Rugate notch filter fabricated by atomic layer deposition Yanghui Li,1,* Weidong Shen,2,3 Xiang Hao,2 Tingting Lang,1 Shangzhong Jin,1 and Xu Liu2 1
College of Optical and Electronic Technology, China Jiliang University, Hangzhou 310018, China
2
State Key Laboratory of Modern Optical Instrumentation, Zhejiang University, Hangzhou 310027, China 3
e-mail: [email protected]
*Corresponding author: [email protected] Received 22 August 2013; accepted 21 November 2013; posted 26 November 2013 (Doc. ID 195988); published 15 January 2014
A Rugate notch filter is fabricated by atomic layer deposition. By regulating the thickness ratio of TiO2 and Al2 O3 in a nanoscale layer, the refractive index is tailored between the refractive indices of the two materials. With the method of equivalent refractive index, the continuously variable refractive index of the designed Rugate filter is dispersed into several discrete ones, so that it can be realized by the refractive index tailoring. To coincide with the thickness, the nanoscale layer is iteratively deposited in the equivalent layer. The experimental reflectance matches the designed one well, and the average reflectance is 86.7% (510–590 nm). © 2014 Optical Society of America OCIS codes: (310.1860) Deposition and fabrication; (310.4165) Multilayer design; (310.6805) Theory and design. http://dx.doi.org/10.1364/AO.53.00A270
1. Introduction
Atomic layer deposition (ALD), based on sequential self-terminating gas–solid reactions, is a kind of optimized chemical vapor deposition method. During deposition, precursors enter into the reactor chamber and react with the active surface moieties while excess precursors and reaction byproducts are eliminated by intermediate purge steps with inert gases. Through ligand exchange, the solid film is produced, and the byproduct is carried out by inert gas [1]. In the past, ALD has been widely used in the area of semiconductors. Because ALD has incomparable conformity and deposition precision at the atomic scale, the applications of ALD in optics has attracted more and more attention around the world [2–4]. Unlike semiconductors, the thickness of the coating in optics is much thicker, and its requirement to the stability 1559-128X/14/04A270-06$15.00/0 © 2014 Optical Society of America A270
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of the ALD system is stricter, which blocks its application in optical coating. At present, ALD has only confirmed its reliability in several simple cases, such as antireflection coating at single point and broadband antireflection coating [5–7]. It is still generally regarded as a great challenge to use ALD for complex optical coating fabrication [7–10]. The Rugate filter is a kind of filter, designed by Fourier transform, whose refractive index varies as a sine curve along the direction of thickness [11–14]. Compared with homogeneous coating, this coating design method can realize extremely narrow passband and steep edge [15]. Simultaneously, it has good performances on mechanical stress. In most situations, a Rugate filter consists of very complicated quasi-sublayers, and its typical thickness reaches the micron micrometer scale, which puts forward the high requirement on the stability and precision of deposition equipment. Hence, to realize a Rugate filter, the refractive index tailoring is necessary. Previously, composite film with the preparation of mixed
materials is the most common manner to fabricate Rugate filter, and its refractive index is determined by adjusting composition ratio of different component [16,17]. Generally, both magnetron sputtering and ion beam sputtering are proper to deposit such composite films. However, its control accuracy is comparatively low. On the other hand, films with adjustable refractive index can also be obtained by alternant thin layers of high and low refractive indices, if the thicknesses of these layers are much smaller than light wavelength [18–20]. To explicitly tailor the value of refractive index as expected, the straightforward solution is adjusting the thickness ratio between the high and low refractive index layers. In each cycle of ALD, the growth rate approximates to the thickness of monoatomic layer theoretically. Its thickness control can be very precise without monitoring, and make it possible to fabricate Rugate filter. In this paper, ALD is adopted to fabricate a Rugate notch filter. With the refractive indices of TiO2 and Al2 O3 deposited by ALD, the Rugate notch filter at 510–590 nm is designed, and its total physical thickness reaches 1.2 μm. The method of equivalent layer is applied to disperse the continuously varied refractive index, which makes the manufacturing possible. To obtain these discrete refractive indices in practice, the comparative thickness of TiO2 and Al2 O3 in nanoscale layers is adjusted. The deposition is carried out without monitoring, and the experimentally average reflectance is above 86% at 510–590 nm, which coincides well with the designed one. 2. Design of Rugate Notch Filter
For the traditional Needle method, the refractive indices of materials are constants, while only the thicknesses and numbers of layers are variable during the optimization procedure. The Rugate filter is a technique of coating design that directly combines refractive index with the spectrum properties by Fourier transform in Eq. (1). Z 2 ∞ Qk sinϕk − kx · dk : nx exp π 0 k
(1)
Here n is the refractive index, x is twice the optical path, λ is the reference wavelength, k 2π∕λ, and ϕk is a phase factor that must be an odd function to ensure that nx is real. Qk is a suitable even function of the desired transmittance T expressed as a function of the wave vector k. The exact expression of Qk is difficult to derive, but the approximate value of Qk can be obtained in many ways [21]. In this paper, to get better results, the Sossi’s definition in Eq. (2) is adopted here during coating design [11]: 1∕2 1 − Tk Qk 1∕2 : Tk
(2)
Compared with the Needle method, the refractive indices in Rugate filter can be regulated between the
