Hassan A. Abu-Safia, Abdullah 1. Al-Sharif, and Ibrahim 0. Abu Aijarayesh. Rugate filters reflectance spectra were calculated for various index profiles with ...
Rugate filter sidelobe suppression using half-apodization Hassan A. Abu-Safia, Abdullah 1.Al-Sharif, and Ibrahim 0. Abu Aijarayesh
Rugate filters reflectance spectra were calculated for various index profiles with half-Gaussian modulation; graded-index as well as step-index profiles were used. The results show excellent sidelobe suppression around the stopbands, relatively high optical density, and good flatness in the reflectance band. Key words: Optical minus filters, rugate filter, apodization, gradient-index modulation, step-index modulation.
Introduction
An ideal minus filter is defined as that which reflects all the incident waves in a given spectral region and transmits all the incident waves elsewhere.1-3 The simplest type of minus filter is a multilayer stack of high-low dielectric thin films.4 Unfortunately in such filters the sidelobe ripple around the reflected region, called the stopband, is too high. The equivalent-layer concept has been used by Thelen 5 to develop a method for designing multilayer interference filters with relatively smooth high-transmittance regions on both sides of the stopband. An important way to design minus filters is to use graded-index profiles.6 The most sophisticated type of these filters is called the rugate filter. Rugate filters are optical thin-film interference filters with periodic index profiles that produce controlled stopband widths with relatively small sidelobes. The refractive-index profile of a simple rugate filter is usually expressed as n(x) = na + 2 np sin
)X
(1)
where na is the average index, np is the peak-to-peak refractive-index modulation, p is the period, and x is the depth in the profile. The bandwidth of the filter is controlled by np and na. 7 In the limit of small refractive-index modulation and by using the coupledwave theory, 7 the following expressions for the band-
The authors are with the Department of Physics, Yarmouk University, Irbid, Jordan. Received 4 November 1991. 0003-6935/93/254831-05$06.00/0.
3 1993 Optical Society of America.
width (BW) and the optical density (OD) (which is a measure of the strength of reflectance peak R) are given by A n BW= A = 2P, X
2n,,
(2)
OD = logio( 1 - R) = (1.36 x BW x N) - logl 0(4/n,),
(3)
where N is the number of cycles in the rugate, and n, is the substrate index. The main goals of any minus filter design are to control the position, bandwidth, shape, and optical density of the stopband, and, of course, to minimize or even eliminate the sidelobes around the stopband.1 Southwell and Hall8 have succeeded in further reducing the sidelobe intensity by using quintic (polynomial of the fifth degree) functions to match both the substrates and the medium with the profile. 9 In fact, when the quintic matching layers are superimposed on the rugate cycles, the optical density increases with no loss of sidelobe suppression. Southwell and Hal18 have also shown that Gaussian, linear, and quintic apodizations when imposed on the rugateindex profiles result in improving the spectral response (i.e., maintain high stopband reflectance and sharp stopband) and appreciably suppress the sidelobe. The sidelobe intensity depends on the matching and apodizing functions that are used. The effect of Gaussian apodization and matching is demonstrated in Fig. 1. In fact, the sidelobes can be almost eliminated by a complete smooth matching of the 1 September 1993 / Vol. 32, No. 25 / APPLIED OPTICS
4831
2.5
2.5
2.0
2.0
n
n 1.5 ,
1.5
1 .0
1.0
-T
0
_r_
1.0
1.0
0.8 -
0.8
R0.6 _
R
0.4
(b)
0.6
0.2 0.0
0.5 1.0 1.5 Wavelength (m)
nn
2.0
0.0
0.5
1.0
1.5
2.0
Wavelength (um)
Fig. 1. (a) 50-cycle rugate filter with Gaussian apodization and Gaussian matchingon each side. (b)Reflectance ofthe corresponding rugate, filter.
profile with the substrate and with the medium, as shown in Fig. 2. The very low sidelobe intensity at the right of the stopband in Fig. 2 might be retained for the matching function. The best match that could ever be obtained would be to set n = no = ns, which is demonstrated in Fig. 3. 2.5 2.0
n 1.5
-
0.4
0.2 0.0
-
-
Fig. 3. (a) 50-cycle rugate filter with Gaussian apodization and no matching regions. The substrate index and the medium index are the same as the average profile index, which is equal to 1.52. (b) Reflectance of the corresponding rugate filter.
As is clear from Eq. (2), in order to increase the bandwidth, na must be as low as possible and np must be as large as possible. Also the sidelobe intensity of the filter must be proportional to the difference between the average index and the substrate index from one side and the difference between the average index and the medium index on the other side. To have good sidelobe suppression the above differences must be at minimum values. However these requirements cannot be satisfied easily by using index profiles such as those shown in Figs. 1-3. To illustrate this point, if one uses a substrate of refractive index (e.g., n = 1.52), the lower limit of the refractive-index profile shown in Fig. 3 reaches a nonphysical value (i.e., na - np/2).
