Production Methods and Thin-Film Materials
`
`519
`
`there was a
`This process appears to be a wanderingof the voids through the material of
`the thin film butis really a processofsurface diffusion aroundtheinterior of
`the voids. Oncet
`
`us the voids simply increase
`
`ally in size as they reduce in number. The reason for the findings of
`Richmond and Lee, and probably also Title and Meaburn, now becomeclear.
`After deposition, the pore-shaped voids in the material are quite irregular
`in shape, especially at the interfaces between the layers. The annealing or
`baking process tends to removetherestrictions in the pores so that although
`their volume is unchanged their regular shape implies a muchfaster fill-
`ing by capillary condensation when exposed to humidity. This means that
`equilibrium is reached much morerapidly andthe filter appears much more
`stable when the environmental conditions are stable. In the case of already
`cementedfilters, the effective environmentis quite stable althoughthefilter
`stability may be disturbed by changes in temperature. However, when the
`temperaturestabilizes, equilibrium is rapidly established once again.
`The improvedstability of the integral laser mirror is probably also derived
`at least partly from this decrease in the time constant for it to reach equi-
`librium. Any drift of the mirrors after alignment in the laser would imme-
`diately cause fluctuations, almost invariably reductions, in laser output. If
`the mirror can reach equilibrium before the final alignment, then, since the
`environmentwithin the laser is reasonably stable from the point of view of
`moisture and consequentadsorption, the laser will be stable.
`Miiller [47] has also explained w.
`re if the bonds that bind
`poor adhesio
`atoms together across an interface are weaker than those that bind similar
`atomstogether in either material, then there is an energetic advantage for a
`void that reaches an interface to remain there. Voids therefore collect at such
`an interface and gradually weaken the adhesion further.
`There are a few morerecent studies of baking in connection with telecom
`quality filters, primarily using energetically deposited materials. Prins and
`colleagues [48] found a curious effect that they termed creep, although, as
`they pointedout, it is not creep in the normalsenseof the word,in the baking
`of narrow-bandfilters that had been energetically deposited The particular
`materials were not identified, butit is likely that they were SiO, and Ta,O;
`or, possibly, the chemically similar Nb,O;, because at the time, they were the
`preferred materials in that application. These materials become amorphous
`
`whenenergetically deposited. Exposure toahightemperature(1minuteat_340°C,forexample)offilters on high expansion coefficient substrates caused
`an expected immediate shift
`owever, on cooling
`back down to room temperature, the original wavelength was not immedi-
`ately restored. Instead, in a gradual recovery that occupied around 5 minutes
`the wavelength gradually returned toward the original value, although very
`slight shifts could continuefor a period of days. A small permanentshift due MATERION
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`520
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`Thin-Film OpticalFilters
`
`to the baking could also be observed.It seems that relaxation of the filmsis
`much slowerat lower temperatures. The authors referred to the property of
`the films as a viscosity that reduced as the temperature increased, and the
`
`
`
`behavior can certainly be interpreted in that way.BecingPERaceeSTRESSEDA particularly useful and informa-Cie
`tivestudyofthe changesinfilters due tosenis dueto Brown [49]. Here the
`filters were definitely constructed fro
`nd the process was
`ion-beam sputtering, frequently used for telecom quality filters. Annealing
`
`
`
`
`attemperatures ofaround500°Cinduceashiftinthefilterstowardlonger
`wavelengths.Energetic processes such as ion-beam sputtering induce hig
`
`levels of compressivestrain in thefilms.
`alin:
`
`
`
`reduction in strain also induces a drop in refractive index since strain bire-
`fringence is reduced.
`i
`i
`i
`i
`n suggested
`also a packing density
`effect in which void volumeplaysa part.It is difficult to say whether these
`are true voids in the sense of actual empty spaces in the film, or just an
`expression of the spacing of the elements of the films but the paper repays
`close study because of the accurate quantitative nature ofthe results.-
`Much moreworkis required on the whole matter of baking and consequent
`
`
`filter stability before all becomes completely clear,berteoceriieadyan!
`
`Wereturn to the matter of moisture adsorption in Chapter 12.
`
`(ERSPRE EE
`
`11.2 Measurementof the Optical Properties
`
`Once a suitable method of producing the particular thin film has been deter-
`mined, the next step is the measurement of the optical properties. Many
`methods for this exist and a useful earlier account is given by Heavens[50].
