JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Multiple Internal Reflections in Photographic Color Prints*tF. C. WILLIAMS AND F. R. CLAPPER

Eastman Kodak Company, Rochester, New York

(Received February 13, 1953)

The apparent reflectance of a submerged object is strongly influenced by refraction and reflection at thesurface of the submerging medium. These actions affect stain and color purities in photographic colorprints, the reflecting bases of which are submerged in gelatin. Print reflection densities can be calculatedfrom base reflectance and gelatin transmittance; a family of graphs of the relationships is presented. By asimple modification, reflectometers can be made to measure directly the reflectance of objects submerged intransparent media.

A PHOTOGRAPHIC color print usually consistsof a thin, colored-gelatin transparency over-

laying a diffusely reflecting white support. The trans-parency and support are in optical contact; therefore,the reflecting support is effectively submerged, at adepth of about 0.001 inch, in gelatin, which has a re-fractive index of about 1.53. The submergence of thereflector in the high-index absorbing medium has astrong influence on the appearance of color prints.Slight stains in "white" areas are intensified, and thepurities of darker colors are reduced. These effectscannot be avoided without radical change of printstructure.

The primary effect of immersing a diffuse reflector iswell known. Figure 1 shows side by side, (a) a photo-graphic color print and (b) its white base only, in air.Both are illuminated at 45-degree incidence and viewedfrom a position along a normal to the surfaces. The basein air, by simple diffuse reflection, directs some of thelight incident on it into the small geometrical conesubtended by the entrance pupil of the observer's eye;by means of this light the base is viewed. In the photo-graphic print, the base receives almost as much incidentlight as the base in air and reflects it with almost thesame efficiency and diffusion. But now much of thelight directed into the same small cone as before is notintercepted by the observer's pupil. In passing from thehigh-index gelatin into the air, the light is spread byrefraction into a wider cone, the angular subtense ofwhich obeys Snell's refraction law. The consequence isreduction of brightness by a factor equalling the squareof the ratio of the refractive indices at the optical inter-face. In a photographic print this index ratio is 1.53,the square of which is 2.34.

By this consideration alone, then, it would seem thatcovering a white photographic base with clear gelatinshould reduce its brightness by the factor 2.34, makingit appear to be a 43-percent reflecting gray. Theapparent reflectance should be reduced further by a 5.5-percent surface loss at entrance and 4.4-percent loss atemergence to make a total factor of 2.59, equivalent to areflectance of about 39 percent. Since clear areas in

* Communication No. 1540 from the Kodak Research Labora-tories.

t Read at the Cleveland Meeting of the Optical Society ofAmerica in October, 1950.

photographic prints are not 39- percent grays but aregood whites, there obviously, and fortunately, arecompensating factors at work.

Figure 2 is a more realistic picture of the opticalsystem of a photographic print. A great deal of thelight reflected by the base is also internally reflected bythe gelatin-air interface; unless it is absorbed, it returnsto the base and is used again. Judd' has calculated theinternal reflectance of spherically distributed lightincident at interfaces of various index ratios. He lists0.614 as the internal reflectance for an index ratio 1.53.Since the reflection at a photographic base is almostcompletely diffuse, the light returned by the interfaceis spherically distributed after its second base reflectionjust as after first base reflection. Therefore, 61.4 percentof this additional light will return for third reflection,and so on, to generate, in a nonabsorbing system, apower series the sum of which is 1/(1-0.614)=2.59.The total base illuminance, therefore, is 2.59 times thatdue to first incidence. Since 2.59 also is the factor bywhich the refraction effect decreases the base brightness,the net effect of a perfectly transparent gelatin overlayon the brightness of a perfectly reflecting base should bezero, as ordinary experience indicates.

Absorption, however, whether in the gelatin or at thebase, will decrease the compensating action of multiplereflections. Prediction of the total effect of these absorp-tions is useful in the study and design of color photo-

b a

"\I \"S

27o32 4

a= 1.00

39 % ' =.53

FIG. 1. Partial optical systems of (a) a photographic colorprint, and (b) its base only, in air. Both are illuminated at 45-degree incidence and viewed from the same position along anormal to the surfaces.

1 D. B. Judd, J. Research Natl. Bur. Standards 29, 329 (1942).

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VOLUME 43, NUMBER 7 JULY, 1953

F. C. WILLIAMS AND F. R. CLAPPER

/

/ q 1.00

r = 1.53

///$ // b 0.03in

FIG. 2. Internal reflections in a photographic color print.

graphic papers. Closely approximate prediction ispossible by mathematical analysis of the system.

If one considers only the first-incidence illuminationof the base and the re-illumination resulting from thefirst interface reflection, the following equation expressesthe ratio between B, the brightness of a gelatin-covereddiffuse reflector, and B', the brightness of a diffuse whitereflector identically illuminated but in air:

B (0.945) (0.956)2_ 13RB

B1 (1.53)2 /

X [1+2RB f /2 sCCOrO sinG cosOdO].

