Estimating the number of layers required and
`other properties of blocker and dichroic optical
`thin films
`
`Ronald R. Willey
`
`Empirically derived formulas are given that allow the thin-film designer to estimate in advance the
`number of layers needed to meet various thin-film optical performance requirements. The estimation of
`peak optical density and width of higher-order blocker bands and their suppression are also discussed.
`© 1996 Optical Society of America
`Key words: Thin-film properties, thin-film design, estimating thin films, higher-order reflectance
`bands.
`
`these designs to be able to estimate how many layers
`will be required for attaining the desired reflection or
`blocking and how wide the blocked band will be.
`The optical density 关OD ⫽ log共1兾transmittance兲兴 in-
`creases almost linearly with the number of layers in
`a stack. The width of the blocking band increases
`with the ratio of the indices of the high- and low-index
`materials in the stack. The relative width of the
`blocking band is less with higher orders of the
`reflection-band central wavelength. We can use all
`these facts to estimate how many pairs of a given
`material combination will be required for achieving a
`given result.
`
`Introduction
`1.
`Before a coating is designed, it is helpful to have some
`idea of whether the goal of the design is achievable.
`The ability to estimate performance limits can avoid
`fruitless design efforts and avoid the neglect of poten-
`tial performance gains or simplifications.
`In the
`case of antireflection coatings, a broad range of em-
`pirical data has been collected and reported1 on op-
`timized designs to provide a formula to estimate what
`can be expected in typical cases. Dobrowolski et al.2
`have recently reported an extension of Ref. 1. The
`thrust of this paper is to aid in the estimation of the
`number of layers and other properties of dichroic,
`bandpass, and blocking filters.
`
`2. Bandpass and Blocker Coatings
`Bandpass, long-wave pass, and short-wave pass, fil-
`ters can be made by the proper positioning of stacks
`of layers that have a quarter-wave optical thickness
`共QWOT兲 to block or reflect the unwanted wave-
`lengths. Thelen3 appears to have been the first to
`discuss minus filters, and he references related work
`by Young,4 wherein the blocked band is in the middle
`of two passbands, one on each side. Dobrowolski5
`applied them in detail, and Thelen discusses them in
`his book.6
`It is helpful when working with any of
`
`The author is with LexaLite Scientific Center, P.O. Box 498,
`Charlevoix, Michigan 49720.
`Received 8 November 1995; revised manuscript received 3 April
`1996.
`0003-6935兾96兾254982-05$10.00兾0
`© 1996 Optical Society of America
`
`4982
`
`APPLIED OPTICS 兾 Vol. 35, No. 25 兾 1 September 1996
`
`A. Estimating the Width of a Blocking Band
`Macleod7 gives Eq. 共1兲 and its derivation for the es-
`timated half-width 关2⌬g from Eq. 共1兲兴 of the blocking
`band in the frequency-related units of g, which equals
`␭0兾␭, where ␭0 is the wavelength at which the layers
`of the stack are of one QWOT:
`
`2 ␲
`
`⌬g ⫽
`
`nH ⫹ nL
`For materials such as TiO2 and SiO2, this gives a
`half-width of 0.138 for nH ⫽ 2.26 and nL ⫽ 1.46.
`If
`such a stack were centered at 539 nm, the edges of
`the reflecting band would be at approximately 474
`and 626 nm.
`In the infrared, where Ge and ThF4
`can be used with nH ⫽ 4.0 and nL ⫽ 1.35, one might
`get ⌬g ⫽ 0.330. With a stack centered at 10 ␮m, this
`would imply band edges at approximately 6.7 and
`13.3 ␮m.
`The spectral distributions tend to be symmetrical
`
`arcsin冉nH ⫺ nL
`
`冊
`
`(1)
`
`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)12(cid:3)
`
`0001
`
`

`
`In the typical case, using the high index as the first
`layer gives a lower OD than starting with the low
`index first does. Actually, it is because the last layer
`of the stack is of low index and acts as an antireflec-
`tion coating. The first layer of low index next to the
`substrate can usually be eliminated if it is not much
`different from the substrate index. Therefore the
`greatest ODP for the fewest layers comes from start-
`ing and finishing the stack with the high-index ma-
`terial.
`The change in OD with the addition of each new
`pair given in approximation 共3兲 can be derived from
`approximation 共2兲:
`
`⌬OD ⬇ 2 log冉nH
`
`nL
`
`冊 .
`
`(3)
`
`This is sufficiently correct as long as there are more
`than a few layers.
