Abstract

The problem of interreflections for Lambertian surfaces of arbitrary shape and with varying reflectance is of interest for many practical applications. We present a general method to approach this problem. We define photometric modes that are uncoupled in the sense that each mode may be assigned a (pseudo) reflectance and that interreflections among modes vanish. Then the problem is formally identical with that of a convex body, in which interreflections are of no importance. The photometric modes depend on the shape of the body. In many practical cases one or a few modes dominate, and the reflected radiance depends more on the shape of the body (the dominant mode) than on the precise irradiance distribution. A few examples are treated explicitly. The redistribution of radiation described by the modes is treated by means of the net vector flux and the space density of radiation. Knowledge of these fields for the dominant mode yields considerable intuitive insight in the physical situation and provides the means to estimate the effects of painting part of the surface or of the introduction of screens.

© 1983 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. A. Jacquez and H. F. Kuppenheim, “Theory of the integrating sphere,” J. Opt. Soc. Am. 45, 460–470 (1955).
    [Crossref]
  2. P. F. O’Brien, “Interreflections in rooms by a network method,” J. Opt. Soc. Am. 45, 419–424 (1955).
    [Crossref]
  3. A. Gershun, “The light field,” J. Math. Phys. 18, 51–151 (1939).
  4. J. J. Koenderink and A. J. van Doorn, “Photometric invariants related to solid shape,” Opt. Acta 27, 981–996 (1980).
    [Crossref]

1980 (1)

J. J. Koenderink and A. J. van Doorn, “Photometric invariants related to solid shape,” Opt. Acta 27, 981–996 (1980).
[Crossref]

1955 (2)

1939 (1)

A. Gershun, “The light field,” J. Math. Phys. 18, 51–151 (1939).

Gershun, A.

A. Gershun, “The light field,” J. Math. Phys. 18, 51–151 (1939).

Jacquez, J. A.

Koenderink, J. J.

J. J. Koenderink and A. J. van Doorn, “Photometric invariants related to solid shape,” Opt. Acta 27, 981–996 (1980).
[Crossref]

Kuppenheim, H. F.

O’Brien, P. F.

van Doorn, A. J.

J. J. Koenderink and A. J. van Doorn, “Photometric invariants related to solid shape,” Opt. Acta 27, 981–996 (1980).
[Crossref]

J. Math. Phys. (1)

A. Gershun, “The light field,” J. Math. Phys. 18, 51–151 (1939).

J. Opt. Soc. Am. (2)

Opt. Acta (1)

J. J. Koenderink and A. J. van Doorn, “Photometric invariants related to solid shape,” Opt. Acta 27, 981–996 (1980).
[Crossref]

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Phenomenon of vignetting. No ray may travel from A to B directly because it must be intercepted by the object. But a ray emitted from B may reach A after having been scattered at C.

Fig. 2
Fig. 2

A hemispherical boss on a plane is irradiated from a distant light source L. An eye views the boss from E. There is a body shadow αβ and a cast shadow βγ in the primary irradiance. Note that the interreflections are not symmetrical: B throws a much larger reflex on A than A throws on B because A receives no primary irradiation whereas B does. The graph shows the resulting reflected radiance that is due to the once- and twice-scattered rays. The reflex can often be noticed in nature and is often depicted by painters.

Fig. 3
Fig. 3

Axial gain for cylinders of different reflectances as a function of the spatial wave number k of rotationally symmetric modes (m = 0).

Fig. 4
Fig. 4

Gains for even and odd modes of parallel plates with unit reflectance.

Fig. 5
Fig. 5

Contours of equal space density of radiation for a sinusoidal (pseudo) radiance distribution on a flat plate. The curves are drawn for equal increments.

Fig. 6
Fig. 6

Net vector flux field for the same case as is depicted in Fig. 5. The density of the field lines is proportional to the magnitude of the net vector flux.

Fig. 7
Fig. 7

Net radiation density [averaged over 0 < x < (λ/4d), 0 < y < d] for even and odd modes.

Fig. 8
Fig. 8

Net flux that leaves a plate [averaged over z = 0, 0 < x < (λ/4d), 0 < y < d] for even and odd modes.

Fig. 9
Fig. 9

Ratio of the flux parallel to the plates [averaged over x = 0, 0 < y < d, 0 < z < (d/2) to the net flux leaving a plate [averaged over z = 0, 0 < x <(λ/4d), 0 < y <d] for an odd mode.

