Design optimization of volume holographic gratings for wavelength filters

Transcription

1 Design optimization of volume holographic gratings for wavelength filters WANG Bo *a, Chang Liang b, TAO Shiquan a a College of Applied Sciences, Beijing University of Technology, Beijing 1, China b College of Laser Engineering, Beijing University of Technology, Beijing 1, China ABSTACT Volume holography is promising for devices such as wavelength filters. However, in previously reported work with these holographic devices the diffraction efficiency and wavelength selectivity were not so satisfactory, which affected the insertion loss and channel spacing of the device respectively. In order to investigate the performances for most of the volume holographic devices which are of finite size and with 9 degree geometry, two-dimensional (-D) coupled-wave theory is more accurate than that based on the well-known Kogelnik s coupled-wave theory. In this paper a close-form analytical solution to -D coupled wave theory for -D restricted gratings is presented firstly. Then in order to achieve the optimum insertion loss and channel spacing for dense wavelength division multiplexing (DWDM) filters, diffraction properties, especially effects of the grating strength and grating size ratio on the peak diffraction efficiency and wavelength selectivity are researched based on the -D coupled-wave theory and its solution. The results show that this solution is capable of design optimization of volume holographic gratings for various devices, including wavelength filters. And the design optimization is given in order to gain the optimum peak diffraction efficiency and wavelength selectivity. Finally, some experimental results showing the angular selectivity for different grating size ratio are given, which agree well with the -D coupled-wave theory. Key words: dense wavelength division multiplexing, insertion loss, channel spacing, volume holography, two-dimensional coupled-wave theory, volume grating with finite size, diffraction efficiency, wavelength selectivity. 1. INTODUCTION ecently, a new method that volume holographic gratings written in the crystal, glass or polymer are used for wavelength filters is proposed [1-5]. Because of rigorous wavelength selectivity of Bragg diffraction in volume holographic gratings and ability that multiple holograms can be written in one medium, many signal channels can be demultiplexed simultaneously with one holographic material. The efficiency will be improved greatly. Therefore, volume holography is promising for wavelength filters. However, experimental results reported so far are not so optimistic. Problems such as serious insertion loss, cross-talk * wbwsx@ s.bjut.edu.cn or wb_wsx@yahoo.com.cn; Tel ; 1, 941#, Beijing University of Technology, Beijing, China. Holography, Diffractive Optics, and Applications II, edited by Yunlong Sheng, Dahsiung Hsu, Chongxiu Yu, Byoungho Lee, Proceedings of SPIE Vol (SPIE, Bellingham, WA, 5) X/5/$15 doi: /

2 and wide channel spacing exist. Wavelength resolution representing channel capacity is far from the need for dense wavelength division multiplexing (λ/ λ 16/.4 = 4, where λ is channel spacing, which is.4 nm for DWDM; λ is the wavelength used in communication, which is near 16 nm). Also, diffraction efficiency needs to be improved. The analyses in most of the previous work were based on the well-known Kogelnik s coupled-wave theory [6]. However, due to the one-dimensional (1-D) nature of this theory, it is best suitable for transmission or reflection gratings, in which the surface size is much larger than the grating thickness. Since most of the holographic devices are of finite size and with 9 degree geometry [7], a more accurate theoretical analysis is required. Hence -D grating theory has long been a subject of considerable attention [8-11]. The overlap grating is formed only in the area where two plane waves of finite widths intersect in the interior of the holographic material. In this paper, we study the diffraction properties of overlap gratings, especially peak diffraction efficiency and wavelength selectivity, based on the closed form solution to -D coupled-wave theory [11]. Design optimization of volume holographic gratings for wavelength filters is then given. In Sect. we review the -D coupled-wave theory and its solution. Wavelength selectivity is discussed in Sect. 3. Design optimization is given in order to gain the optimum peak diffraction efficiency and wavelength selectivity. Our experimental results of angular selectivity are given in Sect. 4. It is shown that the tendency of experimental results coincides well with the calculation based on the -D coupled-wave theory. So, design of volume holographic gratings for wavelength filters can be optimized base on the closed form solution to -D coupled-wave theory. Finally, conclusions are drawn in Sect. 5.. TWO-DIMENSIONAL COUPLED-WAVE THEOY.1 The form of refraction index gratings As shown in Fig. 1, a grating is written by two plane waves, where W and W S are the widths of the reference and signal beams respectively. The coordinate system (x, y) is chosen such that the two beams are incident at angles of φ and φ relative to the x axis. Signal beam y Transmitted beam E 1 W W φ φ eference beam x W S φ 1 φ u S u eadout beam 1 Diffracted beam E Fig. 1. Formation of a -D restricted volume grating. Fig.. eadout of a -D restricted volume grating. The electric field of the recording beams takes the form [9] 68 Proc. of SPIE Vol. 5636

