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Optics Express

Optics Express

  • Editor: Martijn de Sterke
  • Vol. 16, Iss. 20 — Sep. 29, 2008
  • pp: 15531–15539
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Dichroic rugate filters based on birefringent porous silicon

Nobuyuki Ishikura, Minoru Fujii, Kohei Nishida, Shinji Hayashi, and Joachim Diener  »View Author Affiliations


Optics Express, Vol. 16, Issue 20, pp. 15531-15539 (2008)
http://dx.doi.org/10.1364/OE.16.015531


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Abstract

Rugate filters made of anisotropically nanostructured birefringent silicon have been fabricated and studied by polarization-resolved transmission measurements. Electrochemical etching of a (110) oriented Si wafer results in porous silicon layers which exhibit a strong in-plane birefringence. We demonstrate that a sinusoidal refractive index variation of birefringent porous silicon combined with index-matching layers and apodization results in a dichroic rugate filter having a stop-band dependent on the polarization direction of the incident light without higher-order harmonics and sidelobes. We also demonstrate that the combination of different dichroic rugate filters allow us to realize filters with more complex properties in a single preparation step.

© 2008 Optical Society of America

1. Introduction

The combination of electronic and photonic elements in an integrated circuit requires Si processing technology that allows optical elements to be integrated on a Si wafer. Polarizing elements are one of the most important optical elements in the optoelectronic circuits. One problem with realizing polarizing elements from crystalline Si is that its intrinsic birefringence and hence its ability to influence polarized light is small. Therefore, birefringence of Si-based materials has been achieved by the anisotropic nanostructuring. For example, periodic nanostructures like two-dimensional photonic crystals [1

1. F. Genereux, S.W. Leonard, H. M. van Driel, A. Birner, and U. Gösele, “Large birefringence in two-dimensional silicon photonic crystals,” Phys. Rev. B 63, 161101 (2001). [CrossRef]

] and gratings [2

2. F. Xu, R. C. Tyan, P. C. Sun, Y. Fainman, C. C. Cheng, and A. Scherer, “Fabrication, modeling, and characterization of form-birefringent nanostructures,” Opt. Lett. 20, 2457–2459 (1995). [CrossRef] [PubMed]

] exhibit large birefringence. Another approach leading to birefringence of Si is electrochemical etching of a low symmetry, e.g. (110), surface of a Si wafer. Electrochemical etching of a (110) Si wafer results in the formation of Si nanowire skeletons (porous silicon: PSi) aligned in a certain direction. The alignment of Si nanowire skeletons exhibit a strong in-plane birefringence in a very wide wavelength range covering the visible to near infrared region [3

3. N. Künzner, D. Kovalev, J. Diener, E. Gross, V. Yu. Timoshenko, G. Polisski, F. Koch, and M. Fujii, “Giant birefringence in anisotropically nanostructured silicon,” Opt. Lett. 26, 1265–1267 (2001). [CrossRef]

]. Because of the reduced size of Si nanowires and voids smaller than several tenth of nm, birefringent PSi is considered to be a quasicontinuous optical medium. The important advantage of birefringent PSi to realize various optical components is its simplicity to modulate the refractive index. The refractive index and the thickness can be controlled precisely by the etching current density and time. Therefore, a stack of birefringent PSi layers with alternating refractive indices and thicknesses can easily be produced and they result in a dichroic Bragg reflector with a stop-band depending on the polarization direction of the incident light [4

4. J. Diener, N. Künzner, D. Kovalev, E. Gross, V. Yu. Timoshenko, G. Polisski, and F. Koch, “Dichroic Bragg reflectors based on birefringent porous silicon,” Appl. Phys. Lett. 78, 3887–3889 (2001). [CrossRef]

, 5

5. J. Diener, N. Künzner, D. Kovalev, E. Gross, F. Koch, and M. Fujii, “Dichroic behavior of multilayer structures based on anisotropically nanostructured silicon,” J. Appl. Phys. 91, 6704–6709 (2002). [CrossRef]

].

