Abstract

An irregular grid model was employed to describe the refractive index distribution of an aerodynamically heated optical dome according to the theories of thermo-optical and elasto-optical effects. Optical transmission through the dome was simulated using the ray-tracing program based on a fourth-order Runge–Kutta algorithm. Two kinds of imaging quality evaluation parameters were presented, wave aberration of the exit pupil and a modulation transfer function. To validate the ray-tracing program, a ray trace through a regular gradient medium was performed. Results were compared with those obtained from the analytic solution. The program was shown to possess great accuracy by using the appropriate parameters.

© 2010 Optical Society of America

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References

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  1. G. W. Sutton, “Parametric study of optical distortion due to window heating,” Proc. SPIE 3151, 131–137 (1997).
    [CrossRef]
  2. E. Frumker and O. Pade, “Generic method for aero-optic evaluations,” Appl. Opt. 43, 3224–3228 (2004).
    [CrossRef] [PubMed]
  3. Y. P. Zhang and Z. G. Fan, “Study on the optical path difference of aero-optical window,” Optik (Jena) 118, 557–560(2007).
    [CrossRef]
  4. C. A. Klein and R. L. Gentilman, “Thermal shock resistance of convectively heated infrared windows and domes,” Proc. SPIE 3060, 115–129 (1997).
    [CrossRef]
  5. J. E. Pond, C. T. Welch, and G. W. Sutton, “Side mounted IR window aero-optic and aerothermal analysis,” Proc. SPIE 3705, 266–275 (1999).
    [CrossRef]
  6. M. V. Parish, M. R. Pascucci, and W. H. Rhodes, “Aerodynamic IR domes of polycrystalline alumina,” Proc. SPIE 5786, 195–205 (2005).
    [CrossRef]
  7. R. M. Sullivan, “A historical view of germanium as an infrared window material,” Proc. SPIE 7302, 73020L (2009).
    [CrossRef]
  8. Y. C. Yiu and A. R. Meyer, “Computation of optical errors in transparent optical elements due to three dimensional photoelastic effect,” Proc. SPIE 1303, 206–216 (1990).
    [CrossRef]
  9. K. B. Doyle, V. L. Genberg, and G. J. Michels, “Numerical methods to compute optical errors due to stress birefringence,” Proc. SPIE 4769, 34–42 (2002).
    [CrossRef]
  10. J. F. Nye, Physical Properties of Crystals (Oxford Univ. Press, 1985).
  11. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
  12. D. C. Harris, Materials for Infrared Windows and Domes (SPIE, 1999).
    [CrossRef]
  13. D. Yang, M. E. Thomas, and S. G. Kaplan, “Measurement of the infrared refractive index of sapphire as function of temperature,” Proc. SPIE 4375, 53–63 (2001).
    [CrossRef]
  14. W. H. Yu and W. Y. Liu, Crystal Physics (University of Science and Technology of China Press, 1998).
  15. Y. F. Qiao, Gradient Index Optics (Science Press, 1991).

2009 (1)

R. M. Sullivan, “A historical view of germanium as an infrared window material,” Proc. SPIE 7302, 73020L (2009).
[CrossRef]

2007 (1)

Y. P. Zhang and Z. G. Fan, “Study on the optical path difference of aero-optical window,” Optik (Jena) 118, 557–560(2007).
[CrossRef]

2005 (1)

M. V. Parish, M. R. Pascucci, and W. H. Rhodes, “Aerodynamic IR domes of polycrystalline alumina,” Proc. SPIE 5786, 195–205 (2005).
[CrossRef]

2004 (1)

2002 (1)

K. B. Doyle, V. L. Genberg, and G. J. Michels, “Numerical methods to compute optical errors due to stress birefringence,” Proc. SPIE 4769, 34–42 (2002).
[CrossRef]

2001 (1)

D. Yang, M. E. Thomas, and S. G. Kaplan, “Measurement of the infrared refractive index of sapphire as function of temperature,” Proc. SPIE 4375, 53–63 (2001).
[CrossRef]

1999 (1)

J. E. Pond, C. T. Welch, and G. W. Sutton, “Side mounted IR window aero-optic and aerothermal analysis,” Proc. SPIE 3705, 266–275 (1999).
[CrossRef]

1997 (2)

G. W. Sutton, “Parametric study of optical distortion due to window heating,” Proc. SPIE 3151, 131–137 (1997).
[CrossRef]

C. A. Klein and R. L. Gentilman, “Thermal shock resistance of convectively heated infrared windows and domes,” Proc. SPIE 3060, 115–129 (1997).
[CrossRef]

1990 (1)

Y. C. Yiu and A. R. Meyer, “Computation of optical errors in transparent optical elements due to three dimensional photoelastic effect,” Proc. SPIE 1303, 206–216 (1990).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Doyle, K. B.