1.0
1.0
1l
0.8
(b)
R0.6 0.4 0.2 0.0
Half-Apodized Rugate Filters
z~~~~~~~~~~~~~~~~ ' . . . . ~j. . . . I . .. . . . 0.0
0.5 1.0 1.5 Wavelength (m)
The Gaussian half-apodized index profile is given by n(x)
= nL +
2np[1 + siny )exp(-ax2)
(4)
2.0
Fig. 2. (a) 50-cycle rugate filter with Gaussian apodization and Gaussian matchingon each side. (b) Reflectance ofthe corresponding rugate filter. 4832
To overcome the problems created by these requirements, we introduce new index profiles and refer to them as half-apodized profiles. We computed the reflectance curves for different half-apodized rugate filters and for their stack-equivalence filters. The results obtained are discussed and compared with those obtained by Southwell.
APPLIED OPTICS / Vol. 32, No. 25 / 1 September 1993
Note that na is replaced by nL, which is the lowest index in the profile, and auis a constant and was chosen such that the refractive index reaches nL at both ends of the profile. The other symbols have
already been defined. The spectral reflectance for a profile of the form in Eq. (4) was computed by using a computer program that is based on the standard characteristic matrix approach. 4 This program was first used to reproduce the results published by other investigators. 9 10 For comparison we consider a 50cycle rugate filter on a glass substrate with an average index of na = 1.8, a peak-to-peak index variation of np = 0.7, and Gaussian apodization and Gaussian matching on both sides; the resulting index profile is shown in Fig. 1(a). The corresponding spectral reflectance for this index profile was computed by dividing each cycle into 20 films; the results are shown in Fig. 1(b). Doubling the number of films does not change the spectral reflectance. The average sidelobe intensity is 0.04 and the optical density is 5. By using an index profile with the same substrate as the medium (n = 1.52) and also equal to the average index profile, the remaining parameters are the same as those used in Fig. 1(a). All the residual ripples were eliminated as shown in Fig. 3(b). These spectra can be considered as the optimum solution. However, the refractive index reaches a small value (1.17) in the middle part of the profile, and the medium is the same as the substrate, which is not common. To solve these problems, one is faced by practical limitations such as the availability of appropriate material with such a low index (i.e., n = 1.17). For all the above-mentioned reasons, we use Eq. (4) and repeat the same calculations with the same parameters as those used in Fig. 1(a). The computed index profile is shown in Fig. 4(a) and the corresponding spectral reflectance is shown in Fig. 4(b). The
overall spectral reflectance is identical to the one obtained in Fig. 1(b) except for the fluctuation in ripple intensity, which is smoother in Fig. 4(b). In addition, the ripple intensity adjacent to the stopband is lower in Fig. 4(b). This is retained for the matching area that is not needed in half-apodized profiles. The average ripple intensity in both figures is 0.04, which is expected since this is the reflectance at normal incidence of a glass substrate (n = 1.52). The optical density is 5. By the same parameters as in Fig. 4(a) but with a LiF substrate (n = 1.38) and repeating the calculations for the index profile and its corresponding spectral reflectance (Fig. 5), we observe a lower ripple intensity with an average of 0.025. To reduce the ripple intensity further a lower refractive-index substrate is needed. Stack Equivalence (Step-index Modulation)
For the following calculations we used an index profile of the form n(x) = na + /2np[1 + (-1)i]exp(- xaX2 ),
(5)
where i is the cycle order. This index profile is similar to that given by Eq. (4), but the sinusoidal part is replaced by its extremum value. This index profile is probably easy to control and to fabricate experimentally. We used the same parameters as those in Fig. 4(a) to compute the index profile that consists of 100 layers (equivalent to 50 cycles in the rugate) with the same optical thicknesses. The total optical thickness of this profile is equal to that of Fig.
2.5 2.5 -
.
(a) 2.0 2.0
n
1.5
: 1.5 -
1.01.0
.. ...l 6
O......5 T lo .
fi s i |
......... 15
Ei |
|
s |
|
|
|
a |
20 ........25
Optical Thickness (um)
1.0()
(b)
1.0
0.8
0.8 -
R
0.4
0.6
0.4
0.2
0.2 0.0 -
0.0
....... .
0.5 1.0 1.5 Wavelength (pm)
2.0
Fig. 4. (a) 50-cycle rugate filter with Gaussian half-apodization and no matching regions. The substrate index is the same as the lower index in the profile (n = 1.52), and the medium is air. (b) Reflectance of the corresponding rugate filter.
J
0.0 rr' al r ' 0.5 1.0 1.5 0.0 Wavelength (m)
2.0
Fig. 5. (a) 50-cycle rugate filter with Gaussian half-apodization and a LiF substrate (n = 1.38). (b) Reflectance of the corresponding rugate filter. 1 September 1993 / Vol. 32, No. 25 / APPLIED OPTICS
4833
2.5
2.5 -
(a) 2.0
2.0
n
n
111111
1.5 1.0
I
6.........5 .......
