© 2001 OSA/OIC 2001
`
`Fourier Transform approach for the estimation of optical
`thin film thickness.
`
`P.G. Verly
`Institute for Microstructural Sciences, National Research Council of Canada, Ottawa ON, KIA 0R6, Canada
`Pierre.Verly@NRC.CA
`
`Abstract: A parallel is established between an existing empirical procedure for the estimation of
`thin film thickness by means of optical-density-bandwidth products (ODBWP), and new results
`derived from a Fourier Transform thin film synthesis approach.
`©2001 Optical Society of America
`OCIS Codes: (310.0310) Thin films; (310.6860) Thin films, optical properties; (230.4170) Multilayers
`
`An interesting idea was proposed in Ref.[1] for the thickness estimation of blocker and dichroic optical coatings by
`means of optical-density-bandwidth products (ODBWP). Empirical formulas were derived from the observation
`that, when the transmittance T of a quarter-wave (QW) stack of reference wavelength λ0 is plotted on a logarithmic
`scale versus normalized frequency
`
`(1)
`
`,
`
`λλ0
`
`=g
`
`the area under the curve scales almost linearly with the number of layers. The number of layer pairs in the stack is
`given approximately by the total ODBWP in the reflectance band divided by the contribution a layer pair, namely
`σ
`MAX
`∫
`σ
`MIN
`
`OD
`σ
`
`σ
`
`d
`
`H
`
`H
`
`
`
`
`
`−
`1
`
`sin
`
`
`
`
`
`LH
`nn
`
`
`
`
`log2
`
`≈
`
`N
`
`(2)
`
`,
`
`
`
`
`
`L
`
`L
`
`+−
`
`n
`n
`n
`n
`where σ=1/λis the inverse wavelength, nH and nL the low and high refractive indices, and OD = log(1/T). The
`optical thickness of the coating is estimated from the number of layers and their nominal thickness λ0/4.
`The present paper provides a link between this empirical procedure and a Fourier Transform (FT) technique
`used in thin film synthesis [2-4]. New results are also discussed. The core of the FT method is a Fourier transform
`relationship
`
`(3)
`
`~
`σ
`)
`
` ,(TQi
`σ
`π
`
`FT
` →←
`
` 
`
`
` )(xn
`n
`
`o
`
` 
`
`ln
`
`between the logarithm of an inhomogeneous refractive index profile n(x) and a complex function of the
`~
`σφσ
`σ
`=
`)
`(
`ie
`. In Eq.(3), x is a double optical thickness and n0 the average refractive index.
`transmittance
`
` ,(TQ
`)
`
`)
` ,(TQ
`Transmittance T is replaced by the specified transmittance in thin film synthesis. The accuracy of Eq.(3) is linked to
`~
`the accuracy of Q
`because this function is not known exactly: only approximations of the modulus Q exist in the
`literature. Phase φis also usually unknown in thin film synthesis, and it plays an important role when n(x) is forced
`~
`to improve
`to fit within specified thickness or refractive index limits. Iterative corrections can be added to Q
`progressively its magnitude and phase, thus compensating for the residual errors in transmittance.
`
`It is possible to eliminate the complex phase by applying Parseval’s identity to Eq.(3):
`
`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)16
`
`0001
`
`

`
`© 2001 OSA/OIC 2001
`
`(4)
`
`σ
`
`d
`
`.
`
`2
`
`~
`σ
`)
`,(
`TQ
`σ
`
`∞ ∞
`
`∫
`−
`
`=
`
`1
`π
`2
`
`
`
`
`dx
`
`)(
`xn
`n
`
`o
`
`
`
`
`
`∫
`
`2
`
`ln
`
`Assuming that the coating is composed in equal proportion of layers of high and low refractive index nH and nL, one
`can also replace the left hand side of Eq.(4) by
`
`(
`∑
`
`≈
`
`2
`
`)
`
`nt
`
`2
`
`ln
`
`
`
`
`dx
`
`
`
`
`
`2
`
`ln
`
`∫
`
`(5)
`
`.
`
`LH
`nn
`
`)(
`xn
`n
`o
`Equations (5) and (6) provide two different ways of estimating the optical thickness Σnt. For example, introducing
`the following Q-function proposed by Bovard [5] ,
`
`,
`
`(6)
`
`(7)
`
`
`
`ln( OD)10
`
`
`
`
`
`=
`
`1
`T
`
`
`
`ln
`
`~
`=
`QQ
`
`=
`
`one finds the following estimate for the thickness of the coating:
`
`.
`
`OD
`σ
`2
`
`σ
`
`d
`
`∫∞
`
`0
`
`LH
`nn
`
`2
`
`ln
`
`∑
`
`nt
`
`≈
`
`93.0
`
`The symmetry of Q with respect to σ=0 was used to obtain this result. Equation (8) has an interesting similarity with
`Eq. (2).
