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)6(cid:3)
`(cid:3)
`
`0001
`
`

`
`Published in 2001 by
`Taylor & Francis Group
`270 Madison Avenue
`New York, NY 10016
`
`Published in Great Britain by
`Taylor & Francis Group
`2 Park Square
`Milton Park, Abingdon
`Oxon OXI4 4RN
`
`
`
`© 1986, 2001 by H A Macleod
`
`No claim to original U.S. Government works
`Printed in the United States of America on acid-free paper
`
`International Standard Book Number-I0: 0-7503-0688-2 (Hardcover)
`International Standard Book Number-13: 97845030688-1 (Hardcover)
`
`Consultant Editor: Professor W T Welford, Imperial College, London
`
`This book contains information obtained from authentic and highly regarded sources. Reprinted material is
`quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts
`have been made to publish reliable data and information, but the author and the publisher cannot assume
`responsibility for the validity of all materials or for the consequences of their use.
`
`No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic,
`mechanical, or other means, now known or hereafter invented,
`including photocopying, microfilming, and
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`
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`
`
`
`Library of Congress Cataloging-in-Publication Data
`
`Catalog record is available from the Library of Congress
`
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`http://www.taylorandfrancis.com
`
`0002
`
`

`
`Contents
`
`
`Foreword to the third edition
`
`Foreword to the second edition
`
`1
`
`Apologia to the first edition
`Symbols and abbreviations
`Introduction
`1.1 Early history
`1.2 Thin-film filters
`References
`
`2 Basic theory
`2.1 Maxwell’s equations and plane electromagnetic waves
`2.1.1
`The Poynting vector
`2.2 The simple boundary
`2.2.1 Normal incidence
`2.2.2 Oblique incidence
`2.2.3
`The optical admittance for oblique incidence
`2.2.4 Normal incidence in absorbing media
`2.2.5 Oblique incidence in absorbing media
`2.3 The reflectance of a thin film
`2.4 The reflectance of an assembly of thin films
`2.5 Reflectance, transmittance and absorptance
`2.6 Units
`2.7
`Summary of important results
`2.8
`Potential transmittance
`2.9 Quarter— and half-wave optical thicknesses
`2.10 A theorem on the transmittance of a thin-film assembly
`2.11 Admittance loci
`2.12 Electric field and losses in the admittance diagram
`2.13 The vector method
`2.14 Incoherent reflection at two or more surfaces
`2.15 Other techniques
`
`xiii
`
`xv
`
`xix
`xxiii
`1
`1
`5
`9
`
`12
`12
`17
`18
`20
`23
`27
`29
`34
`37
`40
`43
`46
`46
`50
`52
`53
`55
`60
`66
`67
`72
`
`0003
`
`

`
`viii
`
`Contents
`
`2.15.1 The Herpin index
`2.15.2 Alternative method of calculation
`2.15.3 Smith’s method of multilayer design
`2.15.4 The Smith chart
`2.15.5 Reflection circle diagrams
`References
`
`3 Antireflection coatings
`3.1 Antireflection coatings on high-index substrates
`3.1.1
`The single-layer antireflection coating
`3.1.2 Double-layer antireflection coatings
`3.1.3 Multilayer coatings
`3.2 Antireflection coatings on low-index substrates
`3.2.1
`The single-layer antireflection coating
`3.2.2
`Two-layer antireflection coatings
`3.2.3 Multilayer antireflection coatings
`3.3 Equivalent layers
`3.4 Antireflection coatings for two zeros
`3.5 Antireflection coatings for the visible and the infrared
`3.6
`Inhomogeneous layers
`3.7
`Further information
`References
`
`4 Neutral mirrors and beam splitters
`4.1 High—reflectance mirror coatings
`4.1.1 Metallic layers
`4.1.2
`Protection of metal films
`4.1.3 Overall system performance, boosted reflectance
`4.1.4 Reflecting coatings for the ultraviolet
`4.2 Neutral beam splitters
`4.2.1 Beam splitters using metallic layers
`4.2.2 Beam splitters using dielectric layers
`4.3 Neutral—density filters
`References
`
`5 Multilayer high-reflectance coatings
`5.1 The Fabry—Perot interferometer
`5.2 Multilayer dielectric coatings
`5.2.1
`All—dielectric multilayers with extended high-reflectance
`zones
`5.2.2 Coating uniformity requirements
`5.3 Losses
`References
`
`72
`73
`75
`77
`80
`85
`
`86
`87
`87
`92
`102
`108
`110
`111
`118
`135
`139
`144
`152
`156
`156
`
`158
`158
`158
`160
`164
`167
`169
`169
`172
`176
`177
`
`179
`179
`185
`
`193
`200
`204
`208
`
`0004
`
`