`Approximation 共3兲 combined with the integral over
`the bandwidth given above allows us to estimate the
`OD bandwidth product 共ODBWP兲 of each additional
`pair. This is approximated by
`
`ODBWP ⬇ 1.57⌬g⌬OD
`
`⬇ 2 log冉nH
`
`nL
`
`冊arcsin冉nH ⫺ nL
`
`nH ⫹ nL
`
`冊 .
`
`(4)
`
`As far as I am aware, the ODBWP has not been
`proposed before as an estimating tool.
`
`C. Estimating the Number of Layers and Thickness
`Needed
`As an example of the application of approximation
`共4兲, let us take the case of a requirement for a 99%
`reflector from 400 to 700 nm with nH ⫽ 2.25 and nL
`⫽ 1.45. How many pairs would we estimate are
`required for such a design? Such a design would
`probably consist of gradually increasing or decreas-
`ing the layer thicknesses in the stack to give smooth
`coverage over the reflection band. The real issue is
`how much reflection needs to be generated by the
`layer pairs to cover the band. We can work out
`that this band has a ⌬g of 0.273 about a ␭0 of 509
`nm, and the OD over the band must be 2.0 共1%
`transmittance兲. This gives a total ODBWP of 2 ⫻
`0.273 ⫻ 2.0 or 1.092 required. By using approxi-
`mation 共4兲, we find that the ODBWP per layer pair
`would be ⬃0.0832. Dividing this into the 1.092
`required gives us the estimate that 13.125 pairs
`would be required, or ⬃26 layers. We also know
`that each of these layers would average approxi-
`mately one QWOT at 509 nm. Dividing the optical
`thicknesses for a QWOT of 0.12725 ␮m by the in-
`dices of the high and low layers and multiplying by
`13 layers of each index, we get an estimated phys-
`ical thickness of 1.876 ␮m. Note that estimation of
`the physical thickness required is expected to be
`applicable only when the designs are nearly
`quarter-wave stacks. The estimation of the num-
`ber of layers to produce a given ODBWP as de-
`
`1 September 1996 兾 Vol. 35, No. 25 兾 APPLIED OPTICS
`
`4983
`
`Fig. 1. Example with different numbers of layer pairs with indi-
`ces of 2.3 and 1.46 plotted on a linear frequency scale that also
`illustrates that the peak OD increases almost linearly with addi-
`tional layer pairs after the first few pairs. The values calculated
`from the approximation, cos0.5共1.315␦g兾⌬g兲 are plotted on the
`right-hand side of the 9 PAIRS curve.
`
`when plotted versus frequency or g values. Figure 1
`is an example with different numbers of layer pairs of
`2.3 and 1.46 indices plotted in linear frequency or
`wave numbers. The ⌬g predicted by Eq. 共1兲 would
`be 0.1434, but the measured width is wider than this
`for small numbers of layer pairs. Equation 共1兲 pre-
`dicts a number for the width based on a high number
`of pairs. There is an approximation of a pivot point
`at approximately the OD ⫽ 0.7 or 80% reflectance
`level. As the number of layer pairs is increased, this
`point does not change very much, but the slope at the
`point gets progressively steeper. Figure 1 also illus-
`trates how linearly the peak OD increases with ad-
`ditional layer pairs after the first few pairs.
`Also note that the shape of the OD curve is approx-
`imately cos0.5共␲␦g兾2⌬g兲, where ␦g is the distance
`from the QWOT central wavelength in g units.
`Points calculated with this formula are plotted on the
`right-hand side in Fig. 1. However, note that the
`␲兾2 is not precise and for the example of 9 pairs is
`actually 1.315 for best fit. This approximation is
`also valid only up to ␦g ⫽ ⌬g, of course. The inte-
`grated area under this curve from ⫺⌬g to ⫹⌬g is
`approximately 1.57⌬g. This times the OD added at
`the peak by each new pair will give an estimate of the
`OD times bandwidth contribution of each pair.
`
`B. Estimating the Optical Density of a Blocking Band
`The optical density at the maximum point of the
`QWOT stack is given in approximation 共2兲, where p is
`the number of layer pairs in the stack. This will
`depend on whether the stack starts with a high- or a
`low-index layer, but as long as p is more than a few
`pairs, it gives a good approximation of the average
`ODP at the peak:
`
`ODP ⬇ 2 log1兾2冋冉nH
`
`nL
`
`冊p
`
`⫹ 冉nL
`
`nH
`
`冊p册 .
`
`(2)
`
`0002
`
`

`
`Fig. 2. Higher-order reflectance bands for one QWOT each of
`high- and low-index material per pair 共equal thicknesses兲 showing
`that the block band repeats at each odd multiple of a QWOT and
`has an equal width of the band in ⌬g. The first cycle of the curve
`generated by approximation 共5兲 for this case is superimposed on
`the plot.
