- Email: [email protected]
Theoretical analysis of optical coupling properties of the waveguide grating with novel rectangular structure
ARTICLE IN PRESS
Optics & Laser Technology 39 (2007) 1089–1093 www.elsevier.com/locate/optlastec
Theoretical analysis of optical coupling properties of the waveguide grating with novel rectangular structure Yong-Bin Yanga,, Yu-Rong Wanga, Xiao-Qiang Yub a
Optics Department, School of Information Science and Engineering, Shandong University, Jinan 250100, Shandong Province, China b The National Key Lab of Crystal Materials, Shandong University, Jinan 250100, China Received 17 June 2005; accepted 12 April 2006 Available online 19 June 2006
Abstract For the waveguide grating photocoupler with novel structure [Yu XQ, Zhang X, Wong KS, Xu GB, Xu XG, Ren Y, et al. A fabrication of coupling grating in the polymeric waveguide by using two-photon initiated photopolymerization. Mater Lett 2004;58:3879–83. [1]; Yu XQ, Zhang X, Xu GB, Zhao HP, He W, Shao ZS, et al. Fabrication of grating waveguide and coupling grating using two-photon initiated photopolymerization. Chem J Chin Univ 2004;25(10):1931–3 (in Chinese). [2]], the electric fields of the TE guided wave and the TE radiating wave are obtained by solving the Helmholtz equation in the spatial rectangular coordinates. And then the relations between the loss coefficient and the different structure parameters of the waveguide and grating are analyzed by using the mode coupling theories, and their corresponding numerical simulation results are given. In the end the result obtained for this novel structure and that for conventional rectangle structure are compared, and the difference and the sameness are obtained. r 2006 Elsevier Ltd. All rights reserved. Keywords: Waveguide grating; Mode coupling theories; Photocoupler
1. Introduction Since the waveguide grating was reported in 1970s, the research of its application in integrated optics has been more and more extensive, for example, input/output coupler, filter, wavelength division multiplexers, deflector, mode converter, and polarizer, etc. When the waveguide grating is used as input/output coupler, compared with the prism coupler, it has smooth surface and small volume, and is not limited by the refractive index, but lower coupling efficiency of the grating coupler has limited its range of application. In order to raise the coupling efficiency of it and increase further its range of application, many scholars have made a large amount of work in researching the effect of waveguide grating shape, depth and structure on the coupling efficiency [3–7]. There have been many methods for fabricating waveguide grating, for example, ion beam etching, holographic lithography, electronic beam direction writing, phase mask technique [4,8–10]. In recent years, as Corresponding author. Tel.: +86 531 8836 1208.
E-mail address: [email protected] (Y.-B. Yang). 0030-3992/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2006.04.004
people are getting more and more research in various organic photo-polymeric materials, in the field of integrated optics devices, the fabrication and research of optics performance of various polymer type waveguides and waveguide gratings receive much attention [11]. At present, scholars have already fabricated polymeric waveguide grating by ultraviolet ray photo-bleaching [12], photo induced two colors [13] and two-photon initiated photopolymerization [14–16]. In the ultraviolet ray photobleaching, the refractive index of the material is varied by saturation absorption of the material. In the photo induced two colors, the periodic microstructure is fabricated by photo alignment of dye molecules in the material. The two methods are basically physical method, and do not induce natural chemical change within the material, so it is very difficult to guarantee the long-term stability of the instrument fabricated by the two methods. However, the instrument fabricated by using the two-photon initiated photo-polymerization does not have the disadvantage. The basic principle of two-photon initiated photo-polymerization can be expressed as follows [14–16]. The polymer material which comprises the two-photon initiator can
ARTICLE IN PRESS Y.-B. Yang et al. / Optics & Laser Technology 39 (2007) 1089–1093
1090
n3
x n1 z
n2
(a)
y
x
a 2 La 2 n þ n0 . (1) L g L The electric field of the TE wave which propagates in the waveguide in the spatial rectangular coordinates must satisfy the following Helmholtz equation:
Λ
n21 ¼
n3
n0
ng
d z
n2
0
In Fig. 1, n2 and n3 are the refractive index of the substrate and the cladding, respectively. d is the thickness of the waveguide grating. a and L are the width of grating slot and the grating period, respectively. n0 and ng are the refractive indexes of the waveguide in unpolymerized part and in polymeric part, respectively. The equivalent refractive index of the polymer waveguide grating is
a
(b) Fig. 1. (a) Diagram of the waveguide grating fabricated by using twophoton initiated photopolymerization, (b) section diagram of the grating part of the (a) diagram in the xz coordinates.
