Abstract

Thermo-optical simulation is a mandatory enhancement of classical ray tracing, since nowadays many fields in the branch of optical technology have to deal with thermal effects. This paper discusses an approach for coupling the finite element method (FEM) and ray tracing simulation by processing finite element (FE) data using scattered data approximation techniques, particularly with an adaptive weighted least squares approximation algorithm in two dimensions. The validation of the implemented interface is being conducted by comparing approximated data to analytical functions. Finally, FEM data are being processed by the developed algorithm to demonstrate the applicability on appropriate problems.

© 2012 Optical Society of America

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  1. E. Langenbach, “Ray tracing in gradient-index materials,” Proc. SPIE 1780, 486–490 (1992).
    [CrossRef]
  2. J. Chmelfk and J. E. Barth, “An interpolation method for ray tracing in electrostatic fields calculated by the finite element method,” Proc. SPIE 2014, 133–143 (1993).
    [CrossRef]
  3. V. L. Genberg, “Ray tracing from finite element results,” Proc. SPIE 1998, 72–82 (1993).
    [CrossRef]
  4. M. Weck, C. Hermanns, H. Ostendarp, and K. Wermeyer, “Betriebsverhalten transmissiver Optiken bei der Laser-Materialbearbeitung,” Laser Optoelektron. 26, 67–72(1994).
  5. L. H. J. F. Beckmann, “Modelling of and design for thermal radial gradients in lenses for use with high power laser radiation,” Weld. World 41, 97–104 (1998).
  6. D. Bürckner-Koydl, H. Heckmann, U. Ruhnau, and R. Birkner, “Trigonometrische Durchrechnung von thermisch belasteten Optiksystemen,” in DGaO Proceedings (DGaO, 2010).
  7. M. J. Moritz, “Radial distribution of temperature in a thin lens due to absorption of light and heat conduction,” Optik 122, 1050–1057 (2011).
    [CrossRef]
  8. A. Sharma, D. V. Kumar, and A. K. Ghatak, “Tracing rays through graded-index media: a new method,” Appl. Opt. 21, 984–987 (1982).
    [CrossRef]
  9. A. Gatej, U. Thombansen, and P. Loosen, “Kombinierte Thermo-Optische Simulation für Optische Systeme,” in DGaO Proceedings (DGaO, 2011).
  10. P. Loosen, M. C. Funck, A. Gatej, V. Morasch, and J. Stollenwerk, “Integrative production of micro-lasers,” in Integrative Production Technology for High-Wage Countries, 1st ed., C. Brecher, ed. (Springer, 2012), pp. 995–1038.
  11. A. Gatej, U. Thombansen, and P. Loosen, “Simulation des thermischen Linseneffekts in hochbelasteten Lasersystemen,” Photonik 3, 58–60 (2012), http://photonik12.photonik.de/?p=8337 .
  12. A. Nealen, “An as-short-as-possible introduction to the least squares, weighted least squares and moving least squares methods for scattered data approximation and interpolation” (2004), http://www.nealen.com/projects/ .
  13. H. Wendland, Scattered Data Approximation (Cambridge University, 2005).
  14. G. Coombe, Practical Surface Light Fields (Proquest, 2007).
  15. D. Shepard, “A two-dimensional interpolation function for irregularly-spaced data,” in Proceedings ACM National Conference (ACM, 1968), pp. 517–524.
  16. A. Gatej, J. Wasselowski, and P. Loosen, “Thermo-optical (TOP) analysis by coupling FEM and ray tracing,” Proc. SPIE 8429, 84290E (2012).
    [CrossRef]
  17. “LAPACK—Linear Algebra PACKage,” http://www.netlib.org/lapack/ .
  18. “Armadillo—C++ linear algebra library,” http://arma.sourceforge.net/ .
  19. “Boost—C++ libraries,” http://www.boost.org/ .
  20. R. J. Tangelder, L. H. J. F. Beckmann, and J. Meijer, “Influence of temperature gradients on the performance of ZnSe lenses,” Proc. SPIE 1780, 294–302 (1993).
    [CrossRef]

2012 (2)

A. Gatej, U. Thombansen, and P. Loosen, “Simulation des thermischen Linseneffekts in hochbelasteten Lasersystemen,” Photonik 3, 58–60 (2012), http://photonik12.photonik.de/?p=8337 .

A. Gatej, J. Wasselowski, and P. Loosen, “Thermo-optical (TOP) analysis by coupling FEM and ray tracing,” Proc. SPIE 8429, 84290E (2012).
[CrossRef]

2011 (1)

M. J. Moritz, “Radial distribution of temperature in a thin lens due to absorption of light and heat conduction,” Optik 122, 1050–1057 (2011).
[CrossRef]

1998 (1)

L. H. J. F. Beckmann, “Modelling of and design for thermal radial gradients in lenses for use with high power laser radiation,” Weld. World 41, 97–104 (1998).

