Abstract

An analytic solution is derived for radiation transfer between flat quasi-Lambertian surfaces of arbitrary orientation, i.e., surfaces that radiate in a Lambertian fashion but within a numerical aperture smaller than unity. These formulas obviate the need for ray trace simulations and provide exact, physically transparent results. Illustrative examples that capture the salient features of the flux maps and the efficiency of flux transfer are presented for a few configurations of practical interest. There is also a fundamental reciprocity relation for quasi-Lambertian exchange, akin to the reciprocity theorem for fully Lambertian surfaces. Applications include optical fiber coupling, fiber-optic biomedical procedures, and solar concentrators.

© 2010 Optical Society of America

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References

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  1. M. Young, Optics and Lasers Including Fibers and Optical Waveguides, 5th ed. (Springer, 2000), pp. 265–315.
  2. A. Katzir, Lasers and Optical Fibers in Medicine (Academic, 1993), pp. 107–207.
  3. J. M. Gordon, E. A. Katz, D. Feuermann, M. Huleihil, “Toward ultrahigh-flux photovoltaic concentration,” Appl. Phys. Lett. 84, 3642–3644 (2004).
    [CrossRef]
  4. M. F. Modest, Radiative Heat Transfer, 2nd ed. (McGraw-Hill, 2003), pp. 131–158.
    [CrossRef]
  5. R. Winston, J. C. Miñano, P. Benítez, N. Shatz, J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 7–43, 415–420.
  6. A. Rabl, Active Solar Collectors and Their Applications (Oxford U. Press, 1985), pp. 114–145.

2004 (1)

J. M. Gordon, E. A. Katz, D. Feuermann, M. Huleihil, “Toward ultrahigh-flux photovoltaic concentration,” Appl. Phys. Lett. 84, 3642–3644 (2004).
[CrossRef]

Benítez, P.

R. Winston, J. C. Miñano, P. Benítez, N. Shatz, J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 7–43, 415–420.

Bortz, J.

R. Winston, J. C. Miñano, P. Benítez, N. Shatz, J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 7–43, 415–420.

Feuermann, D.

J. M. Gordon, E. A. Katz, D. Feuermann, M. Huleihil, “Toward ultrahigh-flux photovoltaic concentration,” Appl. Phys. Lett. 84, 3642–3644 (2004).
[CrossRef]

Gordon, J. M.

J. M. Gordon, E. A. Katz, D. Feuermann, M. Huleihil, “Toward ultrahigh-flux photovoltaic concentration,” Appl. Phys. Lett. 84, 3642–3644 (2004).
[CrossRef]

Huleihil, M.

J. M. Gordon, E. A. Katz, D. Feuermann, M. Huleihil, “Toward ultrahigh-flux photovoltaic concentration,” Appl. Phys. Lett. 84, 3642–3644 (2004).
[CrossRef]

Katz, E. A.

J. M. Gordon, E. A. Katz, D. Feuermann, M. Huleihil, “Toward ultrahigh-flux photovoltaic concentration,” Appl. Phys. Lett. 84, 3642–3644 (2004).
[CrossRef]

Katzir, A.

A. Katzir, Lasers and Optical Fibers in Medicine (Academic, 1993), pp. 107–207.

Miñano, J. C.

R. Winston, J. C. Miñano, P. Benítez, N. Shatz, J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 7–43, 415–420.

Modest, M. F.

M. F. Modest, Radiative Heat Transfer, 2nd ed. (McGraw-Hill, 2003), pp. 131–158.
[CrossRef]

Rabl, A.

A. Rabl, Active Solar Collectors and Their Applications (Oxford U. Press, 1985), pp. 114–145.

Shatz, N.

R. Winston, J. C. Miñano, P. Benítez, N. Shatz, J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 7–43, 415–420.

Winston, R.

R. Winston, J. C. Miñano, P. Benítez, N. Shatz, J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 7–43, 415–420.

Young, M.