Here t is the transmittance of the gelatin along a singlebase-to-interface path normal to the base; RB is re-flectance of the base, and r is the internal Fresnelreflectance of the interface at angle 0 to the normal.Light is assumed to be externally incident on the inter-face at 45 degrees to the normal; the base is viewedalong a normal to its surface. It is assumed that diffusionby the base is perfect. The transmittance of the entrancesurface at 45 degrees external incidence is 0.945; thetransmittance of the surface for light emerging at nor-mal incidence is 0.956. The 45-degree incident light isrefracted to 27°32'; its path length to base is the normalgelatin thickness times secant 27°32'= 1.13. The totalpath length of the first emergent light is therefore 2.13times the normal gelatin thickness. The base-to-basepath length, for light leaving the base at angle 0 to thenormal, is 2 secantO. (See Fig. 3.)

The integral defines the fraction of base-reflectedlight which each time returns to the base. Since theseries of these diminishing returns has the sum

7. /2 -[1- 2RBJf t

2 sccr sinG cosd]

the expression for relative brightness considering allreflections becomes

E 1 ,,r/2B/B'= 0.193t2 13

- t2

°ro sinG cosOd]

Evaluation of 120 of these ratios, with various values ofRB and , has been effected by numerical integration.

These ratios are reflectances; their cologarithms arereflection densities. Figure 4 shows the reflectiondensities plotted as functions of transmission density ofthe gelatin overlay, for each of a series of base re-flectances.

Experiments have shown that the curve of Fig. 4 forRB= 1.00 closely approximates the relationship existingin glossy photographic color prints on white baryta-coated paper base. At high densities, absorptions pre-vent appreciable contributions from any but first-incidence reflection. The reflection density then issimply the result of darkening due to reflection losses,the change of index between origin and observationspaces, and the attenuation of light by the gelatin alonga single path from surface to base and back to thesurface again. At reflection density 1.2, contributionfrom a second base illumination begins to be noticeable.At reflection density 0.3, multiple reflections havecompensated for more than half the index-change loss,and at zero density compensation is complete.

It is obvious that very small transmission densitiescause disproportionately high reflection densities; it isfor this reason that it is difficult to achieve brilliant

1.00 \ /.1 S I

Bprnt= (0.945) (0.956)1213 1B 1+2RB /2 Sp-ro sinO cos~d9]B~td. (1.53)2 B[B 0 1 sc, in ooo

FIG. 3. Factors in the equation for computing print reflectances.

whites in photographic color prints. The effect is due,not to the multiple internal reflections, but rather to thefact that minute amounts of absorption strongly di-minish the compensating action of the internalreflections.

Similarly, reduction of color purities in color prints isa result of the brightness reduction caused by interfacerefraction; internal reflections effect a compensation,but the presence of any absorption makes the compensa-tion incomplete. Figure 5 demonstrates the action. Thespectral-transmission-density distribution of a dyedgelatin sheet is shown in the lower right-hand quadrantof the figure. This gelatin sheet, together with a perfectdiffuse reflector, will form reflection material that hasthe reflection-density distribution shown by the solidline in the upper right-hand quadrant. The dotted linein this quadrant is the original (transparency's) distri-bution scaled to the same peak density as the reflectiondistribution. It is apparent that for the same peakdensity the reflection material has appreciably higherunwanted densities in the blue and red spectral regions.But if multiple internal reflections were not operating toincrease the base illuminance, the density distributionof the reflection material would be that shown in the

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PHOTOGRAPHIC COLOR PRINTS

dashed curve; here the unwanted densities are obviouslymuch higher.

The broadening of absorption peaks in reflectionimages was investigated several years ago by W. T.Hanson and R. M. Evans.2 Although the effect was atfirst ascribed to integration- of transmission over therange of path lengths taken by the emergent light, itbecame apparent that the broadening was too great tobe explained by this means, and the effect was ascribedto multiple internal reflections.

Figure 6 illustrates the effects of this absorptionbroadening in photographic color reproduction. InSection (a) are shown the spectral-density distributionsof gray images of visual density 1.5, formed by thethree dyes of a (hypothetical) stain-free color process.The solid line represents the transparency image. If thesame dyes were used to form an image of the samedensity in a reflection print, the resulting spectraldistribution would be that shown by the dashed line.

RB'0l)0 0.20 0.30 0.40.50.60 0.80 1.00

0.00 0.20 0.40 0.60Transmission density of unit thickness

FIG. 4. Reflection densities of photographic color-print imagesas functions of transmission density of the gelatin overlay, foreach of a series of base reflectances RB.

In the same way in Sections (b), (c), and (d) are showncomparisons of the spectral-density distributions of thesaturated colors which would result from removing,respectively, the red-, green-, and blue-absorbing dyefrom the neutral images. Section (e) is a plot, in CIEcoordinates, of the chromaticities of the six images; in allcases, illumination by a 4000-K blackbody is assumed.The reflection images, indicated by crosses, are atpositions of appreciably lower purity than are thetransparency images indicated by circles.

The nonlinear relationship between reflection densityof a color print and the transmission density of itsabsorbing component contributes to the difficulties ofsensitometric study of photographic papers. In sensi-tometry of color films,3 4 the multidye images are fre-quently subjected to a kind of optical analysis to deter-mine how much of each of the component dyes is present.