2. Experimental details

PSi structures were produced by electrochemical etching of (110) oriented p+ Si wafers (1-5 mΩ·cm). The etching was performed at room temperature in a solution of 1:1 by volume mixture of HF (46wt.% in water) and ethanol. To vary the porosity of the layers (and therefore their mean refractive indices) the current density was changed from 20 mA/cm 2 to 100 mA/cm2 by computer-controlled current source (Agilent 6612C). The rugate refractive index profiles were realized by stepwise current variation with the minimum step of 0.5 mA/cm 2. During the etching process, several etch-stop steps of zero current were inserted to recover the HF concentration at the dissolution front. At the end of the etching process a high current pulse (400 mA/cm2, 1.6 s) was applied to detach PSi structures from the substrate. Transmittance spectra of the prepared samples were measured by a UV-visible-NIR spectrophotometer (Shimadzu UV-3101PC) over the spectral range of 350 to 2300 nm with the spectral resolution of 1 nm. The incident light was linearly polarized with a Glan-Thompson prism and directed normally onto the sample with polarization vector (E) aligned either parallel to the [001] or to the [1̄10] crystallographic directions in the (110) surface plane.

3. Results and discussion

3.1. Optical constants of a single optically anisotropic PSi layer

For design and fabrication of a PSi based rugate filter, the correlation between the optical constants and electrochemical etching condition is necessary. To obtain etching rates and refractive indices as a function of the etching current density, (110) PSi monolayers were prepared at constant current densities ranging from 20 mA/cm2 to 100 mA/cm2. The etching rates (Fig. 1(a)) are obtained from the layer thickness estimated by cross-sectional scanning electron microscopy (SEM).

Fig. 1. (a) Etching rate of a (110) oriented p+ Si wafer as a function of current density. (b) Refractive indices and optical anisotropy (Δn) for single layers of (110) PSi estimated from the interference fringes for E ‖[001] and E‖ [1̄10] as a function of current density.

To determine the refractive index, the polarization-resolved transmittance spectra of the single layers were recorded. From the spectral position of the interference fringes for E‖[001] and E‖[1̄10] the corresponding refractive indices (n[001] and n[1̄10]) are deduced according to the simple relation;

n=12d(1λr1λr+1)1
(1)

where d denotes the thickness of the PSi layer, n is the refractive index, and λr is the wavelength of the rth fringe. The calculated refractive indices (at wavelength of 2µm) and the anisotropy (Δn=n[1̄10] -n[001]) are shown in Fig. 1(b). The in-plane optical anisotropy of the layers is evident: the refractive index for E parallel to the [1̄10] direction is substantially larger than that to the [001] direction. As seen in Fig. 1(b), the value of the optical anisotropy (Δn) rises as the current density increases and the maximum value reaches ≃0.17 comparable to that of natural birefringent crystals such as calcite.

3.2. Cross-sectional TEM observation of a dichroic rugate filter

3.3. Dichroic rugate filters

The rugate refractive index profile used in this work is expressed as

n(x)=exp[lnnH+lnnL2+lnnHlnnL2sin(4πxλ0+ϕ)]
(2)

where x is the optical path length, nH and nL are maximum and minimum refractive indices used in the layer, λ 0 is the wavelength of the stop-band position, and ϕ is the phase angle [7

7. B. G. Bovard, “Rugate filter theory: an overview,” Appl. Opt. 32, 5427–5442 (1993). [CrossRef] [PubMed]

, 9

9. E. Lorenzo, C. J. Oton, N. E. Capuj, M. Ghulinyan, D. Navarro-Urrios, Z. Gaburro, and L. Pavesi, “Porous silicon-based rugate filters,” Appl. Opt. 44, 5415–5421 (2005). [CrossRef] [PubMed]

].

Fig. 2. Cross-sectional TEM images of a rugate filter made from a (110) Si wafer observed from a [1̄12] direction (a). The magnification factors are (b)104, (c)5×104 and (d)4×105, respectively.

Such a sinusoidal profile would result in the reflection band without higher-order harmonics. However, as a result of the sharp truncation of the refractive index profile at the boundaries of the filter, a simple sinusoidal index profile would also generate interference oscillation (sidelobes) appearing on both sides of the stop-band. To reduce these unnecessary sidelobes, we introduce apodization that is a well-known technique for eliminating sidelobes coming from truncation [12

12. W. H. Southwell, “Using apodization functions to reduce sidelobes in rugate filters,” Appl. Opt. 28, 5091–5094 (1989). [CrossRef] [PubMed]

]. We chose a sin(x) apodization function from 0 to π, which shows good sidelobes suppression properties with high stop-band rejection level. A half-apodization technique [13