absorb two photons at the same time under the excitation of the intense laser. The non-linear interaction of laser and polymer material, namely two-photon absorption, makes the polymer material occur polymerization which is a chemical reaction. In this process the refraction of the material is also changed. In addition, the two-photon absorption makes the polymeric part of the polymer material be limited in a very small region where is nearby the focus. When the polymer material is scanned by the laser, parts of the material can be polymerized, which causes the change of the refraction. However, the refraction of the unpolymerized parts is unchanged. So the polymeric waveguide grating can be fabricated in polymer membrane which has the nature of waveguide. The two-photon initiated photo-polymerization can fabricate the novel structure waveguide grating shown in Fig. 1, in which the refractive index distribution is periodic and its structure is composed of a polymeric part and an unpolymerized part. The grating and the waveguide are in the same membrane and have the same thickness. In this paper, for this novel waveguide grating structure the electric fields of the TE guided wave and the TE radiating wave are obtained by solving the Helmholtz equation in the spatial rectangular coordinates. And then the relations between the loss coefficient and the different structure parameters of the waveguide grating are analyzed by using the mode coupling theories, and their corresponding numerical simulation results are given. 2. Theory method The structure diagram of novel structure waveguide grating fabricated by the two-photon initiated photopolymerization is shown in Fig. 1.
q2 E y þ ðn2 k20 b2 ÞE y ¼ 0, qx2
(2)
where n represents the refractive index of each layer of waveguide grating, k0 ¼ 2p=l is the wave number in vacuum, and b is the TE wave’s propagation constant. The solution of Eq. (2) is decided by the expression n2 k20 b2 (namely Eigen equation’s root). So we can obtain the electric fields of the TE guided wave Ey and the TE radiating wave E 0y in each layer of waveguide by fixing on the relation between the nk0 and b. For the TE guided wave, the relation between nk0 and b is n3 k0 on2 k0 obon1 k0 . One solution (namely the electric field of the TE guided wave Ey in each layer of waveguide) of Eq. (2) can be obtained by using the above inequality, it can be expressed by 8 > < E 3 exp½qðx dÞ; x4d; 0oxod; (3) E y ¼ exp½jðot bzÞ E 1 cosðhx f1 Þ; > : E expðpxÞ; xo0; 2 where the E1, E2, E3, h, p, q and f1 are constants needed to be calculated, among these constants, the h, p, q and b satisfy the following relation: h2 ¼ k20 n21 b2 ;
p2 ¼ b2 k20 n22 ;
p2 ¼ b2 k20 n23 .
(4)
And then according to the continuum condition of the electric field Ey and its derivative qE y =qx in the boundary, and the relation between the peak value of guided wave’s electric field E1 and their power P sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4om0 P E1 ¼ , (5) bd eff where the deff is effective thickness 1 1 d eff ¼ d þ þ . p q
(6)
We can fix on the electric field of the TE guided wave Ey in each layer of waveguide. For the TE radiating wave which propagate in the substrate, the relation between nk0 and b is n3 k0 o n2 k0 obon1 k0 . Another solution (namely the electric field of the TE radiating wave E 0y in each layer of waveguide) of Eq. (2) can be obtained by using the above inequality, it
ARTICLE IN PRESS Y.-B. Yang et al. / Optics & Laser Technology 39 (2007) 1089–1093
can be expressed by 8 0 E exp½Qðx dÞ; > < 3 0 0 0 E y ¼ exp½iðot b zÞ E 1 cosðHx þ c1 Þ; > : E 0 cosðrx þ c Þ; 2
2
xXd; 0pxpd; xp0; (7)
E 01 ,
E 02 ,
2
H 2 ¼ k20 n21 b0 ;
av ¼ pjk^iv j2 .
r2 ¼ k20 n22 b0 ;
The expression of k^iv is Z oe0 þ1 2 k^iv ¼ ðn n20 Þ eut evt dx, 4 1
2
2
Q2 ¼ b0 k20 n23 . (8)
And then according to the continuum condition of the electric field E 0y and its derivative qE 0y =qx in the boundary, and the relation between the peak value of radiated wave’s electric field E 01 and their power P0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4P0 om0 cos2 c2 0 E1 ¼ . (9) pb0 cos2 c1 We also can fix on the electric field of the TE radiating wave E 0y in each layer of waveguide. In Fig. 1 the slab waveguide with periodic structure can be equivalent to the waveguide whose refractive index is perturbed, the refractive index subtraction between the perturbed waveguide’s and the uniform waveguide’s is expressed by n2 ðx; zÞ n20 ðxÞ 8 < ðn2 n2 Þ g 0 ¼ : 0;
of the grating must be varied for coupling the incident wave into the waveguide. In Eq. (11), cv and ci are the amplitudes of the incident field and the guided mode in the waveguide respectively; k^iv and av are the coupling coefficient and the radiant loss coefficient, respectively, their relation is
E 03 ,
where the H, r, Q, c1 and c2 are constants needed to be calculated, among these constants, the H, r, Q and b0 satisfy the following relation:
sinðgap=LÞ gp
exp jg 2p L z ;
0oxod; ð10Þ xo0; x4d;