`Measurementof the optical constants of thin filmsis also included in the book
`by Liddell [51]. A more recent survey is that of Borgogno [52]. Recently, the
`measurementof the optical properties of thin films has increased in impor-
`tance to the extent that special purpose instruments are now available. These
`normally include the extraction software andare essentially push-button in
`operation. As always, however, even when automatic tools are available some
`understandingof the nature ofthe processandits limitationsis still necessary.
`Here weshall be concerned with just a few methodsthat are frequently used.
`In all of this, it is important to understand that we never actually mea-
`sure the optical constants n and k directly. Although thickness, d, is more
`susceptible to direct measurement,its value too is frequently the product of
`an indirect process. The extraction of these properties, and others, involves
`measurements of thin-film behavior followed by a fitting process in whichon
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`
`the parametersof a film model are adjusted so that the calculated behavior
`of the model matches the measured data. The adjustable parameters of the
`model are then taken to be the corresponding parameters of the real film.
`The operation is dependent on a mode] that correspondsclosely to the real
`film. The appropriateness of the model would be of less importance were
`we simply trying to recast the measurements in a more convenient form.
`Even an inadequate model with parameters appropriately adjusted can be
`expected to reconstitute the original measurements. However, the param-
`eters extracted are rarely used in that role. Rather they are used for predic-
`tions of film performance in different situations where film thickness may
`be quite different and where thefilm is part of a much more complex struc-
`ture. This leads to the ideaof stability of optical constants, a rather different
`concept from accuracy. Accurate fitting of measured data using an inappro-
`priate model may reproduce the measurements with immenseprecision yet
`yield predictions for other film thicknesses that are seriously in error. Such
`parametersare lacking in stability. Stable optical constants might reproduce
`the measured results with only satisfactory precision but would have equal
`successin a predictive role. A good example mightbe a case wherea film that
`is really inhomogeneousandfree from absorption is modeled by a homoge-
`neous and absorbingfilm. The extracted film parameters in this case can be
`completely misleading. It must always be rememberedthat the film modelis
`of fundamental importance.
`Almost as important as the model is the accuracy of the actual measure-
`ments. Calibration verification is an indispensable step in the measurement
`of the performance that will be used for the optical constant extraction.
`Rememberthat only two parameters are requiredto define a straightline but
`to verify linearity requires more. Small errors in measurementcan have espe-
`cially serious consequencesin the extinction coefficient and/or assessmentof
`inhomogeneity of the film. The samples themselves should besuitable for the
`quality of measurement. For example, a badly chosen substrate may deflect
`the beam partially out of the system so that the measurementis deficient orit
`may introducescattering losses that are not characteristic of the film.
`Thecalculation of performance given the design of an optical coating is a
`straightforward matter. Optical constant extraction is quite different. Each
`film is a separate puzzle. It may be necessaryto try different techniques and
`different models. Repeat films of different thicknesses or on different sub-
`strates may be required. Somefilms may appearto defy rational explanation.
`A common film defect is a cyclic inhomogeneity that produces measure-
`ments that the usual simpler film models are incapableoffitting with sensible
`results.It is always worthwhile attempting to recalculate the measurements
`using the model and extracted parameters to see where deficiencies might
`lie. Because ofall the caveats in this and the previous paragraphs, exact cor-
`respondence, however, does not necessarily indicate perfect extraction.
`As wesaw in Chapter2, given the optical constants and thicknesses of any
`series of thin films on a substrate, the calculation of the optical properties MATERION
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`
`is straightforward. The inverse problem,that of calculating the optical con-
`stants and thicknessesof evenasingle thin film, given the measured optical
`properties, is much moredifficult and there is no general analytical solution
`to the problem of inverting the equations. For an ideal thin film, there are
`three parameters involved, n, k, and d, the real and imaginarypartsof refrac-
`tive index and the geometrical thickness, respectively. Both n and k vary
`with wavelength, which increases the complexity. The traditional methods
`of measuring optical constants, therefore, rely on special limiting cases that
`have straightforward solutions.
