Optical monitoring of nonquarterwave multilayer filters
`
`B. Vidal, A. Fornier, and E. Pelletier
`
`Many spectral filtering problems require the use of assemblies of layers having thicknesses which bear no ob-
`to each other. Successful production of these multilayers requires films with thicknesses
`vious relationship
`approximating theoretical values. We show that the optical methods currently used in the production of
`layer thicknesses, which are based on the use of just one single wavelength,
`film assemblies of quarterwave
`In contrast we show
`thickness multilayers.
`are poorly adapted to monitoring the deposition of nonintegral
`thickness errors can be reduced.
`that with an optical control system utilizing a broad spectral bandwidth,
`Transmittance measurements with the precision necessary to achieve this improved thickness control are
`attainable with existing instrumentation. This result is established by a computer simulation of the con-
`struction of a specific multilayer and remains valid for other nonquarterwave multilayer filters.
`
`1.
`
`Introduction
`Many spectral filtering problems require assemblies
`of layers having thicknesses which bear no obvious re-
`lationship with each other. This is often the case when
`precisely defined properties are required over a wide
`spectral range.
`Methods of automatic calculation permitting the
`choice of those layers best adapted to the problem have
`been the subject of much study. With recent progress
`in this field' we can now derive theoretical designs to
`satisfy almost any filtering requirement.
`Production of multilayers thus defined requires layers
`with thicknesses approximating theoretical values. It
`is this area which forms the object of the present study.
`We present a brief review of the principles of optically
`monitoring layer thickness currently used in the pro-
`duction of films with quarterwave layer thicknesses or
`exact multiples of quarterwave thicknesses. We then
`go on to consider the extension of those methods to the
`production of multilayers containing nonintegral
`thicknesses. We will show that such methods, which
`invariably use a limited spectral range, lose their prin-
`cipal advantages, and this leads us to the idea of a more
`suitable method of control using a wide spectral re-
`gion.
`
`The authors are with Universite d'Aix-Marseille, Laboratoire
`d'Optique, Centre d'Etude des Couches Minces, Saint Jerome, F13397
`Marseille Cedex 4, France.
`Received 24 May 1977.
`0003-6935/78/0401-1038$0.50/0.
`© 1978 Optical Society of America.
`
`1038
`
`APPLIED OPTICS / Vol. 17, No. 7 / 1 April 1978
`
`II. Survey of Layer Thickness Monitoring Methods
`(Optical)
`It has already been shown that to eliminate the
`principal sources of error, either systematic or random,
`it is generally preferable to use an optical method of
`control which is performed during deposition directly
`on the multilayer which is to be produced rather than
`on a series of separate test glasses.2
`In this way varia-
`tions of the observed signal depend directly on the
`change in reflectance or transmittance of the stack
`during deposition of each layer. The transmittance
`measured during the formation of the ith layer depends
`on its index ni, on its thickness at the instant of mea-
`surement (O < e < ei), on the indices and thicknesses of
`the (i-1) layers already deposited, and finally on the
`wavelength T(X, n1, e1,. . . , nj-1, ei-1, ni, ei) we will de-
`note in what follows by Ti(X,e). When the value of
`thickness e attains the required value ei, we can write
`Ti(X,ei) = Ti 0(X). Similar notation can be used for
`reflectance R.
`Generally the methods used for direct monitoring of
`thicknesses layers are simply derived from
`nonintegral
`those which are traditionally used for layers of quar-
`terwave optical thickness X/4, or integral multiples.
`But we will show that these methods, extended in this
`way, lose their principal advantages because their
`simplicity of operation and their high performance for
`multilayers of integral optical thicknesses are entirely
`due to the special properties of such multilayers.
`
`A. Control of Integral Thickness Layers
`During the deposition of a single transparent thin
`film, the transmittance, measured at wavelength No,
`
`Edmund Optics(cid:15)(cid:3)(cid:44)(cid:81)(cid:70)(cid:17)(cid:3)
`(cid:40)(cid:91)(cid:75)(cid:76)(cid:69)(cid:76)(cid:87)(cid:3)(cid:20)(cid:19)(cid:19)8(cid:3)
`
`0001
`
`

`
`passes through an extremum each time the optical
`thickness reaches a value k XO/4, where k is an integer.