Equations (52)

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

K ( x ; x ) = Pos [ n ( x ) · ( x - x ) ] · Pos [ n ( x ) · ( x - x ) ] x - x 2 ,
Pos [ a ] = a + a 2 .
1 π K ( x ; x ) 2 d x d x 1.
N ( x ) - ρ ( x ) π K ( x ; x ) N ( x ) d x = ρ ( x ) π H ( x ) .
H * ( x ) [ ρ ( x ) ] 1 / 2 = H ( x ) ,
1 π [ ρ ( x ) ] 1 / 2 N * ( x ) = N ( x ) ,
π K * ( x ; x ) [ ρ ( x ) ρ ( x ) ] 1 / 2 = K ( x ; x ) ,
N * ( x ) - K * ( x ; x ) N * ( x ) d x = H * ( x ) .
K 1 = K π ,
K m ( x ; x ) = K ( x , y ) π K m - 1 ( y , x ) d y             ( m 2 ) ,
N ( x ) = ρ H ( x ) π + m = 1 ρ m K m ( x ; x ) ρ H ( x ) π d x .
1 π K ( x ; x ) P k ( x ) d x = μ k P k ( x ) ,
P k ( x ) 2 d x = 1.
N ( x ) = k n k P k ( x ) ,
H ( x ) = k h k P k ( x ) ,
n k = ρ π α k h k ,             α k = 1 1 - ρ μ k .
α k 1 1 - ρ .
N = R π · H ,
[ N N ( x ) - N ] = 1 π ( ρ 1 - ρ 0 0 ρ ) [ H H ( x ) - H ] .
N ( x ) = ρ π [ H ( x ) + ρ 1 - ρ H ] .
K ( 0 , 0 ; ρ , ϕ ) = [ Pos ( sin 2 ϕ 2 ) ] 2 + 0 ( ρ 2 ) .
H ( x ) = π [ 1 - ξ ( x ) ] N * .
ξ ( x ) = 1 π K ( x , x ) d x .
K ( x , x ) = 4 sin 4 ϕ 1 - ϕ 2 2 4 sin 2 ϕ 1 - ϕ 2 2 + ( z 1 - z 2 ) 2 2 .
Φ k m ( z , ϕ ) = exp [ i ( m ϕ + k z ) ] .
μ k m Φ k m = 1 π K ( x , x ) Φ k m ( x ) d x = 1 π - + 0 2 π 4 sin 4 ϕ 1 - ϕ 2 2 exp [ i ( m ϕ + k z ) ] 4 sin 2 ϕ 1 - ϕ 2 2 + ( z 1 - z 2 ) 2 2 d ϕ d z
μ k m = 0 π / 2 cos ( 2 m ν ) ( 1 + 2 k sin ν ) sin ν exp ( - 2 k sin ν ) d ν .
α 0 m = [ 1 - ρ 0 1 T 2 m ( z ) d z ] - 1
α 00 = 1 1 - ρ , for ρ 1 , α 00 , α 01 = 1 1 + ρ 3 , α 01 0.75 , α 02 = 1 1 + ρ 15 , α 02 0.9375 , α 03 = 1 1 + ρ 35 , α 03 0.9722 .
α k 0 = [ 1 - ρ 0 π / 2 ( 1 + 2 k sin θ ) sin θ exp ( - 2 k sin θ ) d θ ] - 1 .
λ c = 2 π ( 8 / 3 ) 1 / 2 = 10.26 = 5.13 D ,
α k 0 = 3 8 k 2 , α k 1 = 3 4 ( 1 - 6 5 k 2 + ) , α k 2 = 15 16 ( 1 + 3 14 k 2 + ) , α k 3 = 35 36 ( 1 - 50 243 k 2 + ) .
Δ N N α k 0 α 00 Δ H H ang
K ( r , r ) = [ 1 + ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2 ] - 2 if r = ( x 1 , y 1 , 0 ) r = ( x 2 , y 2 , 1 ) or r = ( x 1 , y 1 , 1 ) r = ( x 2 , y 2 , 0 ) .
α k + = [ 1 - ρ k K 1 ( k ) ] - 1 , α k - = [ 1 + ρ k K 1 ( k ) ] - 1 .
N ( r ) = ρ π { 1 - { ρ [ k K 1 ( k ) ] } 2 } - 1 H ( r ) = ρ π { 1 + 2 ρ 2 [ k K 1 ( k ) ] 2 + } H ( r ) .
N ( x , s ) = 1 = 0 m = - 1 + 1 a 1 m ( x ) Y 1 m ( s ) = 1 4 π u ( x ) + 3 4 π D ( x ) · s + quadrupole and higher - order terms .
u ( x ) = ( 4 π ) N ( x , s ) d ω ,
D ( x ) = ( 4 π ) N ( x , s ) d ω .
u t + · D = source density .
Δ u = 0.
D · × D = 0 ,
D = Ψ ϕ .
E ( x , y ) = E 0 sin k x .
N = E 0 π sin k x * .
u ( x , y , z ) = N d Ω = 2 E 0 sin k x exp ( - k z ) .
D = N d Ω = E 0 π [ k z K 1 ( k z ) sin k x e z - k z K 0 ( k z ) cos k x e x ] .
D = - E 0 z π [ K 0 ( k z ) sin k x ] . }
u ¯ = 8 E 0 π k d [ 1 - exp ( - k d ) ]             for the even modes = 8 E 0 π k d [ 1 - exp ( - k d 2 ) ] 2             for the odd modes .
Φ = 2 E 0 π 2 [ 1 - k K 1 ( k ) ]             for the even modes = 2 E 0 π 2 [ 1 + k K 1 ( k ) ]             for the odd modes .
d d x ( x K 0 x ) x = k d 2 < 0
λ > 5.2 d .