2 Unpublished work.3 F. C. Williams, J. Opt. Soc. Am. 40, 104 (1950).4Color Sensitometry Subcommittee, Principles of Color Sensi-

tometry, Society of Motion Picture and Television Engineers,New York, 1950.

In the transparency images involved in this work, thespectral densities of the multidye deposit can safely beassumed equal to the sums of the spectral densities ofthe component absorbers; that is, in transparencies,spectral densities are linearly additive. Linear systemsare easily analyzed. But in color paper images, internalreflections destroy the linear additivity of the compo-nent densities. Linear analysis can be performed only ifinternal reflections are avoided.

Figure 7 illustrates an attachment for a reflectiondensitometer which provides means of avoiding practi-cally all re-illumination of a submerged reflector byinternally reflected light. Most simply, it is a glass diskto be placed in the instrument at the aperture plane.The entire disk may be black-lacquered, excepting onlythe face through which light enters and leaves, and aspot on the opposite face which serves as sample windowThe thickness of the disk should be about twice thediameter of the sample window; the diameter of the diskshould be at least twice its thickness. If, now, thesample is sealed to the sample window with an oil ofindex approximating that of glass and gelatin (= 1.54)and the sample is then illuminated, no appreciable re-flection can take place between the sample and theexposed face of the disk. Because of the relatively greatthickness of the disk, light reflected at its exposed facewill be displaced laterally before it returns to thesample plane; it will be lost in the black lacquer. Re-illumination of the sample will be less than 0.1 percentof the initial illumination.

Figure 8 shows the result of using a densitometer somodified to determine the relationship between thetransmission density of a gelatin sheet and the densityof the reflection material which it forms by combinationwith a photographic base. The relationship is shown tobe essentially linear. The reflection densities were

I Wove length in millimicrons

FIG. 5. The gelatin overlay of transmission-density distributiona forms a print as shown by b. Without the compensating actionof internal reflections, the print would be as shown by c. If neitherrefraction nor reflection took place at the emergence interface, aprint as shown by d could be made.

i - - - - - - - -,-, -- -112 1-11 - 'I"I-, - I

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71A X 'I,I40

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July 1953 597

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F. C. WILLIAMS AND F. R. CLAPPER

.0' 2.0

cB

y

0.0 0.t 0.2 0.3 0.4 05 0.6 0.7

FIG. 6. (a) Visually neutral combinations of a set of dyes in a transparency and of the same dyes in acolor-print image; in both cases, dye amounts have been adjusted to form a visual density equal to 1.5.(h) Spectral-density distributions of the red images resulting from removing all of the cyan dye fromthe combinations shown in Fig. 6(a). (c) Spectral-density distributions of the green images resultingfrom removing all of the magenta dye from the combinations shown in Fig. 6(a). (d) Spectral-densitydistributions of the blue images resulting from removing all of the yellow dye from the combinationsshown in Fig. 6(a). (e) Chromaticity coordinates (CIE) of the red, green, and blue images of Fig. 6(b),6(c), and 6(d). It is shown that the purities of the reflection images are lower than those of the trans-mission images; their luminances also are lower.

Vol. 43598

PHOTOGRAPHIC COLOR PRINTS

determined by use of a Beckman Model DU spectro-photometer, the reflection head of which was modified toprovide in the sample position a glass disk 0.25 in.thick and 0.5 in. in diameter. Zero density referencewas established by sealing, with a liquid of index 1.54, agelatin-coated baryta base to this sample window andby adjusting the slit width to obtain zero-densityindication with a wavelength setting of 565 mu. Each ofa series of gelatin filters was then sealed between theglass disk and the baryta reflector; some pairs also were

FIG. 7. An attachment for a reflectometer, providing means ofdirect measurement of submerged-condition reflectances.

used. Measurement was then made of the reflectiondensities for X 565 m of the synthetic reflectionmaterial thus formed. The density readings obtainedwere corrected for the light scattered into the readingsystem by the glass-air surface, the amount of whichwas determined by using in the synthetic reflectionprint a filter opaque to X 565 m/u. Transmission densitiesas plotted are not corrected for surface losses. The rateat which reflection density increases with transmissiondensity (m=2.11) agrees exactly with the theoretical

02 03 0.4Transmission density

FIG. 8. Relationship between reflection density of a syntheticcolor print and transmission density of its absorbing component,as determined by a reflection densitometer equipped with thickglass sample window.

rate calculated from the geometrical characteristics ofthe Beckman reflection head.

Reflectometers modified as described may be gener-ally useful in the measurement of reflectances of mater-ials immersed in transparent media. Frequently,immersion of a material in a high-index mediumchanges its reflection characteristics per se; by meansof the modified reflectometer, the submerged-conditionreflectance can be measured directly. The refractiveindex of the glass disk should be always at least as highas that of the immersion medium. If the sealing liquidis not the same as the immersion medium, its indexshould be intermediate between the indexes of theimmersion medium and the glass.

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