## Abstract

The Fourier transform thin film synthesis method often results in solutions that call for indices that lie outside the range of values of the available materials. To make the resulting refractive index profiles always realizable in our meta-mode sputtering machine, a modified Fourier transform synthesis method is proposed with which the reflectance spectra can be accurately synthesized with controllable and predictable refractive index profiles. In our procedure, an optimal phase function is explored to yield acceptable refractive index profiles. Then the overall thickness is estimated using the Parseval theorem. Finally, several errors inherent to the Fourier transform method, including the imprecision of the spectral function, the truncation of the film and the apodization of the refractive index profiles, are compensated by successive corrections to the magnitude of the spectral function. An explicit iterative formula based on the derivative of the magnitude function is proposed for the compensation of the spectral mismatches. We show with a number of examples that, with the proposed method, it is possible to synthesize gradient-index optical filters with almost any desired spectral performance using experimentally realizable refractive indices.

©2008 Optical Society of America

## Corrections

Xinbin Cheng, Bin fan, J. A. Dobrowolski, Li Wang, and Zhanshan Wang, "Gradient-index optical filter synthesis with controllable and predictable refractive index profiles: erratum," Opt. Express**16**, 8902-8903 (2008)

https://opg.optica.org/oe/abstract.cfm?uri=oe-16-12-8902

## 1. Introduction

The Fourier transform (FT) method is an analytical approach for the synthesis of gradient-index filters [1–3]. Essentially, it is based on the FT relationship between the logarithmic derivative of the refractive index profile and a complex spectral function. The spectral accuracy is limited since the analytical forms of the spectral function so far are only approximate. Various forms of the modified spectral function [4–6], including an approach that makes corrections to it [7], have been explored to improve the spectral fit and they have been shown to work quite well. Modern fabrication techniques [8–10], such as ion beam sputtering, reactive plasma deposition or meta-mode sputtering, make possible the realization of such inhomogeneous films, the realizability of the resulting refractive index profiles is what should be first taken into consideration at the initial stage of the synthesis. Our newly developed meta-mode sputtering machine can fabricate SiO_{2}-Nb_{2}O_{5} intermediate index materials with a continuously variable index range from 1.75 to 2.15. To make the synthesized refractive index profile always realizable in our meta-mode sputtering machine, we propose a modified FT synthesis framework that attaches more importance to the control of the refractive index profiles than to the spectral accuracy. In our procedure, the synthesized refractive indices are first confined within the experimentally realizable range. Then, the total thickness is roughly estimated by applying Parseval’s identity [11, 12] and kept as small as possible. Finally, within these limiting conditions, the spectral mismatches are compensated with iterative process.

The most satisfactory method of controlling the refractive index is by means of the phase function. Several phase functions [3, 7, 13] have been shown to work well for tailoring the shape of the refractive index profile. Here we will focus on the phase function computed by a method called Stored Waveform Inverse Fourier Transform (SWIFT). The refractive index profiles generated by the SWIFT-phase factor can make full use of the maximum index ratio along most parts of the film. The thin-film maximum principle [14] has shown that such refractive index structures result in a minimized total thickness. It is therefore desirable to obtain such refractive index profiles.

The errors in the FT method result mainly from the imprecision of the spectral function, the truncation of the film or the apodization of the refractive index profiles. As long as exact explicit expressions for the magnitude and the phase of the spectral function are not found for more general applications, an iterative process is the only approach that can compensate simultaneously for the errors inherent in the FT method. Although there are two degrees of freedom in the mapping of spectra by refractive index profiles, the magnitude and the phase, we only make successive corrections to the magnitude and fix the phase factor generated by the SWIFT-phase function. This procedure enables us to achieve the spectral accuracy with controllable and predictable refractive index profiles. The reasons and feasibility for such an implementation strategy are given below. The relationship between the phase modulation and the spectral accuracy is equivocal, which is due in part to the truncation of the film. This is why it is very difficult to make explicit iterative corrections to the phase. Because the refractive index profile is closely related to the phase, the control of the refractive index profiles will become relatively unpredictable if the phase is incorporated into iterations. On the other hand, the magnitude is highly correlated to the errors in the transmittance. The iterations of the magnitude are sufficient to compensate the spectral mismatches for most applications. More importantly, the final refractive index profile will not deviate markedly from the initial one as long as only corrections to the magnitude are made. So the spectral accuracy can be achieved with a fairly close resemblance to the initial acceptable refractive index profiles generated with the SWIFT-phase function.