`Perhaps the simplest caseof all is represented by a quarter-wave of mate-
`rial on a substrate, both of which are lossless and dispersionless, thatis, k is
`zero and n is constant with wavelength. The reflectance is given by
`
`nS|
`
`1+n%/n‘sub
`
`ay
`
`wherenis the index of thefilm and n,,,, that of the substrate, and the incident
`medium is assumedto have an index of unity. Then n,is given by
`
`12 Y2
`
`
`ny =F ]
`
`14+ RY
`
`(11.2)
`
`wherethe refractive index of the substrate, n,,,, must, of course, be known.
`The measurement of reflectance must be reasonably accurate.
`If, for
`instance, the refractive index is around 2.3, with a substrate of glass, then
`the reflectance should be measured to around one-third of a percent (abso-
`lute AR of 0.003) for a refractive index measurementaccurate in the second
`decimalplace.
`It is sometimes claimed that this method gives a more accurate value for
`refractive index than the original measure of reflectance since the square
`root of R is used in the calculation. This may be so, but the value obtained for
`refractive index will be used in the subsequentcalculation of the reflectance
`of a coating, and therefore the computedfigure can be only as good as the
`original measurementofreflectance.
`In the absence of dispersion, the curve of reflectance versus wavelength
`of the film will be similarto that in Figure 11.20. The extrema correspond to
`integral numbers of quarter-waves, even numbersbeing half-wave absentees
`and giving reflectance equal to that of the uncoated substrate, and odd cor-
`responding to the quarter-waveof Equations 11.1 and 11.2. Thus,it is easy to
`pick out those valuesof reflectance that correspond to the quarter-waves.
`The technique can be adapted to give results in the presence of slight
`dispersion. The maxima in Figure 11.20 will now no longer be at the same
`heights, but, provided the index of the substrate is known throughout the
`range, the heights of the maxima can be used to calculate valuesforfi]
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`523
`
`At these wavelengthsthefilm is an odd
`integral numberof quarter-wavelengths
`
`Prwe \
`
`
`
`Reflectance of
`z
`
`uncoated substrate
`“o
`Gee NZow ee AAWMAL fe.
`§Oouv
`muv
`[<A
`
`Wavelength (4) ———>
`
`At these wavelengthsthe film
`is an integral numberofhalf-
`wavelengthsin optical thickness
`
`FIGURE 11.20
`
`The reflectance of a simple thin film.
`
`index at the corresponding wavelengths. Interpolation can then be used to
`construct a graph of refractive index against wavelength. Results obtained
`by Hall and Ferguson [53] for MgF, are shownin Figure 11.21.
`This simple method yields results that are usually sufficiently accurate
`for design purposes. If, however, the dispersion is somewhatgreater, orif
`rather more accurate results are required, then the slightly more involved
`formulae given by Hassetal. [54] must be applied.It is still assumedthat the
`absorptionis negligible. If the curve of reflectance or transmittance of a film
`possessing dispersion is examined, it will easily be seen that the maxima
`corresponding to the odd quarter-wave thicknesses are displaced in wave-
`length from the true quarter-wave points, while the half-wave maxima are
`unchanged. This shift is due to the dispersion, and measurementof it can
`yield a more accurate valuefor the refractive index. In the absence of absorp-
`tion the turning valuesof R, T,1/R, and 1/T mustall coincide. Assumingthat
`the refractive index of the incident mediumis unity, that of the substrate n,,,,
`andof the film nf then their expression for T becomes
`
`4
`7 n,, +2+n,,, + 0.53,(n; -1-n+n, ny\[1-cos(4xn,d,/A)]
`
`Since the turning values of T and 1/T coincide, the positions of the turning
`values can be foundin termsof d/A by differentiating the expression for 1/T
`and equatingit to zero as follows:
`
`aTourFEFMeuMoo,1(w-1-12 +72 nA a 4)
`
`+24 vA
`4
`
`8n
`
`‘sub
`
`iT
`
`‘sub
`
`sub’ f
`
`4an.d
`
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`
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`
`
`
`0.04
`
`&w 0.03
`
`0.02
`
`Q&§Q =
`
`vo
`Ss
`
`A 1 hour old
`—— 1 week old
`— — Substrate alone (calculated)
`
`400
`500
`600
`700
`800
`
`3
`
`Wavelength (nm)
`
`1.40
`
`b)
`(b)
`
`ws
`
`.
`-.e
`etre
`
`13s
`
`*.
