Interpretation of induced color in polychromatic glassesN. F. Borrolli, J. B. Chodak, D. A. Nolan, and T. P. Seward, III
Corning Glass Works, Research and Development Laboratory, Technical Staffs Division, Corning, New York 14830(Received 2 January 1979)
The variable colors produced in polychromatic glasses by an optical/thermal treatment are ana-lyzed in terms of a specific geometric model. The calculations, based on small-particle scatteringtheory, involve the specific shape of the silver on the sodium halide microcrystal that is formed in theprocess and account for the color shift with the optical/thermal treatment. In addition, absorptioncalculations were made to explain the formation of the microcrystalline phase.
INTRODUCTION
Polychromatic glasses are a special class of photosensitivematerials in which a variety of colors can be produced by aseries of controlled thermo-optical treatments. The mech-anism of variable controlled coloration in polychromatic glasshas been proposed by Stookey et al.' to involve a photosen-sitized formation of metallic silver on a sodium halide mi-crocrystalline phase. The halide phase can be either NaF orNaBr. The sequence of steps leading to the colored state isthought, first, to involve an ultraviolet-light sensitized for-mation (photoreduction plus aggregation) of small silverspecks within the glass matrix which then serve as the nucleifor the growth of sodium halide microcrystals during thethermal treatment that follows the ultraviolet exposure. Thehabit of the sodium halide crystals has been shown,' by elec-tron microscopy, to be that of a cubic base with varying de-grees of pyramidal protrusions from one cube face. Theconcentration of pyramidal microcrystals seems to be con-trolled by the length of time of the first ultraviolet exposure.A second ultraviolet exposure and heat treatment are thoughtto produce silver on the tips of the pyramidal extensions; theextent of the silver formation correlates with the inducedcolor. Figure 1 shows some typical transmission spectra afterthe second UV exposures and heat treatment.
The polychromatic process clearly involves a number ofinteresting physical phenomena, but in this paper we will re-strict ourselves to only two, those being: (i) the formation ofthe initial silver speck and the subsequent growth of the halidemicrocrystal; and (ii) the nature of the silver decoration on thehalide tip and how it relates to the induced color.
1. FORMATION OF SILVER NUCLEI AND HALIDEGROWTH
It was mentioned above that the first step in the polychro-matic process is the photosensitized formation of a small silverspeck after an exposure and thermal treatment. This processis equivalent to that observed in most photosensitive glasses.2' 3
The second step, the growth of a halide microcrystal on thesilver nucleus, is similar to that in certain photosensitiveglasses.3 A number of studies have been carried out involvinglight scattering from the photosensitively generated nuclei4' 5
and the optical spectra of glasses containing small metal col-orant particles.6' 7' 8 We propose here to use the optical ab-sorption due to the silver specks to track the phenomenon ofhalide growth on the silver nuclei. The absorption due to verysmall free-electron-type metal particles is a relatively sharpresonance-type absorption with the wavelength position ofthe rmaximum depending on the optical constants of the metal,
the size of the metal particles, and the refractive index of themedium in which it is dispersed. For particles whose size ismuch smaller than a wavelength of light, and where the con-centration is low enough so that they do not interact, the ab-sorption cross section per particle a can be expressed as9
a = (87r2 nj/X) Im(p), (1)
where n, is the refractive order of the surrounding medium,X the free-space wavelength, and p the polarizability. Thepolarizability for a spherical particle assuming no interactionbetween particles can be expressed as
p = a3 (E - e)/(E + 2ej), (2)
where e = El + iE2 is the complex optical dielectric constantof silver, Ef = 2 is the optical dielectric constant of the sur-rounding medium, and a is the radius of the particle. Sub-stituting Eq. (2) into Eq. (1) one obtains for the cross sectionper particle:
a = 167r 2 n3a3 62 /A[(E1 + 2e 8 )2
+ 42j. (3)
We see from this expression that a peak in the absorption willoccur when e1 = -2E3 and the absorption half-width will bedetermined by the magnitude of E2 . When the particle sizeis smaller than the electron mean-free-path in the metal, acorrection to the imaginary part of the dielectric constant isnecessary. There are a number of approaches to this cor-rection but they all assume a form
E2 = e2b + c/a, (4)
where 62b is the bulk value of silver and a is the radius of the
0.9
0.78tas
z
30 450 1556070
o_ Q5-
ir0.3-
0.2-
0.I
I I I350 450 550 650 750
WAVELENGTH (nm)
FIG. 1. Transmission spectra of polychromatic glass as a function ofthermo-optic treatment after Ref. 1. Numbers refer to the time of first ul-traviolet exposure.