K. B. Doyle, V. L. Genberg, and G. J. Michels, “Numerical methods to compute optical errors due to stress birefringence,” Proc. SPIE 4769, 34–42 (2002).
[CrossRef]

Fan, Z. G.

Y. P. Zhang and Z. G. Fan, “Study on the optical path difference of aero-optical window,” Optik (Jena) 118, 557–560(2007).
[CrossRef]

Frumker, E.

Genberg, V. L.

K. B. Doyle, V. L. Genberg, and G. J. Michels, “Numerical methods to compute optical errors due to stress birefringence,” Proc. SPIE 4769, 34–42 (2002).
[CrossRef]

Gentilman, R. L.

C. A. Klein and R. L. Gentilman, “Thermal shock resistance of convectively heated infrared windows and domes,” Proc. SPIE 3060, 115–129 (1997).
[CrossRef]

Harris, D. C.

D. C. Harris, Materials for Infrared Windows and Domes (SPIE, 1999).
[CrossRef]

Kaplan, S. G.

D. Yang, M. E. Thomas, and S. G. Kaplan, “Measurement of the infrared refractive index of sapphire as function of temperature,” Proc. SPIE 4375, 53–63 (2001).
[CrossRef]

Klein, C. A.

C. A. Klein and R. L. Gentilman, “Thermal shock resistance of convectively heated infrared windows and domes,” Proc. SPIE 3060, 115–129 (1997).
[CrossRef]

Liu, W. Y.

W. H. Yu and W. Y. Liu, Crystal Physics (University of Science and Technology of China Press, 1998).

Meyer, A. R.

Y. C. Yiu and A. R. Meyer, “Computation of optical errors in transparent optical elements due to three dimensional photoelastic effect,” Proc. SPIE 1303, 206–216 (1990).
[CrossRef]

Michels, G. J.

K. B. Doyle, V. L. Genberg, and G. J. Michels, “Numerical methods to compute optical errors due to stress birefringence,” Proc. SPIE 4769, 34–42 (2002).
[CrossRef]

Nye, J. F.

J. F. Nye, Physical Properties of Crystals (Oxford Univ. Press, 1985).

Pade, O.

Parish, M. V.

M. V. Parish, M. R. Pascucci, and W. H. Rhodes, “Aerodynamic IR domes of polycrystalline alumina,” Proc. SPIE 5786, 195–205 (2005).
[CrossRef]

Pascucci, M. R.

M. V. Parish, M. R. Pascucci, and W. H. Rhodes, “Aerodynamic IR domes of polycrystalline alumina,” Proc. SPIE 5786, 195–205 (2005).
[CrossRef]

Pond, J. E.

J. E. Pond, C. T. Welch, and G. W. Sutton, “Side mounted IR window aero-optic and aerothermal analysis,” Proc. SPIE 3705, 266–275 (1999).
[CrossRef]

Qiao, Y. F.

Y. F. Qiao, Gradient Index Optics (Science Press, 1991).

Rhodes, W. H.

M. V. Parish, M. R. Pascucci, and W. H. Rhodes, “Aerodynamic IR domes of polycrystalline alumina,” Proc. SPIE 5786, 195–205 (2005).
[CrossRef]

Sullivan, R. M.

R. M. Sullivan, “A historical view of germanium as an infrared window material,” Proc. SPIE 7302, 73020L (2009).
[CrossRef]

Sutton, G. W.

J. E. Pond, C. T. Welch, and G. W. Sutton, “Side mounted IR window aero-optic and aerothermal analysis,” Proc. SPIE 3705, 266–275 (1999).
[CrossRef]

G. W. Sutton, “Parametric study of optical distortion due to window heating,” Proc. SPIE 3151, 131–137 (1997).
[CrossRef]

Thomas, M. E.

D. Yang, M. E. Thomas, and S. G. Kaplan, “Measurement of the infrared refractive index of sapphire as function of temperature,” Proc. SPIE 4375, 53–63 (2001).
[CrossRef]

Welch, C. T.

J. E. Pond, C. T. Welch, and G. W. Sutton, “Side mounted IR window aero-optic and aerothermal analysis,” Proc. SPIE 3705, 266–275 (1999).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Yang, D.

D. Yang, M. E. Thomas, and S. G. Kaplan, “Measurement of the infrared refractive index of sapphire as function of temperature,” Proc. SPIE 4375, 53–63 (2001).
[CrossRef]

Yiu, Y. C.

Y. C. Yiu and A. R. Meyer, “Computation of optical errors in transparent optical elements due to three dimensional photoelastic effect,” Proc. SPIE 1303, 206–216 (1990).
[CrossRef]

Yu, W. H.