`e
`tee e
`Forgecot
`
`33
`
`& IdUo Lhour oh
`
`
`2 138
`x
`1 week old
`S&uo
`[<4
`
`
`
`1.37
`
`1.36
`
`400
`
`500
`
`600
`
`700
`
`Wavelength (nm)
`
`FIGURE 11.21
`Therefractive index of magnesium fluoride films. (a) Thereflectance of a single film. (b)
`The reflectance result transforms into refractive index. The curves are formed by the
`results from manyfilms. x denotes bulk indices of the crystalline solid. (After Hall, J-F,, Jr.
`and Ferguson, W.F.C., Journal of the Optical Society ofAmerica, 45, 74-75, 1955.)
`
`Le,
`
`= a)
`~ d(d/A)= 0.25n;5 (MestMeys — Magln;?) 1—cos
`
`4nn,d,
`
`d
`4nn,d
`+ 0.52 (05,1; — My — May + Maggy}]ty +1, — |sin} —££
`sub’ f
`sub"! f
`f i
`7
`
`where nj = dn,/d(d/A). That the equationis satisfied exactly at all half-wave
`positions can easily be seen since both sin(@nn,d,/A) and (1 — cos 4n,d,/d) are
`zero. At wavelengths corresponding to odd quarter-waves, a shift does occur
`and this can be determined by manipulating the above equationinto
`,.--pion
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`
`A
`
`5
`
`2
`
`3
`
`2
`
`2
`4
`nf ~ Mop
`
`an _ny—(1+n2,)n3+NMy H+)
`
`525
`
`(11.3)
`
`’
`ny
`
`A
`
`Of course, it is impossible to solve this equation immediately for n; because
`there are too many unknowns. Generally, the most useful approach is by
`successive approximations using the simpler quarter-wave formula (11.2) to
`obtain a first approximation for the index and the dispersion. It should be
`rememberedthatthe reflection of the rear surface ofthe test glass should be
`taken into accountin the derivation of the reflectance curve.It is also impor-
`tant that the test glass should be free from dispersion to a greater degree
`than the film; otherwise, it must also be taken into account with consequent
`complication of the analysis.
`If absorption is present, then Formula 11.3 cannot be used.In the case of
`heavy absorption, it can safely be assumedthat there is no interference and
`the value of the extinction coefficient can be calculated from the expression
`
`IER exp TK pO ¢
`
`4nkd
`
`A
`
`-
`
`T
`
`(Arkyd,/A because weare dealing with energies not amplitudes) which gives
`[54] for k,
`
`k,=——*—tog{ £8
`f
`4nd, loge BT
`
`(11.4)
`
`where the two logarithmsare to the samebase, usually 10.
`The thin-film designer is not too concerned with very accurate values
`of heavy absorption. Oftenit is sufficient merely to know that the absorp-
`tion is high in a given region andthe result given by Expression 11.4 will be
`morethansatisfactory. In regions where the absorptionis significant but not
`great enough to weaken thesingle-film interference effects, a more accurate
`method can be used.
`Equations 2.122 and 2.125 are valid for any assembly of thin films on a
`transparent substrate, n,,,, and give
`
`
`_T__ Re(nsw)
`1-R Re(BC)
`
`(11.5)
`
`For a single film on a transparent substrate, the values of B and C are
`given by
`
`cos 6,
`B
`C|- iN, sind,
`
`(isind,)/N,
` cosd,
`
`1
`oy |
`Sul
`
`cosd, +i(n,,,/N,)sind,
`n,., cosd, +iN,sind,
`
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`526
`
`Now’
`
`
`
`Thin-Film OpticalFilters
`
`2an,d,
`
`iyLj
`
`.2nk,d
`
`(11.6)
`
`Weshall assumethat k is small compared with n and this implies that y will
`be small compared with @. Nowfor yw sufficiently small
`
`cosé = cosgcosh y +isingsiny = cosg+iy sing
`
`and
`
`sin 6 = sin gcosh y —icosg sinh y = sing ~iy cos@
`
`yielding the following expressions for B and C
`
`|_"—(n,,,/1,YY]cosp—(n,,,k,/1?)sinp+ily+(n,,/n,)]sine
`
`(1, +n,y)cosp+k, sing +i(n, +n,,W)sinp
`
`c|*
`
`|
`
`(11.7)
`
`At wavelengths wherethe optical thickness is an integral number of quarter
`wavelengths, sing or cos@ is zero, and we can neglect terms in cosgsing. The
`valueof the real part of (BC*) is then given by
`
`Re(BC")=cos”1+Satvf+n,y)+sin’iGtaptNW)
`oa| etn,hy
`
`f
`
`f
`
`
`ne
`
`ms
`
`and when substituted in Equation 11.5 yields
`
`1-Rn1s(teesey
`
`giving for k, (using Expression 11.6 in Expression 11.9)
`
`oeamram] (A=k-")
`
`1-R-T
`
`A
`
`(11.8)
`
`(11.9)
`
`(11.10)
`
`This expression is accurate only close to the turning valuesof the reflectance
`or transmittance curves.