metal particle. We used the Kawabata-Kubo1 0 form of thecorrection term in Eq. (4) which takes the form
(2 = (2b + (2.7/a) g(v), (5)
where g(v) is a function defined in Ref. 10, Eq. (4.13).Equation (3) together with Eq. (5) permits us to calculate thewavelength position of the absorption maximum as a functionof the particle radius and the refractive index of the sur-rounding medium. The data used for the optical constantsof silver, (lb and (2b, respectively, were obtained from a com-pilation of the literature data. A least-squares fit to the datawas applied and the following equations result:
1515 J. Opt. Soc. Am., Vol. 69, No. 11, November 1979
100-
z°28C'nn2 sZ(I
Erz
60 -4/
/-/
0.35
A1
/6
045WAVELENGTH (pm)
0.55
FIG. 2. Transmission spectrum of bromine-free glass (A) and bromine-containing glass (B), respectively, after 1-h ultraviolet exposure and 5600Cheat treatment.
was the only halogen present. The expected phase to benucleated by the silver speck would then be NaBr. The sameglass was also made without bromine. Table I lists the com-position of these two glasses.
Table II lists the measured wavelengths corresponding tothe peak absorption for the two glasses listed in Table I, as afunction of the temperature of the heat treatment, after aone-hour ultraviolet exposure. Figure 2 shows a typicaltransmission spectra after a 560°C heat treatment for the glasswithout bromine and with bromine, respectively. For thebromine-free glass, the absorption peak was seen to shift onlyslightly in wavelength as a function of the temperature of thethermal treatment. This indicated that only a small effectdue to the change in the silver speck size occurred. Thevariation of the absorption peak was used to estimate thesilver speck size as a function of the thermal treatment by theuse of Eq. (3). The size effect on the absorption is due to thesize dependence of E2 expressed by Eq. (5). The absorptionpeak shift was used rather than the change in the absorptionband half-width because the latter was significantly alteredby the absorption edge owing to the cerium contained in theglass. The inset to Fig. 3 shows the computed radius versusthe heat-treatment temperature. Now, using this computedvariation of the silver particle size with heat-treating tem-perature, the absorption data for the bromine-containing glass(Table II) were analyzed to determine the value of the re-fractive index of the surrounding medium required to matchthe observed wavelength of maximum absorption at eachtemperature. The result of the calculation is shown in Fig.3. It is seen that the index of the surrounding medium risessharply at slightly above 5000C and assumes a value essen-tially that of NaBr at about 5600C. This can be explained bythe mechanism that, at just above 5000C, a NaBr enrichmentof the region about the silver speck begins to occur and, withhigher heat-treating temperature, leads ultimately to a puresodium bromide phase. The transition could depend on ei-ther size or purity of the NaBr phase. In either case, thiscalculation confirms the proposed idea of the silver nucleationof a sodium halide phase and that temperatures above 5600Care required to fully develop the microcrystal.
Borrelli et al. 1515
(a) PYRAMIDAL TIP
1.55H-
INDEX OF GLASS
(b) ELLIPSOID (c) HALFELLIPSOID
.20 1.25 1.30 1.351000/T
500TEMPERATURE (0C)
600
FIG. 3. Refractive index of the region surrounding the silver sphere, cal-culated from the wavelength of peak absorption vs heat-treatment tem-perature for bromine-containing polychromatic glass. Inset: calculatedsilver sphere size (A) vs heat treatment calculated from the wavelength ofpeak absorption of the bromine-free version of the same glass.
II. SILVER FORMATION ON HALIDE CRYSTAL
Stookey et al.1 proposed, on the basis of electron micro-graphs of colored glasses after various polychromatic treat-ments, that the first ultraviolet exposure controlled thenumber and size of the pyramidal sodium halide crystals,which in this case was NaF. The color appeared then to resultfrom the degree of silver decoration on the pyramidal tips. Inthis section we will calculate what the absorption would be forvarious shapes approximating the silver formation geometry.Beall's estimations of the typical dimensions of the silver cap
(d) ELLIPTICAL SHELL
FIG. 4. Schematic representations of specific models used to approximatethe pyramidal tip geometry: (a) actual apparent geometry; (b) solid ellipsoid;(c) half ellipsoid; (d) elliptical shell.
in the microcrystals corresponding to various colors are listedin Table III. The dimensions listed should be viewed asrepresentative rather than the actual dimensions and shouldprovide only a guide to the subsequent calculations.