`
`As an example of application of this approximation, let us consider the case of a wide band 99% reflector
`in the wavelength region 0.4 – 0.7µm, with nH=2.25 nL=1.45. Equation (7) predicts an optical thickness of at least
`2.9um. This value is a minimum because an actual coating will certainly have some reflectance outside the specified
`rectangular band, a contribution which was neglected in the calculation. Note however that the 1/σ2 weighting factor
`ensures that this extra contribution arises mostly from the low-frequency side of the reflectance band. For
`comparison, the estimates obtained from Eq.(1) are N=13 and Σnt =3.3um [1]. These values agree reasonably well
`with those obtained by the FT method.
`
`As mentioned the accuracy of the FT estimation is linked to the accuracy of the Q-function, which is
`known to deteriorate as the reflectance increases. It is possible to avoid this restriction by using a known solution of
`the problem. The latter solution does not have to be realistic: one can remove the refractive index and thickness
`limits to facilitate this step. The question is then to estimate the thickness of a two-material multilayer of refractive
`indices nH and nL, which has the same reflectance as the first solution in the region of interest. Assuming that the
`other regions of the spectrum do not contribute significantly to the right hand side of Eq.(4), or at least that they
`have comparable contributions, one can write:
`
`,
`
`(8)
`
`dx
`
`
`
`
`
`)(
`x
`A
`n
`
`LHo
`nn
`
`n
`
`
`
`
`
`∫
`
`2
`
`ln
`
`2
`
`ln
`
`∑
`
`≈
`
`2
`
`nt
`
`where nA(x) refers to the thick preliminary solution and Σnt is the optical thickness of the desired solution. This
`approach is best suited to reflecting coatings because the main contribution to the spectral integral on the RHS of
`Eq.(4) should arise from the reflectance in the region of interest. The same restrictions apply to the ODBWP
`procedure.
`
`Figure (1) illustrates the application of Eq.(8) to the 99% reflector problem discussed previously. A thick
`preliminary solution nA(x) was first computed, in a few seconds, by the FT method with iterative corrections. Phase
`φwas assumed to be constant and the refractive indices were allowed to vary freely. Substitution of nA(x) in Eq.(8)
`lead to an estimated optical thickness of 4.50um. A suitable 33-layer solution was obtained by refinement of a
`
`0002
`
`

`
`© 2001 OSA/OIC 2001
`
`chirped QW stack. Its reflectance plotted on a dB scale in the figure below agrees well with the initial specification
`in the region of interest (T=-20dBs corresponds to OD=2). It is interesting to observe that the large ripples outside
`the main reflectance band have a small effect on the estimation because they are strongly damped by the 1/σ2
`weighing factor in Eq.(4).
`
`In summary, a parallel was established between an existing empirical ODBWP procedure for the estimation
`of optical thin film thickness, and results derived from a Fourier transform thin film synthesis approach. Two
`different ways of estimating the thickness of reflecting coatings by the FT method were proposed and illustrated by
`a numerical example. The influence of important assumptions on the accuracy of the estimation was discussed, in
`particular the influence of the reflectance outside the region of interest. Other factors affect the required thickness,
`such as the fine spectral structure in the region of interest, edge steepness, etc. Nevertheless it is useful to evaluate
`the minimum thickness necessary to produce the proper ODBWP in the region of interest. This estimate is certainly
`informative as a first approximation, and it is directly applicable to suitable optical coatings as in the present
`numerical example.
`
`Fig. 1. Application of Eq.(8) to optical thickness estimation. (a), thick preliminary solution nA(x) obtained by
`the FT method without constraints on the refractive indices.
`(b),
`refined two-material multilayer
`corresponding to the 4.5um optical thickness estimated by Eq.(8). (c), transmittance of the multilayer shown
`in (b), represented on a dB scale [T(dB) = 10log(T)=-10OD]. The target transmittance is represented by the
`horizontal line at T=-20dB (OD=2).
`
`References
`1R. R. Willey, "Estimating the number of layers required and other properties of blocker and dichroic optical thin films," Appl. Opt. 35 (25),
`4982-4986 (1996).
`2P. G. Verly, J. A. Dobrowolski, W.J. Wild et al., "Synthesis of high rejection filters with the Fourier transform method.," Appl. Opt. 28 (14),
`2864-75 (1989).
`3P. G. Verly and J. A. Dobrowolski, "Iterative correction process for optical thin film synthesis with the Fourier transform method.," Appl. Opt.
`29 (25), 3672-84 (1990).
`4P. G. Verly, "Design of inhomogeneous and quasi-inhomogeneous optical coatings at the NRC," in Inhomogeneous and quasi-inhomogenous
`optical coatings, J.A. Dobrowolski and P.G. Verly, eds., Proc. SPIE 2046, 36-45 (1993).
`5B. G. Bovard, "Derivation of a matrix describing a rugate dielectric thin film," Appl. Opt. 27 (10), 1998-2005 (1988).
`
`0003