Abstract

An analytical treatment is given which shows that the peak luminance gain and lobe width of a back or front projection screen material can be used as coordinates on a graph to determine the transmission or reflection efficiency. The analysis assumes that the angular distribution of the gain can be expressed as a power of a cosine function. Also, a screen photometer is described which is based upon the adaptation of a spectrometer to measure screens having retroreflective characteristics. The photometer is capable of measuring luminance on the retroreflective axis and out to 15° on either side of the incident beam. Screen samples, 2.54 cm sq, can be oriented through angles of ±30° to the incident beam and luminances from 1 ft-L to 10,000 ft-L (from about 3.5 cd/m2 to 35,000 cd/m2) can be handled. The instrument has three modes of operation: visual, photoelectric, and photographic. A graph is presented illustrating the application of the analytical technique to provide a rapid performance evaluation and comparison of screen efficiencies.

© 1970 Optical Society of America

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Figures (12)

Fig. 1
Fig. 1

A typical polar distribution of screen gain G(θ), where the angle θ is measured from the direction of the peak gain G0.

Fig. 2
Fig. 2

A cartesian representation of a polar distribution lobe expressed by G(θ) = G0(cosθ)P. Point Q, at which 2γ is the angular width, is defined by the 10% level of the peak gain G0.

Fig. 3
Fig. 3

A family of straight lines expressing Eq. (10) with selected values for the reflection efficiency m. Spot values of γ are indicated above the abscissa axis.

Fig. 4
Fig. 4

A family of parallel straight lines expressing Eq. (13) with values for the percentage reflection efficiency m covering two orders of magnitude.

Fig. 5
Fig. 5

A schematic diagram of the photometer which utilizes a modified spectrometer and is shown adapted for the photoelectric mode.

Fig. 6
Fig. 6

A schematic diagram of a part of the photometer, showing the box type camera which replaces the spectrometer telescope for operation in the photographic mode.

Fig. 7
Fig. 7

The gain distribution of a sample of retroreflective material A as measured with the photometer for angles of incidence of 0 deg of arc and 5 deg of arc. No specular reflective component was detected for this material.

Fig. 8
Fig. 8

The gain distribution of a sample of retroreflective material B as measured with the photometer, at normal incidence.

Fig. 9
Fig. 9

The gain distribution of a sample of retroreflective material B as measured with the photometer at an angle of incidence of 5 deg of arc. This shows clearly the separation of the specular and retroreflective components.

Fig. 10
Fig. 10

Display of peak gain G0, spread γ, and reflection efficiency m of various screen material at normal incidence. This includes contributions from both specular and retroreflective components.

Fig. 11
Fig. 11

Display of peak gain G0, spread γ, and reflection efficiency m of various screen materials at an angle of incidence of 5 deg of arc for the retroreflective component only.

Fig. 12
Fig. 12

Equation 3 plotted for three values of p; p = 0 (i.e., lambertian surface), p = 500, and p ≑ 6000. The gain distribution of a sample of retroreflective material A as measured with the photometer is also shown. The abscissa scale has been weighted so that the area beneath the curve is proportional to the reflected radiation flux.

Equations (27)

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L ( θ ) = ( E / π ) G ( θ ) .
F ( β ) = 0 A 0 β L ( θ ) ( cos θ d A ) ( 2 π sin θ d θ ) .
F ( β ) = 2 0 A E d A β 0 G ( θ ) cos θ d ( cos θ ) .
G ( θ ) = G 0 ( cos θ ) p ,
p 0 ,
π / 2 θ π / 2 .
F ( β ) = 2 G 0 0 A E d A β 0 ( cos θ ) p + 1 d ( cos θ ) .
F ( β ) = 2 G 0 [ 1 ( cos β ) p + 2 p + 2 ] 0 A E d A .
F ( π 2 ) = 2 G 0 p + 2 0 A E d A .
f ( β ) = F ( β ) F ( π / 2 ) = 1 ( cos β ) p ( cos β ) 2 .
( cos β ) p = G ( β ) / G 0 , f ( β ) = 1 [ G ( β ) / G 0 ] ( cos β ) 2 .
G ( γ ) / G 0 = 0.1.
f ( γ ) = 1 0.1 ( cos γ ) 2 .
m = Total light flux reflected in lobe Total incident light flux = F ( π 2 ) / 0 A E d A .
m = 2 G 0 / ( p + 2 ) m 1.
p = ln ( G ( β ) / G 0 ) / ln ( cos β ) .
G 0 m + 1 / [ 2 ln ( cos β ) ] 1 / ln [ G 0 / G ( β ) ] = 1 ,
β = γ G ( γ ) / G 0 = 0.1 ,
G 0 m + 1 / [ 2 ln ( cos γ ) ] 1 / ln ( 10 ) = 1.
ϕ ( γ ) = 1 / [ 2 ln ( cos γ ) ] , G 0 / m + ϕ ( γ ) / 0.4343 = 1.
G 0 m 1 / γ 2 0.4343 = 1.
G 0 / m 1.
G 0 / m = 1 / 0.4343 γ 2 .
log G 0 = 2 log γ + log ( 75.6 m ) ,
E = ( 18.5 / R ) 1.155 0.02 < E < 10 ,
G ( θ ) = L ( θ ) / L S = 0.99 [ R S / R ( θ ) ] 1.155 ,
F ( β ) = 0 A E d A 0 β G ( θ ) d ( sin 2 θ ) .

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