`This property remains true during the formation of each
`layer of a stack provided all previous layers have an
`optical thickness exactly equal to a multiple of the same
`quantity Xo/4. This forms the basis for all the moni-
`toring methods discussed in this section.
`The method which is commonly used for the moni-
`toring of such multilayers consists of following the
`changes in Ti (Xo,e) during the deposition of each layer
`and of terminating
`the evaporation as soon as Ti is ob-
`served to reach the desired extremum.
`But this simple method presents several difficulties
`because the direct determination of an extremum is
`never easy, and it is preferable to observe the changes
`in one of the derivatives [Ti(Xo,e)]/(6e) or [Ti(Ao,
`e)]/(OX). In the first case the signal reaches a null when
`the required thickness is attained. In fact, with certain
`precautions, it is sufficient to observe the derivative
`with respect to time [Ti(Xoe)]/(at).
`The light beam
`which is used for the control is then monochromatic of
`wavelength X0.
`Under the same conditions, we can show that the
`derivative of the transmittance with respect to the
`wavelength measured at = Xo, that is, [bTi(Xe)]/
`(oX)}x0 reaches a null when e is very close to Xo/4. As
`the number of layers increases, the thickness for which
`this relationship is satisfied tends still closer to Xo.4,3
`We can therefore use the condition Ti/oX = 0 as our
`termination criterion, and it is this principle which is
`used in the maximetre4 which produces a signal pro-
`portional to Ti/oX by modulating the measuring
`wavelength around the value Xo. Thus in this case we
`no longer use a strictly monochromatic light beam, but
`rather one which comprises a relatively narrow spectral
`band around Xo.
`These monitoring systems are fairly easy to put into
`operation, particularly the first one mentioned. This
`result is due to the fact that control of all layers in the
`film assembly uses the same wavelength (or narrow
`spectral region in the case of the maximetre). It is ap-
`propriate to choose a wavelength at which the com-
`pleted film system must exhibit well defined properties
`(peak wavelength of a narrowband interference filter,
`for example).
`Another advantage of these methods is that they do
`not require any precise knowledge of the refractive in-
`dices of the materials which are being used, because it
`is the optical thickness ne which is important and di-
`rectly controlled.
`So far we have supposed that during the formation
`of a multilayer, each layer has exactly the required
`thickness. This never happens, and we must take into
`account not just the effect of errors on the final perfor-
`mance of the filter but also the effect which the errors
`have on each other. If there are errors in the (i - 1)
`preceding layers, the ith layer will not have the correct
`thickness even if it is controlled perfectly. Because the
`control is carried out on the entire multilayer as it is
`deposited, the thicknesses of the layers are no longer
`
`independent.
`Indeed one might expect under such
`conditions that the combination of successive errors
`would rapidly lead to unacceptable results. In fact this
`result is not always the case, and the reverse result is
`observed in the case of multilayers of integral optical
`thickness.5 6 We have been able to show6 that if an
`error is committed it is automatically compensated
`during the deposition of the following layer when one
`uses the method based on the detection of zeros of
`Tiloe We say in this case that the control is stable,
`which is evidently a fundamental advantage because it
`assures, in a relatively simple way, the production, with
`excellent precision, of multilayers with specified optical
`properties.
`The monitoring methods which we have described are
`therefore particularly well adapted to the monitoring
`of multilayers having integral thickness layers.
`Let us now consider under what conditions we can
`extend the use of these methods to multilayers in which
`the layers have optical thicknesses which are not simply
`related.
`
`B. Monitoring of Multilayers of Nonintegral
`Thicknesses
`The monitoring of the first layer should not present
`any special problem. If it consists of a thickness el of
`index nl, it is sufficient to choose, within the spectral
`range available, a wavelength X1 for which the termi-
`nation criterion OT/be = 0 (or T l/-A = 0) is satisfied
`when the required thickness is attained. But the
`thickness e2 of the second layer of index n 2 has no sim-
`ple relationship with el, and thus the wavelength XI has
`no particular significance as far as the transmittance
`T 2(X1, e) is concerned.