3 E i = a i A i exp(-γ p i ) (1) where i = 1, represents the reference and signal beams respectively, γ = α + jβ, α corresponds to the average loss in the holographic material, β is the propagation constant in the material, a i is the suitably normalized amplitude distribution, A i is a constant characterizing the relative intensities of the recording beams, p i is the plane phase front of the beams, which may be written as Si xcosφ - (-1) i ysinφ () The gratings can be specified by the modulated dielectric constant of the medium ε r, written as ε r = ε r + ε r1a 1 a cos[β (p 1 - p )] (3) where ε r is the average dielectric constant, and ε r1 is the modulation depth, which may be real( complex) to indicate a pure phase( phase and absorption) grating.. The diffraction of gratings (Bragg regime) econstruction is by a third plane wave whose phase front is approximately that of the reference beam. Because of the Bragg diffraction, there are only readout beam E 1 and diffracted signal E in the medium. The total electrical field E is the sum of E 1 and E : E = E 1 + E = a 1 A 1 exp(-γp 1 ) + a A exp(-γp ) (4) Where γ = α + jβ. Here α is the absorption coefficient after development, and β is the propagation constant of readout beam. Wave propagation is described by the scalar wave equation: E + ω µε ε E = (5) r where ε is the dielectric constant in vacuum. Combining equations (3-5), and using similar approximations used in ef. [1, 13], leads to the coupled-wave equations expressed as follows: aaa1 A1 p1 + jκ exp a1 aa1a1 A p + jκ exp a [ ( G + jk )] [( G + jk )] A = A = 1 where G = α(p - p 1 ), K = β (p 1 - p ) - β(p 1 - p ), κ = ε r1 β /(4ε r) (7) (6) A new normalized coordinate system (u, u S ) is introduced, which vectors are at right angles to the readout beam and diffracted beam respectively (as shown in Fig. ). Proc. of SPIE Vol

4 u u S 1 = W s sinφ sinφ cosφ x cosφ y (8) So equation (6) can be rewritten in the new coordinate system A1 aaa1 = jκ WS exp us a1 A aa1a1 = jκ WS exp u a [ ( G + jk )] [( G + jk )] A A 1 where G = α W S sin φ (u + u S ),K = δw S (u + u S ), α = α/sinφ, κ = κ/sinφ. (9) The Bragg mismatching term due to the angle deviation φ and wavelength deviation β when the grating is 1 reconstructed can be described as δ = β φ sec φ + β tanφ. By adopting iemann method [1] with proper variable substitution, the solutions to the electric fields of two waves are [11] E1 = exp( α WSuS ){ a1 ( u ) exp( α WSu cos φ ) κ WS exp[ ( α WS + jδws ) u ] a1( u ) u M a ( τ ) a ( τ ) J ( κ W LM ) exp[ τ ( α W sin φ + jδw )] dτ } for the amplitude of transmitted field, and E = jκ W a for the diffracted field S S S L [ S S + jδwsus ] () τ a () τ J ( κ W LM ) exp[ τ ( α W sin φ + jδw )] dτ ( u ) exp α W ( u u ) u S a1 1 S S where L a1 ( ξ ) dξ = M a ( ) = τ u u S ξ dξ. The diffracted efficiency is defined as the ratio of diffracted to incident power, S S (1) (11) 1 (, u ) E W S dus η diff = (1) W E du 1 ( u,) where W is the ratio of reference beam width to object beam width. Diffraction properties of volume gratings with finite size in complicated conditions, especially Bragg selectivity can be researched based on equations (1-1). 3. WAVELENGTH SELECTIVITY OF THE VOLUME HOLOGAPHIC GATING 3.1 Peak diffraction efficiency Peak diffraction efficiency is important to the performance of the volume holographic gratings. It is responsible for the insertion loss when the grating is used as a dense wavelength division multiplexing filter. Grating strength κw S, 7 Proc. of SPIE Vol. 5636