1.5 -
..... - II.. I-
-il ........ 119
Optical Thickness (m)
R
(a)
rr'r
Z5
20
0
1.0
1.0 -
0.8
0.8
0.6
R
0.4
5 10 15 20 Optical Thickness (m)
25
(b)
0.6 -
0.4
0.2
0.2 -
0.0
0.0
-
......................................
0.5 1.0 1.5 Wavelength (m) Fig. 6. (a) Stack equivalence of the rugate filter shown in Fig. 4. (b) Reflectance of the corresponding stack.
0.5 1.0 1.5 2.0 Wavelength (m) Fig. 8. (a) Same index profile as that in Fig. 6(a) but with n, = 0.5 and n, = 1.38. (b) Reflectance of the corresponding stack.
4(a) for comparison. The spectral reflectance for this index profile is shown in Fig. 6(b). As shown in Fig. 6(b), the optical density increases (OD = 6.6) because the difference between the refractive index and its average value at each point is greater in this index profile than its corresponding rugate profile. In other words, the deviations in refractive index are larger in the stack equivalence than in the rugate equivalence. The sidelobe intensities are almost iden-
tical to those in Fig. 4(a). Moreover, the second harmonic is lower in Fig. 6(b), but higher harmonics have appeared. When a substrate of lower index (n = 1.38) is used, the OD increases to 7.3, and the sidelobe intensity is suppressed further. Figure 7 shows a 50-layer stack-equivalence filter having a total optical thickness equal to half of that of Fig. 6 and using LiF (n = 1.38) as the substrate. The stopband is less uniform than that using a 100-layer stack and the OD is reduced to 3.4. A rugate filter with this total optical thickness gives a nonsatisfying result. As shown in Fig. 8, when np is reduced from 0.7 to 0.5 and the rest of the parameters in Fig. 5(a) are retained, we obtain the same spectral reflectance except for higher harmonics. Thus by controlling np one can use step-index modulation to replace gradedindex modulation when necessary. Finally, it is relevant to mention that all our computations were carried out at normal incidence. However, when the computer program was modified to take into account other incident angles, we observed that, for Oi < 20, the spectral reflectance obtained is not noticeably changed. For O > 20 the optical density decreases for both s andp polarizations. In addition, the average ripple intensity increases and the stopbands shift toward shorter wavelengths.
92.5;
(a) 2.0 -
n 1.5.
.~rL~iflhtf~ftft~ffR~rf . . _.
1.0 ....... aTn m 6.... 5 10 15 20 Optical Thickness (m) -
2... 25
(b)
I
1.0-
l.,lllli
0.8
R
0.6 0.4 -
Conclusion
0.2 0.0v 0.0
...
'
0.5
. ... .. .. .. . .1 1 1
1.0
Wavelength (m)
1.5
2.0
Fig. 7. (a) Stack equivalence filter of 50 layers with Gaussian half-apodization. (b) Reflectance of the corresponding stack. 4834
0.0
APPLIED OPTICS / Vol. 32, No. 25 / 1 September 1993
In this numerical study we have demonstrated that rugate filters, based on the proposed index profiles, have a relatively high optical density with excellent sidelobe suppression and good stopband reflectance. Moreover, these profiles have several advantages over the kinds of profile shown in Fig. 1(a), namely,
flexibility in choosing a substrate, the ability of having large np without exceeding the refractiveindex limits, and matching layers are not needed if a substrate of low refractive index were used, which reduces the total thickness of the corresponding filter. References 1. W. H. Southwell, "Spectral response calculations of rugate filters using coupled-wave theory," J. Opt. Soc. Am. A 5, 1558-1564(1988). 2. W. H. Southwell, "Coating design using very thin high- and low-index layers," Appl. Opt. 24, 457-460 (1985). 3. P. G. Veryl, 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).
4. M. Born and E. Wolf, Principlesof Optics, 6th ed. (Pergamon, New York, 1980). 5. A. Thelen, "Design of optical minus filters," J. Opt. Soc. Am. 61, 365-369 (1971). 6. E. P. Donovan, D. Van Vechten, A. D. F. Kahn, C. A. Carosella, and G. K. Hubler, "Near infrared rugate filter fabrication by ion beam assisted deposition of Si(1 X)NX films," Appl. Opt. 28, 2940-2944(1989). 7. W. J. Gunning, R. L. Hall, F. J. Woodberry, W. H. Southwell, and N. S. Gluck, "Codeposition of continuous composition rugate filters," Appl. Opt. 28, 2945-2948 (1989). 8. W. H. Southwell and R. L. Hall, "Rugate filter sidelobe suppression using quintic and rugated quintic matching layers," Appl. Opt. 28, 2949-2950 (1989). 9. W. H. Southwell, "Using apodization functions to reduce sidelobes in rugate filters," Appl. Opt. 28, 5091-5094 (1989). 10. B. G. Bovard, "Rugate filter design: the modified Fourier transform technique," Appl. Opt. 29, 24-30 (1990).
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