high and low refractive indices, thereby supplying additional flexibility to achieve the designing goal. The interfaces of materials with high and low refractive indices are not as evident as the general ones, which is propitious to release the stress in multilayer coating. To fabricate Rugate filters, tailoring refractive index is always considered as the primary step. Generally, real-time adjustment of the ratio of high and low refractive index materials during coating deposition is the most widely used method to realize this target, but many factors will affect their deposited precision [16,17]. As an alternative, here ALD is proposed to solve this problem. In ALD, a single growth cycle corresponds to a monoatomic layer, which makes it possible to precisely fabricate the layer consisted of alternating high and low refractive index materials in nanoscale. The refractive index of this nanoscale layer can be manipulated by adjusting the thickness ratio of the two materials. With linear interpolation, the refractive index of this layer is describe as [18]: n nH × dH nL × dL ∕dH dL :
(3)
Here nH and nL represent the refractive indices of high and low refractive index materials at reference wavelength. dH and dL are the physical thicknesses of high and low refractive index materials in this nanoscale layer, respectively. In this paper, TiO2 and Al2 O3 deposited by ALD are chosen as high and low refractive index materials. When the substrate temperature is 120°C, the refractive index is about 2.42 for TiO2, while 1.61 for Al2 O3 at 550 nm, respectively. To evaluate the accuracy of linear interpolation and provide initial value of optimization, mixed coatings consisting of alternating nanoscale layers are deposited. For convenience, the number of cycles for Al2 O3 in every nanoscale layer is set as 10, while the number of cycles for TiO2 is varied corresponding to the required refractive index. With the Cauchy dispersion model, these refractive indices are calculated in Fig. 1. The solid line and dashed curves represented theoretical and experimental refractive indices, respectively. Both of the two data sets coincide with each other well, but some deviations still exist, which are mainly due to the measured errors of ellipsometry. These results verify that the accuracy of linear interpolation in nanoscale layer is reliable, and the precision of ALD can satisfy the requirement of Rugate filter fabrication. The Rugate notch filter is designed with the experimental refractive indices mentioned above. To efficiently suppress the sidelobe ripple, a quintic apodization function is adopted during optimization. In theory, the thickness increasing of the Rugate notch filter is propitious to obtain sharper edge of stopband and narrower bandwidth [17,22]. However, for ALD, its growth rate and the capacity of its precursor temporarily limit its application on 1 February 2014 / Vol. 53, No. 4 / APPLIED OPTICS
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3. Equivalency of Refractive Index
Fig. 1. Refractive indices with different thickness ratios. N T and N A separately represent the number of cycles for TiO2 and Al2 O3 in the nanoscale layer. Solid curves correspond to experimental data and dashed curves to calculated data according to the linear interpolation.
excessively thick optical coatings. Therefore, as a new attempt, the designing bandwidth here is limited within a range of 510–590 nm [12,21], and the stability of deposition is guaranteed at the expense of the optical performance. Some optical properties of the filter are moderately weakened to restrict the thickness of the Rugate notch filter no larger than 1500 nm. The ultimate thickness reaches 1216 nm, and the refractive index profile is presented in Fig. 2. Different from common coating type, the refractive index along the thickness direction is sine distributed, and no evident sandwich-like structure exists in this filter. The index profile contains 17 semi-periods of sine curve and is symmetrical about the center of the thickness. The maximal and minimum refractive indices correspond to the refractive indices of TiO2 and Al2 O3 . The reflectance of this notch filter is shown in Fig. 3. In the range of 510–590 nm, the average reflectance is above 90%. Because of the constraint on thickness during optimization, the performances of this notch filter on reflectance and sidelobe have some discrepancy from the ones whose thicknesses are up to several tens micron.
It is practically difficult to deposit continuous refractive index in Fig. 2 by ALD. The method of equivalent refractive index is adopted to implement this refractive index. With the average value of this refractive index (1.96), the sine curve is divided into 17 semiperiods. In each semi-period, the sine curve is centrosymmetric. It indicates that each semi-period has the property of single layer coating, and its refractive index can be substituted by a constant. The specific solution is shown in Fig. 4, along the direction of thickness; the refractive index is sliced into 101 sublayers. The precision of equivalency can be enhanced by increasing the slice number. The refractive index of the sublayer is determined by its average value conveniently. With such dispersing, the refractive indices of these sublayers are symmetrical about the center of the semi-period. Meanwhile, the thicknesses of sublayers are the same, which demonstrates that the sublayers in every semi-period are symmetrical. According to Herpin’s theorem, a symmetrical thin-film combination is equivalent, at one wavelength, to a single film. The matrix applied to describe such single homogeneous film has M 11 M 22 [23,24]. In this way, the symmetrical thin-film combination can be characterized by an equivalent index (N) and equivalent thickness (γ) [23,24]: M 11 M 22 cos γ;
(4)
N i sin γ∕M 12 −iM 21 ∕ sin γ:
(5)
For a symmetrical film system with three layers (pqp), the elements in the matrix are illustrated as follows: M 11 M 22 cos 2φp cos φq 1 − nq ∕np np ∕nq sin 2φp sin φq ; 2
Fig. 2. Refractive index profile of the Rugate notch filter. A272
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(6)
Fig. 3. Theoretical reflectance, equivalent reflectance, and experimental reflectance of the Rugate notch filter.