`It is only by using a computer
`that we can, in general, determine a wavelength X2 for
`which the chosen termination criterion is satisfied when
`the thickness e2 is attained. This operation must be
`repeated for each layer making up the multilayer. In
`this way we are led to the idea of simulating by com-
`puter the formation of the multilayer and to choose the
`wavelength for each layer which is best suited for con-
`trol. The product of this simulation is the control
`program which must be rigorously followed during the
`production of the filter. Thus the use of the criteria
`based on OT/Ze and 6T/OA leads, in the case of assem-
`blies of layers of nonintegral thickness, to a complicated
`monitoring method which has none of the advantages
`possessed in the case of integral quarterwave thick-
`nesses. In fact the shifting of the control wavelength
`for each layer makes the whole operation much more
`complicated and increases the chance of error. Further,
`the complete multilayer does not exhibit any special
`properties for the control wavelengths.
`In addition, the derivation of the control program for
`nonintegral film assemblies requires a more precise
`knowledge of the indices of refraction of the materials
`which are used. Obviously the values of these indices
`must also be reproducible, which imposes tighter control
`of the conditions under which the layers are deposited.
`Finally no reason exists for the presence of any benefi-
`
`1 April 1978 / Vol. 17, No. 7 / APPLIED OPTICS
`
`1039
`
`0002
`
`

`
`cial compensation of errors, and in fact experiments
`have shown that cumulative effects of successive errors
`can rapidly become disastrous.2
`The extension of the classical methods of control to
`multilayers of nonintegral thickness is therefore of
`doubtful value. Such multilayers often have optical
`properties precisely defined over a wide spectral range,
`and the monitoring methods which use monochromatic
`or quasi-monochromatic light are poorly adapted to deal
`with them. It appears necessary to control the depo-
`sition of such multilayers by examining the optical
`properties and the way in which they change over the
`whole spectral region of interest.
`
`Ill. Wideband Monitoring
`This consists of comparing continuously, over the
`whole useful spectral range, the actual spectral profile
`of the assembly Ti (X,e) during the formation of the ith
`layer, with the spectral profile Ti(X,ei) which the as-
`sembly should possess when the thickness of the ith
`layer reaches its correct value. We could simply observe
`the two profiles simultaneously by eye and thus observe
`directly their coincidence. But the accuracy of such a
`method would be poor because it is difficult to detect
`by eye the instant at which the closest fit of the two
`curves is obtained.
`The operation of a method based on this principle
`assumes that we can evaluate the distance between the
`two functions Ti(Xei) and TA(X,e). We define this by
`the value
`
`ITiei) - Ti(Ne)IdX,
`
`fi= f
`which we call the merit function. We must therefore
`calculate this function continuously during the forma-
`tion of the layer and terminate deposition when it
`reaches a null.
`The principle of this method is therefore simple, but
`its operation requires the solution of a number of
`problems and compliance of several conditions which
`appear contradictory. The use of this system cannot
`be recommended unless it complies with two essential
`preconditions. It must first and foremost be completely
`reliable, and its operation must be as simple as the usual
`methods. But it must above all detect the terminal
`points with sufficient sensitivity and with as great ac-
`It is this last point which is ob-
`curacy as possible.
`viously fundamental because it is the sole justification
`for this new method, and it is this aspect which we will
`now examine. The problems found in the practical
`operation of this system and a description of the first
`trials will form the subject of a later publication. Here
`we will examine the accuracy and sensitivity of the
`method in comparison with a classical method using a
`single control wavelength.
`
`IV. Comparison of Wideband and Monochromatic
`Methods
`Computer simulation allows us to observe, step by
`step, the formation of film assemblies. Simulation
`techniques are therefore particularly suited for the
`
`1040
`
`APPLIED OPTICS / Vol. 17, No. 7 / 1 April 1978
`
`evaluation of control techniques. We can apply them
`to the problem posed by the construction of a prede-
`termined multilayer and thus compare the wide band-
`width method and the classical method of control. Our
`aim is to know the accuracy with which the merit
`function should be determined so that the wideband
`method will lead to the construction of a filter more
`reliably than the classical method. The precision with
`which f is known depends particularly on the position
`and extent of the spectral zone which is used and on the
`quality of the photometric measurements. Obviously
`it is fundamental to verify that the requirements in this
`zone are compatible with the experimental possibili-
`ties.
`A. Choice of Multilayer-Constructional Tolerances
`As an example we have chosen the case of a beam
`splitter at normal incidence for this study. Its ideal
`profile corresponds to a transmittance T(X) equal to
`the 400-800-nm spectral region.