5 absorption αw S, and ratio of reference beam width to object beam width W can affect it. In this paper, pure phase gratings are assumed to exist in lossy materials. So for a given crystal, the absorption coefficient is certain. So effects of grating strength and grating size ratio on diffraction properties are paid more attentions. Fig. 3 shows the peak diffraction efficiency versus grating strength κw S with prescribed grating size ratio W and absorption αw S. It indicates that strongly coupled gratings can lead to high diffraction efficiency. These properties coincide with those of reflection gratings predicted by Kogelnik theory essentially. 5Z η κ W S Fig. 3. Bragg-matched diffraction efficiency vs. grating strength, αw S =.5. (φ = 45 deg.) However, grating size ratio can also affect the diffraction efficiency. Fig. 4 shows the peak diffraction efficiency versus grating size ratio W with different grating strength, where absorption αw S is prescribed. η N:V N:V N:V W Fig. 4. Bragg-matched diffraction efficiency vs. grating size ratio with different grating strength, αw S =.5. (φ = 45 deg.) One can see from Fig. 4 that the relationship between diffraction efficiency η and grating size ratio W is not Proc. of SPIE Vol

6 monotonic. The grating can lead to the highest peak diffraction efficiency (~.8) when the grating size ratio is 1.4 with αw S =.5 and κw S = 1. For the same absorption, the grating can also lead to the highest peak diffraction efficiency (~.5) when the grating size ratio is.6 with κw S =. As a result, the volume holographic grating can lead to the highest diffraction efficiency when the grating is designed at the optimum grating size ratio W_opt. Deviation from W_opt will result in the fall of diffraction efficiency. The optimum size ratio decreases with the increase of grating strength and will be affected by the absorption, αw S. In order to obtain the optimum diffraction efficiency, the grating size ratio should be chosen near the optimum value. 3. Wavelength selectivity of volume gratings Wavelength selectivity is also important to the performance of the volume holographic gratings. It corresponds to the channel spacing of dense wavelength division multiplexing filters. Wavelength resolution power λ/ λ representing channel capacity should reach 4 in order to realize DWDM, i.e. λ/λ =.5. Grating strength κw S, absorption αw S, and ratio of reference beam width to object beam width W will affect the wavelength selectivity too. oughly speaking, the 3 db width of a wavelength selectivity curve (a curve of normalized diffraction efficiency versus wavelength deviation from Bragg wavelength) represents the bandwidth of a channel, the 13 db width of this curve can be used for the cross-talk assessment between neighbor channels, and 1 db width represents wavelength resolution power. So in this paper, half 1 db width corresponding resolution power, λ 1dB /λ, is discussed in detail. Figure 5 shows λ 1dB /λ versus grating strength κw S with prescribed different grating size ratio W and absorption αw S, where W S /λ = 3. From this figure one can see that wavelength resolution power can reach 4 when κw S =1. for W =1, and κw S =1.7 for W =. So in order to reach high wavelength resolution power, the grating coupling should be stronger with increasing grating size ratio. Figure 6 shows λ 1dB /λ versus grating size ratio W for different grating strengths, where material absorption αw S is prescribed, W S /λ = 3. It shows that wavelength resolution power can reach 4 when W =1 for κw S =1, and W =4.75 for κw S =. So in order to obtain high wavelength resolution power, the grating size ratio should increase for strongly coupled gratings. λ 5Z 5Z λ κ W S Fig. 5 λ 1dB /λ as a function of grating strength κw S with different grating size ratio W, αw S =.5. (φ =45 deg.) 7 Proc. of SPIE Vol. 5636

