Abstract

When parallel light is reflected from a small perfect reflecting sphere the illumination is approximately independent of the angle of incidence and reflection. A formula is developed to study this approximation. Also a formula is given for the case of a sphere which is not supposed to be perfectly reflecting but reflects according to some law like Fresnel’s reflection law.

© 1923 Optical Society of America

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Figures (2)

Fig. 1
Fig. 1

A diagram for showing the reflection of parallel light from a sphere.

Fig. 2
Fig. 2

Showing the reflection of parallel light from a perfect reflecting sphere of radius, r for the three different distances 10r, 100r and 1000r.

Equations (14)

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I = I 0 d H d A
θ = < AOD = < QCD = 1 2 < AOB = 1 2 < QCP . Let α = < AOQ = < POB
then s = ( 2 θ - α ) R and d s d h = R ( 2 d θ d h - d α d h ) .
            d θ d h = 1 r cos θ .
Also sin α = h / R             d α d h = 1 R cos α .
d s d h = 2 R r cos θ ( 1 - r cos θ 2 R cos α ) d h d s = r cos θ 2 R ( 1 - r cos θ 2 R cos α ) - 1 .
We have             d H d A = d H d h d h d s d s d A .
Now             H = π h 2 d H / d h = 2 π h = 2 π r sin θ .
d A = 2 π PM d s = 2 π R sin ( s R ) d s ,
d A d s = 2 π R sin 2 θ ( cos α - cot 2 θ sin α )
I = I 0 r 2 4 R 2 Ψ ( θ )
where             1 Ψ ( θ ) = 1 - r 2 R 2 sin 2 θ - r R ( 1 2 cos θ + sin θ cot 2 θ ) + r 2 cos 2 2 θ 4 R 2 1 - r 2 R 2 sin 2 θ .
1 Ψ ( θ ) = 1 - r R ( 1 2 cos θ + sin θ cot 2 θ ) + ( r R ) 2 ( 1 4 cos 2 θ - 1 2 sin θ ) + ( r R ) 4 ( 1 8 cos 2 θ sin θ - 8 sin θ )
I = I 0 r 2 4 R 2 F ( θ ) Ψ ( θ ) .