`
`
`
`Chapter 6
`
`Edge filters
`
`Filters in which the primary characteristic is an abrupt change between a region
`of rejection and a region of transmission are know as edge filters. Edge filters are
`divided into two main groups, longwave-pass and shortwave pass. The operation
`may depend on many different mechanisms and the construction may take a
`number of different forms. The following account is limited to thin-film edge
`filters. These rely for their operation on absorption or interference or both.
`
`6.1 Thin-film absorption filters
`
`A thin-film absorption filter consists of a thin film of material which has an
`absorption edge at the required wavelength and is usually longwave-pass in
`character. Semiconductors which exhibit a very rapid transition from opacity to
`transparency at the intrinsic edge are particularly useful in this respect, making
`excellent 1ongwave—pass filters. The only complication which usually exists is
`a reflection loss in the pass region due to the high refractive index of the film.
`Germanium, for example, with an edge at 1.65 am, has an index of 4.0, and,
`as the thickness of germanium necessary to achieve useful rejection will be at
`least several quarter—waves, there will be prominent interference fringes in the
`pass zone showing variations from substrate level, at the half-wave positions, to a
`reflectance of 68% (in the case of a glass substrate) at the quarter-wave position.
`The problem can be readily solved by placing antireflection coatings between the
`substrate and the germanium layer, and between the germanium layer and the
`
`
`
`Fig:
`4.25
`
`spec
`
`for
`
`of (
`the
`wav
`
`sulp
`laye
`
`incl
`
`silic
`
`givi
`
`also
`wav
`
`thicl
`
`air. Single quarter-wave antireflection coatings are usually quite adequate. For
`optimum matching the values required for the indices of the antireflecting layers
`are 2.46 between glass and germanium, and 2.0 between germanium and air. The
`index of zinc sulphide, 2.35, is sufficiently near to both values and, with it, the
`reflectance near the peak of the quarter-wave coatings will oscillate between
`
`
`
`(1 — (2.354)/(42 x 1.52))2 _ 1 %%
`
`1+ (2.354)/(42 x 1.52)
`
`” "
`
`210
`
`0005
`
`

`
`Interference edge filters
`
`21 1
`
`30 x M4 M4 ZnS
`M4
`CaF
`substrate ZnS PbTe
`
`
`
`
`
`0.9
`0.8
`0_7
`
`
`
`Transmittance(%) O 01
`
`
`
`2
`
`
`
`
`Monitoring wavelength 3.0 pm
`
`3
`
`4
`
`5
`
`Wavelength (urn)
`
`Figure 6.1. The measured characteristic of a lead telluride filter. The small dip at
`4.25 ,u.m is probably due to atmospheric CO2 causing a slight unbalance of the measuring
`
`spectrometer. (Courtesy of Sir Howard Grubb, Parsons & Co. Ltd.)
`
`for wavelengths where the germanium layer is equal to an integral odd number
`of quarter—waves, and 4%, that is the reflectance of the bare substrate, where
`the germanium layer is an integral number of half-waves thick (for at such a
`wavelength the germanium layer acts as an absentee layer and the two zinc
`sulphide layers combine also to form a ha1f—wave and, therefore, an absentee
`layer).
`Other materials used to form single—layer absorption filters in this way
`include cerium dioxide, giving an ultraviolet rejection—visible transmitting filter,
`silicon, giving a longwave-pass filter with an edge at 1 ,um, and lead telluride,
`giving a1ongwave—pass filter at 3.4 ,um.
`A practical lead telluride filter characteristic is shown in figure 6.1, which
`also gives the design. The two zinc sulphide layers were arranged to be quarter-
`waves at 3.0 nm. Better results would probably have been obtained if the
`thicknesses had been increased to quarter—waves at 4.5 nm.
`
`6.2
`
`Interference edge filters
`
`6.2.1 The quarter-wave stack
`
`The basic type of interference edge filter is the quarter-wave stack of the previous
`chapter. As was explained there,
`the principal characteristic of the optical
`transmission curve plotted as a function of wavelength is a series of high-
`reflection zones, i.e. low transmission, separated by regions of high transmission.
`The shape of the transmission curve of a quarter-wave stack is shown in figure 6.2.
`The particular combination of materials shown is useful in the infrared beyond
`2 ,um, but the curve is typical of any pair of materials having a reasonably high
`ratio of refractive indices.
`
`0006
`
`