`
`Fig. 3. Reflectance bands with a 3:1 ratio between the overall
`thickness of the layer pairs to the thinnest layer of the pair, which
`adds the second and fourth harmonics but suppresses the third and
`multiples of it, such as the sixth, etc. The first cycle of the curve
`generated by approximation 共5兲 for this case is superimposed on
`the plot.
`
`scribed above, on the other hand, is not particularly
`dependent on the thickness of the layers involved.
`
`D. Estimating More Complex Coatings
`If the higher-order reflection bands do not come into
`play, it may be practical to divide the spectral band to
`be covered into subsections and just consider the sum
`of the layers needed to meet the ODBWP of each
`subsection. This has been found to be practical for
`estimation purposes before a design is started. An-
`gles other than normal incidence will probably add to
`the number of layers required.
`There may be more complex coatings to be esti-
`mated when there are multiple bands to be blocked.
`In such cases, it would be logical to look first to see if
`higher-order reflectances of the QWOT stack can be
`useful to the requirements.
`Macleod7 also illustrates, as in Fig. 2, that, for one
`QWOT each of high- and low-index materials per pair
`共equal thicknesses兲, the reflectance or block band re-
`peats at each odd multiple of a QWOT. The width of
`each of these block bands is the same in ⌬g. This
`then leads to the fact that the third-harmonic fre-
`quency will have 1兾3 the relative width at that wave-
`length as at the fundamental QWOT wavelength.
`Similarly, the fifth harmonic will be 1兾5 as wide.
`This then allows the creation of narrower bands when
`needed, but it is at the expense of 3 or 5 times as thick
`a stack for a given block band wavelength. Using
`high- and low-index materials that are closer to each
`other can accomplish the same thing with thinner
`stacks if the appropriate materials are usable. The
`free width between block bands also needs to be con-
`sidered.
`If any of these higher-order reflection bands con-
`tribute to the reflection needed, they do not add to the
`layer count because they already exist from some
`other part of the coating.
`If, however, a high-order
`reflection band is in a place where transmission is
`needed, the design would have to be changed to sup-
`
`4984
`
`APPLIED OPTICS 兾 Vol. 35, No. 25 兾 1 September 1996
`
`press that band. Baumeister8 showed how different
`bands can be suppressed by using something other
`than a unity ratio between the thicknesses of the
`high- and low-index layers. For example, a 3:1 ratio
`between the overall thickness of the pair to the thin-
`nest layer will add the second and fourth harmonics
`but suppress the third, sixth, etc., as seen in Fig. 3.
`A 4:1 ratio will add the second but not the fourth, etc.,
`as seen in Fig. 4.
`If we call the ratio A:1, it can be seen that the Ath
`harmonic of g0 has a zero value and that the peaks of
`the harmonics have an envelope that is approxi-
`mately a sine function of g from 0 to ␲. An empirical
`fit to the data yields
`
`
`
`ODN ⬇冏ODE sin1.2冉␲Ng冊冏 ,
`
`A
`Here ODN is the OD of the peak of the Nth harmonic
`block band, and ODE is the OD of the peak achieved
`
`N ⫽ 1, 2, . . . .
`
`(5)
`
`Fig. 4. Reflectance bands as seen in Fig. 3 but with a 4:1 ratio,
`which also adds the second harmonic but suppresses the fourth,
`eighth, etc. The first cycle of the curve generated by approxima-
`tion 共5兲 for this case is superimposed on the plot.
`
`0003
`
`

`
`the above approach should be satisfactory. How-
`ever, it is also of great interest to know how many
`layers are needed to achieve a certain edge slope
`between the pass and block bands. This is often the
`determining factor, rather than the OD, in such fil-
`ters.
`The steepness of the side of an edge filter is in
`inverse proportion to the number of layers or pairs.
`The spectral distance from the high to the low trans-
`mittance region is usually the important factor for
`the designer. This might be from 80% to 20% T
`共approximately 0.1 to 0.7 OD兲 or some other choice of
`limits.
`If we call the spectral distance dg and the
`peak density at the QWOT wavelength ODP, the ef-
`fect of steepness may be approximated by the empir-
`ically developed approximation 共6兲:
`dg ⬇ 2
`3
`ODP
`We know the ⌬g from Eq. 共1兲 and the ODP from
`approximation 共2兲. The 2兾3 factor would need to be
`changed if the specific OD range from high to low
`transmittance or reflectance were changed from the
`80% to 20% T used above. Because we know that
`the peak OD is directly proportional to the number of
`layers in the stack, the dg will be inversely propor-
`tional to the number of layers 共to some power兲. This
`shows how adding layers for steepness has a strong
`effect at a low total number of layers, but a weak
`effect if there are already many layers.