where g is the diffraction order, here its value is 1, ng ; n0 ; a; L have been shown in Fig. 1. When a light beam is incident on the waveguide grating shown in Fig. 1, the mode coupling equation which the light wave needs to satisfy is X sinðgap=LÞ dcv ci expðjBzÞ av cv , ¼ j k^iv (11) gp dz g where B ¼ bi bv gð2p=LÞ represents the phase-matching condition, in this condition, bi and bv are propagation constants of the incident electric field and the guided mode, respectively. Only when the condition is satisfied, the incident electric field can be effective coupled into the waveguide, that is to say bi bv gð2p=LÞ ¼ 0, so we can obtain 2p , (12) L where y is incident angle, the superscript 0 and 2 of y indicate that the light wave is incident on the waveguide from the air and the substrate, respectively. The other parameters in Eq. (12) have been discussed in the front of this section. That is to say, when the incident angle and the wavelength of the incident light wave are varied, the period ð2Þ n3 k sin yð0Þ g ¼ n2 k sin yg ¼ bv þ g
1091
(13)
(14)
where eut and evt are the electric fields of the guided mode and the radiating mode, respectively; n is perturbed waveguide’s refractive index n(x,z) in Eq. (10). According to the analysis of Eq. (2), we can obtain the electric fields of the TE0 guided wave and the TE radiating wave. And then we substitute the two electric fields and Eq. (10) into Eq. (14) and integrate them, we can obtain sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 o m e cos c PP0 H2 0 0 2 k^iv ¼ 0 cos2 c1 bb pd eff H 2 h2 1 ½cosðhd f1 Þ sinðHd þ c1 Þ cos f1 sin c1 H h 2 ½sinðhd f1 Þ cosðHd þ c1 Þ H sin q La p 2 2 . ð15Þ þ sin f1 cos c1 ðng n0 Þ qp And then we can obtain av by substituting Eq. (15) into Eq. (13). 3. Numerical simulation If we want to obtain the maximal coupling efficiency in the waveguide grating importing coupler, loss coefficient av and the width wG of the incident Gaussian light beam must satisfy the relation [3] av wG sec yg ¼ 1:36. So when the incident field has been known, and after the yg also has been obtained from Eq. (12), if there is proper av , we can obtain the maximal coupling efficiency. So we numerically simulate the variation of av with each structure parameter of the waveguide grating, for example, grating thickness d, the ratio a/L between the width of grating slot and the grating period, the absolute value jn2g n20 j of the subtraction of permittivity between the waveguide polymeric part and its unpolymerized part, and the sum n2g þ n20 of the permittivity between the waveguide polymeric part and its unpolymerized part. The variations of av with these parameters are shown from Figs. 2–5 with solid lines. We also analyze the relation between av and each parameter of conventional rectangle structure [3], which is shown from Figs. 2 to 4 with dashed lines (in these figures, the values of the waveguide grating’s parameters we use are the same to those used in Ref. [3], among these parameters, ew is the waveguide permittivity, ea is the air permittivity, es
ARTICLE IN PRESS Y.-B. Yang et al. / Optics & Laser Technology 39 (2007) 1089–1093
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0.035
1.4
0.030
1.2
0.020
εr=3.0
n0=1.6
εw=3.0
ng=1.7
εa=1.0
n2=1.5
εs=2.3
0.015
0.000 0.0
a/Λ=0.5
d=0.3μm
εa=1.0
n2=1.5
εs=2.3
n3=1.0
tw=λ/π
a/Λ=0.5
Λ= λ/2
λ=0.632μm
d= λ/π a/Λ=0.5
0.4
a/Λ =0.5
Λ=λ/2 0.005
0.8 0.6
n3=1.0
tw= λ/π
0.010
1.0
αν
αν
0.025
εw=3.0
0.2
λ=0.632μm
0.0 0.0 0.4
0.8
1.2
1.6
2.0
0.4
0.8
2.4
d (μm) Fig. 2. Solid line illustrates the variation of the loss coefficient av with the waveguide grating thickness d for the novel structure, the parameters of the right column marked in the figure are used for the novel structure. Dashed line illustrates the variation of the normalized loss coefficient av l with the grating thickness d/l for the conventional rectangle structure, the parameters of the left column marked in the figure are used for the conventional structure.
1.2
1.6
2.0
2 2 |ng-n0|
Fig. 4. Solid line illustrates the variation of loss coefficient av with the absolute value jn2g n20 j of the subtraction of permittivity between the waveguide polymeric part and its unpolymerized part for the novel structure, the parameters of the right column marked in the figure are used for the novel structure. Dashed line illustrates the variation of the normalized loss coefficient 100av l with the grating permittivity er for the conventional rectangle structure, the parameters of the left column marked in the figure are used for the conventional structure.
0.05 d=0.3μm
0.020
d=0.3∝m
εw=3.0
n0=1.6
εa=1.0
ng=1.7
0.04
εs=2.3
αν
0.015
n2=1.5
λ=0.632μm
0.03
n2=1.5
tw=λ/π
n0=1.0 a/Λ=0.5
αν
0.025
εr=3.0
0.02
n3=1.0
Λ=λ/2 0.010
λ=0.632μm
d=λ/2π
0.01
0.00
0.005
4
6
8
10
12 2
14
16
18
20
2
n0+ng 0.000 0.0
0.2
0.4
0.6
0.8
1.0
a/Λ Fig. 3. Solid line illustrates the variation of loss coefficient av with the ratio a/L between the width of grating slot and the grating period for the novel structure, the parameters of the right column marked in the figure are used for the novel structure. Dashed line illustrates the variation of loss coefficient av with the ratio a/L between the width of grating slot and the grating period for the conventional rectangle structure, the parameters of the left column marked in the figure are used for the conventional structure.