`
`212
`
`Edge filters
`
`Transmittance
`
`4
`
`6
`5
`Wavelength (um)
`
`7
`
`8
`
`9
`
`10
`
`Figure 6.2. Computed characteristic of a 13-layer quarter—wave stack of germanium
`(index 4.0) and silicon monoxide (index 1.70) on a substrate of index 1.42. The reference
`wavelength, A0, is 4.0 mm.
`
`The system of figure 6.2 can be used either as a longwave-pass filter with
`an edge at 5.0 urn or a shortwave-pass filter with an edge at 3.3 ,um. These
`wavelengths can be altered at will by changing the monitoring wavelength.
`It sometimes happens that the width of the rejection zone is adequate for
`the particular application, as, for example, where light of a particularly narrow
`spectral region only is to be eliminated, or where the detector itself is insensitive
`to wavelength beyond the opposite edge of the rejection zone.
`In most cases,
`however, it is desirable to eliminate all wavelengths shorter than, or longer than,
`a particular value. The rejection zone, shown in figure 6.2, must somehow be
`extended. This is usually done by coupling the interference filter with one of the
`absorption type.
`Absorption filters usually have very high rejection in the stop region, but,
`as they depend on the fundamental optical properties of the basic materials, they
`are inflexible in character and the edge positions are fixed. Using interference
`and absorption filters together combines the best properties of both, the deep
`rejection of the absorption filter with the flexibility of the interference filter. The
`interference layers can be deposited on an absorption filter, which acts as the
`substrate, or the interference section can sometimes be made from material which
`itself has an absorption edge within the interference rejection zone. Within the
`absorption region the filter behaves in much the same way as the single layers of
`the previous section.
`Other methods of improving the width of the rejection zone will be dealt
`with shortly, but now we must turn our attention to the more difficult problem
`created by the magnitude of the ripple in transmission in the pass region. As the
`
`curv.
`
`WOU‘
`
`its a
`
`in lh
`
`the p
`
`6.2.2
`
`The
`
`of a
`
`of w
`
`imag
`in w
`mull
`
`be sl
`simi
`
`This
`
`equi
`can I
`
`forn'
`lhe<
`
`malt
`
`give
`
`(Wilt
`rcfrz
`
`0007
`
`

`
`Interference edge filters
`
`213
`
`curve of figure 6.2 shows, the ripple is severe and the performance of the filter
`would be very much improved if somehow the ripple could be reduced.
`Before we can reduce the ripple we must first investigate the reason for
`its appearance, and this is not an easy task, because of the complexity of the
`mathematics. A paper published by Epstein [1] in 1952 is of immense importance,
`in that it lays the foundation of a method which gives the necessary insight into
`the problem to enable the performance to be not only predicted but also improved.
`
`6.2.2 Symmetrical multilayers and the Herpin index
`
`The paper written by Epstein [l] in 1952 dealt with the mathematical equivalent
`of a symmetrical combination of films and a single layer, and was the beginning
`of what has become the most powerful design method to date for thin—film filters.
`Any thin—film combination is known as symmetrical if each half is a mirror
`image of the other half. The simplest example of this is a three—layer combination
`in which a central layer is sandwiched between to identical outer layers.
`If a
`multilayer can be split into a number of equal symmetrical periods, then it can
`be shown that it is equivalent in performance to a single layer having a thickness
`similar to that of the multilayer and an optical admittance that can be calculated.
`This is a most important result. Unfortunately, the accurate calculation of the
`equivalent optical admittance is rather involved, but the basic form of the result
`can be established relatively easily and used as a qualitative guide. Once the basic
`form of a filter has been established, computer techniques can be used to finalise
`
`the design.
`Consider first a symmetrical three-layer period pqp, made up of dielectric
`materials free from absorption. The characteristic matrix of the combination is
`given by
`
`M21 M22
`
`[M11 M12] :[ cosap
`X
`. cosrlp
`17]], sin 8,,
`
`inp sin 61,
`
`cos 8,,
`
`(isin81,)/n,,:|[ c0s8q
`(1s1n6,,)/17,,
`cos 6,,
`
`inq sin Sq
`
`cos 6,,
`
`(isin8q)/nq:l
`(6.1)
`
`(where we have used the more general optical admittance 17 rather than the
`refractive index 11). By performing the multiplication we find:
`
`.
`
`.
`
`2?
`
`"P
`
`n
`
`liq
`
`M1] = cos 25,, cos3,, — %(—i + l) s1n28,, Sln5,,
`M12 = -1-
`sin 28,,cos6,, + %<31 + -7713) cos 28,, sinaq
`+ §<% —
`sin5,]
`
`rip
`
`7712
`
`7761
`
`9
`
`P
`
`(6.2a)
`
`(6.219)
`
`0008
`
`