M 12
8 9 sin 2φp cos φq > > > > < = 1 i∕np 2 np ∕nq nq ∕np cos 2φp sin φq ; > > > > : 1 n ∕n − n ∕n sin φ ; 2
p
q
q
p
q
(7)
M 21
8 9 sin 2φp cos φq > > > > < = 1 inp 2 np ∕nq nq ∕np cos 2φp sin φq : (8) > > > > : − 1 n ∕n − n ∕n sin φ ; 2
p
q
q
p
q
Through consecutive iteration with this method, a symmetrical film with more layers can also be replaced by a single layer coating [23]. In our experiment, the equivalence starts from the middle sublayer in every semi-period. The 50 sublayer, the 52 sublayer, and the 51 layer are first chosen to be equivalent to a single layer with Eqs. (6)–(8). Moreover, the refractive indices of this equivalent single layer, the 49 layer, and the 53 layer are also centrosymmetric and can be equivalent to a new single layer. In this way, the 101 sublayers in every semiperiod can be iteratively equivalent to a single layer with a constant refractive index, illustrated in Fig. 5.
Fig. 4. Equivalent method of symmetrical film system.
After the simplification above, the continuously varied refractive index in Fig. 2 is substituted for by 17 discrete values shown in Fig. 5. Actually, the 17 refractive indices are centrosymmetric, so only eight diverse refractive indices are left. Though such equivalent method simplifies the practical fabrication, slicing and dispersing the continuous refractive index inevitably introduce reflectance errors. To estimate its accuracy, the reflectance of the notch filter with the equivalent refractive indices is calculated in Fig. 3. Compared with the Rugate notch filter, the reflectance with equivalent refractive index matches the original one well, but some discrepancies still exist. The main reason stems from dispersion. The equivalent refractive index is only for the central wavelength 550 nm, and the refractive indices at other wavelengths are not taken into consideration. The effect of dispersion depends on numbers of sampling points, namely the sublayer in each semi-period. A smaller effect originates from a larger sampling number. However, excessively dense sampling points will increase the computational complexity. Therefore, it is important to make a tradeoff between them. 4. Experiment
The experiment is carried out by a TFS200 reactor made by Beneq, Finland. Trimethyl aluminum [TMA, AlCH3 ], titanium tetrachloride (TiCl4 ), and deionized water are selected as aluminum, titanium, and oxygen reactant precursors, respectively. During coating deposition, the temperature of the precursors are kept at 20°C. To obtain the amorphous TiO2, 120°C is set as deposition temperature for both TiO2 and Al2 O3 [25]. Because the reaction in ALD has the characteristic of self-limiting, the deposition has very excellent repeatability, and time monitoring is adopted in most of the equipment. In ALD, the growth rate is defined by the growing thickness in each cycle. Through counting the numbers of deposited cycle, the thicknesses of TiO2 (dTiO2 ) and Al2 O3 (dAl2 O3 ) are calculated: dTiO2 RTiO2 × N TiO2 ;
(9)
Fig. 5. Equivalent refractive indices. 1 February 2014 / Vol. 53, No. 4 / APPLIED OPTICS
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Table 1.
Layer 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Deposited Parameters for Equivalent Refractive Index of the Rugate Notch Filter
Equivalent Refractive Index
Numbers of Cycle for TiO2
T
2.00 1.83 2.07 1.76 2.14 1.71 2.20 1.68 2.22 1.68 2.20 1.71 2.14 1.76 2.07 1.83 2.00
20 8 29 5 41 3 57 2 65 2 57 3 41 5 29 8 20
21 46 20 51 17 55 13 58 12 58 13 55 17 51 20 46 23
dAl2 O3 RAl2 O3 × N Al2 O3 :
(10)
Here the growth rates of TiO2 and Al2 O3 are expressed as RTiO2 and RAl2 O3 . The growth rate of TiO2 is 0.59 nm∕cycle while 0.13 nm∕cycle for Al2 O3 . The time for depositing each TiO2 cycle is 4.4 seconds while 5.2 seconds for Al2 O3. Apparently, the growth rate is in atomic scale, which provides the possibility for the tailoring of refractive index in nanoscale. N TiO2 and N Al2 O3 are the numbers of cycle to deposit TiO2 and Al2 O3 in one nanoscale layer. To obtain the refractive index in Fig. 5, the ratio of N TiO2 to N Al2 O3 is correspondingly varied. For convenience, N Al2 O3 is usually fixed, while N TiO2 is adjusted according to the specific refractive index of the nanoscale layer. Theoretically, the experiment presents better accuracy if the layer is thinner. To make a tradeoff between the precision and efficiency, N Al2 O3 is 10, and dAl2 O3 is about 1.31 nm in our experiment. From Eq. (3), we know that, to acquire the equivalent refractive indexn in Fig. 5, the only undecided parameter dTiO2 is easily calculated. The sum of dAl2 O3 and dTiO2 is the thickness of the nanoscale layer in the equivalent refractive index layer. Combined with the thicknesses of each equivalent refractive index (d) in Fig. 5, the numbers of nanoscale layer (T) in every equivalent refractive index layer are confirmed: T d∕dTiO2 dAl2 O3 :
(11)