Fig. 5. Relation between av and the sum n2g þ n20 of the permittivity between the waveguide polymeric part and its unpolymerized part for the novel structure, the other parameters are also marked in the figure.
is the substrate permittivity, L is the grating period, tw is the waveguide thickness, a/L is the ratio between the width of grating slot and grating period, d is the grating thickness, er is the grating permittivity). In the conventional rectangle structure, the relation between av and each parameter obtained in this paper with mode coupling
ARTICLE IN PRESS Y.-B. Yang et al. / Optics & Laser Technology 39 (2007) 1089–1093
theory are basically the same to the result reported in Ref. [3]. After comparing the conventional rectangle structure’s relation between av and each parameter with the novel structure’s, we can know that in the novel structure analyzed in this paper, when the grating thickness is increased, the av first gets large and then falls, the av’s value also fluctuates when it is falling, in the end the av gets to a very little value which is close to zero; but in conventional rectangle structure, when grating thickness is increased, the av first gets large and then when it gets the max, the av only fluctuates in the position of the max with a certain period, it no more gets small, this is the difference between the novel structure and the conventional rectangle structure. The relation between av and the ratio a/L between the width of grating slot and the grating period is the same to conventional rectangle structure’s. In conventional rectangle structure, the relation between av and the grating permittivity er (which is shown in Fig. 4 by dashed line) is actually equivalent to the relation between av and jn2g n20 j (which is shown in Fig. 4 by solid line) of the novel structure analyzed by this paper, and their variation trend are the same. In addition, we also analyze the relation between av and the sum n2g þ n20 of the permittivity between the waveguide polymeric part and its unpolymerized part, which adds a condition for the grating’s fabrication and practical applications.
4. Conclusion The electric fields of the TE guided wave and the TE radiating wave are obtained by solving the Helmholtz equation in the spatial rectangular coordinates. And then the coupling between the incident wave and the novel structure waveguide grating shown in Fig. 1 is analyzed by using the mode coupling theories, and the relations between the loss coefficient and the different structure parameters of the waveguide grating are numerical simulated; we also numerically simulated the relations between the loss coefficient and the different structure parameters of the waveguide grating in the conventional rectangle structure, and then we compare and analyze the two results and obtain their sameness and differences.
1093
Acknowledgements The authors acknowledge the support by the Natural Science Foundations of Shandong Province, China, Grant no. Y2004G01. References [1] Yu XQ, Zhang X, Wong KS, Xu GB, Xu XG, Ren Y, et al. A fabrication of coupling grating in the polymeric waveguide by using two-photon initiated photopolymerization. Mater Lett 2004;58: 3879–83. [2] Yu XQ, Zhang X, Xu GB, Zhao HP, He W, Shao ZS, et al. Fabrication of grating waveguide and coupling grating using twophoton initiated photopolymerization. Chem J Chin Univ 2004; 25(10):1931–3 (in Chinese). [3] Tamir T, Peng ST. Analysis and design of grating couplers. Appl Phys 1977;14(3):235–54. [4] Bates KA, Li LF, Roncone RL, Burke JJ. Gaussian beams from variable groove depth grating couplers in planar waveguides. Appl Opt 1993;32(12):2112–6. [5] Emmons RM, Hall DG. Comparison of film thickness tolerances in waveguide grating couplers. Opt Lett 1991;16(13):998–1000. [6] Haus HA, Schmidt RV. Approximate analysis of optical waveguide grating coupling coefficients. Appl Opt 1976;15(3):774–81. [7] Avrutsky IA, Svakhin AS, Sychugov VA, Parriaux O. High-effiency single-order waveguide grating coupler. Opt Lett 1990;15(24):1446–8. [8] Xu ZQ, Jin GL. Grating apparatus’s application in integrated optical circuit. Opt Tech 1988;2:21–7 (in Chinese). [9] Ma SJ, Li Y, Xu M, Lin JL. Fabricating of waveguide grating by phase mask technique. Chin J Lumin 1998;19(1):77–9 (in Chinese). [10] Jia ZH. Study on polymer PMMA/DR1 Chirped grating coupler. Acta Photon Sin 2003;32(1):86–8 (in Chinese). [11] Lee JM, Park S, Kim M, Ahn JT, Lee MH. Birefringence as a function of upper-cladding layers in polymeric arrayed waveguide gratings. Opt Commun 2004;232:139–44. [12] Ahn SW, Shin SY. Post-fabrication tuning of a polymeric gratingassisted codirectional coupler filter by photobleaching. Opt Commun 2001;194:309–12. [13] Toussaere E, Labbe´ P. Linear and non-linear gratings in DR1 side chain polymers. Opt Mater 1999;12:357–62. [14] Yu XQ, Wang C, Zhao X, Wang XM, Yan YX, Fang Q, et al. Twophoton absorption photoinitiated polymerization. Chem J Chin Univ 2000;21(12):1953–5 (in Chinese). [15] Maruo S, Nakamura O, Kawata S. Three-dimensional microfabrication with two-photon-absorbed photopolymerization. Opt Lett 1997; 22(2):132–4. [16] Cumpston BH, Ananthavel SP, Barlow S, Dyer DL. Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication. Nature 1999;398:51–4.