7 λ kws=1. kws=1.5 kws=. λ W Fig. 6 λ 1dB /λ as a function of grating size ratio W with different grating strength κw S, αw S =.5. (φ = 45 deg.) 3.3 Design optimization of volume holographic gratings for wavelength filters Parameters that affect performances of the volume holographic filters are shown clearly in Sec. 3.1 and Sec. 3.. Effects of the grating strength and grating size ratio on the diffraction efficiency and channel spacing are researched based on the closed form solution to -D coupled-wave theory by calculation with prescribed absorption. According to these diffraction properties, the grating size ratio or the grating strength can be chosen in order to gain high diffraction efficiency provided that the wavelength resolution power can meet the requirement for DWDM. The design of volume holographic gratings for wavelength filters is shown in Table 1. Table 1: The design of volume holographic gratings for wavelength filters, αw S =.5. Prescribed parameters Parameter optimization Diffraction efficiency Wavelength resolution power W =1 κw S = W = κw S = κw S = 1 W = κw S = W = Table 1 shows the design optimization with different grating size ratio or grating strength and prescribed absorption. In practice, design of volume holographic gratings for wavelength filters can be optimized base on the closed form solution to -D coupled-wave theory according to the conditions and requirements. 4. EXPEIMENTAL MEASUEMENT Proc. of SPIE Vol

8 In the previous sections, design optimization of volume holographic gratings for wavelength filters is given based on the closed form solution to -D coupled-wave theory. In this section, experimental results are shown to demonstrate the agreement with the -D coupled-wave theory and its solution in this section. Because of the limit of equipments, it is very difficult to vary the wavelength of the readout beam precisely to demonstrate wavelength selectivity. However, it can be seen by theoretic analysis that wavelength deviation is equivalent to angle deviation in effect on diffraction efficiency. So, we choose to demonstrate angular selectivity, i.e. selective angles with different grating size ratio. We use an experimental set-up shown in Fig. 7. The light for writing and reading volume holographic gratings is from a He-Ne laser (λ = 63.8 nm). A doubly doped lithium niobate crystal Fe: In: LiNbO 3 (.3 mol.-% Fe-doped,.5 mol.-% In-doped) is used in the experiments. Firstly, a grating was written in the crystal. Then, the diffraction efficiency was measured with the angle scanning of reference beam when the grating was reading. According to the diffraction curve of diffraction efficiency versus angle deviation from Bragg angle, selective angles can be achieved. Selective angles with different grating size ratio (.5, 1,, and 3 respectively) are obtained by experimental measurement. The object beam width is fixed to mm during the experiments. Fig. 8 shows the four group experimental results and theoretic calculation results based on the -D coupled-wave theory and its solution, where αw S =.3, κw S =.. (φ =45Ü) Mirror 1/ wave plate Beam Polarizing Aperture 3 Aperture 1 expander beam splitter Laser He-Ne Crystal Shutter 1 Lens Lens 1/ wave plate Aperture Shutter 4f system Power meter otation mirror Computer Fig. 7. Schematic diagram showing the experimental set-up for investigating the selective angle with different grating size ratio. Figure 8 shows measured selective angles with different grating size ratio (.5, 1,, and 3 respectively). Calculated results based on the -D coupled-wave theory are shown in the same figure in order to compare with experimental results. From Fig. 8, one can see that the tendency of experimental results coincides well with the calculation. It can be expected that wavelength selectivity with different grating size ratio will also coincide well with the calculation based on the -D coupled-wave theory. So, design of volume holographic gratings for wavelength filters can be optimized base on the closed form solution to -D coupled-wave theory. A possible reason that experimental results deviate from 74 Proc. of SPIE Vol. 5636