1994 (1)

M. Weck, C. Hermanns, H. Ostendarp, and K. Wermeyer, “Betriebsverhalten transmissiver Optiken bei der Laser-Materialbearbeitung,” Laser Optoelektron. 26, 67–72(1994).

1993 (3)

J. Chmelfk and J. E. Barth, “An interpolation method for ray tracing in electrostatic fields calculated by the finite element method,” Proc. SPIE 2014, 133–143 (1993).
[CrossRef]

V. L. Genberg, “Ray tracing from finite element results,” Proc. SPIE 1998, 72–82 (1993).
[CrossRef]

R. J. Tangelder, L. H. J. F. Beckmann, and J. Meijer, “Influence of temperature gradients on the performance of ZnSe lenses,” Proc. SPIE 1780, 294–302 (1993).
[CrossRef]

1992 (1)

E. Langenbach, “Ray tracing in gradient-index materials,” Proc. SPIE 1780, 486–490 (1992).
[CrossRef]

1982 (1)

Barth, J. E.

J. Chmelfk and J. E. Barth, “An interpolation method for ray tracing in electrostatic fields calculated by the finite element method,” Proc. SPIE 2014, 133–143 (1993).
[CrossRef]

Beckmann, L. H. J. F.

L. H. J. F. Beckmann, “Modelling of and design for thermal radial gradients in lenses for use with high power laser radiation,” Weld. World 41, 97–104 (1998).

R. J. Tangelder, L. H. J. F. Beckmann, and J. Meijer, “Influence of temperature gradients on the performance of ZnSe lenses,” Proc. SPIE 1780, 294–302 (1993).
[CrossRef]

Birkner, R.

D. Bürckner-Koydl, H. Heckmann, U. Ruhnau, and R. Birkner, “Trigonometrische Durchrechnung von thermisch belasteten Optiksystemen,” in DGaO Proceedings (DGaO, 2010).

Bürckner-Koydl, D.

D. Bürckner-Koydl, H. Heckmann, U. Ruhnau, and R. Birkner, “Trigonometrische Durchrechnung von thermisch belasteten Optiksystemen,” in DGaO Proceedings (DGaO, 2010).

Chmelfk, J.

J. Chmelfk and J. E. Barth, “An interpolation method for ray tracing in electrostatic fields calculated by the finite element method,” Proc. SPIE 2014, 133–143 (1993).
[CrossRef]

Coombe, G.

G. Coombe, Practical Surface Light Fields (Proquest, 2007).

Funck, M. C.

P. Loosen, M. C. Funck, A. Gatej, V. Morasch, and J. Stollenwerk, “Integrative production of micro-lasers,” in Integrative Production Technology for High-Wage Countries, 1st ed., C. Brecher, ed. (Springer, 2012), pp. 995–1038.

Gatej, A.

A. Gatej, J. Wasselowski, and P. Loosen, “Thermo-optical (TOP) analysis by coupling FEM and ray tracing,” Proc. SPIE 8429, 84290E (2012).
[CrossRef]

A. Gatej, U. Thombansen, and P. Loosen, “Simulation des thermischen Linseneffekts in hochbelasteten Lasersystemen,” Photonik 3, 58–60 (2012), http://photonik12.photonik.de/?p=8337 .

A. Gatej, U. Thombansen, and P. Loosen, “Kombinierte Thermo-Optische Simulation für Optische Systeme,” in DGaO Proceedings (DGaO, 2011).

P. Loosen, M. C. Funck, A. Gatej, V. Morasch, and J. Stollenwerk, “Integrative production of micro-lasers,” in Integrative Production Technology for High-Wage Countries, 1st ed., C. Brecher, ed. (Springer, 2012), pp. 995–1038.

Genberg, V. L.

V. L. Genberg, “Ray tracing from finite element results,” Proc. SPIE 1998, 72–82 (1993).
[CrossRef]

Ghatak, A. K.

Heckmann, H.

D. Bürckner-Koydl, H. Heckmann, U. Ruhnau, and R. Birkner, “Trigonometrische Durchrechnung von thermisch belasteten Optiksystemen,” in DGaO Proceedings (DGaO, 2010).

Hermanns, C.

M. Weck, C. Hermanns, H. Ostendarp, and K. Wermeyer, “Betriebsverhalten transmissiver Optiken bei der Laser-Materialbearbeitung,” Laser Optoelektron. 26, 67–72(1994).

Kumar, D. V.

Langenbach, E.

E. Langenbach, “Ray tracing in gradient-index materials,” Proc. SPIE 1780, 486–490 (1992).
[CrossRef]

Loosen, P.