13. H. A. Abu-Safia, A. I. Al-Sharif, and I. O. Abu Alijarayesh, “Rugate filter sidelobe suppression using halfapodization,” Appl. Opt. 32, 4831–4835 (1993). [CrossRef] [PubMed]

] is also used to reduce total etching time [9

9. E. Lorenzo, C. J. Oton, N. E. Capuj, M. Ghulinyan, D. Navarro-Urrios, Z. Gaburro, and L. Pavesi, “Porous silicon-based rugate filters,” Appl. Opt. 44, 5415–5421 (2005). [CrossRef] [PubMed]

]. Furthermore, to remove sidelobes caused by large refractive index mismatch between the rugate structure and the surrounding media, the quintic index-matching layers are added to the front and back of the filter. The refractive index profile of the quintic index-matching layer [14

14. W. H. Southwell, “Gradient-index antireflection coatings,” Opt. Lett. 8, 584–586 (1983). [CrossRef] [PubMed]

] that matches indices form n 1 to n 2 is expressed as

n(t)=n1+(n2n1)(10t315t4+6t5)
(3)

where t is the normalized layer thickness, which varies from 0 to 1.

The etching rate and the refractive index shown in Fig. 1 are used to realize a rugate refractive index profile. In birefringent PSi, however, the refractive index in the [001

1. F. Genereux, S.W. Leonard, H. M. van Driel, A. Birner, and U. Gösele, “Large birefringence in two-dimensional silicon photonic crystals,” Phys. Rev. B 63, 161101 (2001). [CrossRef]

] and [1̄10] directions cannot be controlled independently and thus optimization of the structure in both directions is very difficult. Furthermore, strong etching current density dependence of Δn makes the structure design more difficult. In this work, as a compromise, we employ an average refractive index nmid=(n[1̄10]+n[001])/2 to design the structure.

Fig. 3. A dichroic rugate filter with 30 periods: (a) refractive index profiles versus physical depth, (b) calculated and (c) measured polarization-resolved transmittance spectra for E ‖[001] and E‖[1̄10]. The maximum optical density at the center of the stop-band is 2.7 for both polarization directions.

Figure 3(a) shows the refractive index profiles of a 30 periods (110) rugate filter for E parallel to the [1̄10] and [001

1. F. Genereux, S.W. Leonard, H. M. van Driel, A. Birner, and U. Gösele, “Large birefringence in two-dimensional silicon photonic crystals,” Phys. Rev. B 63, 161101 (2001). [CrossRef]

] crystallographic directions. Figures 3(b) and (c) show the calculated and measured polarization-resolved transmittance spectra of the (110) rugate filter, respectively. The calculation is made by a transfer matrix method [15

15. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).

] by taking into account the wavelength dispersion of the refractive index. Contrary to standard PSi-based rugate filters [8

8. M. G. Berger, R. Arens-Fischer, M. Thönissen, M. Krüger, S. Billat, H. Lüth, S. Hilbrich, W. Theiβ, and P. Grosse, “Dielectric filters made of PS: advanced performance by oxidation and new layer structures,” Thin Solid Films 297, 237–240 (1997). [CrossRef]

, 9

9. E. Lorenzo, C. J. Oton, N. E. Capuj, M. Ghulinyan, D. Navarro-Urrios, Z. Gaburro, and L. Pavesi, “Porous silicon-based rugate filters,” Appl. Opt. 44, 5415–5421 (2005). [CrossRef] [PubMed]

, 10

10. M. S. Salem, M. J. Sailor, T. Sakka, and Y. H. Ogata, “Electrochemical preparation of a rugate filter in silicon and its deviation from the ideal structure,” J. Appl. Phys. 101, 063503 (2007). [CrossRef]

] the spectral position of a stop-band depends on the polarization direction of the incident light. For E‖[001] the stop-band appears around 1430 nm but is significantly shifted towards longer wavelength (~1540 nm) for E‖[1̄10]. On the both sides of the stop-band, one can see strong sidelobes in both the measured and the calculated spectra. The sidelobes for E‖[1̄10] are slightly larger because the refractive index mismatch between the rugate structure and the surrounding media is larger than that for E‖[001]. A very weak second-harmonic peak and good coincidence between the experimental and calculated transmittance spectra indicate that the etching process is well controlled.