9 calculated results is that the grating written in crystal is not so homogeneous, i.e. grating strength may be not a constant. However, the closed form solutions are based on the condition that grating strength is a constant. In addition, the measurement errors in aperture width, scattering noise from the crystal and so on may also lead to an increase of measured selective angle. Calculated esults Experimental esults Θ W Fig. 8. A comparison of the measured selective angle Θ vs. grating size ratio W with calculated results. 5. CONCLUSIONS In this paper, diffraction properties, especially effects of the grating dimension and absorption coefficient on the peak diffraction efficiency and angle/wavelength selectivity are researched based on the -D coupled-wave theory and its solution. Design optimization of volume holographic gratings for wavelength filters is given in order to gain the optimum peak diffraction efficiency and wavelength selectivity. It was found that with prescribed grating dimension and material absorption the selectivity can meet the requirement for DWDM when the grating strength is weaker than a threshold value. And the value may be larger with the increase of grating size ratio. However, weakly coupled gratings receive low diffraction efficiency. So, the grating strength should be chosen near the threshold value to gain the best diffraction properties. Correspondingly exposure time should be controlled properly when the grating is written in the material. On the other hand, with prescribed grating strength and material absorption, the wavelength resolution power can meet the need for DWDM when the grating size ratio is greater than a threshold value. And the value may be larger Proc. of SPIE Vol

10 with the increase of grating strength. However, there is a optimum grating size ratio which can lead to the highest diffraction efficiency under the same conditions. So, the grating size ratio should be chosen near the optimum value to gain high diffraction efficiency provided that the wavelength resolution power can reach the need for DWDM. In order to demonstrate -D coupled-wave theory and its solution, experiments of angular selectivity are done. Selective angles are measured for with different grating size ratio. The tendency of experimental results coincides well with the calculation based on the -D coupled-wave theory. So it indicates that the closed form solutions of -D coupled-wave theory can be used for design optimization of volume holographic gratings for wavelength filters. ACKNOWLEDGEMENTS This work is supported by the National esearch Fund for Fundamental Key Project of China under Grant No. (G199933). EFEENCES 1. S. Breer and K. Buse, Wavelength demultiplexing with volume phase holograms in photorefractive lithium niobate, Appl. Phy. B: Lasers Opt., 66, pp , X Deng, F. Zhao, Zhenhai Fu, Jizuo Zou, Jie Qiao, G. Kim, and ay T. Chen, linearity of volume hologram out-coupling for wavelength-division demultiplexing, Proceedings of SPIE, 3949, pp ,. 3. Charles C. Zhou, Sean Sutton, ay T. Chen, Boyd Hunter, and Paul Dempewolf, Four channel multimode wavelength division demultiplexer (WDM) system based on surface-normal volume holographic gratings and substrate-guided waves, Proceedings of SPIE, 388, pp. 89-9, J.-W. An, N. Kim, and K.-W. Lee, Volume holographic wavelength demultiplexer based on rotation multiplexing in the 9 geometry, Optics Communications, 197, pp , Pierpaolo Boffi, Maria Chiara, Ubaldi, Davide Piccinin, Claudio Frascolla, and Mario Martinelli, 155-nm volume holography for optical communication devices, IEEE Photonics Technology Letters, 1(1), pp ,. 6. H. Kogelnik, Coupled wave theory for thick hologram gratings, The Bell. Syst. Tech, 48(9), pp , F. H. Mok, Angle-multiplexed storage of 5 holograms in lithium niobate, Opt. Lett., 18(11), pp , L. Solymar, A general two-dimensional theory for volume holograms, Appl. Phys. Lett. 31(1), pp. 8-8, L. Solymar and D. J. Cooke, Volume holography and volume gratings, pp , Academic Press, New York, P. St. J. ussell and L. Solymar, The properties of holographic overlap gratings, Opt. Acta, 6(3), pp , Shiquan Tao, Bo Wang, Geoffrey W. Burr, and Jiabi Chen, Diffraction efficiency of volume gratings with finite size: corrected analytical solution, Journal of Modern Optics, 51(8), pp , Courant and D. Hilbert, Methods of mathematical physics, Vol., Chap. V, Interscience Publishers, New York, Proc. of SPIE Vol. 5636