Briefly, our method uses the SWIFT-phase function to yield the acceptable refractive index profiles and then makes successive corrections to the magnitude to compensate the spectral mismatches. In our iterative procedure, a modified iterative formula based on the derivative of the magnitude function is proposed. It can be both easily and exactly implemented in the iterative routines for the improvement of the spectral accuracy. A number of examples presented here show that good results are possible with the proposed method.

## 2. Theory

The Fourier transform relation is derived from Maxwell’s equation by making several approximations. The film is assumed to be sandwiched between two identical media with additional assumptions that there is no dispersion, no absorption, and that the light is incident at normal incidence. Sossi [2] showed that a simple relationship

exists between the refractive index *n(x)* and a complex spectral function *Q*̃(*σ*)=*Q*(*σ*)·exp[*iϕ*(*σ*)]. Here σ is the wave-number *1*/λ, *n _{0}* is the refractive index of the identical external media, and

*x*is two times the optical thickness.

*Q*(

*σ*) is a suitable even function of the desired transmittance

*T*and

*ϕ*(

*σ*) is an odd phase function, which ensures that

*n(x)*is real.

Several forms for *Q*(*σ*) and *ϕ*(*σ*), including an approach that makes corrections to them, have been explored in the past in order to compensate the errors inherent to the FT method. Here, we present a modified FT method with which the reflectance spectra can be accurately synthesized with predicted acceptable refractive index profiles. Our method follows fairly closely the SWIFT method proposed by Druessel and, in addition, it combines an iterative process to compensate the spectral mismatches.

The mathematical basis for the SWIFT method has been fully described by Guan and McIver [15, 16], to whom the interested reader is referred to for details. In principle, the SWIFT method is based on Parseval’s theorem and the time-shifting theorem of Fourier analysis. The SWIFT-phase function is

Here, *x _{1}*-

*x*is referred to as the power spread thickness, over which the refractive index is distributed uniformly. The second term

_{0}*x*shifts the location of the index changes to lie between

_{0}σ*x*and

_{0}*x*. The constant term is usually set to zero. As the power spread thickness for the phase calculation is increased, the index variation is more evenly distributed throughout the total thickness and the refractive index contrast is reduced. However, the even distribution of the refractive index always leads to considerable loss of the “active” portions of the film since the film must be truncated at a finite thickness. Usually the total thickness has to be increased to compensate the deterioration of the spectral performance caused by the lost “active” index portions, but it is also possible to compensate such spectral deteriorations by an iterative process without increasing the overall thickness.

_{1}Since our procedure is based on the iterative process [17], several different definitions of *Q*[*T*(*σ*)] or their combinations can be used for the synthesis with design results that are almost the same. For the convenience of comparison, we use the same *Q*[*T*(*σ*)] that was proposed by Druessel:

Here we propose a modified iterative formula based on the derivative of the magnitude function. It has been observed that *Q*[*T*(*σ*)] is a monotonically decreasing function of *T* within the range (0, 1). So we can use derivatives *dQ*/*dT* and differences between synthesized *T* and desired transmittance *T _{D}* to make successive corrections to

*Q*[

*T*(

*σ*)], as shown in equation 4.1. Moreover, the step size

*dQ*/

*dT*is automatically adjusted to control the magnitude of the corrections for various transmittances.

*Q*[

*T*(

*σ*)] is highly nonlinear with

*T*and the derivative

*dQ*/

*dT*varies as the

*T*changes. The accuracy of

*Q*[

*T*(

*σ*)] is poor for low transmittances, the large derivatives of

*dQ*/

*dT*help the spectral fit to be achieved quickly. On the contrary, the accuracy of

*Q*[

*T*(

*σ*)] is good for high transmittances, and so small derivatives

*dQ*/

*dT*are used to avoid over-corrections. The implementation of the iterative process is given by the expressions:

The subscripts *i* in Eq. (4) refer to iterative processes, the operation can be repeated until results within the required tolerance are found. Thus *Q*[*T*(*σ*)] is corrected by the iterations, improved refractive index profiles are generated and the spectral mismatches are compensated.

## 3. Numerical tests

A computer program for the synthesis of gradient-index optical filters based on the above equations has been written using the FORTRAN language. The spectral performance of the gradient-index optical filter is computed by approximating the inhomogeneous film as a discrete film with many thin layers. Discrete layers on the order of 5nm thick give a good approximation for the reflectance of a gradient-index film for visible light. Several numerical examples are presented that show that the proposed method of synthesis can yield excellent results with controllable and predictable refractive index profiles. In all examples, the refractive index range is from 1.75 to 2.15, the indices of the incident and emergent media are same and are set to 1.94, and the tolerance in the reflection band is set to be 0.2%.