Fig. 4. An apodized dichroic rugate filter with 30 periods and index-matching layers: (a) refractive index profiles versus physical depth, (b) calculated and (c) measured polarization-resolved transmittance spectra for E‖[001] and E‖[1̄10]. The optical density is 2.5 for E‖[001] and 2.2 for E‖[1̄10] at the center of the stop-band. Inset shows polarizationresolved transmittance spectra of the same structure in the spectral range from 1355 to 1450 nm.

Figure 4(a) shows a refractive index profiles of an apodized (110) rugate filter with index-matching layers. The calculated and measured transmittance spectra are shown in Figs. 4(b) and (c), respectively. We can see strong suppression of the sidelobes because of the apodization and the presence of index-matching layers. The width of the stop-band decreases by about 30% compared with that of the nonapodized sample, because apodization usually reduces the bandwidth. Inset of Fig. 4(c) shows the polarization-resolved transmission spectra in the spectral range from 1355 to 1450 nm. For this specific spectral range light polarized along the [001

1. F. Genereux, S.W. Leonard, H. M. van Driel, A. Birner, and U. Gösele, “Large birefringence in two-dimensional silicon photonic crystals,” Phys. Rev. B 63, 161101 (2001). [CrossRef]

] direction is reflected while that light polarized along the [1̄10] direction passes the whole structure. Since the [001] and [1̄10] directions are orthogonal in the (110) Si surface plane, the transmitted light is linearly polarized, i.e. this rugate filter acts as a planar linear polarizer. Due to the small sidelobes and high reflectance, the extinction ratio is better than 20 dB (up to 25 dB) over the wavelength range and the transmittance is very high.

3.4. Narrow bandwidth dichroic rugate filter

For high quality dichroic structures it is essential that both stop-bands do not overlap. The separation of the rugate filter stop-bands can be achieved by reducing its bandwidth. The width of the stop-band is determined by the index contrast of the rugate filter. Therefore, we fabricate a rugate filter with smaller contrast to demonstrate the separated stop-bands. Figure 5(a) shows the refractive index profiles of the reduced contrast rugate filter. Compared to the dichroic rugate filter shown in Fig. 4(a) the refractive index contrasts of this narrow bandwidth rugate filter are reduced from 1.4-1.85 (1.56-1.95) to 1.4-1.63 (1.56-1.77) for E‖[001] ([1̄10]). To compensate small reflectance arising from the low refractive index contrast, the periods are increased to 70. The measured polarization-resolved transmittance spectra are shown in Fig. 5(b). The bandwidth for both polarization directions of incident light are successfully reduced and the stop-bands are fully spectrally separated, while sidelobes and harmonics are kept very low. Since the separation of the stop-bands is determined by the birefringence level (Δn) of the structure [5

5. J. Diener, N. Künzner, D. Kovalev, E. Gross, F. Koch, and M. Fujii, “Dichroic behavior of multilayer structures based on anisotropically nanostructured silicon,” J. Appl. Phys. 91, 6704–6709 (2002). [CrossRef]

], the spectral splitting can be adjusted by changing the average refractive index of the structure utilizing the large variation of Δn shown in Fig. 1(b).

Fig. 5. Narrow bandwidth dichroic rugate filter with 70 periods: (a) refractive index profiles versus physical depth, (b) calculated and (c) measured polarization-resolved transmittance spectra for E parallel to the [1̄10] and [001] directions.

3.5. Double band dichroic rugate filter

4. Conclusions

We have succeeded in producing Si-based dichroic rugate filters by the formation of sinusoidal refractive index profiles by electrochemical etching of (110) Si wafers. The filter had a stop-band depending on the polarization direction of the incident light without higher-order harmonics and sidelobes. The filters act as a high quality planar linear polarizer that can be used for high power lasers since there is no absorption of incident light in near infrared region. In the present procedure, the position and the width of stop-bands can easily be controlled. Furthermore, designs of complex structures can be achieved in a single step by combination of different dichroic rugate filters.

Fig. 6. Double band dichroic rugate filter centered around 1500 nm and 1900 nm with 63 and 50 periods, respectively: (a) refractive index profiles versus physical depth, (b) calculated and (c) measured polarization-resolved transmittance spectra for E parallel to the [1̄10] and [001] directions.

Acknowledgment

This work is supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology, Japan.

References and links

1.

F. Genereux, S.W. Leonard, H. M. van Driel, A. Birner, and U. Gösele, “Large birefringence in two-dimensional silicon photonic crystals,” Phys. Rev. B 63, 161101 (2001). [CrossRef]

2.