For the equivalent notch filter, the explicitly deposited parameters of each layer are shown in Table 1. The third column in this table exhibits the numbers of cycle for TiO2 in one nanoscale layer, and the forth column presents the iterative times of nanoscale layer to attain the thickness of each equivalent layer. The experimental result is illustrated in Fig. 3. Except a little decrease of reflectance in reflection A274
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bandwidth, the curve basically coincides well with the design one, and the average reflectance is 86.7% in the range of 510–590 nm. The deposition mechanism of ALD results in the existence of many interfaces. Generally, the superfluous precursors are applied to guarantee the saturated reaction in ALD; thus it is difficult to avoid the residual precursors on the interfaces. As a result, scattering and refractive index inhomogeneity easily occur at these interfaces, which make the experimental reflectance deviate from the designed one. 5. Conclusion
The feasibility of Rugate notch filter fabricated by ALD was validated in this paper. Compared with other deposition methods, ALD allows a better control of thickness and refractive index to realize Rugate filters because of its growth rate of atomic scale, and the equipment was simpler. Through adjusting the thickness ratio of high and low refractive index materials in nanoscale, the refractive index tailoring was achieved. With this variable refractive index technology, the Rugate filter was fabricated by ALD, and the result was as good as the theoretical prediction. Taking previous multilayer coating fabricated by ALD for reference, this Rugate filter included much more interfaces and was more complicated which promoted the application of ALD in optical coating. The development and maturity of ALD technology will inspire its application on more elaborate optical coatings, and better results will be expected. This work was financially supported by grants from the National High Technology Research and Development Program 863(2012AA040401), National Science & Technology Pillar Program of China (Grant No. 2011BAF06B02-1), Public Program of General Administration of Quality Supervision, Inspection and Quarantine of the People’s Republic of China (Grant No. 201210094). References 1. T. Suntola, “Atomic layer epitaxy,” Thin Solid Films 216, 84–89 (1992). 2. S. Zaitsu, S. Motokoshi, T. Jitsuno, M. Nakatsuka, and T. Yamanaka, “Large-area optical coatings with uniform thickness grown by surface chemical reactions for high-power laser applications,” Jpn. J. Appl. Phys. 41, 160–165 (2002). 3. M. Ritala, M. Leskela, J. P. Dekker, C. Mutsaers, P. J. Soininen, and J. Skarp, “Perfectly conformal TiN and Al2O3 films deposited by atomic layer deposition,” Chem. Vapor. Depos. 5, 7–9 (1999). 4. N. T. Gabriel, S. S. Kim, and J. J. Talghader, “Control of thermal deformation in dielectric mirrors using mechanical design and atomic layer deposition,” Opt. Lett. 34, 1958–1960 (2009). 5. Y. H. Li, W. D. Shen, Y. G. Zhang, X. Hao, H. H. Fan, and X. Liu, “Precise broadband anti-refection coating fabricated by atomic layer deposition,” Opt. Commun. 292, 31–35 (2013). 6. D. Riihelä, M. Ritala, R. Matero, and M. Leskelä, “Introducing atomic layer epitaxy for the deposition of optical thin films,” Thin Solid Films 289, 250–255 (1996). 7. A. Szeghalmi, M. Helgert, R. Brunner, F. Heyroth, U. Gösele, and M. Knez, “Atomic layer deposition of Al2O3 and TiO2 multilayers for applications as bandpass filters and antireflection coatings,” Appl. Opt. 48, 1727–1732 (2009).
8. Y. W. Wei, H. Liu, O. Y. Sheng, Z. C. Liu, S. L. Chen, and L. M. Yang, “Laser damage properties of TiO2/Al2O3 thin films grown by atomic layer deposition,” Appl. Opt. 50, 4720–4727 (2011). 9. Z. C. Liu, S. L. Chen, P. Ma, Y. W. Wei, Y. Zheng, F. Pan, H. Liu, and G. Y. Tang, “Characterization of 1064 nm nanosecond laser-induced damage on antireflection coatings grown by atomic layer deposition,” Opt. Express 20, 854–863 (2012). 10. D. Riihela, M. Ritala, R. Matero, and M. Leskela, “Introducing atomic layer epitaxy for the deposition of optical thin films,” Thin Solid Films 289, 250–255 (1996). 11. P. G. Verly, “Fourier-transform technique with frequency filtering for optical thin-film design,” Appl. Opt. 34, 688–694 (1995). 12. B. G. Bovard, “Rugate filter design—the modified Fouriertransform technique,” Appl. Opt. 29, 24–30 (1990). 13. H. Fabricius, “Gradient-index filters—designing filters with steep skirts, high reflection, and quintic matching layers,” Appl. Opt. 31, 5191–5196 (1992). 14. A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “Application of constrained optimization to the design of quasi-Rugate optical coatings,” Appl. Opt. 47, 5103–5109 (2008). 15. W. H. Southwell, “Using apodization functions to reduce sidelobes in Rugate filters,” Appl. Opt. 28, 5091–5094 (1989). 16. S. Lim, S. Shih, and J. F. Wager, “Design and fabrication of a double bandstop Rugate, filter grown by plasma-enhanced chemical vapor deposition,” Thin Solid Films 277, 144–146 (1996).