A. Gatej, J. Wasselowski, and P. Loosen, “Thermo-optical (TOP) analysis by coupling FEM and ray tracing,” Proc. SPIE 8429, 84290E (2012).
[CrossRef]

A. Gatej, U. Thombansen, and P. Loosen, “Simulation des thermischen Linseneffekts in hochbelasteten Lasersystemen,” Photonik 3, 58–60 (2012), http://photonik12.photonik.de/?p=8337 .

A. Gatej, U. Thombansen, and P. Loosen, “Kombinierte Thermo-Optische Simulation für Optische Systeme,” in DGaO Proceedings (DGaO, 2011).

P. Loosen, M. C. Funck, A. Gatej, V. Morasch, and J. Stollenwerk, “Integrative production of micro-lasers,” in Integrative Production Technology for High-Wage Countries, 1st ed., C. Brecher, ed. (Springer, 2012), pp. 995–1038.

Meijer, J.

R. J. Tangelder, L. H. J. F. Beckmann, and J. Meijer, “Influence of temperature gradients on the performance of ZnSe lenses,” Proc. SPIE 1780, 294–302 (1993).
[CrossRef]

Morasch, V.

P. Loosen, M. C. Funck, A. Gatej, V. Morasch, and J. Stollenwerk, “Integrative production of micro-lasers,” in Integrative Production Technology for High-Wage Countries, 1st ed., C. Brecher, ed. (Springer, 2012), pp. 995–1038.

Moritz, M. J.

M. J. Moritz, “Radial distribution of temperature in a thin lens due to absorption of light and heat conduction,” Optik 122, 1050–1057 (2011).
[CrossRef]

Nealen, A.

A. Nealen, “An as-short-as-possible introduction to the least squares, weighted least squares and moving least squares methods for scattered data approximation and interpolation” (2004), http://www.nealen.com/projects/ .

Ostendarp, H.

M. Weck, C. Hermanns, H. Ostendarp, and K. Wermeyer, “Betriebsverhalten transmissiver Optiken bei der Laser-Materialbearbeitung,” Laser Optoelektron. 26, 67–72(1994).

Ruhnau, U.

D. Bürckner-Koydl, H. Heckmann, U. Ruhnau, and R. Birkner, “Trigonometrische Durchrechnung von thermisch belasteten Optiksystemen,” in DGaO Proceedings (DGaO, 2010).

Sharma, A.

Shepard, D.

D. Shepard, “A two-dimensional interpolation function for irregularly-spaced data,” in Proceedings ACM National Conference (ACM, 1968), pp. 517–524.

Stollenwerk, J.

P. Loosen, M. C. Funck, A. Gatej, V. Morasch, and J. Stollenwerk, “Integrative production of micro-lasers,” in Integrative Production Technology for High-Wage Countries, 1st ed., C. Brecher, ed. (Springer, 2012), pp. 995–1038.

Tangelder, R. J.

R. J. Tangelder, L. H. J. F. Beckmann, and J. Meijer, “Influence of temperature gradients on the performance of ZnSe lenses,” Proc. SPIE 1780, 294–302 (1993).
[CrossRef]

Thombansen, U.

A. Gatej, U. Thombansen, and P. Loosen, “Simulation des thermischen Linseneffekts in hochbelasteten Lasersystemen,” Photonik 3, 58–60 (2012), http://photonik12.photonik.de/?p=8337 .

A. Gatej, U. Thombansen, and P. Loosen, “Kombinierte Thermo-Optische Simulation für Optische Systeme,” in DGaO Proceedings (DGaO, 2011).

Wasselowski, J.

A. Gatej, J. Wasselowski, and P. Loosen, “Thermo-optical (TOP) analysis by coupling FEM and ray tracing,” Proc. SPIE 8429, 84290E (2012).
[CrossRef]

Weck, M.

M. Weck, C. Hermanns, H. Ostendarp, and K. Wermeyer, “Betriebsverhalten transmissiver Optiken bei der Laser-Materialbearbeitung,” Laser Optoelektron. 26, 67–72(1994).

Wendland, H.

H. Wendland, Scattered Data Approximation (Cambridge University, 2005).

Wermeyer, K.

M. Weck, C. Hermanns, H. Ostendarp, and K. Wermeyer, “Betriebsverhalten transmissiver Optiken bei der Laser-Materialbearbeitung,” Laser Optoelektron. 26, 67–72(1994).

Appl. Opt. (1)

Laser Optoelektron. (1)

M. Weck, C. Hermanns, H. Ostendarp, and K. Wermeyer, “Betriebsverhalten transmissiver Optiken bei der Laser-Materialbearbeitung,” Laser Optoelektron. 26, 67–72(1994).