`
`(6)
`
`
`
`冉 ⌬g1.74冊 .
`
`4. Summary
`It has been shown that the ODBWP can be a useful
`tool for estimating the number of layers required, and
`a formula 关approximation 共4兲兴 has been given for its
`application. A formula 关approximation 共5兲兴 has been
`developed to estimate the peak OD of higher-order
`reflection bands of layer stacks of any constant and
`repetitive ratio between the thickness of high- and
`low-index layers. An empirical formula 关approxima-
`tion 共6兲兴 has been given for the estimation of the edge
`steepness between the passbands and block bands of
`a dichroic filter. A new notation, A, different from
`what has been used by Baumeister8 and others up to
`this time, has been employed for the ratio of total
`thickness of a high- and low-index pair of layers to the
`thickness of the thinnest layer of the pair. This is
`somewhat more convenient for the application of ap-
`proximation 共5兲 and indicates where the nodes of the
`harmonic reflectance bands will be, at integral mul-
`tiples of A.
`It has been shown that there is quite a bit that
`can be estimated about most coating designs even
`before the design is started. The estimating pro-
`cess also provides guidance for the design process.
`It might be said that the estimating process is the
`first-order design as in lens design, and computer
`optimization is the rigorous completion of the de-
`sign process. The new material presented here
`has been empirically derived for the aid of optical
`thin-film engineers. More rigorous mathematical
`
`1 September 1996 兾 Vol. 35, No. 25 兾 APPLIED OPTICS
`
`4985
`
`Illustration that shows that approximation 共5兲 gives the
`Fig. 5.
`correct results for an A of 4.75, where the reflectance goes to zero
`at g ⫽ 4.75. The first cycle of the curve generated by approxima-
`tion 共5兲 for this case is superimposed on the plot.
`
`by an equal-thickness pair stack. This is illustrated
`in Figs. 2–6.
`It is interesting to note that approximation 共5兲
`gives the correct results even for noninteger values
`of A, such as 4.75 and even 1.5, as seen in Figs. 5
`and 6.
`These properties can be useful in designing block-
`ers for laser lines with doubled, tripled, quadrupled,
`etc., harmonics. These series of figures suggest
`ways to adjust the relative blocking in the harmon-
`ics by the choice of the A value. Other applications
`for these harmonics may appear in the future in
`areas such as fiber-optic communications filtering,
`etc.
`
`3. Dichroic Reflection Coatings
`The estimation of general coating spectral shapes
`such as color correction filters may be reasonably
`approximated by the above methods in most cases.
`In the case of dichroic filters for color separation, etc.,
`
`Illustration that shows that approximation 共5兲 gives the
`Fig. 6.
`correct results even for a smaller value of A such as 1.5, where the
`reflectance goes to zero at g ⫽ 1.5. The first cycle of the curve
`generated by approximation 共5兲 for this case is superimposed on
`the plot.
`
`0004
`
`

`
`derivations and proofs are left for future investiga-
`tors.
`
`References
`1. R. R. Willey, “Predicting achievable design performance of broad-
`band antireflection coatings,” Appl. Opt. 32, 5447–5451 共1993兲.
`2. J. A. Dobrowolski, A. V. Tikhonravov, M. K. Trubetskov, B. T.
`Sullivan, and P. G. Verly, “Optimal single-band normal-incidence
`antireflection coatings,” Appl. Opt. 35, 644–658 共1996兲.
`3. A. Thelen, “Design of optical minus filters,” J. Opt. Soc. Am. 61,
`365–369 共1971兲.
`
`4. L. Young, “Multilayer interference filters with narrow stop
`bands,” Appl. Opt. 6, 297–315 共1967兲.
`5. J. A. Dobrowolski, “Subtractive method of optical thin-film in-
`terference filter design,” Appl. Opt. 12, 1885–1893 共1973兲.
`6. A. Thelen, Design of Optical Interference Coatings 共McGraw-
`Hill, New York, 1988兲, Chap. 7.
`7. H. A. Macleod, Thin Film Optical Filters, 2nd ed. 共MacMillan,
`New York, 1986兲, p. 171.
`8. P. Baumeister, Military Standardization Handbook, Optical
`Design, MIL-HDBK-141 共Defense Supply Agency, Washington,
`D.C., 1962兲, Chap. 20, p. 45.
`
`4986
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`0005