`0.35 throughout
`Computer synthesis7 leads us to an assembly of seven
`alternate zinc sulfide and cryolite layers with the fol-
`lowing thicknesses (nm): glass, 103.4, 194.9, 89.2, 154.7,
`67.7, 80.1, 42.9 air.
`The corresponding spectral profile T, (X) is shown in
`Fig. 1. The synthesis calculation assumes a perfect
`knowledge of the refractive indices of the deposited
`layers, the values of which have been given elsewhere
`(Figs. 4 and 7 in Ref. 8). In the special case of an ach-
`romatic filter it is convenient to characterize its profile
`by the mean value T, of its transmission over the
`spectral interval considered, by the mean of the absolute
`values of the deviations AT, = T- T, and by the
`maximum deviation I T m. The problem consists of
`
`TRANSMITTANCE
`
`-----------------------------------------------
`
`--------------------------
`
`------------------------
`
`- -- -- -- -- -- -- ---
`
`--- ----- -- - -- -- -------
`
`--- -- -- - --- -- -- -- -- -- --- -- -- -- -- -----------------
`
`n
`
`l
`
`d t~~~~~~~~--------------
`
`1.00
`
`a 80
`
`Oz60
`
`0o40
`
`0o20
`
`400a
`
`500 .
`
`600 a
`
`WAVELENGTH n m)
`
`Fig. 1. Theoretical performance of a seven-layer beam splitter of
`design: glass 103.4, 194.9, 89.2, 154.7, 67.7, 80.1, 42.9 (nm) made
`of ZnS (odd layers) and cryolite (even layers) on glass (n = 1.52). The
`refractive indices are given in Ref. 8.
`
`0003
`
`

`
`Table I. Simulation of the Production by Turning Value Monitoring of the Seven-Layer Beam Splitter shown in
`Fig. 1
`
`Layer number
`
`1
`
`2
`
`3
`
`4
`
`5
`
`6
`
`7
`
`Control wavelength (nm)
`Maximum or minimum
`T initial
`Intermediate extremum of T
`T final
`
`497.3
`max
`0.960
`0.659
`0.957
`
`449.2
`min
`0.884
`0.982
`0.867
`
`456.0
`max
`0.889
`0.744
`0.891
`
`431.2
`742.9
`min
`min
`0.826
`0.643
`0.960 none
`0.820
`0.219
`
`427.2
`max
`0.772
`none
`0.935
`
`427.8
`min
`0.939
`none
`0.323
`
`the construction of a filter which has optical properties
`as close as possible to those predicted by calculations.
`We therefore establish the permissible tolerances of the
`optical properties. We must deduce the maximum
`errors which can be permitted
`in the thicknesses of
`successive layers, and of course the optical monitoring
`must ensure the construction of the assembly with the
`corresponding precision.
`First we shall study construction of the beam splitter
`using one of the common optical methods of thickness
`control. We will then compare these results with those
`which can be obtained using a wideband optical control
`method.
`In these simulation calculations we can as-
`sume as a first approximation that the indices of the
`layers are accurately known. Experimental attempts
`at the construction of such a beam splitter show that
`this hypothesis is valid at least for the dielectric mate-
`rials we will use.
`
`B. Application of Monochromatic Monitoring
`The most common system of monitoring is that which
`consists of measuring directly the transmission for one
`single wavelength, previously calculated, and stopping
`the deposition when an extremum of transmission is
`observed.
`It is this well known method we will apply
`in the case of a beam splitter. Since the assembly does
`not consist of layers of equal thickness, the control
`wavelengths will be different. A simple calculation
`gives the wavelengths in the 400-800-nm spectral range
`for monitoring each layer, and those which give the
`maximum sensitivity will be chosen. Table I lists the
`principal data necessary for the monitoring of this
`multilayer.
`One can form an idea of the manufacturing tolerances
`necessary for this filter by studying the effects on the
`thicknesses, and consequently on the spectral profile
`of the complete multilayer, of errors in the detection of
`the extrema serving as termination criteria for the de-
`position.
`In these calculations one takes account of the
`cumulative errors in the direct monitoring system. The
`method of calculation is analogous to that used in a re-
`cent study which treats quarterwave multilayers.9 The
`sole modification of any importance in the calculation
`program consists of the changing wavelength which
`must be specified separately for each layer.