17. N. Perelman and I. Averbukh, “Rugate filter design: an analytical approach using uniform WKB solutions,” J. Appl. Phys. 79, 2839–2845 (1996). 18. S. Zaitsu, T. Jitsuno, M. Nakatsuka, T. Yamanaka, and S. Motokoshi, “Optical thin films consisting of nanoscale laminated layers,” Appl. Phys. Lett. 80, 2442–2444 (2002). 19. J. C. Zhang, M. Fang, Y. C. Shao, Y. X. Jin, and H. B. He, “Design and fabrication of broadband Rugate filter,” Chin. Phys. B 21, 054219 (2012). 20. P. G. Verly, “Hybrid approach for Rugate filter design,” Appl. Opt. 47, C172–C178 (2008). 21. S. Larouche and L. Martinu, “OpenFilters: open-source software for the design, optimization, and synthesis of optical filters,” Appl. Opt. 47, C219–C230 (2008). 22. L. Zhao, Z. Sui, Q. H. Zhu, Y. Zhang, and Y. L. Zuo, “Design of Rugate filter for gain narrowing compensation,” Acta Phys. Sin. 58, 3977–3982 (2009). 23. F. Aguayo-Rios, F. Villa-Villa, and J. A. Gaspar-Armenta, “One-dimensional photonic crystals of inhomogeneous thin films: band structure of Rugate filters,” Opt. Commun. 244, 259–267 (2005). 24. L. I. Epstein, “The design of optical filters,” J. Opt. Soc. Am. 42, 806–808 (1952). 25. H. Kumagai, K. Toyoda, K. Kobayashi, M. Obara, and Y. Iimura, “Titanium oxide aluminum oxide multilayer reflectors for “water-window” wavelengths,” Appl. Phys. Lett. 70, 2338–2340 (1997).
1 February 2014 / Vol. 53, No. 4 / APPLIED OPTICS
A275
College of Optical and Electronic Technology, China Jiliang University, Hangzhou 310018, China
2
State Key Laboratory of Modern Optical Instrumentation, Zhejiang University, Hangzhou 310027, China 3
e-mail: [email protected]
*Corresponding author: [email protected] Received 22 August 2013; accepted 21 November 2013; posted 26 November 2013 (Doc. ID 195988); published 15 January 2014
A Rugate notch filter is fabricated by atomic layer deposition. By regulating the thickness ratio of TiO2 and Al2 O3 in a nanoscale layer, the refractive index is tailored between the refractive indices of the two materials. With the method of equivalent refractive index, the continuously variable refractive index of the designed Rugate filter is dispersed into several discrete ones, so that it can be realized by the refractive index tailoring. To coincide with the thickness, the nanoscale layer is iteratively deposited in the equivalent layer. The experimental reflectance matches the designed one well, and the average reflectance is 86.7% (510–590 nm). © 2014 Optical Society of America OCIS codes: (310.1860) Deposition and fabrication; (310.4165) Multilayer design; (310.6805) Theory and design. http://dx.doi.org/10.1364/AO.53.00A270
1. Introduction
Atomic layer deposition (ALD), based on sequential self-terminating gas–solid reactions, is a kind of optimized chemical vapor deposition method. During deposition, precursors enter into the reactor chamber and react with the active surface moieties while excess precursors and reaction byproducts are eliminated by intermediate purge steps with inert gases. Through ligand exchange, the solid film is produced, and the byproduct is carried out by inert gas [1]. In the past, ALD has been widely used in the area of semiconductors. Because ALD has incomparable conformity and deposition precision at the atomic scale, the applications of ALD in optics has attracted more and more attention around the world [2–4]. Unlike semiconductors, the thickness of the coating in optics is much thicker, and its requirement to the stability 1559-128X/14/04A270-06$15.00/0 © 2014 Optical Society of America A270
APPLIED OPTICS / Vol. 53, No. 4 / 1 February 2014
of the ALD system is stricter, which blocks its application in optical coating. At present, ALD has only confirmed its reliability in several simple cases, such as antireflection coating at single point and broadband antireflection coating [5–7]. It is still generally regarded as a great challenge to use ALD for complex optical coating fabrication [7–10]. The Rugate filter is a kind of filter, designed by Fourier transform, whose refractive index varies as a sine curve along the direction of thickness [11–14]. Compared with homogeneous coating, this coating design method can realize extremely narrow passband and steep edge [15]. Simultaneously, it has good performances on mechanical stress. In most situations, a Rugate filter consists of very complicated quasi-sublayers, and its typical thickness reaches the micron micrometer scale, which puts forward the high requirement on the stability and precision of deposition equipment. Hence, to realize a Rugate filter, the refractive index tailoring is necessary. Previously, composite film with the preparation of mixed
materials is the most common manner to fabricate Rugate filter, and its refractive index is determined by adjusting composition ratio of different component [16,17]. Generally, both magnetron sputtering and ion beam sputtering are proper to deposit such composite films. However, its control accuracy is comparatively low. On the other hand, films with adjustable refractive index can also be obtained by alternant thin layers of high and low refractive indices, if the thicknesses of these layers are much smaller than light wavelength [18–20]. To explicitly tailor the value of refractive index as expected, the straightforward solution is adjusting the thickness ratio between the high and low refractive index layers. In each cycle of ALD, the growth rate approximates to the thickness of monoatomic layer theoretically. Its thickness control can be very precise without monitoring, and make it possible to fabricate Rugate filter. In this paper, ALD is adopted to fabricate a Rugate notch filter. With the refractive indices of TiO2 and Al2 O3 deposited by ALD, the Rugate notch filter at 510–590 nm is designed, and its total physical thickness reaches 1.2 μm. The method of equivalent layer is applied to disperse the continuously varied refractive index, which makes the manufacturing possible. To obtain these discrete refractive indices in practice, the comparative thickness of TiO2 and Al2 O3 in nanoscale layers is adjusted. The deposition is carried out without monitoring, and the experimentally average reflectance is above 86% at 510–590 nm, which coincides well with the designed one. 2. Design of Rugate Notch Filter