`
`214
`
`Edge filters
`
`M21=in,,[sin28,,cos6q +
`—
`— 21) sin84]
`
`Up
`
`7711
`
`+ 21) cos261,sin6,,
`
`"P
`
`7761
`
`and
`
`M22 = M11.
`
`It is this last relationship which permits the next step.
`Now, let
`
`and if we set
`
`M11 = COS ]/ = M22
`
`'
`_
`1 sin 1/
`E
`
`M12 =
`
`then, since M11M22 — M12M21 = 1
`
`M21 = iE sin y.
`
`(6.2c)
`
`(6.2d)
`
`(6.3)
`
`(6-4)
`
`(6.5)
`
`These quantities have exactly the same form as a single layer of phase
`thickness )1 and admittance E. The equations can be solved for y and E, choosing
`the particular value of y which is nearest to the total phase thickness of the period.
`3/ is then the equivalent phase thickness of the three—layer combination and E
`is the equivalent optical admittance, also known sometimes as the Herpin index.
`M11 does not equal M22 in an unsymmetrical arrangement and such a combination
`cannot, therefore, be replaced by a single layer.
`It can easily be shown that
`this result can be extended to cover any
`symmetrical period consisting of any number of layers. First the central three
`layers which, by definition, will form a symmetrical assembly on their own can
`be replaced by a single layer. This equivalent layer can then be taken along with
`the next layers on either side as a second symmetrical three-layer combination,
`which can, in its turn, be replaced by a single layer. The process can be repeated
`until all the layers have been replaced and a single equivalent layer found.
`The importance of this result lies both in the ease of interpretation (the
`properties of a single layer can be visualised much more readily than those of
`a multilayer) and in the ease with which the result for a single period may be
`extended to that for a multilayer consisting of many periods.
`If a multilayer is made up of, say, S identical symmetrical periods, each of
`which has an equivalent phase thickness 7/ and equivalent admittance E, then
`physical considerations show that the multilayer will be equivalent to a single
`layer of thickness Sy and admittance E. This result also follows because of an
`easily derived result:
`
`[ cosy
`
`iE siny
`
`isiny/E:|5=|: cosSy
`
`iE sin Sy
`
`cosy
`
`isinSy/E].
`
`cos Sy
`
`(6-6)
`
`It Shot
`
`for th1
`mathei
`
`changi
`mullilz
`Ir
`
`thfll [ht
`
`be solx
`
`Signifit
`as the»
`
`unity i
`matrix
`
`which
`
`stop b;
`imagir
`these I
`bands
`
`6.2.2.)
`
`Rcturr
`
`§Ir>p_|y ‘
`is sum
`index
`
`high-ii
`
`and
`
`These
`
`and
`
`rcspci
`
`
`
`0009
`
`

`
`Interference edge filters
`
`2 1 5
`
`It should be noted that the equivalent single layer is not an exact replacement
`for the symmetrical combination in every respect physically.
`It is merely a
`mathematical expression of the product of a number of matrices. The effect of
`changes in angle of incidence, for instance, cannot be estimated by converting the
`multilayer to a single layer in this way.
`In any practical case when the matrix elements are computed it will be found
`that there are regions where M11 < —1, i.e. cos y < -1. This expression cannot
`be solved for real y, and in this region 3/ and E are both imaginary. The physical
`significance of this was explained in the previous chapter, where it was shown that
`as the number of basic periods is increased the reflectance of a multilayer tends to
`unity in regions where |M11 + M22]/2 > 1, M11 and M22 being elements of the
`matrix of the basic period. In the present symmetrical case this is equivalent to
`
`|M11| = lM11| >1
`
`which therefore denotes a region of high reflectance, i.e. a stop band. Inside the
`stop band, the equivalent phase thickness and the equivalent admittance are both
`imaginary. Outside the stop band the phase thickness and admittance are real and
`these regions are known as pass regions or pass bands. The edges of the pass
`bands and stop bands are given by M 11 = —1.
`
`6.2.2.] Application of the Herpin index to the quarter-wave stack
`
`Returning for the moment to our quarter-wave stack, we see that it is possible to
`apply the above results directly if a simple alteration to the design is made. This
`is simply to add a pair of eighth-wave layers to the stack, one at each end. Low-
`index layers are required if the basic stack begins and ends with quarter-wave
`high-index layers and vice versa. The two possibilities are
`
`and
`
`H
`H
`—2—LHLHLH...HL~2——
`
`L
`L
`-HLHLHL...LH——.
`2
`2
`
`These arrangements we can replace immediately by
`
`H H
`H HH HH HH HH H
`-L———L——L——L——L—...——L—
`2 22 22 22 22 2
`2
`2
`
`and
`
`L
`L
`L LL LL LL LL L
`H
`—H——H———H~—— —...—~H—
`22 22 22 2
`22
`2
`2
`
`2
`
`respectively which can then be written as
`
`[%Lf’2—]S
`
`and
`
`[gays
`
`0010
`
`