A number of simple models were investigated to approxi-mate the apparent geometry of the silver formation. In thefirst two, we assume that the silver is solid on the tip in theform of an ellipsoid of revolution or a half-ellipsoid of revo-lution, respectively, as shown in Fig. 4. The polarizabilitiesfor the two geometric forms were obtained by solving Laplace'sequation in prolate-spheroidal coordinates as shown in Fig.5. The actual methods of solution are shown in the Appendix.Owing to the anisotropic shape of the models, the polariz-ability was evaluated along the unique axis and perpendicular,respectively. The absorption cross section was obtained fromthe following equation which represents the average value overall orientations:
a Estimated from electron photomicrographs after Beall.'b Volume fraction taken to be Q = 0.8.c Experimental absorption peaks for same glass and identical thermo-optical treatment with size and aspect ratio not determined.d Inner aspect ratios 8/1, 6/1, and 4/1, respectively.
1516 J. Opt. Soc. Am., Vol. 69, No. 11, November 1979
1.65iNaBr - - - - - - _
1.60-
£__
c
0z
wIt0~
>-
w:
1.50
1 4 1i I
Borrelli et al. 1516
(b)
a S ao, O ',8 '7r/2; e = EAg
a - aO7 7r/2 : -¾ 7r; E = HAL
a - a0 ; c= EGLASS
(c)
a -al ; E= (HAL Ia, a 1 ' a2 ; e = A I[a- a2-;E = EGLASS EJ
R = 7r/ 2
ALL SOLUTIONS DEVELOPED IN FORM
E [A, P, (cosh a ) Bn Q nm (cos h a P (cos/,)e 56
FIG. 5. Representations of the models listed in Fig. 4 in prolate spheroidalcoordinates with appropriate regions specified. Solutions for all threemodels were developed in Legendre functions as shown in the Ap-pendix.
where p refers to the polarizability when the external fieldis along its unique axis and p l when the field is perpendicularto this direction. The other symbols have the same meaningas in Eq. (1). We have again assumed that the particle di-mensions are small compared to the wavelength of light. Thisshould be true for particles with dimensions up to about 200A. For particle dimensions slightly larger, we used a methodsuggested by Skillman" to estimate the effect of scatteringand higher-order multipole absorption that produce only aslight broadening effect.
The wavelengths corresponding to the maximum absorp-tion obtained from the calculations for various aspect ratiosare listed in Table III. Clearly the calculated absorptionmaxima are substantially red-shifted from the electron-mi-crograph estimates of the aspect ratios. The full ellipsoidapproximation seems to be in better agreement with the ex-perimental data; for although the long-wavelength absorptionmaxima are red-shifted from the experimentally observedmaxima, the predicted short wavelength peak agrees quitewell. This is shown in Fig. 6. Another point to be mentionedis that, although the aspect ratios estimated from the electronmicrographs were obtained from the same samples as thoseon which the optical data was measured, nevertheless, otheridentical thermo-optic treatments performed on this glassproduced absorption peaks shifted to the infrared. Thesedata are listed in parentheses in Table III.
1517 J. Opt. Soc. Am., Vol. 69, No. 11, November 1979
I_r
II
e-e -Q eAg-eSE+2ES eAg+ 2 es
(8)
For any of the geometric models used above, we can use Eq.(8) for e in the polarizability expressions to calculate the ab-sorption cross section. Using a value of Q = 0.8 for the solidellipsoidal model, we obtained the wavelengths of peak ab-sorption for the various aspect ratios listed in Table I. Thisporous silver model does fit the experimental peak absorptionwavelength data somewhat better than the continuous silverapproach; however, the uncertainty of the estimated aspectratios precludes any definite conclusion concerning the natureof the silver deposit.
Ill. CONCLUSIONS
In general, it has been found that the proposed explanationby Stookey et al.' of the polychromatic process is basicallysupported by quantitative calculations. In particular, thenucleation of a sodium halide phase on the silver speck isverified by calculations and the decoration of the tip of the
Borrelli et al. 1517
(a)
I ac a ;E6=Ag
I a-a ; E=EGLASS
0
2
02 A
I I I350 450 550 650 750
WAVELENGTH (nm)
FIG. 6. Comparison of experimental transmission spectrum (A) and thatcalculated from the solid ellipsoidal model (B): experimental was for 30min first ultraviolet exposure; calculated was for particle 160 X 40 A.
