Abstract

Conventional ray-tracing methods fail for non-Lambertian sources. To address this deficiency, we introduce a radiometric ray-tracing (R2T) method, applicable to quasi-homogeneous sources of arbitrary spatial coherence. Based on Fourier optics, applied to physical radiometry in the radiance transfer function second-order approximation, the R2T method retains the standard ray-tracing codes but modifies them to include phase-space weighting factors attached to conventional geometric rays.

© 1996 Optical Society of America

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References

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  1. W. T. Welford, R. Winston, The Optics of Non-Imaging Concentrators (Academic, New York, 1978).
  2. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), Sect. 4.8.
  3. D. Marcuse, Light Transmission Optics (Academic, New York, 1970).
  4. J. W. Goodman, Statistical Optics (Wiley, New York, 1980).
  5. E. Wolf, J. Jannson, T. Jannson, “Analog of the Van Cittert–Zernike theorem for statistically homogeneous wave fields,” Opt. Lett. 15, 1032–1034 (1990).
    [CrossRef] [PubMed]
  6. T. Jannson, “Radiance transfer function,”J. Opt. Soc. Am. 70, 1544–1549 (1980).
    [CrossRef]
  7. J. T. Foley, E. Wolf, “Radiometry as a short-wavelength limit of statistical wave theory with globally incoherent sources,” Opt. Commun. 55, 236–241 (1985).
    [CrossRef]
  8. K. Kim, E. Wolf, “Propagation law for Walther’s first generalized radiance function and its short wavelength limit with quasi-homogeneous sources,” J. Opt. Soc. Am. A 4, 1233–1236 (1987).
    [CrossRef]
  9. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  10. T. Jannson, “Self-imaging effect in physical radiometry,”J. Opt. Soc. Am. 73, 402–409 (1983).
    [CrossRef]
  11. A. Walther, “Radiometry and coherence,”J. Opt. Soc. Am. 58, 1256–1259 (1968).
    [CrossRef]
  12. A. Walther, “Radiometry and coherence,”J. Opt. Soc. Am. 63, 1622–1623 (1973).
    [CrossRef]
  13. A. Walther, “Propagation of the generalized radiance through lenses,”J. Opt. Soc. Am. 68, 1606–1611 (1978).
    [CrossRef]
  14. E. W. Marchand, E. Wolf, “Radiometry with sources of any state of coherence,”J. Opt. Soc. Am. 64, 1219–1226 (1974).
    [CrossRef]
  15. E. W. Marchand, E. Wolf, “Walther’s definition of generalized radiance,”J. Opt. Soc. Am. 64, 1273–1274 (1974).
    [CrossRef]
  16. W. H. Carter, E. Wolf, “Coherence properties of Lambertian and non-Lambertian sources,”J. Opt. Soc. Am. 65, 1067–1071 (1975).
    [CrossRef]
  17. W. H. Carter, E. Wolf, “Coherence and radiometry with quasihomogeneous planar sources,”J. Opt. Soc. Am. 67, 785–796 (1977).
    [CrossRef]
  18. E. Wolf, “The radiant intensity from planar sources of any state of coherence,”J. Opt. Soc. Am. 68, 1597–1605 (1978).
    [CrossRef]
  19. E. Wolf, “Coherence and radiometry,”J. Opt. Soc. Am. 68, 6–17 (1978).
    [CrossRef]
  20. H. P. Baltes, J. Geist, A. Walther, “Radiometry and coherence,” in Topics in Current Physics, H. P. Baltes, ed. (Springer, Berlin, 1978), Vol. 9.
    [CrossRef]
  21. K. J. Kim, “Brightness, coherence and propagation characteristics of synchrotron radiation,” Nucl. Instrum. Methods A246, 71–76 (1986).
  22. T. Jannson, I. Tengara, “Radiometric ray tracing,” in Proceedings of the 10th Symposium on Energy Engineering Sciences (Argonne National Laboratory, Argonne, Illinois, 1992), pp. 241–250.
  23. T. Jannson, L. Sadovnik, T. Aye, I. Tengara, “Radiometric ray tracing,” in Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper MCC4.

1990 (1)

1987 (1)

1986 (1)

K. J. Kim, “Brightness, coherence and propagation characteristics of synchrotron radiation,” Nucl. Instrum. Methods A246, 71–76 (1986).

1985 (1)

J. T. Foley, E. Wolf, “Radiometry as a short-wavelength limit of statistical wave theory with globally incoherent sources,” Opt. Commun. 55, 236–241 (1985).
[CrossRef]

1983 (1)

1980 (1)

1978 (3)

1977 (1)

1975 (1)

1974 (2)

1973 (1)

1968 (1)

Aye, T.