Optics & Laser Technology 39 (2007) 1089–1093 www.elsevier.com/locate/optlastec
Theoretical analysis of optical coupling properties of the waveguide grating with novel rectangular structure Yong-Bin Yanga,, Yu-Rong Wanga, Xiao-Qiang Yub a
Optics Department, School of Information Science and Engineering, Shandong University, Jinan 250100, Shandong Province, China b The National Key Lab of Crystal Materials, Shandong University, Jinan 250100, China Received 17 June 2005; accepted 12 April 2006 Available online 19 June 2006
Abstract For the waveguide grating photocoupler with novel structure [Yu XQ, Zhang X, Wong KS, Xu GB, Xu XG, Ren Y, et al. A fabrication of coupling grating in the polymeric waveguide by using two-photon initiated photopolymerization. Mater Lett 2004;58:3879–83. [1]; Yu XQ, Zhang X, Xu GB, Zhao HP, He W, Shao ZS, et al. Fabrication of grating waveguide and coupling grating using two-photon initiated photopolymerization. Chem J Chin Univ 2004;25(10):1931–3 (in Chinese). [2]], the electric fields of the TE guided wave and the TE radiating wave are obtained by solving the Helmholtz equation in the spatial rectangular coordinates. And then the relations between the loss coefficient and the different structure parameters of the waveguide and grating are analyzed by using the mode coupling theories, and their corresponding numerical simulation results are given. In the end the result obtained for this novel structure and that for conventional rectangle structure are compared, and the difference and the sameness are obtained. r 2006 Elsevier Ltd. All rights reserved. Keywords: Waveguide grating; Mode coupling theories; Photocoupler
1. Introduction Since the waveguide grating was reported in 1970s, the research of its application in integrated optics has been more and more extensive, for example, input/output coupler, filter, wavelength division multiplexers, deflector, mode converter, and polarizer, etc. When the waveguide grating is used as input/output coupler, compared with the prism coupler, it has smooth surface and small volume, and is not limited by the refractive index, but lower coupling efficiency of the grating coupler has limited its range of application. In order to raise the coupling efficiency of it and increase further its range of application, many scholars have made a large amount of work in researching the effect of waveguide grating shape, depth and structure on the coupling efficiency [3–7]. There have been many methods for fabricating waveguide grating, for example, ion beam etching, holographic lithography, electronic beam direction writing, phase mask technique [4,8–10]. In recent years, as Corresponding author. Tel.: +86 531 8836 1208.
E-mail address: [email protected] (Y.-B. Yang). 0030-3992/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2006.04.004
people are getting more and more research in various organic photo-polymeric materials, in the field of integrated optics devices, the fabrication and research of optics performance of various polymer type waveguides and waveguide gratings receive much attention [11]. At present, scholars have already fabricated polymeric waveguide grating by ultraviolet ray photo-bleaching [12], photo induced two colors [13] and two-photon initiated photopolymerization [14–16]. In the ultraviolet ray photobleaching, the refractive index of the material is varied by saturation absorption of the material. In the photo induced two colors, the periodic microstructure is fabricated by photo alignment of dye molecules in the material. The two methods are basically physical method, and do not induce natural chemical change within the material, so it is very difficult to guarantee the long-term stability of the instrument fabricated by the two methods. However, the instrument fabricated by using the two-photon initiated photo-polymerization does not have the disadvantage. The basic principle of two-photon initiated photo-polymerization can be expressed as follows [14–16]. The polymer material which comprises the two-photon initiator can
ARTICLE IN PRESS Y.-B. Yang et al. / Optics & Laser Technology 39 (2007) 1089–1093
1090
n3
x n1 z
n2
(a)
y
x
a 2 La 2 n þ n0 . (1) L g L The electric field of the TE wave which propagates in the waveguide in the spatial rectangular coordinates must satisfy the following Helmholtz equation:
Λ
n21 ¼
n3
n0
ng
d z
n2
0
In Fig. 1, n2 and n3 are the refractive index of the substrate and the cladding, respectively. d is the thickness of the waveguide grating. a and L are the width of grating slot and the grating period, respectively. n0 and ng are the refractive indexes of the waveguide in unpolymerized part and in polymeric part, respectively. The equivalent refractive index of the polymer waveguide grating is
a
(b) Fig. 1. (a) Diagram of the waveguide grating fabricated by using twophoton initiated photopolymerization, (b) section diagram of the grating part of the (a) diagram in the xz coordinates.
absorb two photons at the same time under the excitation of the intense laser. The non-linear interaction of laser and polymer material, namely two-photon absorption, makes the polymer material occur polymerization which is a chemical reaction. In this process the refraction of the material is also changed. In addition, the two-photon absorption makes the polymeric part of the polymer material be limited in a very small region where is nearby the focus. When the polymer material is scanned by the laser, parts of the material can be polymerized, which causes the change of the refraction. However, the refraction of the unpolymerized parts is unchanged. So the polymeric waveguide grating can be fabricated in polymer membrane which has the nature of waveguide. The two-photon initiated photo-polymerization can fabricate the novel structure waveguide grating shown in Fig. 1, in which the refractive index distribution is periodic and its structure is composed of a polymeric part and an unpolymerized part. The grating and the waveguide are in the same membrane and have the same thickness. In this paper, for this novel waveguide grating structure the electric fields of the TE guided wave and the TE radiating wave are obtained by solving the Helmholtz equation in the spatial rectangular coordinates. And then the relations between the loss coefficient and the different structure parameters of the waveguide grating are analyzed by using the mode coupling theories, and their corresponding numerical simulation results are given. 2. Theory method The structure diagram of novel structure waveguide grating fabricated by the two-photon initiated photopolymerization is shown in Fig. 1.