For the traditional Needle method, the refractive indices of materials are constants, while only the thicknesses and numbers of layers are variable during the optimization procedure. The Rugate filter is a technique of coating design that directly combines refractive index with the spectrum properties by Fourier transform in Eq. (1). Z 2 ∞ Qk sinϕk − kx · dk : nx exp π 0 k
(1)
Here n is the refractive index, x is twice the optical path, λ is the reference wavelength, k 2π∕λ, and ϕk is a phase factor that must be an odd function to ensure that nx is real. Qk is a suitable even function of the desired transmittance T expressed as a function of the wave vector k. The exact expression of Qk is difficult to derive, but the approximate value of Qk can be obtained in many ways [21]. In this paper, to get better results, the Sossi’s definition in Eq. (2) is adopted here during coating design [11]: 1∕2 1 − Tk Qk 1∕2 : Tk
(2)
Compared with the Needle method, the refractive indices in Rugate filter can be regulated between the
high and low refractive indices, thereby supplying additional flexibility to achieve the designing goal. The interfaces of materials with high and low refractive indices are not as evident as the general ones, which is propitious to release the stress in multilayer coating. To fabricate Rugate filters, tailoring refractive index is always considered as the primary step. Generally, real-time adjustment of the ratio of high and low refractive index materials during coating deposition is the most widely used method to realize this target, but many factors will affect their deposited precision [16,17]. As an alternative, here ALD is proposed to solve this problem. In ALD, a single growth cycle corresponds to a monoatomic layer, which makes it possible to precisely fabricate the layer consisted of alternating high and low refractive index materials in nanoscale. The refractive index of this nanoscale layer can be manipulated by adjusting the thickness ratio of the two materials. With linear interpolation, the refractive index of this layer is describe as [18]: n nH × dH nL × dL ∕dH dL :
(3)
Here nH and nL represent the refractive indices of high and low refractive index materials at reference wavelength. dH and dL are the physical thicknesses of high and low refractive index materials in this nanoscale layer, respectively. In this paper, TiO2 and Al2 O3 deposited by ALD are chosen as high and low refractive index materials. When the substrate temperature is 120°C, the refractive index is about 2.42 for TiO2, while 1.61 for Al2 O3 at 550 nm, respectively. To evaluate the accuracy of linear interpolation and provide initial value of optimization, mixed coatings consisting of alternating nanoscale layers are deposited. For convenience, the number of cycles for Al2 O3 in every nanoscale layer is set as 10, while the number of cycles for TiO2 is varied corresponding to the required refractive index. With the Cauchy dispersion model, these refractive indices are calculated in Fig. 1. The solid line and dashed curves represented theoretical and experimental refractive indices, respectively. Both of the two data sets coincide with each other well, but some deviations still exist, which are mainly due to the measured errors of ellipsometry. These results verify that the accuracy of linear interpolation in nanoscale layer is reliable, and the precision of ALD can satisfy the requirement of Rugate filter fabrication. The Rugate notch filter is designed with the experimental refractive indices mentioned above. To efficiently suppress the sidelobe ripple, a quintic apodization function is adopted during optimization. In theory, the thickness increasing of the Rugate notch filter is propitious to obtain sharper edge of stopband and narrower bandwidth [17,22]. However, for ALD, its growth rate and the capacity of its precursor temporarily limit its application on 1 February 2014 / Vol. 53, No. 4 / APPLIED OPTICS
A271
3. Equivalency of Refractive Index
Fig. 1. Refractive indices with different thickness ratios. N T and N A separately represent the number of cycles for TiO2 and Al2 O3 in the nanoscale layer. Solid curves correspond to experimental data and dashed curves to calculated data according to the linear interpolation.
excessively thick optical coatings. Therefore, as a new attempt, the designing bandwidth here is limited within a range of 510–590 nm [12,21], and the stability of deposition is guaranteed at the expense of the optical performance. Some optical properties of the filter are moderately weakened to restrict the thickness of the Rugate notch filter no larger than 1500 nm. The ultimate thickness reaches 1216 nm, and the refractive index profile is presented in Fig. 2. Different from common coating type, the refractive index along the thickness direction is sine distributed, and no evident sandwich-like structure exists in this filter. The index profile contains 17 semi-periods of sine curve and is symmetrical about the center of the thickness. The maximal and minimum refractive indices correspond to the refractive indices of TiO2 and Al2 O3 . The reflectance of this notch filter is shown in Fig. 3. In the range of 510–590 nm, the average reflectance is above 90%. Because of the constraint on thickness during optimization, the performances of this notch filter on reflectance and sidelobe have some discrepancy from the ones whose thicknesses are up to several tens micron.