First, a simple 50% reflector is designed with a total optical thickness of 10 µm. The refractive index profile for this 50% reflector calculated with *ϕ*(*σ*)=0 and reflectance spectrum computed with standard matrix methods are shown in Fig. 1(a). The profile cannot be produced experimentally and the Gibbs oscillations are obvious in the reflection band. The refractive index and reflectance profiles obtained with a SWIFT-phase and a 7.5 µm power spread thickness are shown in Fig. 1(b). The indices are precisely located within the acceptable range of 1.75 to 2.15. Moreover, the SWIFT-phase actually helps to improve the spectral fit for such a rectangular spectral shape, as Druessel has shown in his paper. However, the improvement of the spectral performance is limited due to the considerable loss of the “active” index portions when the film is truncated at 10 µm. Iterative corrections of the magnitude can always compensate the residual spectral mismatches quickly when such a good starting design is used. The 50% reflector is synthesized within the required tolerance in 20 iterations and the indices still lie within acceptable limits, as shown in Fig. 1(c).

The versatility of the proposed method is further illustrated in the design of a more complicated filter with the shape of a house. The refractive index profiles and the reflectance spectra calculated with *ϕ*(*σ*)= 0 are shown in Fig. 2(a).

The index ranges from 1.3 to 3.0, which cannot be realized in practice. The resulting reflectance performance is also poor. It is worth noting that the SWIFT-phase is never an optimal phase for such an irregular spectral shape. The lost “active” index portions lead to a further deterioration of the spectral performance when the refractive index is evenly distributed over an optical thickness of 12 µm, which is shown in Fig. 2(b). The question might be raised whether the iterative process can converge properly for such an ill-behaved starting design. A solution is feasible because the SWIFT-phase factor is a continuously variable phase function and because the resulting reflectance spectrum usually varies smoothly with s for a suitable choice of the power spread- and the overall thicknesses. Provided that the spectrum synthesized with the SWIFT-phase function does not differ significantly from the desired spectrum by having sharp peaks or valleys, it is possible to correct such discrepancies by successive corrections to the magnitude. Figure 2(c) shows that, after 170 iterations, an excellent fit between the desired and synthesized spectra has been obtained with an acceptable refractive index profile.

The drawback of the design of Fig. 1(c) is that there are some fluctuations in the transmittance band, which result from the truncation of the film. Apodization, which consists of the multiplication of the Fourier transform by a window function, is an effective technique for eliminating such side lobes. A further deterioration of the spectral performance usually occurs at this point because apodization of the refractive index profiles results in the loss of “active” portions of the film. But it is also possible to compensate these defects by the iterative process. Here we will redesign the 50% reflector to illustrate the efficiency of an apodization based on the Hann window to illustrate the efficiency of the proposed method. After performing the apodization of the refractive index profile that was calculated with a SWIFT-phase function, the resulting reflectance was greatly deformed, as will be seen in Fig. 3(a).

Since the resulting spectrum varies smoothly with λ, it is possible to compensate the spectral mismatch by successive corrections to the magnitude. After 154 iterations the 50% reflector again meets the required tolerances. The final index profile deviates markedly from the initial value, but it is still quite acceptable. And the side lobes have been eliminated since the long tails of the refractive index profile act as good antireflection coatings. This is illustrated in Fig. 3(b).

Since the refractive index of the surrounding media is equal to the central index of the gradient-index film in this paper, it is safe to say that it is possible to design gradient-index optical filters with almost any desired spectral performance using the proposed method. The degree of fit to the target depends considerably on the overall thickness of the coating. However, it is our experience that the proposed iterative process does not always work well for the case that the reflectivity is higher than 0.99 or lower than 0.01.

## 4. Conclusion

A modified FT synthesis method has been presented which attaches more importance to the control of the refractive index profiles than to spectral accuracy. The SWIFT-phase function is used to generate acceptable refractive index profiles. These index profiles are then regarded as an initial solution for further corrections to compensate the discrepancies between the target and calculated spectral performances. The correction process described is fairly well behaved. In general, it yields excellent results. However, the final performance still depends on the initial reflectance spectra calculated with SWIFT-phase function, where a smooth spectral variation without significant discrepancies from the design spectrum is essential for a proper convergence of the design process. This procedure enables us to accurately synthesize the required reflectance profiles with controllable and predictable refractive index profiles. This is very desirable from the practical point of view. Future work will be done to adapt the above technique for use with dispersive and weakly absorbing media.