`The principal inconvenience of this monitoring
`method is tied to the difficulty of detecting with accu-
`racy the instant when the transmission Ti (i,e) passes
`
`through an extremum. One in general has to adopt a
`systematic overshoot with an inevitable associated
`random error. The evaporation will therefore be ter-
`minated for a value of transmittance which differs from
`the turning value by the quantity AT = a +
`(Fig. 2),
`where a represents the systematic overshoot and f the
`random error. The consequences of the errors a and
`can be very different.
`One can very rapidly put a figure on the order of ac-
`curacy which must be obtained in the detection of each
`transmittance extremum. As a first step we will sup-
`pose that fi is negligible by comparison with a and that
`a has the same value for every layer. Simulation shows
`that it is then extremely difficult to retain control right
`up to the final layer. If the systematic error a is of the
`order of 0.005 the thicknesses which are actually de-
`posited differ considerably from the correct theoretical
`values, and control cannot beretained beyond the third
`layer. For a = 0.003 control is lost during the fifth
`layer. Control throughout the entire multilayer is im-
`possible unless a is less than 0.001. With a = 0.001 the
`thicknesses which are obtained are: 105.0, 199.5,98.3,
`152.4, 67.9, 71.9, 43.9 (nm). They differ considerably
`;
`
`Fig. 2. Diagram illustrating the effect in turning value monitoring
`of an overshoot on the thickness error in a layer.
`
`Layer thickness
`
`1 April 1978 / Vol. 17, No. 7 / APPLIED OPTICS
`
`1041
`
`0004
`
`

`
`TRANSMITTANCE
`
`1I00
`
`080
`
`0.60
`
`0o40
`
`0a20
`
`400 e
`
`i
`100~ e
`800X
`WAVELENGTH nm)
`
`Fig. 3. Simulated production run of the turning value monitoring
`of the beam splitter of Fig. 1 with a systematic error of a = 0.001 in
`determining the turning values: - perfect filter; 0 filter with sys-
`tematic error.
`
`from the true theoretical values, and, for the third and
`sixth layers especially, the thickness error is of the order
`of 10%. Figure 3 shows the spectral profile of the cor-
`responding filter. In certain regions the deviation of
`the transmission with respect to that of the perfect filter
`reaches more than 15%.
`The consequences of systematic errors on the detec-
`tion of the extrema of T are therefore serious. How-
`ever, one could think of eliminating them by using a
`derivative method. In this case the residual error is
`, and we have simulated the
`completely random, AT =
`deposition of several filters having such errors.
`Table II summarizes the performance of filters cal-
`culated for the case where a(flj) = 0.005. We can
`compare them with the performance of the perfect filter
`for which the mean transmittance over the spectral
`range used is T8 = 0.34; the mean deviation AT, is 0.046,
`and the maximum deviation I ATS I m = 0.09. The value
`of mean transmittance (Fig. 4) varies greatly from one
`filter to another, and we are often far from obtaining a
`correct achromatic response over the whole spectral
`region required.
`
`Table II. Simulated Production Runs of the Beam Splitter of Fig. 1 Using Turning Value Monitoring with Random
`Errors of 0.005 In Determing the Turning Values
`
`Simulationno
`
`1
`
`2
`
`3
`
`4
`
`5
`
`6
`
`7
`
`8
`
`9
`
`0.345
`0.352
`0.568
`0.341
`0.340
`0.427
`0.413
`0.389
`0.274
`Ts
`0.043
`0.098
`0.163
`0.104
`0.086
`0.065
`0.074
`0.107
`0.072
`AvT,
`0.157
`IATsm
`0.223
`0.342
`0.245
`0.198
`0.393
`0.466
`0.210
`0.210
`Note: The transmittance of the simulated filters is given in Fig. 4. The performance of the perfect
`filter is: Mean transmittance T, = 0.343, mean deviation AT, = 0.046, and the maximum deviation
`I ATs m= 0.094.
`
`TRANSMITTANCE
`
`1.00
`
`0 80
`
`060
`
`0.40
`
`0.20
`
`0.
`
`400.
`
`500.
`
`6000
`
`S00 o00.