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`Production Methods and Thin-Film Materials
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`527
`
`In the case of low absorption, the index should also be corrected. Hall and
`Ferguson [55] give the following expressions.
`/2
`
`-BytiR, |
`
`
`a
`
`[1-vR “™
`
`;
`
`Ney,(1+VR)
`
`41-JR
`
`(11.11)
`
`whereR is the value of reflectanceof thefilm at the reflectance maximum.
`In the methods discussedso far, we have been assumingthat the thickness
`of the film is unknown,except inasmuchasit can be deduced from the mea-
`surements of reflectance and transmittance, and the extrema have been the
`principalindicatorof film thickness. However, it is possible to measure film
`thickness in other ways, such as multiple beam interferometry, or electron
`microscopy, or by using a stylus step-measuring instrument.If there could be
`an independent accurate measure of physical thickness, then the problem of
`calculating the optical constants would become muchsimpler. This was the
`basis of a technique devised by Hadley (see Heavens[50] for a description).
`Since two optical constants, Np and k, are involved at each wavelength, two
`parameters must be measured, and these can most conveniently be R andT.
`In the ideal form of the technique, if now a value of n, is assumed, then by
`trial and error one value of ky can be found, which, together with the known
`geometrical thickness and the assumed Ny, yields the correct measured value
`of R, and then a second valueof k, that similarly yields the correct valueof T.
`A different value of n, will give two further valuesof k,, and so on.
`Proceeding thus, we can plot two curvesof k-against n,, one corresponding
`to the T values and the other to the R values, and, where they intersect, we
`havethe correct values of n, andk, for the film. The angle of intersection of
`the curves gives an indication of the precision of the result.
`Hadley, at a time when such calculations were exceedingly cumbersome,
`produced a bookof curves giving the reflectance and transmittanceoffilms
`as a function of the ratio of geometrical thickness to wavelength, with n;
`and k;as parameters, which greatly speeded up the process. Nowadays, the
`method can be readily programmedandprecision estimates incorporated.
`This method can be applied to any thicknessof film, not just at the extrema,
`although maximum precision is achieved, as we might expect, near optical
`thicknesses of odd quarter-waves, while, at half-wave optical thicknesses,it
`is unable to yield any results. As with many other techniques, it suffers from
`multiple solutions, particularly when the films are thick, and in practice a
`range of wavelengths is employed, which adds an element of redundancy
`andhelpsto eliminate someof the less probable solutions.
`Hadley’s method involves simple iteration and does not require any very
`powerful computingfacilities. Even in the absence of Hadley’s precalculated
`curves, it can be accommodated on a programmable calculator of modest
`capacity. It does, however, involve the additional measurementof film thick-
`ness, whichis of a different character from the measurements of R and T. This jyaterion
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`
`Thin-Film OpticalFilters
`
`is the primary disadvantage. There is a problem withvirtually all techniques
`that make independent measurements of thickness. Unless the thickness
`is very accurately determined and the model used for the thin film is well
`chosen,the values of optical constants that are derived may have quite seri-
`ous errors. The sourceofthe difficulty is that the extremaof the reflectance
`or transmittance curves are essentially fixed in position by the value of ny
`and d,. There is only a very smallinfluenceon thepart of k. Should the value
`for d; be incorrect, then there is no way in which a correct choice of n, can
`satisfy both the value andthe position of the extremum. What happens,then,
`is that the extremum position is ensured by an apparentdispersion, usually
`enormousand quite false, and the values of n, are then seriously in error,
`sometimes showing abrupt gapsin the curve. Thesituation is often worse
`at the half-wave points than at the quarter-wave ones but, even in between
`the extrema, there are clearerrorsin level that tend to be alternately too high
`and then too low in between successive extremum pairs. A technique that
`has been used to avoid this difficulty is to permit some small variation of
`d, around the measured value andto search for a value that removes to the
`greatest extent the incorrect features of the variation of n,.