Another possible silver geometry is that in which the silvercoats the crystal tip. This we approximated by an ellipticalshell as shown in Fig. 3(d); Fig. 4(c) shows the prolate-sphe-roidal coordinate representation. In this geometry, an ad-ditional parameter comes into play, namely, the shell thick-ness. We used three inner ellipsoid aspect ratios with a fixedouter aspect ratio of 3/1. The wavelengths corresponding topeak absorption were calculated for these cases and are listedin Table III. The predicted positions of the peak absorptionwavelength were even further red-shifted than those of thesolid models for the same ratio.
In the above three geometric approximations, the silver-microcrystalline halide tip is assumed to be solid silver or tohave a continuous silver coating. Another possibility is thatthe silver forms in a granular fashion. One can approximatethis situation by appealing to the Maxwell-Garnett12 formu-lation for scattering by a collection of spherical particles,where there are many particles per unit volume equal to awavelength cubed of light. 1 ' The expression for the dielectricconstant e of a material that is composed of a given volumefraction Q of dipole radiators (EAg), dispersed in a medium (eS),is given by
z
x=sinha sing coscf
y=sinha sinl sin4
/ = 7r z=cosha cos/3
FIG. 7. Prolate-spheroidal coordinate system.
microcrystal with silver can account for the color progres-sions.
APPENDIX: CALCULATION OF POLARIZABILITIES
The calculation of the polarizabilities for the models shownin Fig. 5 were carried out in prolate-spheroidal coordinatesas shown in Fig. 7. Laplace's equation was solved in thiscoordinate system for each region with the appropriatematching conditions applied at the boundaries.
For the solid ellipsoidal particle shown in Fig. 5(a), thepotential in the two regions with the external field appliedalong the Z axis (long axis) is found to be
where P1 and Q1 are the associated Legendre functions of the
1518 J. Opt. Soc. Am., Vol. 69, No. 11, November 1979
first and second kind, respectively, and
PI(cosha) = (1 - cosh2 a)11 /2 - P(cosha),dcx
Qi(cosha) =-(1 - cosh2 a)l/ 2 Ql(cosha). (A6)da
Using the boundary conditions, Eq. (A-3), the polarizabilityfor the perpendicular field direction is obtained:
(ej - ejj) sinh 2ao coshao
= (1/2)(Ei - eII)[1 - sinh 2 aoQl (cosha)] + 1
The polarizability for the elliptical shell, shown in Fig. 5(c),is carried out in a similar way where now we must considerthree regions each matched through the continuity of thepotential and displacement. For the field parallel to the Zdirection we have
Using the boundary conditions as in Eq. (A3) over eachboundary, we can evaluate the coefficient B, which is the po-larizability.
The evaluation of the polarizability for the half-ellipsoidof revolution as shown in Fig. 3(b) is somewhat more involvedthan the two cases above owing to the lower symmetry.Solving the Laplacian in the prolate-spheroidal coordinateone obtains the following expressions for the potential, for thecase where the field is along the Z axis:
bJ = a () P, (cosha)P, (cost),
(DI, = E a(2 ) P (cosha)P, [cos(7r - f)], (A8)
(A2) where here the v's are not necessarily integers and must be(M) determined. For the outside region, we have
4),ll = E bn Q, (cosha)P, (cosf) + EoP (cosha)Pl (cosO),n
(A9)
3) where the n's are integers. The boundary conditions are
The last two conditions of Eq. (A10) were used to determine
Borrelli et al. 1518
the values of the v's in Eq. (A8) as well as determining therelationship between a l2) and aD. Using the rest of theboundary conditions of (A10) allows the a(1) and b,') to bedetermined. The numerical evaluation of the above yieldedthe value of N), which is the polarizability.
IS. D. Stookey, G. H. Beall, and J. E. Pierson, "Full-color photosen-sitive glass," J. Appl. Phys. 49, 5114-5213 (1978).
2S. D. Stookey, "Photosensitive Glass," Ind. Eng. Chem. 41,856-861(1949).
3S. D. Stookey, "Chemical Machining of Photosensitive Glass," Ind.Eng. Chem. 45, 115-118 (1953).