`WAVELENGTH nm)
`
`found with successive
`Fig. 4. Envelopes showing the transmittance
`runs of the beam splitter of Fig. 1 with random errors of 0.005 in de-
`termining the turning values. The performance of these simulated
`filter with
`filters is summarized in Table II: - perfect filter; -
`random errors.
`
`1042
`
`APPLIED OPTICS / Vol. 17, No. 7 / 1 April 1978
`
`Table III was obtained assuming o(fli) = 0.003. On
`average the results are better but still not good enough.
`In fact, even with a(fl3) = 0.001 (Table IV) there is a high
`probability that multilayers will be obtained which do
`not have satisfactory performance over the whole
`400-800-nm spectral range. Figure 5 shows the spectral
`profiles of the filters, and they are all rather different
`from the perfect one.
`It still remains to define precisely what should be the
`effect of a simultaneous systematic overshoot and a
`random error. In fact in the case of quarterwave mul-
`tilayers we have been able to show9 that a certain com-
`pensation can occur between these two types of errors.
`We now take AT = a + dl with ai = 0.001 for each layer
`and the same random distribution with standard de-
`viation c(0j) = 0.001. The filters in this case are even
`poorer than those obtained when only either a or / was
`present by itself (Table V). In contrast to the quar-
`terwave case, no compensation exists here.
`Summarizing, the production of this simple multi-
`layer composed of unequal thickness layers poses a
`difficult problem which can be resolved with a moni-
`
`0005
`
`

`
`Table lii. Simulated Production Runs of the Beam Splitter of Fig. 1 Using Turning Value Monitoring with
`Random Errors of 0.003 in Determining the Turning Values
`
`Simulation no
`
`1
`
`2
`
`3
`
`4
`
`5
`
`6
`
`7
`
`8
`
`T__
`"ATS
`I ATs Im
`
`0.281
`0.068
`0.191
`
`0.368
`0.068
`0.448
`
`0.382
`0.094
`0.191
`
`0.404
`0.058
`0.382
`
`0.334
`0.090
`0.211
`
`0.334
`0.075
`0.185
`
`0.494
`0.149
`0.268
`
`0.346
`0.083
`0.207
`
`9
`
`0.324
`0.043
`0.125
`
`Table IV. Simulated Production Runs of the Beam Splitter of Fig. 1 Using Turning Value Monitoring with Random
`Errors of 0.001 in Determining the Turning Values
`
`Simulation no
`
`1
`
`2
`
`3
`
`4
`
`5
`
`6
`
`7
`
`8
`
`__
`
`AT8
`IATsIm
`
`0.298
`0.059
`0.156
`
`0.371
`0.066
`0.154
`
`0.352
`0.056
`0.367
`
`0.375
`0.050
`0.286
`
`0.330
`0.063
`0.154
`
`0.331
`0.071
`0.165
`
`0.420
`0.086
`0.178
`
`0.339
`0.067
`0.171
`
`9
`
`0.318
`0.047
`0.110
`
`Note: The transmittance of the simulated filters is given in Fig. 5.
`
`TRANSMITTANCE
`
`1000
`
`0.80
`
`0060
`
`0040
`
`0020
`
`400.
`
`500.
`
`600. o
`
`00
`800O
`WAVELENGTH nm)
`
`Fig. 5. Envelopes showing the transmittance
`found with successive
`runs of the beam splitter of Fig. 1 with random errors of 0.001 in de-
`termining the turning values. The performance of these simulated
`filters is summarized in Table IV: - perfect filter; -
`filter with
`random errors.
`
`toring system which is both accurate and specific.
`Strictly, the differentiation procedures for detecting the
`zeros of either T/?Ie or ?T/bA can be used with great
`difficulty. The difficulty arises because in order to
`manufacture a filter for which the optical properties are
`sufficiently close to the theoretical properties it is nec-
`essary to proceed with great numbers of systematic
`trials in order to eliminate little by little the imperfec-
`tions of production.2 The least error committed in the
`thickness of any layer has no chance of compensation
`during deposition of subsequent layers, and the resul-
`tant filter is often discrepant.