q2 E y þ ðn2 k20 b2 ÞE y ¼ 0, qx2
(2)
where n represents the refractive index of each layer of waveguide grating, k0 ¼ 2p=l is the wave number in vacuum, and b is the TE wave’s propagation constant. The solution of Eq. (2) is decided by the expression n2 k20 b2 (namely Eigen equation’s root). So we can obtain the electric fields of the TE guided wave Ey and the TE radiating wave E 0y in each layer of waveguide by fixing on the relation between the nk0 and b. For the TE guided wave, the relation between nk0 and b is n3 k0 on2 k0 obon1 k0 . One solution (namely the electric field of the TE guided wave Ey in each layer of waveguide) of Eq. (2) can be obtained by using the above inequality, it can be expressed by 8 > < E 3 exp½qðx dÞ; x4d; 0oxod; (3) E y ¼ exp½jðot bzÞ E 1 cosðhx f1 Þ; > : E expðpxÞ; xo0; 2 where the E1, E2, E3, h, p, q and f1 are constants needed to be calculated, among these constants, the h, p, q and b satisfy the following relation: h2 ¼ k20 n21 b2 ;
p2 ¼ b2 k20 n22 ;
p2 ¼ b2 k20 n23 .
(4)
And then according to the continuum condition of the electric field Ey and its derivative qE y =qx in the boundary, and the relation between the peak value of guided wave’s electric field E1 and their power P sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4om0 P E1 ¼ , (5) bd eff where the deff is effective thickness 1 1 d eff ¼ d þ þ . p q
(6)
We can fix on the electric field of the TE guided wave Ey in each layer of waveguide. For the TE radiating wave which propagate in the substrate, the relation between nk0 and b is n3 k0 o n2 k0 obon1 k0 . Another solution (namely the electric field of the TE radiating wave E 0y in each layer of waveguide) of Eq. (2) can be obtained by using the above inequality, it
ARTICLE IN PRESS Y.-B. Yang et al. / Optics & Laser Technology 39 (2007) 1089–1093
can be expressed by 8 0 E exp½Qðx dÞ; > < 3 0 0 0 E y ¼ exp½iðot b zÞ E 1 cosðHx þ c1 Þ; > : E 0 cosðrx þ c Þ; 2
2
xXd; 0pxpd; xp0; (7)
E 01 ,
E 02 ,
2
H 2 ¼ k20 n21 b0 ;
av ¼ pjk^iv j2 .
r2 ¼ k20 n22 b0 ;
The expression of k^iv is Z oe0 þ1 2 k^iv ¼ ðn n20 Þ eut evt dx, 4 1
2
2
Q2 ¼ b0 k20 n23 . (8)
And then according to the continuum condition of the electric field E 0y and its derivative qE 0y =qx in the boundary, and the relation between the peak value of radiated wave’s electric field E 01 and their power P0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4P0 om0 cos2 c2 0 E1 ¼ . (9) pb0 cos2 c1 We also can fix on the electric field of the TE radiating wave E 0y in each layer of waveguide. In Fig. 1 the slab waveguide with periodic structure can be equivalent to the waveguide whose refractive index is perturbed, the refractive index subtraction between the perturbed waveguide’s and the uniform waveguide’s is expressed by n2 ðx; zÞ n20 ðxÞ 8 < ðn2 n2 Þ g 0 ¼ : 0;
of the grating must be varied for coupling the incident wave into the waveguide. In Eq. (11), cv and ci are the amplitudes of the incident field and the guided mode in the waveguide respectively; k^iv and av are the coupling coefficient and the radiant loss coefficient, respectively, their relation is
E 03 ,
where the H, r, Q, c1 and c2 are constants needed to be calculated, among these constants, the H, r, Q and b0 satisfy the following relation:
sinðgap=LÞ gp
exp jg 2p L z ;
0oxod; ð10Þ xo0; x4d;
where g is the diffraction order, here its value is 1, ng ; n0 ; a; L have been shown in Fig. 1. When a light beam is incident on the waveguide grating shown in Fig. 1, the mode coupling equation which the light wave needs to satisfy is X sinðgap=LÞ dcv ci expðjBzÞ av cv , ¼ j k^iv (11) gp dz g where B ¼ bi bv gð2p=LÞ represents the phase-matching condition, in this condition, bi and bv are propagation constants of the incident electric field and the guided mode, respectively. Only when the condition is satisfied, the incident electric field can be effective coupled into the waveguide, that is to say bi bv gð2p=LÞ ¼ 0, so we can obtain 2p , (12) L where y is incident angle, the superscript 0 and 2 of y indicate that the light wave is incident on the waveguide from the air and the substrate, respectively. The other parameters in Eq. (12) have been discussed in the front of this section. That is to say, when the incident angle and the wavelength of the incident light wave are varied, the period ð2Þ n3 k sin yð0Þ g ¼ n2 k sin yg ¼ bv þ g