M. Young, Optics and Lasers Including Fibers and Optical Waveguides, 5th ed. (Springer, 2000), pp. 265–315.

Appl. Phys. Lett. (1)

J. M. Gordon, E. A. Katz, D. Feuermann, M. Huleihil, “Toward ultrahigh-flux photovoltaic concentration,” Appl. Phys. Lett. 84, 3642–3644 (2004).
[CrossRef]

Other (5)

M. F. Modest, Radiative Heat Transfer, 2nd ed. (McGraw-Hill, 2003), pp. 131–158.
[CrossRef]

R. Winston, J. C. Miñano, P. Benítez, N. Shatz, J. Bortz, Nonimaging Optics (Elsevier, 2005), pp. 7–43, 415–420.

A. Rabl, Active Solar Collectors and Their Applications (Oxford U. Press, 1985), pp. 114–145.

M. Young, Optics and Lasers Including Fibers and Optical Waveguides, 5th ed. (Springer, 2000), pp. 265–315.

A. Katzir, Lasers and Optical Fibers in Medicine (Academic, 1993), pp. 107–207.

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Figures (12)

Fig. 1
Fig. 1

Radiative exchange between infinitesimal source and target surface elements. The arrows indicate the normals to the surfaces.

Fig. 2
Fig. 2

Exchange between parallel surfaces. The shaded region on the source is a virtual circle of radius R v . The distance between the source element d A s at ( x s , y s , h ) ) and the target element d A t at ( x t , 0 , 0 ) is l = [ ( x t x s ) 2 + y s 2 + h 2 ] 1 / 2 . The projection angles are cos ( θ 1 ) = cos ( θ 2 ) = h / l .

Fig. 3
Fig. 3

Illustration of the region of a disk source that contributes flux to a parallel target element, to wit, the overlap of the virtual circle R v and the source.

Fig. 4
Fig. 4

Configuration for two flat nonparallel surfaces.

Fig. 5
Fig. 5

Flux maps: target flux density relative to that of the source as a function of (nondimensional) target radial position along the line y t = 0 . (a) Results for several large values of (nondimensional) source–target separation at fixed NA = 0.6 . The lower X t / R s regime in which the far-field cos 4 ( θ 1 ) approximation is valid is indicated by open symbols, and the remainder of the curve (solid curve) constitutes the general near-field solution. (b) As in (a), but for far lower source–target separations toward illustrating the uniform core region in the target flux map. (c) Sensitivity to NA at fixed h = R s .

Fig. 6
Fig. 6

Parallel source–target configurations with a disk source of radius R s . (a) Coaxial disk target of radius R t . (b) Misaligned disk target of radius R t = R s . (c) Square target of side L t coaxial with the source. The virtual circle of radius R v (see Figs. 2, 3) is included for clarity.

Fig. 7
Fig. 7

Fraction of power emitted by a disk source received by a coaxial disk target [geometry of Fig. 6a] for several target radii. (a) As a function of NA at fixed source–target separation h = R s . (b) As a function of (nondimensional) source–target separation at fixed NA = 0.6 . For comparison, the corresponding result for fully Lambertian exchange ( NA = 1 ) between source and target disks of equal radii is included. In both instances, 100% of the emitted power is accepted by the target when R t > R s + R v . For the specific case R t = R s , this condition is achieved only within the limit of vanishing h or NA. For smaller targets, as long as R t < R s R v , the fraction of source emission accepted at the target is equal to the ratio between the areas of the target and the source.

Fig. 8
Fig. 8

Fraction of power emitted by a disk source received by a misaligned disk target of equal radius [geometry of Fig. 6b] as a function of the (nondimensional) misalignment. (a) Sensitivity to NA at fixed h / R s = 1 . (b) Sensitivity to h / R s at fixed NA = 0.6 . In all cases, the accepted power vanishes when the misalignment exceeds 2 R s + R v .