`
`C. Application of Wideband Optical Monitoring
`To operate a wideband monitoring system it is nec-
`essary to measure the optical properties of the filter
`continuously during construction and to compare the
`values obtained with those furnished by the control
`program which has been previously calculated. This
`implies the absolute measurement of transmittance (or
`reflectance), and these measurements are difficult to
`attain with great precision. Under these conditions the
`detection of the zero of the merit function, which should
`
`Table V. Simulated Production Runs of the Beam Splitter of Fig. 1 Using Turning Value Monitoring with
`Random Errors of 0.001 as in Table IV and a Simultaneous Systematic Overshoot of 0.001
`
`Simulation no
`
`1
`
`2
`
`3
`
`4
`
`5
`
`6
`
`7
`
`8
`
`T_
`AT,
`jATslm
`
`0.353
`0.049
`0.358
`
`0.402
`0.090
`0.258
`
`0.420
`0.076
`0.453
`
`0.410
`0.069
`0.398
`
`0.340
`0.059
`0.223
`
`0.375
`0.054
`0.415
`
`0.450
`0.105
`0.339
`
`0.352
`0.059
`0.331
`
`9
`
`0.337
`0.047
`0.182
`
`1 April 1978 / Vol. 17, No. 7 / APPLIED OPTICS
`
`1043
`
`0006
`
`

`
`Table VI. Error (in nm) In the Thickness of the ith Layer of the Beam
`Splitter which Occurs when the Transmittance at Termination of
`Deposition Overshoots the Correct Value of the Spectral Profile over
`400-800 nm by bTI
`
`Layer i
`5Ti
`
`1
`
`2
`
`3
`
`4
`
`5
`
`6
`
`7
`
`1.5
`1.0
`0.5
`
`TRANSMITTANCE
`
`400
`
`0.80
`
`0060
`
`0040
`
`0020
`
`4.7
`1.5
`0.015
`3.2
`1.0
`0.010
`1.5
`0.5
`0.005
`Note: The thicknesses of the layers of the perfect filter are 103.4,
`194.9, 89.2, 154.7, 67.7, 80.1, 42.9 nm.
`
`1.5
`1.0
`0.5
`
`6.5
`4.5
`2.5
`
`1.0
`0.5
`0.2
`
`4.0
`3.0
`1.5
`
`mark the coincidence between the measured and ex-
`pected results, can present some difficulties. Thus it
`is necessary for us to determine whether, for the preci-
`sion one can effectively attain in the measurement of T,
`the method leads to satisfactory results.
`First we will assume that the errors in the first (i -
`1) layers are negligible. Then during the deposition of
`the ith layer the spectral profile must, at a certain in-
`stant, coincide with the theoretical one. At this instant
`the merit function will reach a null, but the errors in the
`in the
`measurement of T will cause some uncertainty
`detection of this null. Thus the thickness actually de-
`to
`posited will differ from ideal, and it is necessary
`correlate the corresponding thickness error with the
`errors in the measure of T. bTi will represent the
`standard deviation of the measure of T in the spectral
`region used and bei the corresponding error in es.
`We have assumed for this calculation that 6Ti is the
`same for each layer and have assigned to it the values
`0.015, 0.010, and 0.005. The corresponding values 5ei
`calculated for each layer of the assembly (i = 1,7) are
`given in Table VI as follows:
`This first result demonstrates that even if the un-
`certainty in the individual measurements of T are of the
`order of +0.015, nevertheless, because the measurement
`is over a wide spectral range, the optical thicknesses can
`be controlled with an acceptable precision. Even for
`the sixth layer, which seems most critical, the thickness
`error is less than +6%.
`However it is necessary to take this study further.
`Up until now we have considered that errors in the first
`(i - 1) layers have no effect on the control of the ith
`layer, that is, that we have neglected the cumulative
`effects of errors. This is certainly not justified because
`the error committed in the thickness of any layer
`changes the growth of the spectral profile during the
`
`4000
`
`8000
`'400a
`WAVELENGTH nm)
`
`Fig. 6. The results of a computer simulation of the seven-layer beam
`splitter, - perfect filter, A filter obtained assuming that an absolute
`transmittance error of 0.015 is made in layer monitoring, each mini-
`mum of the monitoring merit function being systematically overshot.
`v filter obtained assuming that an absolute transmittance error of
`0.015 is made in layer monitoring, the deposition of each layer being
`terminated before the minimum of the merit function.