F. Xu, R. C. Tyan, P. C. Sun, Y. Fainman, C. C. Cheng, and A. Scherer, “Fabrication, modeling, and characterization of form-birefringent nanostructures,” Opt. Lett. 20, 2457–2459 (1995). [CrossRef] [PubMed]

3.

N. Künzner, D. Kovalev, J. Diener, E. Gross, V. Yu. Timoshenko, G. Polisski, F. Koch, and M. Fujii, “Giant birefringence in anisotropically nanostructured silicon,” Opt. Lett. 26, 1265–1267 (2001). [CrossRef]

4.

J. Diener, N. Künzner, D. Kovalev, E. Gross, V. Yu. Timoshenko, G. Polisski, and F. Koch, “Dichroic Bragg reflectors based on birefringent porous silicon,” Appl. Phys. Lett. 78, 3887–3889 (2001). [CrossRef]

5.

J. Diener, N. Künzner, D. Kovalev, E. Gross, F. Koch, and M. Fujii, “Dichroic behavior of multilayer structures based on anisotropically nanostructured silicon,” J. Appl. Phys. 91, 6704–6709 (2002). [CrossRef]

6.

J. Diener, N. Künzner, E. Gross, D. Kovalev, and M. Fujii, “Planar silicon-based light polarizers,” Opt. Lett. 29, 195–197 (2004). [CrossRef] [PubMed]

7.

B. G. Bovard, “Rugate filter theory: an overview,” Appl. Opt. 32, 5427–5442 (1993). [CrossRef] [PubMed]

8.

M. G. Berger, R. Arens-Fischer, M. Thönissen, M. Krüger, S. Billat, H. Lüth, S. Hilbrich, W. Theiβ, and P. Grosse, “Dielectric filters made of PS: advanced performance by oxidation and new layer structures,” Thin Solid Films 297, 237–240 (1997). [CrossRef]

9.

E. Lorenzo, C. J. Oton, N. E. Capuj, M. Ghulinyan, D. Navarro-Urrios, Z. Gaburro, and L. Pavesi, “Porous silicon-based rugate filters,” Appl. Opt. 44, 5415–5421 (2005). [CrossRef] [PubMed]

10.

M. S. Salem, M. J. Sailor, T. Sakka, and Y. H. Ogata, “Electrochemical preparation of a rugate filter in silicon and its deviation from the ideal structure,” J. Appl. Phys. 101, 063503 (2007). [CrossRef]

11.

N. Künzner, J. Diener, E. Gross, D. Kovalev, V. Yu. Timoshenko, and M. Fujii, “Form birefringence of anisotropically nanostructured silicon,” Phys. Rev. B 71, 195304 (2005). [CrossRef]

12.

W. H. Southwell, “Using apodization functions to reduce sidelobes in rugate filters,” Appl. Opt. 28, 5091–5094 (1989). [CrossRef] [PubMed]

13.

H. A. Abu-Safia, A. I. Al-Sharif, and I. O. Abu Alijarayesh, “Rugate filter sidelobe suppression using halfapodization,” Appl. Opt. 32, 4831–4835 (1993). [CrossRef] [PubMed]

14.

W. H. Southwell, “Gradient-index antireflection coatings,” Opt. Lett. 8, 584–586 (1983). [CrossRef] [PubMed]

15.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).

16.

J. A. Dobrowolski and D. Lowe, “Optical thin film synthesis program based on the use of Fourier transforms,” Appl. Opt. 17, 3039–3050 (1978). [CrossRef] [PubMed]

17.

A. G. Imenes and D. R. McKenzie, “Flat-topped broadband rugate filters,” Appl. Opt. 45, 7841–7850 (2006). [CrossRef] [PubMed]

18.