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(13)
(14)
where eut and evt are the electric fields of the guided mode and the radiating mode, respectively; n is perturbed waveguide’s refractive index n(x,z) in Eq. (10). According to the analysis of Eq. (2), we can obtain the electric fields of the TE0 guided wave and the TE radiating wave. And then we substitute the two electric fields and Eq. (10) into Eq. (14) and integrate them, we can obtain sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 o m e cos c PP0 H2 0 0 2 k^iv ¼ 0 cos2 c1 bb pd eff H 2 h2 1 ½cosðhd f1 Þ sinðHd þ c1 Þ cos f1 sin c1 H h 2 ½sinðhd f1 Þ cosðHd þ c1 Þ H sin q La p 2 2 . ð15Þ þ sin f1 cos c1 ðng n0 Þ qp And then we can obtain av by substituting Eq. (15) into Eq. (13). 3. Numerical simulation If we want to obtain the maximal coupling efficiency in the waveguide grating importing coupler, loss coefficient av and the width wG of the incident Gaussian light beam must satisfy the relation [3] av wG sec yg ¼ 1:36. So when the incident field has been known, and after the yg also has been obtained from Eq. (12), if there is proper av , we can obtain the maximal coupling efficiency. So we numerically simulate the variation of av with each structure parameter of the waveguide grating, for example, grating thickness d, the ratio a/L between the width of grating slot and the grating period, the absolute value jn2g n20 j of the subtraction of permittivity between the waveguide polymeric part and its unpolymerized part, and the sum n2g þ n20 of the permittivity between the waveguide polymeric part and its unpolymerized part. The variations of av with these parameters are shown from Figs. 2–5 with solid lines. We also analyze the relation between av and each parameter of conventional rectangle structure [3], which is shown from Figs. 2 to 4 with dashed lines (in these figures, the values of the waveguide grating’s parameters we use are the same to those used in Ref. [3], among these parameters, ew is the waveguide permittivity, ea is the air permittivity, es
ARTICLE IN PRESS Y.-B. Yang et al. / Optics & Laser Technology 39 (2007) 1089–1093
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0.035
1.4
0.030
1.2
0.020
εr=3.0
n0=1.6
εw=3.0
ng=1.7
εa=1.0
n2=1.5
εs=2.3
0.015
0.000 0.0
a/Λ=0.5
d=0.3μm
εa=1.0
n2=1.5
εs=2.3
n3=1.0
tw=λ/π
a/Λ=0.5
Λ= λ/2
λ=0.632μm
d= λ/π a/Λ=0.5
0.4
a/Λ =0.5
Λ=λ/2 0.005
0.8 0.6
n3=1.0
tw= λ/π
0.010
1.0
αν
αν
0.025
εw=3.0
0.2
λ=0.632μm
0.0 0.0 0.4
0.8
1.2
1.6
2.0
0.4
0.8
2.4
d (μm) Fig. 2. Solid line illustrates the variation of the loss coefficient av with the waveguide grating thickness d for the novel structure, the parameters of the right column marked in the figure are used for the novel structure. Dashed line illustrates the variation of the normalized loss coefficient av l with the grating thickness d/l for the conventional rectangle structure, the parameters of the left column marked in the figure are used for the conventional structure.
1.2
1.6
2.0
2 2 |ng-n0|
Fig. 4. Solid line illustrates the variation of loss coefficient av with the absolute value jn2g n20 j of the subtraction of permittivity between the waveguide polymeric part and its unpolymerized part for the novel structure, the parameters of the right column marked in the figure are used for the novel structure. Dashed line illustrates the variation of the normalized loss coefficient 100av l with the grating permittivity er for the conventional rectangle structure, the parameters of the left column marked in the figure are used for the conventional structure.
0.05 d=0.3μm
0.020
d=0.3∝m
εw=3.0
n0=1.6
εa=1.0
ng=1.7
0.04
εs=2.3
αν
0.015
n2=1.5
λ=0.632μm
0.03
n2=1.5
tw=λ/π
n0=1.0 a/Λ=0.5
αν
0.025
εr=3.0
0.02
n3=1.0
Λ=λ/2 0.010
λ=0.632μm
d=λ/2π
0.01
0.00
0.005
4
6
8
10
12 2
14
16
18
20
2
n0+ng 0.000 0.0
0.2
0.4
0.6
0.8
1.0
a/Λ Fig. 3. Solid line illustrates the variation of loss coefficient av with the ratio a/L between the width of grating slot and the grating period for the novel structure, the parameters of the right column marked in the figure are used for the novel structure. Dashed line illustrates the variation of loss coefficient av with the ratio a/L between the width of grating slot and the grating period for the conventional rectangle structure, the parameters of the left column marked in the figure are used for the conventional structure.