Fig. 9
Fig. 9

Fraction of power emitted by a disk source received by a collinear square target [geometry of Fig. 6c] for several target sizes. (a) As a function of NA at fixed h / R s = 1 . (b) As a function of h / R s at fixed NA = 0.6 . If L t > 2 ( R s + R v ) , then the target receives 100% of the source emission. For L t = 2 R s , full acceptance is achieved only within the limit of vanishing h or NA. For smaller targets, as long as L t 2 1 / 2 ( R s R v ) , the fraction of source emission accepted by the target is equal to the ratio between the areas of the target and the source.

Fig. 10
Fig. 10

Exchange between coaxial tilted disks. The maximum emission and acceptance half-angles at the source and target are denoted by θ max 1 and θ max 2 , respectively.

Fig. 11
Fig. 11

Flux maps: target flux density as a function of (nondimensional) target position, expressed relative to the source flux density, and measured along the line y t = 0 on a flat target tilted at angle α relative to a disk source (refer to Fig. 10). NA s = 0.6 and h = R s in all cases. Results are shown for NA t = 0.6 and 1, at three tilt angles. Note that the extent of the uniform core region increases with tilt angle, and vanishes for a fully Lambertian target. For NA s = NA t = 0.6 , when α > θ max 1 + θ max 2 ( = 73.74 ° in this case), all rays that reach the target are rejected.

Fig. 12
Fig. 12

Fraction of source emission accepted at the target as a function of tilt angle for exchange between two tilted disks: NA s = 0.6 and h = R s in all cases. Results are shown for NA t = 0.6 and 1, at three target sizes. For the fully Lambertian target, the accepted target flux can vanish at sufficiently large tilt angles only when R t < h R s sin ( α ) tan ( α θ max 1 ) R s cos ( α ) (note that target flux will not vanish at any tilt angle if h < R s , which is intrinsic to this inequality).

Equations (14)

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P d A s d A t = I cos ( θ 1 ) cos ( θ 2 ) l 2 d A s d A t ,
P A s A t = I cos ( θ 1 ) cos ( θ 2 ) l 2 d A s d A t
P d A s d A t = I h 2 [ ( x t x s ) 2 + y s 2 + h 2 ] 2 d A s d A t ,
P A s d A t / d A t = I h 2 [ ( x t x s ) 2 + y s 2 + h 2 ] 2 d x s d y s .
X min = M a x ( R s , x t R v ) ; X max = M i n ( R s , x t + R v ) Y min ( x ) = Y max ( x ) = { [ R v 2 ( x t x ) 2 ] 1 / 2 for     x x int [ R s 2 x 2 ] 1 / 2 for     x > x int 0 for     x t > R v + R s } ,
l = { ( x t x s ) 2 + ( y t y s ) 2 + [ h + x s tan ( α ) ] 2 } 1 / 2 ,
cos ( θ 1 ) = ( 1 / l ) { ( x t x s ) sin ( α ) + [ h + x s tan ( α ) ] cos ( α ) } .
P d A s d A t = I ( x t x s ) [ h + x s tan ( α ) ] sin ( α ) + [ h + x s tan ( α ) ] 2 cos ( α ) { ( x t x s ) 2 + ( y t y s ) 2 + [ h + x s tan ( α ) ] 2 } 2 d A s d A t .
x s 2 ( NA t 2 tan 2 ( α ) 1 NA t 2 1 ) + x s ( 2 h NA t 2 tan ( α ) 1 NA t 2 + 2 x t ) + NA t 2 ( h ) 2 1 NA t 2 x t 2 ( y t y s ) 2 > 0.
x s 2 cos 2 ( α ) + x s [ 2 x t 2 h tan ( α ) ] + [ x t sin ( α ) + h cos ( α ) ] 2 1 NA s 2 x t 2 ( h ) 2 ( y t y s ) 2 > 0.
P A s d A t = I d A t ( x t x s ) [ h + x s tan ( α ) ] sin ( α ) + [ h + x s tan ( α ) ] 2 cos ( α ) { ( x x s ) 2 + ( y y s ) 2 + [ h + x s tan ( α ) ] 2 } 2 d x s d y s .
P i j = π I A i F i j ,
A i F i j = A j F j i .
A i F i j = A j F j i ,

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