W. H. Southwell, “Extended-bandwidth reflector designs by using wavelets,” Appl. Opt. 36, 314–318 (1997). [CrossRef] [PubMed]

OCIS Codes
(160.6000) Materials : Semiconductor materials
(220.4610) Optical design and fabrication : Optical fabrication
(230.4170) Optical devices : Multilayers
(260.1440) Physical optics : Birefringence
(350.2460) Other areas of optics : Filters, interference

ToC Category:
Optical Devices

History
Original Manuscript: July 30, 2008
Revised Manuscript: September 7, 2008
Manuscript Accepted: September 13, 2008
Published: September 17, 2008

Citation
Nobuyuki Ishikura, Minoru Fujii, Kohei Nishida, Shinji Hayashi, and Joachim Diener, "Dichroic rugate filters based on birefringent porous silicon," Opt. Express 16, 15531-15539 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-15531


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References

  1. F. Genereux, S.W. Leonard, H. M. van Driel, A. Birner, and U. Gösele, "Large birefringence in two-dimensional silicon photonic crystals," Phys. Rev. B 63, 161101 (2001). [CrossRef]
  2. F. Xu, R. C. Tyan, P. C. Sun, Y. Fainman, C. C. Cheng, and A. Scherer, "Fabrication, modeling, and characterization of form-birefringent nanostructures," Opt. Lett. 20, 2457-2459 (1995). [CrossRef] [PubMed]
  3. N. Künzner, D. Kovalev, J. Diener, E. Gross, V. Yu. Timoshenko, G. Polisski, F. Koch, and M. Fujii, "Giant birefringence in anisotropically nanostructured silicon," Opt. Lett. 26, 1265-1267 (2001). [CrossRef]
  4. J. Diener, N. Künzner, D. Kovalev, E. Gross, V. Yu. Timoshenko, G. Polisski, and F. Koch, "Dichroic Bragg reflectors based on birefringent porous silicon," Appl. Phys. Lett. 78, 3887-3889 (2001). [CrossRef]
  5. J. Diener, N. Künzner, D. Kovalev, E. Gross, F. Koch, and M. Fujii, "Dichroic behavior of multilayer structures based on anisotropically nanostructured silicon," J. Appl. Phys. 91, 6704-6709 (2002). [CrossRef]
  6. J. Diener, N. Künzner, E. Gross, D. Kovalev, and M. Fujii, "Planar silicon-based light polarizers," Opt. Lett. 29, 195-197 (2004). [CrossRef] [PubMed]
  7. B. G. Bovard, "Rugate filter theory: an overview," Appl. Opt. 32, 5427-5442 (1993). [CrossRef] [PubMed]
  8. M. G. Berger, R. Arens-Fischer, M. Thönissen, M. Krüger, S. Billat, H. Lüth, S. Hilbrich,W. Theiβ, and P. Grosse, "Dielectric filters made of PS: advanced performance by oxidation and new layer structures," Thin Solid Films 297, 237-240 (1997). [CrossRef]
  9. E. Lorenzo, C. J. Oton, N. E. Capuj, M. Ghulinyan, D. Navarro-Urrios, Z. Gaburro, and L. Pavesi, "Porous silicon-based rugate filters," Appl. Opt. 44, 5415-5421 (2005). [CrossRef] [PubMed]
  10. M. S. Salem, M. J. Sailor, T. Sakka, and Y. H. Ogata, "Electrochemical preparation of a rugate filter in silicon and its deviation from the ideal structure," J. Appl. Phys. 101, 063503 (2007). [CrossRef]
  11. N. Künzner, J. Diener, E. Gross, D. Kovalev, V. Yu. Timoshenko, and M. Fujii, "Form birefringence of anisotropically nanostructured silicon," Phys. Rev. B 71, 195304 (2005). [CrossRef]
  12. W. H. Southwell, "Using apodization functions to reduce sidelobes in rugate filters," Appl. Opt. 28, 5091-5094 (1989). [CrossRef] [PubMed]
  13. H. A. Abu-Safia, A. I. Al-Sharif, and I. O. Abu Alijarayesh, "Rugate filter sidelobe suppression using halfapodization," Appl. Opt. 32, 4831-4835 (1993). [CrossRef] [PubMed]
  14. W. H. Southwell, "Gradient-index antireflection coatings," Opt. Lett. 8, 584-586 (1983). [CrossRef] [PubMed]
  15. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).
  16. J. A. Dobrowolski and D. Lowe, "Optical thin film synthesis program based on the use of Fourier transforms," Appl. Opt. 17, 3039-3050 (1978). [CrossRef] [PubMed]
  17. A. G. Imenes and D. R. McKenzie, "Flat-topped broadband rugate filters," Appl. Opt. 45, 7841-7850 (2006). [CrossRef] [PubMed]
  18. W. H. Southwell, "Extended-bandwidth reflector designs by using wavelets," Appl. Opt. 36, 314-318 (1997). [CrossRef] [PubMed]

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