## Acknowledgements

This work is partly supported by the Chinese National Key Basic Research under Grant No. 2007CB613206 and by the Shanghai Key Subject Program under Grant No. 03dz11007. I would like to thank Dr. Haruo Takahashi of Optorun for his helpful discussions on the Fourier transform algorithm.

## References and links

**1. **R. Delano, “Fourier Synthesis of Multilayer Filters,” J. Opt. Soc. Am. **57**, 1529–1533 (1967). [CrossRef]

**2. **L. Sossi and P. Kard, “A Method for the Synthesis of Multilayer Interference Coatings,” Eesti NSV Tead. Akad. Toim. Fuus. Mat.23, 229–237 (1974). (An English translation of this paper is available from the Translation Services of the Canada Institute for Scientific and Technical Information, National Research Council, Ottawa, Ontario K1A OS2, Canada.)

**3. **J. A. Dobrowolski and D. G. Lowe, “Optical Thin Film Synthesis Program Based on the Use of Fourier Transforms,” Appl. Opt. **17**, 3039–3050 (1978). [CrossRef] [PubMed]

**4. **P. G. Verly, J. A. Dobrowolski, W. J. Wild, and R. L. Burton, “Synthesis of High Rejection Filters with the Fourier Transform Method,” Appl. Opt. **28**, 2864–2875 (1989). [CrossRef] [PubMed]

**5. **B. Bovard, “Rugate Filter Design: the Modified Fourier Transform Technique,” Appl. Opt. **29**, 24–30 (1990). [CrossRef] [PubMed]

**6. **H. Fabricius, “Gradient-index filters: designing filters with steep skirts, high reflection, and quintic matching layers,” Appl. Opt. **31**, 5191–5196 (1992). [CrossRef] [PubMed]

**7. **P. G. Verly and J. A. Dobrowolski, “Iterative Correction Process for Optical Thin Film Synthesis with the Fourier Transform Method,” Appl. Opt. **29**, 3672–3684 (1990). [CrossRef] [PubMed]

**8. **C. C. Lee, C. J. Tang, and J. Y. Wu, “Rugate Filter Made with Composite Thin Films by Ion-beam Sputtering,” Appl. Opt. **45**, 1333–1337 (2006). [CrossRef] [PubMed]

**9. **D. Poitras, S. Larouche, and L. Martinu, “Design and Plasma Deposition of Dispersion-Corrected Multiband Rugate Filters,” Appl. Opt. **41**, 5249–5255 (2002). [CrossRef] [PubMed]

**10. **P. V. Bulkin, P. L. Swart, and B. M. Lacquet, “Fourier-transform Design and Electron Cyclotron Resonance Plasma-enhanced Deposition of Lossy Graded-index Optical Coatings,” Appl. Opt. **35**, 4413–4419 (1996). [CrossRef] [PubMed]

**11. **P. G. Verly, “Fourier Transform Approach for Thickness Estimation of Reflecting Interference Filters,” Appl. Opt. **32**, 5636–5641 (2006). [CrossRef]

**12. **P. G. Verly, “Fourier Transform Approach for Thickness Estimation of Reflecting Interference Filters. 2. Generalized Theory,” Appl. Opt. **46**, 76–83 (2007). [CrossRef]

**13. **J. Druessel and J. Grantham, “Optimal Phase Modulation for Gradient-index Optical Filters,” Opt. Lett. **18**, 1583–1585 (1993). [CrossRef] [PubMed]

**14. **A. V. Tikhonravov, “Some Aspect of Thin-film Optics and Their Applications,” Appl. Opt. **32**, 5417–5426 (1993). [CrossRef] [PubMed]

**15. **S. Guan and J. Chem, “General phase modulation method for stored waveform inverse Fourier transform excitation for Fourier transform ion cyclotron resonance mass spectrometry,” J. Chem. Phys. **91**, 775–777 (1989). [CrossRef]

**16. **S. Guan and R. McIver, Jr., “Optimal phase modulation in stored wave form inverse Fourier transform excitation for Fourier transform mass spectrometry. I. Basic algorithm,” J. Chem. Phys. **92**, 5841–5846 (1990). [CrossRef]

**17. **M. Hacker, G. Stobrawa, and T. Feurer, “Iterative Fourier transform algorithm for phase-only pulse shaping,” Opt. Express **9**, 191–199 (2001). [CrossRef] [PubMed]