T. Jannson, L. Sadovnik, T. Aye, I. Tengara, “Radiometric ray tracing,” in Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper MCC4.

Baltes, H. P.

H. P. Baltes, J. Geist, A. Walther, “Radiometry and coherence,” in Topics in Current Physics, H. P. Baltes, ed. (Springer, Berlin, 1978), Vol. 9.
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), Sect. 4.8.

Carter, W. H.

Foley, J. T.

J. T. Foley, E. Wolf, “Radiometry as a short-wavelength limit of statistical wave theory with globally incoherent sources,” Opt. Commun. 55, 236–241 (1985).
[CrossRef]

Geist, J.

H. P. Baltes, J. Geist, A. Walther, “Radiometry and coherence,” in Topics in Current Physics, H. P. Baltes, ed. (Springer, Berlin, 1978), Vol. 9.
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

J. W. Goodman, Statistical Optics (Wiley, New York, 1980).

Jannson, J.

Jannson, T.

E. Wolf, J. Jannson, T. Jannson, “Analog of the Van Cittert–Zernike theorem for statistically homogeneous wave fields,” Opt. Lett. 15, 1032–1034 (1990).
[CrossRef] [PubMed]

T. Jannson, “Self-imaging effect in physical radiometry,”J. Opt. Soc. Am. 73, 402–409 (1983).
[CrossRef]

T. Jannson, “Radiance transfer function,”J. Opt. Soc. Am. 70, 1544–1549 (1980).
[CrossRef]

T. Jannson, I. Tengara, “Radiometric ray tracing,” in Proceedings of the 10th Symposium on Energy Engineering Sciences (Argonne National Laboratory, Argonne, Illinois, 1992), pp. 241–250.

T. Jannson, L. Sadovnik, T. Aye, I. Tengara, “Radiometric ray tracing,” in Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper MCC4.

Kim, K.

Kim, K. J.

K. J. Kim, “Brightness, coherence and propagation characteristics of synchrotron radiation,” Nucl. Instrum. Methods A246, 71–76 (1986).

Marchand, E. W.

Marcuse, D.

D. Marcuse, Light Transmission Optics (Academic, New York, 1970).

Sadovnik, L.

T. Jannson, L. Sadovnik, T. Aye, I. Tengara, “Radiometric ray tracing,” in Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper MCC4.

Tengara, I.

T. Jannson, I. Tengara, “Radiometric ray tracing,” in Proceedings of the 10th Symposium on Energy Engineering Sciences (Argonne National Laboratory, Argonne, Illinois, 1992), pp. 241–250.

T. Jannson, L. Sadovnik, T. Aye, I. Tengara, “Radiometric ray tracing,” in Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper MCC4.

Walther, A.

Welford, W. T.

W. T. Welford, R. Winston, The Optics of Non-Imaging Concentrators (Academic, New York, 1978).

Winston, R.

W. T. Welford, R. Winston, The Optics of Non-Imaging Concentrators (Academic, New York, 1978).

Wolf, E.

J. Opt. Soc. Am. (11)

J. Opt. Soc. Am. A (1)

Nucl. Instrum. Methods (1)

K. J. Kim, “Brightness, coherence and propagation characteristics of synchrotron radiation,” Nucl. Instrum. Methods A246, 71–76 (1986).

Opt. Commun. (1)

J. T. Foley, E. Wolf, “Radiometry as a short-wavelength limit of statistical wave theory with globally incoherent sources,” Opt. Commun. 55, 236–241 (1985).
[CrossRef]

Opt. Lett. (1)

Other (8)

T. Jannson, I. Tengara, “Radiometric ray tracing,” in Proceedings of the 10th Symposium on Energy Engineering Sciences (Argonne National Laboratory, Argonne, Illinois, 1992), pp. 241–250.

T. Jannson, L. Sadovnik, T. Aye, I. Tengara, “Radiometric ray tracing,” in Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper MCC4.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

W. T. Welford, R. Winston, The Optics of Non-Imaging Concentrators (Academic, New York, 1978).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), Sect. 4.8.

D. Marcuse, Light Transmission Optics (Academic, New York, 1970).

J. W. Goodman, Statistical Optics (Wiley, New York, 1980).

H. P. Baltes, J. Geist, A. Walther, “Radiometry and coherence,” in Topics in Current Physics, H. P. Baltes, ed. (Springer, Berlin, 1978), Vol. 9.
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Illustration of optical systems that can be analyzed by using the R2T model: (a) imaging systems, (b) nonimaging systems, (c) holographic systems.