4R. D. Maurer, "Nucleation and growth in a photosensitive glass," J.Appl. Phys. 29, 1-8, (1958).
5 R. D. Maurer, "Effect of catalyst size in heterogeneous nucleation,"J. Chem. Phys. 31, 444-448 (1959).
6R. H. Doremus, "Optical properties of small silver particles," J.
Chem. Phys. 42, 414-417 (1965).7V. Kreibig and C. V. Fragstein, "The Limitation of Electron Mean
Free Path in Small Silver Particles," Z. Phys. 224, 307-323(1969).
8M. A. Smithard and R. Dupree, "The Preparation and Optical Propof Small Silver Particles in Glass," Phys. Status Solidi 11(a),695-703 (1972).
9H. C. Van de Hulst, Light Scattering by Small Particles (Wiley, NewYork, 1957).
10A. Kawabata and R. Kubo, "Electronic Properties of Fine MetallicParticles. II. Plasma Resonance Absorption, "J. Phys. Soc. Jpn.21, 1765-1771 (1966).
'D. C. Skiliman and C. R. Berry, "Spectral extinction of colloidalsilver", J. Opt. Soc. Am. 63, 707-713 (1973).
"J. C. Maxwell-Garnett, "Colours in Metal Glasses and in MetallicFilms," Trans. R. Soc. (London) 203A, 385-420 (1904). See alsoR. W. Wood, Physical Optics (McMillan, New York, 1934), pp.643-645.
Sampling theorems in polar coordinatesHenry Stark
Electrical and Systems Engineering Department, Rensselaer Polytechnic Institute, Troy, New York 12181(Received 25 October 1978; revised 11 May 1979)
We investigate the problem of representing an arbitrary class of real functions ft) in terms of theirsampled values along the radius r and at equal angular increments of the azimuthal angle 0. Twodifferent bandwidth constraints on fr,0) are considered: Fourier and Hankel. The end result is twotheorems which enable images to be reconstructed from their samples. The theorems have potentialapplication in image storage, image encoding, and computer-aided tomography.
INTRODUCTION
In this paper we consider two alternative sampling theoremsfor functions f(r,O) in polar coordinates. Both samplingtheorems enable images to be reconstructed from samples,thereby significantly easing image storage requirements in acomputer. The first sampling theorem, Eq. (5.8), applieswhen the Fourier transform F(p,) of f(r,O) vanishes outsidea circle of radius p = 0. The second theorem, Eq. (6.12), issimilar to the first except that the bandwidth constraint is interms of Hankel rather than Fourier transforms. Specifically,when f(r,O) is expressed as a Fourier series in 0, it is requiredthat the Hankel transform of the Fourier coefficients are zerooutside a circle of radius p = a.
To develop the theorems, we synthesize from known results.The structure of the paper is as follows. First, we review themain properties of the Fourier-Bessel series as they pertainto these theorems; an extensive review of the properties ofFourier-Bessel series is given by Watson.' Next, we brieflydiscuss the Fourier transforms of arbitrary functions f(r,0)which are of bounded variation and absolutely integrable (i.e.,their Fourier transforms exist). This is followed by a briefreview and discussion of sampling theorems for isotropic andband-limited periodic functions. In Secs. V and VI we com-bine the results of the earlier sections to prove the main re-sults. Finally, in Sec. VII we consider both the practicalityof implementing the theorems and some possible applica-tions.
1. FOURIER-BESSEL SERIES
Watson considers an expansion of an arbitrary function f (r)defined over the open interval (0,1) in the form
f(r) = L b.J,(z,.r),n=1
(1.1)
where Jj(.) is the vth-order Bessel function of the first kind,Z,,n is the nth zerio of J,(-), and the coefficients bn are givenby
b,, = 2 l'j ( xf(x)J,(ZvnX) dx.Jv4l (Z vn) f
(1.2)
The Fourier-Bessel theorem states that if v > -1/2 and f (r) isof bounded variation and satisfies
(1.3)I'If(x)fIdx < w,0
then the series yt 1 b,,J,(znr) converges and its sum is (1/2)
If (r + 0) + f (r - 0)j. Equation (1.2) was obtained by multi-plying both sides of Eq. (1.1) by rJ, (zpmr), integrating, andusing the orthogonality relation
rJv(zv,,r)J, (zmr) dr = j2+1 n = m)
2 t n=m.
The above results are easily extended to arbitrary functions