`A different approach that has been developed by Pelletier and his col-
`leagues in Marseille [56], and requires the use of powerful computingfacil-
`ities, retains the measurement of R and T, but, instead of an independent
`measure of film thickness, adds the measurementof R’, the reflectance of the
`film from the substrate side. Now wehave three parameters to calculate at
`each wavelength and three measurements, and it might appear possible that
`all three could be calculated by a processofiteration, rather like the Hadley
`method, but the Marseille group found the possible precision rather poor
`and it broke down completely when there was no absorption.
`To overcome this difficulty, the Marseille method uses the fact that the
`geometrical thickness of the film does not vary with wavelength, and there-
`fore, if information over a spectral region is used, there will be sufficient
`redundancy to permit an accurate estimate of geometrical thickness. Then
`once the thickness has been determined, a computer method akinto refine-
`mentfinds accurate valuesof the optical constants 1, and k, over the whole
`wavelengthregion. For dielectric layers of use in optical coatings, k, will usu-
`ally be small, and often negligible, over at least part of the region and a pre-
`liminary calculation involving an approximatevalueof n, is able to yield a
`value for geometrical thickness, which in mostcasesis sufficiently accurate
`for the subsequent determination of the optical constants. Given the thick-
`ness, R andT, as we have seen, should in fact be sufficient to determine Ny and
`k,. However, this would mean discarding the extra information in R’, and so
`the determination of the optical constants uses successive approximations
`to minimize a figure of merit consisting of a weighted sum of the squares
`of the differences between measured T, R, and R’ and the calculated values
`of the same quantities using the assumed valuesof n, and k, Although sel-
`dom necessary, the new values of the optical constants can then be used_in
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`
`an improved estimate of the geometrical thickness, and the optical constants
`recalculated. For an estimate of precision, the changes in ny and ky to change
`the values of T, R, and R’ by a prescribed amount, usually 0.3%, are calcu-
`lated. Invariably, there are regions around the wavelengths for which the
`film is an integral numberof half-waves thick, where the errors are greater
`than can be accepted andresults in these regionsare rejected.In practice the
`films are deposited over half of a substrate, slightly wedgedto eliminate the
`effects of multiple reflections, and measurements are madeof R and R’ and
`T and T’, the transmittance measured in the opposite direction (theoretically
`identical), on both coated and uncoated portions of the substrate. This per-
`mits the optical constants of the substrate to be estimated; the redundancyin
`the measurements of T and T’ gives a check on the stability of the apparatus.
`A very large numberof different dielectric thin film materials have been
`measured in this way anda typical result is shown in Figure 11.22.
`A particularly useful and straightforward family of techniques are known
`as envelope methods. The results that they yield are particularly stable. The
`envelope method wasfirst described in detail by Manifacier et al. [57] and
`later elaborated by Swanepoel[58].
`Let us imagine that we have a homogeneousdielectric film that is com-
`pletely free from absorption. Let us deposit this film on a transparent
`substrate and gradually increase the thicknessof this film, all the time mea-
`suring the reflectance. Let the maximumreflectance be given by Ryg, and the
`minimum by R,,;,. We can plot the locusof the film at one wavelength as the
`thickness increases and this will appear as Figure 11.23.
`
`
`
`330
`
`400
`
`500
`
`600
`
`700
`
`% (nm)
`
`FIGURE 11.22
`Therefractive index ofa film of zinc sulfide. The slight departure from a smooth curve is due to
`structural imperfections suggesting that even in this case of a very well behaved optical mate-
`ria] there is some veryslight residual inhomogeneity. (After Pelletier, E., Roche, P., and Vidal,
`B. Nouvelle Revue d'Optique, 7, 353-362, 1976.)
`
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`
`
`FIGURE11.23
`
`Thecircles labeled R,,,, and R,,, are isoreflectance circles. The two circles tangent to both of
`them are possible loci for the thin film. Two other possible circular loci, tangent to bothcircles,
`are possible geometrically, but would represent admittances less than yy and so have been
`discarded.(See color insert following page 398.)