Fig. 2
Fig. 2

Illustration of Eq. (22), representing the linear RTF approximation, identical to the conventional radiometry formula. It can be seen that the vector Rr0 is indeed parallel to the vector S.

Fig. 3
Fig. 3

Basic principle of the R2T model in the linear RTF approximation.

Fig. 4
Fig. 4

Solution of two approximate models: Fresnel diffraction (dashed curve) and radiometric ray tracing (solid curve) for output intensity (emissivity) at z/a = 10 and rectangular input intensity distribution. Q = 4, a/σg = 5, σg/λ = 3.3, and ϕF = 1.

Fig. 5
Fig. 5

Same as Fig. 4 but for Q = 78, a/σg = 5, σg/λ = 3, and ϕF = 12.

Fig. 6
Fig. 6

Same as Fig. 4 but for z/a = 50, Q = 98, a/σg = 25, σg/λ = 3, and ϕF = 0.62.

Fig. 7
Fig. 7

Basic principle of the R2T model in the quadratic RTF approximation, applying the conventional ray tracing ( B ˜ 0 B ˜ ) and the Fourier intensity spectrum (I0Î0) into a quasi-homogeneous source with arbitrary degree of spatial coherence μ0

Fig. 8
Fig. 8

Two-dimensional spatial distribution of output emissivity (optical intensity) computed using the R2T model. The output of the NIO truncated cone with input and output radii of 0.15 mm and 0.6 mm, respectively, is z/a = 200. The input is a Gaussian-correlated circular source of radius a with a Gaussian intensity at z/a = 0, 2πσg/λ = 20, 2πa/λ = 1771, and 2πσs/λ = 2000. The total number of rays with the use of up to three total internal reflections is 7,110,861.

Tables (1)

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Table 1 Estimation of Relative Skew Factor Wa

Equations (68)