Optik (1)

M. J. Moritz, “Radial distribution of temperature in a thin lens due to absorption of light and heat conduction,” Optik 122, 1050–1057 (2011).
[CrossRef]

Photonik (1)

A. Gatej, U. Thombansen, and P. Loosen, “Simulation des thermischen Linseneffekts in hochbelasteten Lasersystemen,” Photonik 3, 58–60 (2012), http://photonik12.photonik.de/?p=8337 .

Proc. SPIE (5)

E. Langenbach, “Ray tracing in gradient-index materials,” Proc. SPIE 1780, 486–490 (1992).
[CrossRef]

J. Chmelfk and J. E. Barth, “An interpolation method for ray tracing in electrostatic fields calculated by the finite element method,” Proc. SPIE 2014, 133–143 (1993).
[CrossRef]

V. L. Genberg, “Ray tracing from finite element results,” Proc. SPIE 1998, 72–82 (1993).
[CrossRef]

R. J. Tangelder, L. H. J. F. Beckmann, and J. Meijer, “Influence of temperature gradients on the performance of ZnSe lenses,” Proc. SPIE 1780, 294–302 (1993).
[CrossRef]

A. Gatej, J. Wasselowski, and P. Loosen, “Thermo-optical (TOP) analysis by coupling FEM and ray tracing,” Proc. SPIE 8429, 84290E (2012).
[CrossRef]

Weld. World (1)

L. H. J. F. Beckmann, “Modelling of and design for thermal radial gradients in lenses for use with high power laser radiation,” Weld. World 41, 97–104 (1998).

Other (10)

D. Bürckner-Koydl, H. Heckmann, U. Ruhnau, and R. Birkner, “Trigonometrische Durchrechnung von thermisch belasteten Optiksystemen,” in DGaO Proceedings (DGaO, 2010).

A. Nealen, “An as-short-as-possible introduction to the least squares, weighted least squares and moving least squares methods for scattered data approximation and interpolation” (2004), http://www.nealen.com/projects/ .

H. Wendland, Scattered Data Approximation (Cambridge University, 2005).

G. Coombe, Practical Surface Light Fields (Proquest, 2007).

D. Shepard, “A two-dimensional interpolation function for irregularly-spaced data,” in Proceedings ACM National Conference (ACM, 1968), pp. 517–524.

“LAPACK—Linear Algebra PACKage,” http://www.netlib.org/lapack/ .

“Armadillo—C++ linear algebra library,” http://arma.sourceforge.net/ .

“Boost—C++ libraries,” http://www.boost.org/ .

A. Gatej, U. Thombansen, and P. Loosen, “Kombinierte Thermo-Optische Simulation für Optische Systeme,” in DGaO Proceedings (DGaO, 2011).

P. Loosen, M. C. Funck, A. Gatej, V. Morasch, and J. Stollenwerk, “Integrative production of micro-lasers,” in Integrative Production Technology for High-Wage Countries, 1st ed., C. Brecher, ed. (Springer, 2012), pp. 995–1038.

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Figures (11)

Fig. 1.
Fig. 1.

Thermally loaded optical system with coaxial process control.

Fig. 2.
Fig. 2.

Coupling of FEM with ray tracing simulation by using data approximation techniques.

Fig. 3.
Fig. 3.

Reconstruction of a function based on scattered data by using the WLS algorithm [16].

Fig. 4.
Fig. 4.

Flow chart of numerical experiments for WLS validation in the context of a TOP simulation.

Fig. 5.
Fig. 5.

Radial (r) and axial (z) temperature distribution for an exemplary cylindrical laser crystal.

Fig. 6.
Fig. 6.

Convergence of the temperature’s RMSE at intermediate points.

Fig. 7.
Fig. 7.

RMSE of the radial gradient over sampling points.

Fig. 8.
Fig. 8.

Percentage deviation of the focal point between analytical and approximated functions for different threshold approximation errors.

Fig. 9.
Fig. 9.

Percentage deviation of the focal point using a noisy input signal for approximation.

Fig. 10.
Fig. 10.

Comparison of the variation of deviation for undisturbed (left) and noisy data at 0.1% error threshold.

Fig. 11.
Fig. 11.

Deviation in temperature in radial (r) and axial (z) direction between FEM and analytic solutions.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

n(r⃗(s))dsmin.
dds(n(r⃗)dr⃗ds)=n(r⃗).
dr⃗dτ2=n(r⃗)n(r⃗).
n(T)=n0+dndTΔT.
1flens=1fcurv+1fTL.
T(r,z)={T0+ΔT(1+2ln(r0rB)r2rB2)eαz0rrBT0+2ΔTln(rr0)eαzrB<rr0.
T*(r,x)=ε(r,x)·T(r,x);0.99ε1.01.

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