`
`The maximum and minimum reflectances will each be represented by an
`isoreflectancecircle in the admittance plane. Any possible locusfor the thin
`film mustthen bea circle tangent to both of them. There could be four such
`loci but, since the incident medium will usually be air, two of the loci must
`represent characteristic admittances less than unity, and can therefore be
`discarded. There are then two possible remaining loci, both shownin the
`diagram. Should the addition of the film increase the reflectance above that
`of the uncoated substrate, then the substrate must be represented by the point
`B in the diagram. There is then only one locus that can represent the film,
`that is, the locus with extreme points A and B. Thereflectances at A and B
`can then be converted into admittances, and the square root of their product
`will be the characteristic admittance of the film. Should the film reduce the
`reflectance below that of the substrate then the substrate must be represented
`by the point A, and we nowhavetwopossible solutions for the admittance of
`the film. Provided the minimumreflectanceis not too low, we should be able
`to distinguish the correct solution provided we havea sufficiently reasonable
`idea of the correct value.It is easy to see the danger, however,of a film that is
`acting as a good antireflection coating for the substrate. Then the two possi-
`ble values will be close together andit will be very difficult to separate them.
`In addition, it can be shown thatthe effect of errors in the measurementof
`reflectance have a muchgreatereffect on the extracted value of characteristic
`admittance when the extremum represents a very low value.
`So far, to extract the value of the film characteristic admittance, we do
`not need to calculate the admittance of the substrate separately. Let the film
`now’beslightly inhomogeneousin a simple way wherethe refractive index
`changes uniformly and slowly throughthefilm. The locus will no longer be
`thatofa circle but a slowly contracting or expandingspiral and therewill be
`a gap betweenthe notional half-wave points and the actual substrate admit-
`tance. Now, separate knowledge of the substrate admittance will allow an
`estimate also of the inhomogeneity of the film.
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`Production Methods and Thin-Film Materials
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`531
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`Now let the film be also slightly absorbing. A slightly absorbing film
`showsvery little difference in reflectance when compared with an exactly
`similar but transparent film. However, this is not the case in transmittance.
`Transmittance is sensitive to both inhomogeneity and absorptance. The
`expressionsare a little more complicated in transmittance, nevertheless, if
`we add transmittance measurements, T,,,, and T,,;,, to our corresponding
`reflectance measurements, we will be able to distinguish and separately esti-
`mate both absorptance and inhomogeneity.
`Finally, provided we knowthestarting and finishing points for the film
`locus and the numberof circles, or the numberof exhibited extrema, then
`we can calculate the optical thickness, and hence physical thickness of
`the film.
`Unfortunately, we seldom have the necessary information in this form.
`What we generally have are plots of reflectance and/or transmittance
`of an already deposited film in terms of wavelength, and similar plots of
`an uncoated substrate. Figure 11.24 shows typical fringes as a function of
`wavelength. To these fringes have been added two envelope curves that
`pass throughthe fringe extrema. The basis of the envelope techniqueis the
`assumption that these envelope curves in both reflectance and in transmit-
`tance can be used at any wavelength as substitutes for the Ryraxs Risin Tnaxr aN
`Tinin that would have been available had the growing film been measured
`continuously at every wavelength point.
`Although,atfirst sight, this might seem a somewhatinaccurate technique
`because it depends on envelopes that might be arbitrary, it turns out that
`films that can be modeled as a straightforward, slightly absorbing, slightly
`
`100
`
`90
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`80
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`
`
`Transmittance(%) 70
`
`60
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`200
`
`400
`
`600
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`800
`
`1000
`
`1200
`
`1400
`
`1600
`
`Wavelength (nm)
`
`FIGURE11.24
`Thefringes measured in transmissionofa film of tantalum pentoxide over a substrate of glass.
`The envelopesof the fringes are shown. With well-behavedfilms like this one, the adding of
`MATERION
`the envelopes is a straightforward process.
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`Thin-Film Optical Filters
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`inhomogeneousdielectric film exhibit well-formed fringes that lend them-
`selves to simple envelope curves, such as in Figure 11.24. Films that show
`fringes that are more variable in their extrema are invariably more compli-
`cated in their structure and unable to be represented by a simple model. A
`. great advantageof this techniqueis that the extracted valuesof film param-
`eters are exceptionally stable. Predictions of performance