W. H. Yu and W. Y. Liu, Crystal Physics (University of Science and Technology of China Press, 1998).

Zhang, Y. P.

Y. P. Zhang and Z. G. Fan, “Study on the optical path difference of aero-optical window,” Optik (Jena) 118, 557–560(2007).
[CrossRef]

Appl. Opt. (1)

Optik (Jena) (1)

Y. P. Zhang and Z. G. Fan, “Study on the optical path difference of aero-optical window,” Optik (Jena) 118, 557–560(2007).
[CrossRef]

Proc. SPIE (8)

C. A. Klein and R. L. Gentilman, “Thermal shock resistance of convectively heated infrared windows and domes,” Proc. SPIE 3060, 115–129 (1997).
[CrossRef]

J. E. Pond, C. T. Welch, and G. W. Sutton, “Side mounted IR window aero-optic and aerothermal analysis,” Proc. SPIE 3705, 266–275 (1999).
[CrossRef]

M. V. Parish, M. R. Pascucci, and W. H. Rhodes, “Aerodynamic IR domes of polycrystalline alumina,” Proc. SPIE 5786, 195–205 (2005).
[CrossRef]

R. M. Sullivan, “A historical view of germanium as an infrared window material,” Proc. SPIE 7302, 73020L (2009).
[CrossRef]

Y. C. Yiu and A. R. Meyer, “Computation of optical errors in transparent optical elements due to three dimensional photoelastic effect,” Proc. SPIE 1303, 206–216 (1990).
[CrossRef]

K. B. Doyle, V. L. Genberg, and G. J. Michels, “Numerical methods to compute optical errors due to stress birefringence,” Proc. SPIE 4769, 34–42 (2002).
[CrossRef]

D. Yang, M. E. Thomas, and S. G. Kaplan, “Measurement of the infrared refractive index of sapphire as function of temperature,” Proc. SPIE 4375, 53–63 (2001).
[CrossRef]

G. W. Sutton, “Parametric study of optical distortion due to window heating,” Proc. SPIE 3151, 131–137 (1997).
[CrossRef]

Other (5)

W. H. Yu and W. Y. Liu, Crystal Physics (University of Science and Technology of China Press, 1998).

Y. F. Qiao, Gradient Index Optics (Science Press, 1991).

J. F. Nye, Physical Properties of Crystals (Oxford Univ. Press, 1985).

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

D. C. Harris, Materials for Infrared Windows and Domes (SPIE, 1999).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Irregular refractive index grid model of the optical dome: (a) normal view, (b) cut-away view.

Fig. 2
Fig. 2

Discrete optical path in a refractive index grid of the dome. P is the point of propagation in optical path P P 1 . The hexahedron is the closest refractive index grid encircling P. G 1 , G 2 , G 3 , and G 4 are the nodes located at the top of the refractive index grid. The x axis is parallel to the thickness direction of the dome.

Fig. 3
Fig. 3

Definitions for azimuth and elevation incident angles. Curved arrows indicate positive angles.

Fig. 4
Fig. 4

Ray tracing on the deformed surface of the optical dome. A , B , C , and D are the nodes of the deformed surface grid encircling intersection point E of the incident ray and the deformed surface. A, B, C, and D are the counterparts of the nondeformed surface grid encircling intersection point E.

Fig. 5
Fig. 5

Heat flux distribution on the outside surface of the dome.

Fig. 6
Fig. 6

Temperature field of the optical dome after 15 s : (a) normal view, (b) cut-away view 1, and (c) cut-away view 2. Maximum temperature is 281.28 ° C ; minimum temperature is 28.93 ° C .

Fig. 7
Fig. 7

Sum deformation field of the optical dome after 15 s : (a) normal view, (b) cut-away view 1, and (c) cut-away view 2. Maximum sum deformation is 9.36 × 10 5 m ; minimum sum deformation is 0 m .

Fig. 8
Fig. 8

Equivalent von Mises strain field of the optical dome after 15 s : (a) normal view, (b) cut-away view 1, and (c) cut-away view 2. Maximum equivalent von Mises strain is 3.26 × 10 4 ; minimum equivalent von Mises strain is 1.21 × 10 5 .

Fig. 9
Fig. 9

Absolute errors of the ray-tracing program. Dashed curve depicts the absolute error of the x coordinate ( σ x ); solid curve depicts the absolute error of the y coordinate ( σ y ).

Fig. 10
Fig. 10

Wave aberration results of the optical dome at different incident angles: (a) 0 ° / 90 ° (azimuth/elevation), (b) 0 ° / 89 ° , and (c) 0 ° / 88.5 ° .

