2.
Table 3 Comparison of the simulated and experimental results.
Experiment New method Conventional method
Horizontal direction
Vertical direction
Results (μm)
Error compared with experimental result
Results (μm)
Error compared with experimental result
19.7 19.6 18.9
0 0.5% 4.1%
184.0 184.8 185.5
0 0.4% 0.8%
CCD
4.2. Example 2
4.2.2. Simulation and calculation The numerical method given in this work and the conventional method were used to solve example 2. The calculation steps were the same as those for example 1 although the process was more complex. The amount of beam path disorder after the simulation is shown in Fig. 6. The amount of beam path disorder under three different excitations was calculated using two different methods. The simulation results are shown in Table 4. As shown in Table 4, the amount of beam path disorder increased as the excitation amplitude increased. In the horizontal direction, the amounts of beam disorder calculated by the two methods increased from 24.9 μm and 23.7 μm to 94.9 μm and 79.4 μm, respectively. Additionally, in
0.1m
Laser Generator Reflector
0.1m
Refractor
Reflector
Optical Table 2m
Vibrating Table
Fig. 5. A sketch of the three-mirror optical system for example 2.
3 Axis Accelerometer
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S. Feng, D. Li / Optics Communications 306 (2013) 35–41
Fig. 6. Comparison of the simulation results for example 2. (a), (c), and (e) are the results for the simulation with deformation of the optics under excitation amplitudes of 7 4, 7 8, and 7 12 m/s2, respectively. (b), (d), and (f) are the results for the simulation without deformation of the optics under excitation amplitudes of 7 4, 7 8, and 7 12 m/s2, respectively.
Table 4 Comparison of simulated results for different methods and excitations. Direction
Horizontal misalignment Vertical misalignment
Method
New method Conventional method New method Conventional method
Excitation 1
Excitation 2
Excitation 3
Result (μm)
Deviation (%)
Result (μm)
Deviation (%)
Result (μm)
Deviation (%)
24.9 23.8 229.7 216.9
4.7
62.9 55.3 577.9 521.2
12.1
94.9 79.4 872.6 735.5
16.4
5.6
the vertical direction, the amounts of beam disorder calculated by the two methods increased from 229.7 μm and 216.9 μm to 872.6 μm and 735.5 μm, respectively. For the same excitation, the amount of beam disorder calculated with the method described in this work was larger than the amount of beam disorder calculated with the conventional method. Additionally, as the excitation amplitude increases, the deviations of results calculated with the method based on the assumption of elastic optics and with the conventional method
9.8
15.7
increased. In Table 4, the deviations calculated with the two methods increased from 4.7% for excitation 1 to 16.4% for excitation 3 in the horizontal direction. Additionally, the deviations calculated with the two methods increased from 5.6% for excitation 1 to 15.7% for excitation 3 in the vertical direction. The reason for this phenomenon is that the method in this work includes the amount of beam disorder caused by optical deformation whereas the conventional method does not. If amplitude of the excitation increases, the deformation of the optics becomes greater and
S. Feng, D. Li / Optics Communications 306 (2013) 35–41
influences the amount of beam disorder much more. As a result, the deviations between the results calculated by the two methods increased as the excitation increased. 4.3. Discussion It is necessary to be attentive to the optical mesh because the precision of the method given in this work relies on the scale of the mesh. The finer the mesh is, the more accurate the calculation results are. Additionally, if the amount of bending of the lens is great, the influence of the scale of the mesh is obvious. In this work, an example of a one-reflector optical system under external excitation was given. In example 1, by comparing the calculated results at different mesh scales, how fine the meshing should be was pointed out. Then, to investigate the correlation of the calculation results and the excitation, the simulation results were analysed statistically. The statistical properties of simulation results and the excitation were similar. The reliability of the method in this work was tested. Because of limitations on experimental conditions, we did not perform experimental research. However, to validate the numerical method given in this work, the calculation results were compared with the simulation results and experimental data from Shao. The deviation between the simulation results of the method described in this work and the experimental data was less than that between the simulation results of the conventional method and the experimental data. The results showed that the amount of beam disorder was caused by both the rigid displacement and the elastic deformation of the optics. Hence, the calculation method could be improved by introducing the deformation of optics to beam path tracing. The deviation between the method described in this work and the conventional method under three different excitation conditions was shown with example 2. As the external excitations became larger and larger, the deviation of the calculated results for the two methods became larger and larger. This result arose because the deformation of the optical lens became greater as the amplitude of the external excitation increased. The calculated results for the conventional method that did not consider the deformation of the optics had greater errors under the increased excitation conditions. The results showed that the deformation of optics must be introduced to methods to calculate the beam path disorder to obtain more accurate results when the external excitation is violent. It was also shown that the method described in this work could be applied to a complex optical system. Compared with the conventional method, the method in this work requires the same workload to calculate the deformation of the optical mount and stand but requires more computational
41
time to calculate the deformation of the optics. As a result, the computational cost for the improved accuracy is the computational time for the deformation of the optics. Because the method described in this work is based on a mesh method and does not use an analytical equation for the surface of the optic, this method could be used in optical systems with optics of arbitrary shape, such as optics with aspheric surfaces.
5. Conclusions To increase the calculation accuracy for disordered beam path tracing, this work improved upon the conventional method. The calculation method in this work was based on the assumption of elastic optics and included the optic, the rigid displacement and elastic deformation, which influenced the beam disorder. If the external excitation was violent, the deformation of the optic could not be neglected. Thus, the improved method for disorder in beam tracing in this work has a greater precision than the conventional method. Additionally, compared with the conventional method, the method described in this work does not obviously increase the complexity or the workload and is suitable for calculations of the beam path in precise optical instruments. However, it is necessary to be attentive to the optical mesh because the precision of the method given in this work relies on the scale of the mesh. The finer the mesh is, the more accurate the calculation results are. Additionally, if the amount of bending of the optic is great, the influence of the scale of the mesh is clear. The next step is to use the method described in this work to investigate how the different amounts of bending of the optics influence the precision of the disorder in beam tracing. References [1] X.L. Ji, H.T. Eyyuboglu, Y. Baykal, Optics Express 18 (2010) 6604. [2] Q. Wang, L.Y. Tan, J. Ma, S.Y. Yu, Y.J. Jiang, Optics Express 20 (2012) 1033. [3] M.H. Mahdieh, M. Shirmahi, M. Alavi-Nejad, Beam quality in unstable optical resonators and the effects of misalignment—art. in: W. Gries, T.P. Pearsall (Eds.), Workshop on Laser Applications in Europe, 2006, 61570X. [4] Y.Q. Gao, B.Q. Zhu, D.Z. Liu, Z.Q. Lin, Journal of Optics 12 (2010) 095704. [5] H. Wang, Y. Li, L.J. Zeng, C.Y. Yin, Z.J. Feng, Optics Communications 232 (2004) 61. [6] H.S. Cho, W.H. Kim, H.H. Park, Optics and Lasers in Engineering 42 (2004) 503. [7] C.F. Chen, C.H. Lin, H.T. Jan, Y.L. Yang, Optics Communications 282 (2009) 360. [8] A. Rakich, Calculation of third-order misalignment aberrations with the optical plate diagram, in: J. Bentley, A. Gupta, R.N. Youngworth, (Eds.), International Optical Design Conference 2010. [9] I. Iparraguirre, T. del Rio-Gaztelurrutia, Optics Communications 284 (2011) 418. [10] J. Shao, J. Ye, Z. Hu, Z. Zhang, M. Huang, Optics and Precision Engineering 19 (2011) 7.