`
`216
`
`Edge filters
`
`(H /2)L(H/2) and (L/2)H (L /2) being the basic periods in each case. The
`results in equations (6.l)—(6.6) can then be used to replace both the above stack
`by single layers making the performance in the pass bands and also the extent
`of the stop bands easily calculable. We shall examine first the width of the stop
`bands. As mentioned above, the edges of the stop bands are given by M 11 = — 1.
`Using equation (6.2a) this is equivalent to
`
`cosz Sqe —
`
`+ 23) sinz 843 = -1
`
`77q
`
`77p
`
`which is exactly the same expression as was obtained in the previous chapter for
`the width of the unaltered quarter-wave stack. There, 6 was replaced by (Jr/2)g,
`where g = A0 /A (or v/vo, where v is the wavenumber), and the edges of the stop
`band were defined by
`
`71'
`8e ._ 2 (1 i Ag).
`
`The width is therefore
`
`where, if 171, < nq,
`
`or,ifnq < 711,,
`
`2A — 2A A0
`g _
`2.
`
`Ag = — sin"1(nq W’)
`
`__
`
`774 + '71)
`
`2
`
`7’
`
`(6.7)
`
`2
`
`_
`
`(6.8)
`nq).
`Ag = — sin'1<np
`"P + 7761
`77
`These expressions are plotted in figure 5.7. The width of the stop band is therefore
`exactly the same regardless of whether the basic period is (H /2)L(H /2), or
`(L /2)H (L /2). Of course, it is possible to have other three-layer combinations
`where the width of the central layer is not equal to twice the thickness of the two
`outer layers, and some of the other possible arrangements will be examined, both
`in this chapter and the next, as they have some interesting properties, but, as far
`as the width of the stop band is concerned, it has been shown by Vera [2] that the
`maximum width for a three-layer symmetrical period is obtained when the central
`layer is a quarter-wave and the outer layers an eighth-wave each.
`Let us now turn our attention to the pass band; first the equivalent admittance
`and then the equivalent optical thickness. The expression for the equivalent
`admittance in the pass band is quite a complicated one. From equations (6.2b),
`(6.2c), (6.4) and (6.5)
`
`E = ., (_’‘Z_2.1_)”2
`_ + <n%[s1n2zS,,coséq + %(7]p/7]q + nq/np)cos26,, smaq — %(n,,/nq — nq/np)s1n6,,])
`
`M12
`
`.
`
`.
`
`.
`
`1/2
`
`sin28,, COS3q + %(T}p/7]q + nq/n,,)cos 28,, sin5q + %(np/7]q — nq/n,,)sin6q
`
`(6.9)
`
`0011
`
`

`
`18
`
`Interference edge filters
`
`217
`
`16
`
`14
`
`12
`
`10
`
`540 ]4w
`
`42oJ
`
`ml
`
`300-
`
`240 a‘
`
`180 4
`
`0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00
`
`
`
`
`
`
`Equivalentopticaladmittance(E) OM-bdim
`
`
`Equivalentphasethickness(7)
`
`This curve applies to both
`
`A L and H fl eriods
`2H2
`2L2p
`
`120 J60
`
`0
`
`0
`
`0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00
`
`g = V/V0 = KO/7».
`
`Figure 6.3. Equivalent optical admittance, E, and phase thickness, y, of a symmetrical
`period of zinc sulphide (n = 2.35) and cryolite (n = 1.35) at normal incidence.
`
`This is not a particularly easy expression to handle analytically, but
`evaluation is straightforward, either by computer or even a programmable
`calculator. Figure 6.3 shows the equivalent admittance and optical thickness
`of combinations of zinc sulphide and cryolite. The form of this curve is quite
`typical of such periods. Once the equivalent admittance and thickness have been
`evaluated, the calculation of the performance of the filter in the pass region, and
`its subsequent improvement, become much more straightforward. They are dealt
`with in greater detail later in this chapter. First we shall examine some of the
`properties of the expression for the equivalent optical admittance.
`We can normalise expression (6.9) by dividing both sides by 271,. E /17 I, is
`then solely a function of 81,, 6,1 and the ratio 27I, / 114. Next, we can make the
`further simplification, which we have not so far, that 28 P : 84. The expression
`
`0012
`
`