Fig. 11
Fig. 11

MTF results of the optical dome at different incident angles: (a) 0 ° / 90 ° (azimuth/elevation), (b) 0 ° / 89 ° , and (c) 0 ° / 88.5 ° . Solid curve depicts diffraction-limited MTF. Curve with circles marks depicts meridional MTF. Curve with triangles depicts sagittal MTF.

Tables (3)

Tables Icon

Table 1 Main Physical Properties of the Sapphire Crystal Near 300 K

Tables Icon

Table 2 Refractive Index Variations of the Point with Maximum Temperature and Maximum Sum Deformation Variations at Different Working Times a

Tables Icon

Table 3 Refractive Index Variations of the Point with Maximum Equivalent von Mises Strain Variation at Different Working Times a

Equations (20)

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n i ( λ , T ) = n i ( λ , T 0 ) + Δ n i = n i ( λ , T 0 ) + n i ( λ , T ) T Δ T ( x , y , z ) ,
B 1 x 1 2 + B 2 x 2 2 + B 3 x 3 2 = 1 ,
B 1 = B 2 = n e 2 , B 3 = n o 2 .
Δ B = [ B 1 B 1 B 2 B 2 B 3 B 3 B 4 B 4 B 5 B 5 B 6 B 6 ] = [ P 11 P 12 P 13 P 14 0 0 P 12 P 11 P 13 P 14 0 0 P 31 P 13 P 33 0 0 0 P 41 P 41 0 P 44 0 0 0 0 0 0 P 44 P 41 0 0 0 0 P 14 0.5 ( P 11 P 12 ) ] [ ε x ε y ε z γ y z γ x z γ x y ] ,
Δ n 1 = n 1 n 1 = ( B 1 ) 1 / 2 B 1 1 / 2 0.5 n e 3 ( P 11 ε x + P 12 ε y + P 13 ε z + P 14 γ y z ) , Δ n 2 = n 2 n 2 = ( B 2 ) 1 / 2 B 2 1 / 2 0.5 n e 3 ( P 12 ε x + P 11 ε y + P 13 ε z P 14 γ y z ) , Δ n 3 = n 3 n 3 = ( B 3 ) 1 / 2 B 3 1 / 2 0.5 n o 3 ( P 31 ε x + P 13 ε y + P 33 ε z ) , Δ n 4 = n 4 n 4 = ( B 4 ) 1 / 2 B 4 1 / 2 0.5 ( n o n e ) 3 / 2 ( P 41 ε x P 41 ε y + P 44 γ y z ) , Δ n 5 = n 5 n 5 = ( B 5 ) 1 / 2 B 5 1 / 2 0.5 ( n o n e ) 3 / 2 ( P 44 γ x z + P 41 γ x y ) , Δ n 6 = n 6 n 6 = ( B 6 ) 1 / 2 B 6 1 / 2 0.5 n e 3 [ P 14 γ x z + 0.5 ( P 11 P 12 ) γ x y ] ,
n ( P ) = [ n ( G 1 ) d 4 2 + n ( G 2 ) d 3 2 + n ( G 3 ) d 2 2 + n ( G 4 ) d 1 2 ] ( d 1 2 + d 2 2 + d 3 2 + d 4 2 ) ,
OPL i = n i l i ,
OPL i T = ( n i l i ) T = n i l i T + l i n i T ,
OPL i ε = ( n i l i ) ε = n i l i ε + l i n i ε ,
OPL i = T 0 T 0 + Δ T ( n i l i T + l i n i T ) d T + ε 0 ε 0 + Δ ε ( n i l i ε + l i n i ε ) d ε .
OPL = Σ i OPL i .
OPL 0 = 1 N Σ k OPL k ,
W k ( x , y ) = 2 π λ ( OPL k OPL 0 ) .
W ( x , y ) = Σ k W k ( x , y ) = Σ k 2 π λ ( OPL k OPL 0 ) .
A ( x , y ) = { a ( x , y ) exp [ j W ( x , y ) ] x 2 + y 2 r 2 0 x 2 + y 2 > r 2 ,
U ( x , y ) = A ( x , y ) exp [ j 2 π λ f ( x x + y y ) ] d x d y .
PSF ( x , y ) = | U ( x , y ) | 2 .
n ( x , y ) = n 0 [ 1 + α 2 ( x 2 + y 2 ) ] 1 / 2 ,
x = x 0 cosh ( n 0 α z / L 0 ) + p 0 n 0 α sinh ( n 0 α z / L 0 ) , y = y 0 cosh ( n 0 α z / L 0 ) + q 0 n 0 α sinh ( n 0 α z / L 0 ) ,
σ x = | x R x A | , σ y = | y R y A | ,

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