`
`TRANSMITTANCE
`
`1000
`
`0080
`
`0060
`
`0040
`
`O020
`
`4000
`
`300D
`8000
`WAVELENGTH (nm)
`
`Fig. 7. Effect of 0.015 standard deviation transmittance error on the
`performance of the seven-layer beam splitter using wideband moni-
`toring (400-800 nm): - perfect filter. - Computed filters with
`transmittance errors. The performance is summarized in Table
`VII.
`
`Table VIl. Effect of 0.015 Standard Deviation Transmittance Error on the Performance of the Seven-Layer
`Beam Splitter Using Wideband Monitoring (400-800 nm)
`
`Simulationno
`
`T
`AT
`IATIm
`
`1
`
`0.356
`0.061
`0.190
`
`2
`
`3
`
`4
`
`5
`
`0.339
`0.052
`0.116
`
`0.335
`0.046
`0.141
`
`0.345
`0.052
`0.165
`
`0.362
`0.049
`0.129
`
`6
`
`0.344
`0.047
`0.108
`
`7
`
`8
`
`0.354
`0.046
`0.118
`
`0.349
`0.047
`0.100
`
`9
`
`0.337
`0.050
`0.118
`
`Note: The plots of the simulated filters are given in Fig. 7.
`
`1044
`
`APPLIED OPTICS / Vol. 17, No. 7 / 1 April 1978
`
`0007
`
`

`
`Table Vil. Effect of 0.010 Standard Deviation Transmittance Error on the Performance of the Seven-Layer
`Beam Splitter Using Wideband Monitoring (400-800 nm)
`
`Simulation no
`
`1
`
`2
`
`3
`
`4
`
`5
`
`6
`
`7
`
`8
`
`9
`
`0.355
`0.343
`0.337
`0.343
`0.356
`T
`0.046
`0.051
`0.045
`0.049
`0.056
`A\ T
`jATIm
`0.116
`0.108
`0.107
`0.121
`0.168
`Note: The plots of the simulated filters are given in Fig. 8.
`
`0.355
`0.047
`0.119
`
`0.343
`0.045
`0.108
`
`0.346
`0.047
`0.095
`
`0.340
`0.048
`0.108
`
`TRANSMITTANCE
`
`1000
`
`------------------------
`0080
`
`
`
`--
`
`-
`
`
`
`------------------ ------------
`
`TRANSMITTANCE
`
`____________
`
`7
`
`1
`
`7
`
`4.00
`
`0080
`
`-----------------------
`
`"----------------.--------r------------------------------------------------
`
`0060 I------------------------.-------------------------.------------------------.-------------------------
`
`0060
`
`
`
`------------------------ ------------------------- --------------------------------------------------
`
`040
`
`-------------------------
`
`0040
`
`------------------------
`
`0020
`
`-----------------------------------------------------------------------------
`
`-------------------------
`
`0020
`
`----------------------------------------
`
`---------------------
`
`------------------------
`
`400D
`
`500.
`
`6000
`
`8000
`300 o
`WAVJELENGTH tnm)
`
`400D
`
`500D
`
`8000
`
`SOOo
`300 a
`WAVELENGTH n m)
`
`Fig. 8. Effect of 0.010 standard deviation transmittance error on the
`performance of the seven-layer beam splitter using wideband moni-
`toring (400-800 nm): - perfect filter; -
`computer filters with
`transmittance errors. The performance is summarized in Table
`VIII.
`
`Fig. 9. Effect of 0.005 standard deviation transmittance error on the
`performance of the seven-layer beam splitter using wideband moni-
`toring (400-800 nm): - perfect filter; -
`computed filters with
`transmittance errors. The performance is summarized in Table
`IX.
`
`Table IX. Effect of 0.005 Standard Deviation Transmittance Error on the Performance of the Seven-Layer
`Beam Splitter Using Wideband Monitoring (400-800 nm)
`
`Simulation no
`
`1
`
`2
`
`3
`
`4
`
`5
`
`6
`
`7
`
`8
`
`9
`
`T
`AT
`IATJm
`
`0.349
`0.050
`0.132
`
`0.345
`0.049
`0.097
`
`0.342
`0.046
`0.099
`
`0.349
`0.051
`0.118
`
`0.354
`0.048
`0.116
`
`0.354
`0.049
`0.122
`
`0.341
`0.044
`0.10