`
`218 ‘
`
`Edge filters
`
`for E/ 27 1, then becomes
`
`5 +({1+ %[p + (1/p)]}c0S5qSi“3q ~ %[p — (1/p)] Sin8q>1/2
`{1+%tp+<1/p>1}cosaqsina., + %[p — (1//>)]sin6q
`
`(610)
`'
`
`wherep = np/nq.
`It is now easy to see that the following relationships are true. We write
`(E /27p) (,0,
`(Sq) to indicate that it is a function of the variables p and 8g.
`
`E
`
`'51-J-(,0,
`
`E
`
`1
`
`71’ -651) ——
`84> — .
`
`1
`
`1
`
`(6.12)
`
`These relationships are, in fact, true for all symmetrical periods, even ones which
`involve inhomogeneous layers, and general statements and proofs of these and
`other theorems are given by Thelen [3].
`Thelen has shown how these relationships may be used to reduce the labour
`in calculating the equivalent admittance over a wide range. Figure 6.4 shows
`a set of curves giving the equivalent admittance for various values of the ratio
`of admittances. The vertical scale has been made logarithmic which has the
`advantage of making the various sections of the curve repetitions of the first
`section. This follows directly from the relationships (6.11) and (6.12). The values
`of the ratios of optical admittances which have been used are all greater than unity.
`Values less than unity can be derived from the plotted curves using relation (6.12).
`Again the logarithmic scale means that it is necessary only to reorient the curve
`for 17,,/nq = k to give that for 17,,/nq = l/ k. All the information necessary to
`plot the curves is therefore given in the enlarged version of the first section of
`figure 6.4 which is reproduced in figure 6.5. Figures 6.4 and 6.5 are both taken
`from the paper by Thelen [3].
`It is also useful to note the limiting values of E 2
`
`E tends to (r;,,m,)1/2
`and
`
`as
`
`(Sq tends to zero
`
`(6.13)
`
`E tends to 7]p(77])/77q)]/2
`
`as
`
`(Sq tends to 71'.
`
`The equivalent phase thickness of the period is given by (6.201) and (6.3) as
`
`y = cos‘1 cos26,, cos64 — —(:7£ + n—q) sin 28,, sin (Sq
`
`1
`
`2
`
`nq W
`
`.
`
`(6.14)
`
`This expression for y is multivalued, and the value chosen is that nearest to
`28,, + 8,1, the actual sum of the individual phase thicknesses, which is the most
`easily interpreted value.
`It is clear from the expression for y that it does not
`
`is a 1
`band
`
`mat
`
`phu:
`from
`
`0013
`
`

`
`Interference edge filters
`
`219
`
`1 0
`
`
`
`Equivalentindex
`
`0.1
`
`0
`
`1
`
`2
`
`3
`
`4
`
`5
`
`6
`
`g = V/V0 = ho/7»
`
`Figure 6.4. Equivalent admittance for the system (L /2)H (L /2). n;_ = 1.00 and nH/nL
`
`is a parameter with values 1.23, 1.50, 1.75, 2.0, 2.5, 3.0. The curves with the wider stop
`
`bands have the higher 11;; / n L values. (After Thelen [3].)
`
`10
`
`8 6
`
`l\)-A
`
`
`
`Equivalentindex
`
`0
`
`0.2
`
`0.4
`
`I
`
`0.6
`
`0.8
`
`1.0
`
`g = V/V0 = KO/A
`
`Figure 6.5. Enlarged first part of figure 6.4. (After Thelen [3].)
`
`matter whether the ratio of the admittances is greater or less than unity. The
`phase thickness for ,0 is the same as that for 1 / p. Figure 6.6, which is also taken
`from The1en’s paper, shows the phase thickness of the combinations in figures 6.4
`
`0014
`
`

`
`220
`
`Edge filters
`
`/
`
`J’;
`
`1.0
`
`I5 O-8
`(D
`.C
`E)C
`
`2 %
`
`$ 0.6
`E
`§
`C
`4‘
`E 0.4
`
`.§
`:5
`L3’ 0.2
`
`0
`
`0
`
`0.2
`
`0.4
`
`0.6
`
`0.3
`
`g=v/v0=7t0/7»
`
`Figure 6.6. Equivalent thickness of the system described in figure 6.4. (After Thelen [3].)
`
`and 6.5. Because of the obvious symmetries, all the information necessary for
`the complete curve of the equivalent phase thickness is given in this diagram.
`The equivalent thickness departs significantly from the true thickness only near
`the edge of the high—reflectance zone. At any other point in the pass bands the
`equivalent phase thickness is almost exactly equal to the actual phase thickness of
`the combination.
`
`6.2.2.2 Application of the Herpin index to multilayers of other than quarter-
`waves
`
`All the curves shown so far are for |eighth—wave| quarter-wave |eighth-wavel
`periods. If the relative thicknesses of the layers are varied from this arrangement
`then the equivalent admittance is altered. It has already been mentioned that the
`reflectance zones for a combination other than the above must be narrower. Some
`idea of the way in which the equivalent admittance alters can be obtained from
`
`180°
`
`the Vail
`
`3
`3
`90° §
`35-.
`G)
`3
`,5-
`
`.5
`I3
`
`0°
`
`Now si
`
`Reamu
`
`This re
`
`througl
`This re
`where
`
`replace
`
`El
`admitlz
`
`sulphic
`U
`
`cquiva
`St
`
`value (
`
`26,, ->
`
`E—— —>
`
`7],,
`
`just c0
`
`0015
`
`