It is practically difficult to deposit continuous refractive index in Fig. 2 by ALD. The method of equivalent refractive index is adopted to implement this refractive index. With the average value of this refractive index (1.96), the sine curve is divided into 17 semiperiods. In each semi-period, the sine curve is centrosymmetric. It indicates that each semi-period has the property of single layer coating, and its refractive index can be substituted by a constant. The specific solution is shown in Fig. 4, along the direction of thickness; the refractive index is sliced into 101 sublayers. The precision of equivalency can be enhanced by increasing the slice number. The refractive index of the sublayer is determined by its average value conveniently. With such dispersing, the refractive indices of these sublayers are symmetrical about the center of the semi-period. Meanwhile, the thicknesses of sublayers are the same, which demonstrates that the sublayers in every semi-period are symmetrical. According to Herpin’s theorem, a symmetrical thin-film combination is equivalent, at one wavelength, to a single film. The matrix applied to describe such single homogeneous film has M 11 M 22 [23,24]. In this way, the symmetrical thin-film combination can be characterized by an equivalent index (N) and equivalent thickness (γ) [23,24]: M 11 M 22 cos γ;
(4)
N i sin γ∕M 12 −iM 21 ∕ sin γ:
(5)
For a symmetrical film system with three layers (pqp), the elements in the matrix are illustrated as follows: M 11 M 22 cos 2φp cos φq 1 − nq ∕np np ∕nq sin 2φp sin φq ; 2
Fig. 2. Refractive index profile of the Rugate notch filter. A272
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(6)
Fig. 3. Theoretical reflectance, equivalent reflectance, and experimental reflectance of the Rugate notch filter.
M 12
8 9 sin 2φp cos φq > > > > < = 1 i∕np 2 np ∕nq nq ∕np cos 2φp sin φq ; > > > > : 1 n ∕n − n ∕n sin φ ; 2
p
q
q
p
q
(7)
M 21
8 9 sin 2φp cos φq > > > > < = 1 inp 2 np ∕nq nq ∕np cos 2φp sin φq : (8) > > > > : − 1 n ∕n − n ∕n sin φ ; 2
p
q
q
p
q
Through consecutive iteration with this method, a symmetrical film with more layers can also be replaced by a single layer coating [23]. In our experiment, the equivalence starts from the middle sublayer in every semi-period. The 50 sublayer, the 52 sublayer, and the 51 layer are first chosen to be equivalent to a single layer with Eqs. (6)–(8). Moreover, the refractive indices of this equivalent single layer, the 49 layer, and the 53 layer are also centrosymmetric and can be equivalent to a new single layer. In this way, the 101 sublayers in every semiperiod can be iteratively equivalent to a single layer with a constant refractive index, illustrated in Fig. 5.
Fig. 4. Equivalent method of symmetrical film system.
After the simplification above, the continuously varied refractive index in Fig. 2 is substituted for by 17 discrete values shown in Fig. 5. Actually, the 17 refractive indices are centrosymmetric, so only eight diverse refractive indices are left. Though such equivalent method simplifies the practical fabrication, slicing and dispersing the continuous refractive index inevitably introduce reflectance errors. To estimate its accuracy, the reflectance of the notch filter with the equivalent refractive indices is calculated in Fig. 3. Compared with the Rugate notch filter, the reflectance with equivalent refractive index matches the original one well, but some discrepancies still exist. The main reason stems from dispersion. The equivalent refractive index is only for the central wavelength 550 nm, and the refractive indices at other wavelengths are not taken into consideration. The effect of dispersion depends on numbers of sampling points, namely the sublayer in each semi-period. A smaller effect originates from a larger sampling number. However, excessively dense sampling points will increase the computational complexity. Therefore, it is important to make a tradeoff between them. 4. Experiment
The experiment is carried out by a TFS200 reactor made by Beneq, Finland. Trimethyl aluminum [TMA, AlCH3 ], titanium tetrachloride (TiCl4 ), and deionized water are selected as aluminum, titanium, and oxygen reactant precursors, respectively. During coating deposition, the temperature of the precursors are kept at 20°C. To obtain the amorphous TiO2, 120°C is set as deposition temperature for both TiO2 and Al2 O3 [25]. Because the reaction in ALD has the characteristic of self-limiting, the deposition has very excellent repeatability, and time monitoring is adopted in most of the equipment. In ALD, the growth rate is defined by the growing thickness in each cycle. Through counting the numbers of deposited cycle, the thicknesses of TiO2 (dTiO2 ) and Al2 O3 (dAl2 O3 ) are calculated: dTiO2 RTiO2 × N TiO2 ;
(9)
Fig. 5. Equivalent refractive indices. 1 February 2014 / Vol. 53, No. 4 / APPLIED OPTICS
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Table 1.