Fig. 5. Relation between av and the sum n2g þ n20 of the permittivity between the waveguide polymeric part and its unpolymerized part for the novel structure, the other parameters are also marked in the figure.
is the substrate permittivity, L is the grating period, tw is the waveguide thickness, a/L is the ratio between the width of grating slot and grating period, d is the grating thickness, er is the grating permittivity). In the conventional rectangle structure, the relation between av and each parameter obtained in this paper with mode coupling
ARTICLE IN PRESS Y.-B. Yang et al. / Optics & Laser Technology 39 (2007) 1089–1093
theory are basically the same to the result reported in Ref. [3]. After comparing the conventional rectangle structure’s relation between av and each parameter with the novel structure’s, we can know that in the novel structure analyzed in this paper, when the grating thickness is increased, the av first gets large and then falls, the av’s value also fluctuates when it is falling, in the end the av gets to a very little value which is close to zero; but in conventional rectangle structure, when grating thickness is increased, the av first gets large and then when it gets the max, the av only fluctuates in the position of the max with a certain period, it no more gets small, this is the difference between the novel structure and the conventional rectangle structure. The relation between av and the ratio a/L between the width of grating slot and the grating period is the same to conventional rectangle structure’s. In conventional rectangle structure, the relation between av and the grating permittivity er (which is shown in Fig. 4 by dashed line) is actually equivalent to the relation between av and jn2g n20 j (which is shown in Fig. 4 by solid line) of the novel structure analyzed by this paper, and their variation trend are the same. In addition, we also analyze the relation between av and the sum n2g þ n20 of the permittivity between the waveguide polymeric part and its unpolymerized part, which adds a condition for the grating’s fabrication and practical applications.
4. Conclusion The electric fields of the TE guided wave and the TE radiating wave are obtained by solving the Helmholtz equation in the spatial rectangular coordinates. And then the coupling between the incident wave and the novel structure waveguide grating shown in Fig. 1 is analyzed by using the mode coupling theories, and the relations between the loss coefficient and the different structure parameters of the waveguide grating are numerical simulated; we also numerically simulated the relations between the loss coefficient and the different structure parameters of the waveguide grating in the conventional rectangle structure, and then we compare and analyze the two results and obtain their sameness and differences.
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Acknowledgements The authors acknowledge the support by the Natural Science Foundations of Shandong Province, China, Grant no. Y2004G01. References [1] Yu XQ, Zhang X, Wong KS, Xu GB, Xu XG, Ren Y, et al. A fabrication of coupling grating in the polymeric waveguide by using two-photon initiated photopolymerization. Mater Lett 2004;58: 3879–83. [2] Yu XQ, Zhang X, Xu GB, Zhao HP, He W, Shao ZS, et al. Fabrication of grating waveguide and coupling grating using twophoton initiated photopolymerization. Chem J Chin Univ 2004; 25(10):1931–3 (in Chinese). [3] Tamir T, Peng ST. Analysis and design of grating couplers. Appl Phys 1977;14(3):235–54. [4] Bates KA, Li LF, Roncone RL, Burke JJ. Gaussian beams from variable groove depth grating couplers in planar waveguides. Appl Opt 1993;32(12):2112–6. [5] Emmons RM, Hall DG. Comparison of film thickness tolerances in waveguide grating couplers. Opt Lett 1991;16(13):998–1000. [6] Haus HA, Schmidt RV. Approximate analysis of optical waveguide grating coupling coefficients. Appl Opt 1976;15(3):774–81. [7] Avrutsky IA, Svakhin AS, Sychugov VA, Parriaux O. High-effiency single-order waveguide grating coupler. Opt Lett 1990;15(24):1446–8. [8] Xu ZQ, Jin GL. Grating apparatus’s application in integrated optical circuit. Opt Tech 1988;2:21–7 (in Chinese). [9] Ma SJ, Li Y, Xu M, Lin JL. Fabricating of waveguide grating by phase mask technique. Chin J Lumin 1998;19(1):77–9 (in Chinese). [10] Jia ZH. Study on polymer PMMA/DR1 Chirped grating coupler. Acta Photon Sin 2003;32(1):86–8 (in Chinese). [11] Lee JM, Park S, Kim M, Ahn JT, Lee MH. Birefringence as a function of upper-cladding layers in polymeric arrayed waveguide gratings. Opt Commun 2004;232:139–44. [12] Ahn SW, Shin SY. Post-fabrication tuning of a polymeric gratingassisted codirectional coupler filter by photobleaching. Opt Commun 2001;194:309–12. [13] Toussaere E, Labbe´ P. Linear and non-linear gratings in DR1 side chain polymers. Opt Mater 1999;12:357–62. [14] Yu XQ, Wang C, Zhao X, Wang XM, Yan YX, Fang Q, et al. Twophoton absorption photoinitiated polymerization. Chem J Chin Univ 2000;21(12):1953–5 (in Chinese). [15] Maruo S, Nakamura O, Kawata S. Three-dimensional microfabrication with two-photon-absorbed photopolymerization. Opt Lett 1997; 22(2):132–4. [16] Cumpston BH, Ananthavel SP, Barlow S, Dyer DL. Two-photon polymerization initiators for three-dimensional optical data storage and microfabrication. Nature 1999;398:51–4.