`
`the value as g —> 0. Let 26,,/64 : 1!/. Then, from equation (6.9)
`
`Interference edge filters
`
`221
`
`_
`s1n28
`
`17
`
`’7q
`
`r7
`
`'71’
`
`28‘) cos6q+%(-£+—q)cos28,,—%(l——l
`_
`E: +7712)‘:
`Sm q
`sin 26,,
`1
`27
`77
`17
`17
`x|:
`.
`cos8q+§(-I3+—q)cos26p+%(—E—-1)]
`Sm ‘Sq
`77:1
`7717
`7711
`Up
`
`.
`
`77
`
`'74
`
`77
`
`7729
`
`l/2
`
`4/2
`
`.
`
`(6.15)
`
`Now sin26p/ sinsq —> «p as g —> 0, since (Sq —> 0,81, ——> 0, i.e.
`
`E_,,,p,7,+%(’7_P+fl1)_%(@_"_q)
`
`7761
`
`7712
`
`nq
`
`77p
`
`X ¢,+%('7_P+fl1)+%(Q11_3‘L)
`
`'74
`
`7717
`
`'74
`
`"17
`
`1/2
`
`-1/2
`
`_
`
`Rearranging this we obtain
`
`5 _,
`
`77p
`
`ifl + (771)/77:/)
`
`W
`
`This result shows that, for small g, it is possible to vary the equivalent admittance
`throughout the range of values between 17 p and nq but not outside that range.
`This result has already been referred to in the chapter on antireflection coatings,
`where it was shown how to use the concept of equivalent admittance to create
`replacements for layers having indices difficult to reproduce.
`Epstein [1] has considered in more detail
`the variation of equivalent
`admittance by altering the thickness ratio and gives tables of results of zinc
`sulphide/cryolite multilayers.
`Ufford and Baumeister [4] give sets of curves which assist in the use of
`equivalent admittance in a wide range of design problems.
`Some results which are at first sight rather surprising are obtained when the
`value of the equivalent admittance around g = 2 is investigated. As g ~+ 2,
`28,, —> It and 871 —+ 7T so that, from equation (6.15)
`
` E (-1 ~ %[(17p/nq) + (nq/7712)] ~ %[(np/nq) — <2»,/n,,>1>‘/2: (Q11)1?
`
`E1: —>
`"1 “ 'l2‘[(77p/U4) + (’7q/7712)] + %l’.(77p/77(1) ‘“ (nq/7717)]
`771617)
`This is quite a straightforward result. Now let 261,/Sq 2 1p, as in the case
`just considered where g —> 0. Let g —> 2 so that
`
`28,, + (Sq ——> 27:.
`
`0016
`
`

`
`222
`
`Edge filters
`
`(This is really how, in this case, we define g = A0/A by defining AD as that
`wavelength which makes 2(Sp + 6q = IF.)
`We have, as g —> 2
`
`——> cos(2rr — (Sq) = cos (Sq
`cos 28,,
`sin 25,, —> — sin(27r — 5,) = — sin (sq
`
`and (Sq —+ 2:1/(1+ tlr) so that
`
`7719
`
`E— —> —sin(Sq cos(Sq +
`X l:—- sin(Sq cos(Sq +
`
`7h]
`
`77])
`
`+ -711) cos(Sq sin(Sq —
`+ 171) cos(Sq sin(Sq
`
`774
`
`"P
`
`x —cos5q[1—%(3’-3+"—“)]+%("—”-"—")
`
`7741
`
`771)
`
`77¢]
`
`771)
`
`1/2
`
`— 21-) sin6q
`
`-1/2
`
`(6.18)
`
`where cos (Sq = cos[27t/(1 + 1/1)].
`Whatever the value of 1.//, the quantities within the square root brackets have
`opposite signs, which means that the equivalent admittance is imaginary. Even
`as tp —>
`1, where one would expect the limit to coincide with the result in
`equation (6.17), the admittance is still imaginary.
`An
`follows.
`as
`The
`explanation of
`this
`apparent paradox is
`imaginary equivalent admittance, as we have seen, indicates a zone of high
`reflectance. Consider first the ideal eighth-wave|quarter—wave|eighth—wave stack
`of equation (6.17). At the wavelength corresponding to g = 2, the straightforward
`theory predicts that the reflectance of the substrate shall not be altered by the
`presence of the multilayer, because each period of the multilayer is acting as a
`full wave of real admittance and is therefore an absentee layer. Looking more
`closely at the structure of the multilayer we can see that this can also be explained
`by the fact the all the individual layers are a half-wavelength thick. If the ratio
`of the thicknesses is altered, the layers are no longer a half-wavelength thick and
`cannot act as absentees. In fact, the theory of the above result shows that a zone
`
`of high reflectance occurs.
`The transmission of a shortwave-pass filter at the wavelength corresponding
`to g = 2 is therefore very sensitive to errors in the relative thicknesses of the
`layers. Even a small error leads to a peak of reflection. The width of this spurious
`
`high-re
`of 3 DH
`is quilt
`someu,
`
`6.2.3.]
`
`The Ira
`parame
`
`edges 0
`LL‘
`
`Charact
`
`Al the c
`or 00 (it
`
`so that
`
`at the st
`
`and, dc;
`
`for the
`It‘
`
`then lllt
`
`equaho
`
`0017
`
`