Layer 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Deposited Parameters for Equivalent Refractive Index of the Rugate Notch Filter
Equivalent Refractive Index
Numbers of Cycle for TiO2
T
2.00 1.83 2.07 1.76 2.14 1.71 2.20 1.68 2.22 1.68 2.20 1.71 2.14 1.76 2.07 1.83 2.00
20 8 29 5 41 3 57 2 65 2 57 3 41 5 29 8 20
21 46 20 51 17 55 13 58 12 58 13 55 17 51 20 46 23
dAl2 O3 RAl2 O3 × N Al2 O3 :
(10)
Here the growth rates of TiO2 and Al2 O3 are expressed as RTiO2 and RAl2 O3 . The growth rate of TiO2 is 0.59 nm∕cycle while 0.13 nm∕cycle for Al2 O3 . The time for depositing each TiO2 cycle is 4.4 seconds while 5.2 seconds for Al2 O3. Apparently, the growth rate is in atomic scale, which provides the possibility for the tailoring of refractive index in nanoscale. N TiO2 and N Al2 O3 are the numbers of cycle to deposit TiO2 and Al2 O3 in one nanoscale layer. To obtain the refractive index in Fig. 5, the ratio of N TiO2 to N Al2 O3 is correspondingly varied. For convenience, N Al2 O3 is usually fixed, while N TiO2 is adjusted according to the specific refractive index of the nanoscale layer. Theoretically, the experiment presents better accuracy if the layer is thinner. To make a tradeoff between the precision and efficiency, N Al2 O3 is 10, and dAl2 O3 is about 1.31 nm in our experiment. From Eq. (3), we know that, to acquire the equivalent refractive indexn in Fig. 5, the only undecided parameter dTiO2 is easily calculated. The sum of dAl2 O3 and dTiO2 is the thickness of the nanoscale layer in the equivalent refractive index layer. Combined with the thicknesses of each equivalent refractive index (d) in Fig. 5, the numbers of nanoscale layer (T) in every equivalent refractive index layer are confirmed: T d∕dTiO2 dAl2 O3 :
(11)
For the equivalent notch filter, the explicitly deposited parameters of each layer are shown in Table 1. The third column in this table exhibits the numbers of cycle for TiO2 in one nanoscale layer, and the forth column presents the iterative times of nanoscale layer to attain the thickness of each equivalent layer. The experimental result is illustrated in Fig. 3. Except a little decrease of reflectance in reflection A274
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bandwidth, the curve basically coincides well with the design one, and the average reflectance is 86.7% in the range of 510–590 nm. The deposition mechanism of ALD results in the existence of many interfaces. Generally, the superfluous precursors are applied to guarantee the saturated reaction in ALD; thus it is difficult to avoid the residual precursors on the interfaces. As a result, scattering and refractive index inhomogeneity easily occur at these interfaces, which make the experimental reflectance deviate from the designed one. 5. Conclusion
The feasibility of Rugate notch filter fabricated by ALD was validated in this paper. Compared with other deposition methods, ALD allows a better control of thickness and refractive index to realize Rugate filters because of its growth rate of atomic scale, and the equipment was simpler. Through adjusting the thickness ratio of high and low refractive index materials in nanoscale, the refractive index tailoring was achieved. With this variable refractive index technology, the Rugate filter was fabricated by ALD, and the result was as good as the theoretical prediction. Taking previous multilayer coating fabricated by ALD for reference, this Rugate filter included much more interfaces and was more complicated which promoted the application of ALD in optical coating. The development and maturity of ALD technology will inspire its application on more elaborate optical coatings, and better results will be expected. This work was financially supported by grants from the National High Technology Research and Development Program 863(2012AA040401), National Science & Technology Pillar Program of China (Grant No. 2011BAF06B02-1), Public Program of General Administration of Quality Supervision, Inspection and Quarantine of the People’s Republic of China (Grant No. 201210094). References 1. T. Suntola, “Atomic layer epitaxy,” Thin Solid Films 216, 84–89 (1992). 2. S. Zaitsu, S. Motokoshi, T. Jitsuno, M. Nakatsuka, and T. Yamanaka, “Large-area optical coatings with uniform thickness grown by surface chemical reactions for high-power laser applications,” Jpn. J. Appl. Phys. 41, 160–165 (2002). 3. M. Ritala, M. Leskela, J. P. Dekker, C. Mutsaers, P. J. Soininen, and J. Skarp, “Perfectly conformal TiN and Al2O3 films deposited by atomic layer deposition,” Chem. Vapor. Depos. 5, 7–9 (1999). 4. N. T. Gabriel, S. S. Kim, and J. J. Talghader, “Control of thermal deformation in dielectric mirrors using mechanical design and atomic layer deposition,” Opt. Lett. 34, 1958–1960 (2009). 5. Y. H. Li, W. D. Shen, Y. G. Zhang, X. Hao, H. H. Fan, and X. Liu, “Precise broadband anti-refection coating fabricated by atomic layer deposition,” Opt. Commun. 292, 31–35 (2013). 6. D. Riihelä, M. Ritala, R. Matero, and M. Leskelä, “Introducing atomic layer epitaxy for the deposition of optical thin films,” Thin Solid Films 289, 250–255 (1996). 7. A. Szeghalmi, M. Helgert, R. Brunner, F. Heyroth, U. Gösele, and M. Knez, “Atomic layer deposition of Al2O3 and TiO2 multilayers for applications as bandpass filters and antireflection coatings,” Appl. Opt. 48, 1727–1732 (2009).
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