LIOHT FILTERS 297 



cut down by the same ratio or the same percentage rather than by the same definite 

 amount. 



Another term for the transmission of an object is its transparency; other terms 

 must now also be introduced. The opacity Ox is the reciprocal of the transparency or 

 transmission T\, or 



Ox = Y^ (4) 



The optical density is defined to be the common logarithm of the opacity, or from the 

 relation between opacity and transparency, density is the common logarithm of the 

 reciprocal of the transparency. Thus 



D\ = logio 0\ = logio jT = — logio T\ (5) 



The apparent cutoff and the transparency or transmission curves change with 

 variations in thickness of the filter. In specifying the transmission characteristics 

 for filters, it is therefore evident that the thickness of the selective absorbing medium 

 must be given before the complete filter characteristics are specified. 



It should be noticed that no attention has been paid to the reflection losses at the 

 surface of the fUter. Such reflection losses depend upon the quality of polish of 

 the filter surfaces, the angle of incidence of the light, and the index of refraction of the 

 filter medium. They are usually small enough to be neglected without appreciable 

 error. 



Use of Several Filters.^ — Sometimes a filter transmission characteristic is desired 

 which cannot be accurately or adequately fulfilled by any known dyed gelatin, 

 colored glasses, or liquids. In such cases it is sometimes possible to obtain a close 

 approach to the desired transmission characteristic by using two or more filters 

 simultaneously, one in back of the other so that the light must pass through all fUters 

 in succession. The transmission of this combination of filters, t\, is the product of the 

 transmission characteristics of the separate individual filters Tx, T'^, etc. If we have 

 three filters in use at the same time, the transmission characteristic of this combination 

 in terms of the separate filter transmission characteristics will be, 



rx = T'^T'iT"^ (6) 



The transmission characteristics of the individual fUters are usually expressed 

 graphically or by means of a table from which a transmission curve may be con- 

 structed. If the curves for the three filters are available, the above equation gives 

 the over-all transmission for the three filters, used one behind the other simultaneously. 

 It is evident that the over-all transmission characteristic is obtained by multiplying 

 the transmission of the individual filters, wavelength by wavelength. Since the 

 transmission of any filter can never be greater than unity (and can be unity only in the 

 case of an absolutely perfect filter having zero losses), it follows that the use of several 

 filters behind one another will give an over-all or net transmission for the filter system 

 which will be successively smaller the greater the number of filters employed. For 

 this reason the light available for photographic work is effectively diminished and a 

 longer exposure required; so it is desirable not to use more filters than are necessary 

 to produce the desired spectral transmission characteristic. In practice this number 

 seldom if ever exceeds two. Other objections to the use of more filters than are abso- 

 lutely essential result from multiple reflections from the surfaces of the filter and 

 dimunition of optical quality as the number of filters is increased. Exception to this 

 statement may be made for optically prepared filters intended for uses of this type, 

 but the average photographer seldom encounters such cases. 