`
`In terference edge filters
`
`223
`
`high-reflectance zone is quite narrow if the error is small. Thus the appearance
`of a pronounced narrow dip in the transmission curve of a shortwave-pass filter
`is quite a common feature and is difficult to eliminate. The dip is referred to
`sometimes as a ‘half—wave hole’.
`
`6.2.3 Performance calculations
`
`We are now in a position to make some performance calculations.
`
`6.2.3.1 Transmission at the edge of a stop band
`
`The transmission in the high—reflectance region, or stop band, is an important
`parameter of the filter. Thelen [3] gives a useful method for calculating this at the
`edges of the band. His analysis is as follows.
`Let
`the multilayer be made up of S fundamental periods so that
`characteristic matrix of the multilayer is
`
`the
`
`[M]5=
`
`cosy
`iE sin y
`
`(isiny)/E S:
`cos y
`
`cosSy
`iE sin Sy
`
`(isinSy)/E
`cos Sy
`
`'
`
`At the edges of the stop band we know that cos S y ——> 1, sin Sy —> 0, and E —> 0
`or 00 depending on the particular combination of layers. Now,
`
`'S
`Sm V —>S
`sin 3/
`
`as
`
`siny——>O
`
`so that the matrix tends to
`
`1
`iESsiny
`
`1
`(iSsiny)/E _
`l
`_ SM21
`
`SM12
`l
`
`at the stop band limits. Either M12 or M21 will also tend to zero because
`
`M11M22 - M12M21 =1
`
`and, depending on which tends to zero, we have either
`
`1
`0
`
`SM12
`l
`
`or
`
`1
`SM21
`
`O
`l
`
`for the matrix.
`If no is the admittance of the incident medium and nm of the substrate,
`then the transmittance of the multilayer at the edge of the stop band is given by
`
`equation (2.67):
`
`T :
`
`47}077m
`(7103 + C)(77oB + C)*
`
`it
`
`ifi
`lE
`.1
`
`0018
`
`

`
`224
`
`Edge filters
`
`where
`
`B _ 1
`
`SM12
`
`[cHo 1 Has]
`
`1
`
`or
`
`i.e.
`
`1
`
`iSM21
`
`0
`
`1
`
`1:|i77mi
`
`B = 1+S77mM12
`
`Um
`
`Ci
`
`‘f “H
`
`_
`
`f
`
`.
`
`1
`
`Al =:0
`
`12
`
`or
`
`1
`
`Um‘+'SA421
`
`]
`
`so that, if there is no absorption,
`
`477077m
`T= W“ “F”
`m
`In
`
`“'19)
`
`Or
`
`4770 77m
`T = "T"'—*‘2'
`(770 + 77m) + (3|M21l)
`
`when
`
`M12 = 0
`
`(6.20)
`
`(since M12 and M21 are imaginary in the absence of absorption). For M 12 or M21
`to be zero requires that
`
`sin28,cos8 +1 @+n—q cos28 sin5 =:}:l E-771 sin8.
`1
`4
`2
`P
`61
`2
`4
`7751
`771)
`7'/q
`77])
`
`If M12 is zero we can deduce that
`
`|M21| =
`
`
`
`np(n—p — -751-) sin6q
`
`7/11
`
`77:1
`
`or, if M21 is zero, that
`
`|M12| =
`
`—l—<@ — 29-) sinéq
`
`Up
`
`Wq
`
`Up
`
`
`
`.
`
`(6.21)
`
`(6.22)
`
`At the limits of the high-reflectance zone we have already seen that
`
`i.e.
`
`2
`
`cos26 = (nq — 771’)
`
`"q ‘i"‘ 77p
`
`sin26 =1—cos28 =
`
`477p77q
`(nq + 77p)2
`
`Substit
`
`T(
`
`express
`
`If
`
`If
`
`6.2.3.2
`
`For the
`of the I
`
`multila
`wave 12
`
`reprcse
`
`which i
`
`If there
`centre <
`
`t i'7p/
`
`I!
`
`Let 17,1.
`
`0019
`
`

`
`Substituting this in the expressions (6.21) and (6.22) for [M 21] and |M12| we find
`
`Interference edge filters
`
`225
`
`|M21|2 ‘
`|M12l2 =
`
`4
`
`2
`
`_
`
`‘I
`
`4(
`
`—
`
`>7-
`
`"fine
`
`for
`for
`
`M12 = 0
`M21 = 0.
`
`(6.23)
`(6.24)
`
`To give the transmittance at the edges of the high-reflectance zone, these
`expressions should be used in equations (6.19) and (6.20) according to the rule:
`
`If E, the equivalent admittance, is zero, then M21 is zero.
`If E, the equivalent admittance, is oo, then M 12 is zero.
`
`6.2.3.2 Transmission in the centre of a stop band
`
`For the simple quarter—wave stack an expression for transmittance at the centre
`of the high-reflectance zone has already been given in chapter 5. For the present
`multilayer, the t