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U ˜ ( f ) = F ^ { U ( r ) } = - U ( r ) exp ( - i 2 π f · r ) d 2 r ,
U ( r ) = F ^ - 1 { U ˜ ( f ) } = - U ˜ ( f ) exp ( i 2 π f · r ) d 2 f .
U ( r ) = h ( r - r 0 ) U 0 ( r 0 ) d 2 r 0 ,
U ˜ ( f ) = H ( f ) U ˜ 0 ( f ) ,
B ( r , p ) = 2 ω k s z Re [ U ( r ) U ˜ * ( p / λ ) exp ( - i k p · r ) ] ,
B 0 ( r , p ) = 2 ω k s z I 0 ( r ) μ ˜ 0 ( p / λ ) ,
B 0 ( r , p ) = B 0 * ( r , p ) .
μ 0 ( r ) = μ 0 ( r 1 - r 2 ) = U 0 * ( r 1 ) U 0 ( r 2 ) [ I 0 ( r 1 ) I 0 ( r 2 ) ] 1 / 2 ,
λ f I 1 ,             1 ,
B ( r , p ) = g ( r - r 0 , p ) B 0 ( r 0 , p ) d 2 r 0 ,
g ( r , p ) = F ^ - 1 { G ( f , p ) } ;
B ˜ ( f , p ) = G ( f , p ) B ˜ 0 ( f , p ) ,
G ( f , p ) = 1 2 [ H ˜ ( p λ ) H ˜ * ( p λ - f ) + H ˜ * ( p λ ) H ˜ ( p λ + f ) ] ,
H ˜ ( f ) = exp ( 2 π i z 1 λ 2 - f 2 ) ,
H ˜ ( p / λ ) = exp ( i k s z z ) ,
G ( f , p ) = 1 2 exp [ ( i k s z z ) ( 1 - 1 + a λ 2 + b λ ) ] + 1 2 exp [ ( i k s z z ) ( 1 + a λ 2 - b λ - 1 ) ] ,
a λ 2 ± b λ 1.
G ( 1 ) ( f , p ) = exp ( - 2 π i z s z p · f ) ;
g ( 1 ) ( r , p ) = F ^ { G ( 1 ) ( f , p ) } = δ ( r - z s z p ) .
B ( r , p ) = B 0 ( r - z s z p , p ) .
r - r 0 = z s z p
( R - R 0 ) × S = 0 ,             R 0 = r 0 ,
E ( r ) = B ( r , p ) d 2 p ,
G ( 2 ) ( f , p ) = exp ( - 2 π i z s z p · f ) cos { z π λ s z [ f 2 + ( p , f ) 2 s z 2 ] } .
M ( f , p ) = cos { z π λ s z [ f 2 + ( p · f ) 2 s z 2 ] } .
M ( f , p ) 1 ,
ϕ = z π λ f 2 1.
( r ) = exp ( - r 2 2 σ s 2 ) circ ( r a ) ,
μ 0 ( r ) = exp ( - r 2 2 σ g 2 ) ,             σ g a .
z 1 = Λ 1 2 π λ
Q = ( 2 a ) 3 4 λ π a ( 1 + x a ) 4 1             ( Fresnel diffraction approximation 15 ) ,
a σ g 1             ( quasi - homogeneous source approximation ) ,
ϕ F = π λ z f 1 2 1             ( linear R 2 T model ) ,
σ g λ > 1             ( no evanescent waves ) .
E ( r ) = c k Im U * U z ,
S = c k Im U * U ,
U z i k s z U ,
E ( r ) = c U 2 I ( r ) ,
g ( 2 ) ( r , p ) = F ^ - 1 { G ( 2 ) ( f , p ) } = M ( f , p ) exp [ 2 π i f · ( r - z s z p ) ] d 2 f ,
g p ( 2 ) ( r , p ) = cos [ k 2 z ( r - z s z p ) 2 ] ;
B p ( 2 ) ( r , p ) = cos [ k 2 z ( r - r 0 - z s z p ) 2 ] × B 0 ( r 0 , p ) d 2 r 0 .
lim λ 0 cos [ k 2 z ( r - r 0 - z s z p ) 2 ] = δ ( r - r 0 - z s z p ) ,
I 0 ( r 0 ) = I ^ 0 ( f I ) exp ( 2 π i f I · r 0 ) d 2 f I ,
I 0 ( r 0 ) = A 0 ( f I ) cos [ 2 π f I · r 0 + Φ 0 ( f I ) ] d 2 f I ,
I ^ 0 ( f I ) = A 0 ( f I ) exp [ i Φ 0 ( f I ) ] .
I ^ 0 ( f I ) = I ( r 0 ) exp ( - 2 π i f I · r 0 ) d 2 r 0 ,
A 0 ( f I ) = I ^ 0 ( f I ) ,
Φ 0 ( f I ) = arg [ I ^ 0 ( f I ) ] .
B 0 ( r 0 , p ) = B ˜ 0 ( p ; f I ) d 2 f I ,
B ˜ 0 ( p ; f I ) = 2 ω k s z μ ^ ( p / λ ) A ( f I ) cos [ 2 π f I r 0 + Φ ( f I ) ] .
B ( r , p ) = B ˜ ( p ; f I ) d 2 f I ,
B ˜ ( 2 ) ( r , p ; f I ) = g ˜ ( 2 ) ( r - r 0 , p ; f I ) B ˜ 0 ( p ; f I ) d 2 r 0 .
g ˜ ( 2 ) ( r - r 0 , p ; f I ) = M ( f I , p ) δ ( r - r 0 - z s z p ) ;
B ˜ ( 2 ) ( r , p ; f I ) = 2 ω k s z μ ^ ( p / λ ) M ( f I , p ) , B ˜ 0 ( r - z s z p , f I ) = 2 ω k s z μ ^ ( p / λ ) M ( f I , p ) A ( f I ) × cos [ 2 π f I ( r - z s z p ) ] .
B ˜ ( 2 ) ( r , p ) = 2 ω k s z μ ^ ( p / λ ) A ( f I ) M ( f I , p ) × cos [ 2 π f I ( r - z s z p ) + Φ ( f I ) ] d 2 f I ,
A ( f I ) = A δ ( f I - f I 0 ) ;
( I 0 , μ 0 ) ( E ) Source Output Emittance .
G ( 3 ) ( f , p ) = M ( f , p ) exp [ - 2 π i z s z ( p · f ) ] × exp { [ - π i z λ 2 s z ( p · f ) ] × [ ( p · f ) 2 s z 4 - f 2 s z 2 ] } ,
p = p ( p , f ) = p ( 1 + D λ 2 ) ,
D = D ( p , f ) = 1 2 [ ( p · f ) 2 s z 4 - f 2 s z 2 ] .
G ( 3 ) ( f , p ) = M ( f , p ) exp [ - 2 π i z λ 2 s z ( p · f ) ] .
δ p = p - p .
W δ p p = D λ 2 ,
D max = 1 2 ( p 2 f 2 s z 4 + f 2 s z 2 ) = f 2 2 cos 2 θ ( 1 + tan 2 θ ) ,
D max = 2 f 2 ,
W = δ θ tan θ = δ θ = 2 λ 2 f 2 .
δ θ 0.001°             for θ 45° ,
Λ I 5 λ .

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