Long-period gratings in planar optical waveguides Vipul Rastogi and Kin Seng Chiang We present a theoretical analysis of light propagation in a four-layer planar waveguide that consists of a long-period grating (LPG)having a period of the order of 100 um.By means of the coupled-mode theory,we show that such a structure is capable of coupling light from the fundamental guided mode to the cladding modes at specific wavelengths(resonance wavelengths)and thus results in sharp rejection bands in the transmission spectrum of the waveguide.Our numerical results show that the resonance wavelengths as well as the transmission spectrum can be significantly changed with the waveguide and grating parameters.A waveguide-based LPG should provide a useful approach to the design of a wide range of integrated-optic devices,including wavelength-tunable filters,switches,and environmental sensors.2002 Optical Society of America 0 CIS codes:060.2340,130.0130.130.3120,130.6010,350.2770. 1.Introduction waveguide-based LPG exhibits a much wider range of Recently,there has been considerable research on optical characteristics because of the additional de- long-period fiber gratings for their applications as grees of freedom available in the design of optical gain flatteners of erbium-doped fiber amplifiers,1-4 waveguides.We therefore expect a wide range of wavelength filters,5-10 broadband add/drop multi- applications with waveguide-based LPGs,especially plexers,11 dispersion controllers,12.13 and various in the construction of integrated-optic devices. kinds of sensors.14-20 A long-period grating(LPG) in the core of a single-mode fiber enables light cou- pling from the guided mode to the cladding modes 2.Analysis and thus produces dips at specific wavelengths(res- The waveguide structure is shown in Fig.1,which onance wavelengths)in the transmission spectrum of consists of a thick substrate of refractive index ng,a the fiber.However,optical fibers are exclusively guiding film of refractive index nr and thickness d,a round in shape and are made of silica.The geome- cladding layer of refractive index nc and thickness try and material constraints of a fiber impose signif- icant limitations on the functions that an LPG can da and an external medium of refractive index that extends to infinity,where n>ne>ns,nex. achieve.To remove such constraints,we propose assume that the waveguide supports only the funda- forming LPGs in thin-film optical waveguides,which mental (TEo and TMo)mode with nd No ne can be fabricated into many different geometric where No is the mode index,and an LPG with period shapes with many different kinds of materials.In A is embedded in the guiding film.The LPG allows this paper,we investigate some general properties of light coupling from the fundamental mode to the a waveguide-based LPG by considering an LPG in a cladding(TE and TM)modes whose mode indices slab waveguide with a cladding layer.Although the N(m =1,2,3,...)are smaller than ndl,i.e.,n< light-coupling mechanisms in a waveguide-based <nd. LPG and a fiber LPG are basically the same,a The cladding layer plays a key role in the present study.Without the cladding layer,the structure re- duces to a three-layer slab waveguide.The LPG in a three-layer slab waveguide can at best couple light The authors are with the Department of Electronic Engineering, to the radiation or substrate modes,but the efficiency Optoelectronics Research Centre,City University of Hong Kong, Tat Chee Avenue,Hong Kong,China.V.Rastogi's email address will be low (because of the small field overlap between is vipul.rastogi@cityu.edu.hk. the guided mode and the radiation mode).Even Received 4 January 2002;revised manuscript received 26 July though light coupling can still take place,no distinct 2002. resonance wavelengths will be seen (because the ra- 0003-6935/02/306351-05$15.00/0 diation modes cover a continuum of mode indices). 2002 Optical Society of America For the LPG to function.therefore,it is essential to 20 October 2002/Vol.41,No.30/APPLIED OPTICS 6351
Long-period gratings in planar optical waveguides Vipul Rastogi and Kin Seng Chiang We present a theoretical analysis of light propagation in a four-layer planar waveguide that consists of a long-period grating LPG having a period of the order of 100 m. By means of the coupled-mode theory, we show that such a structure is capable of coupling light from the fundamental guided mode to the cladding modes at specific wavelengths resonance wavelengths and thus results in sharp rejection bands in the transmission spectrum of the waveguide. Our numerical results show that the resonance wavelengths as well as the transmission spectrum can be significantly changed with the waveguide and grating parameters. A waveguide-based LPG should provide a useful approach to the design of a wide range of integrated-optic devices, including wavelength-tunable filters, switches, and environmental sensors. © 2002 Optical Society of America OCIS codes: 060.2340, 130.0130, 130.3120, 130.6010, 350.2770. 1. Introduction Recently, there has been considerable research on long-period fiber gratings for their applications as gain flatteners of erbium-doped fiber amplifiers,1–4 wavelength filters,5–10 broadband adddrop multiplexers,11 dispersion controllers,12,13 and various kinds of sensors.14–20 A long-period grating LPG in the core of a single-mode fiber enables light coupling from the guided mode to the cladding modes and thus produces dips at specific wavelengths resonance wavelengths in the transmission spectrum of the fiber. However, optical fibers are exclusively round in shape and are made of silica. The geometry and material constraints of a fiber impose significant limitations on the functions that an LPG can achieve. To remove such constraints, we propose forming LPGs in thin-film optical waveguides, which can be fabricated into many different geometric shapes with many different kinds of materials. In this paper, we investigate some general properties of a waveguide-based LPG by considering an LPG in a slab waveguide with a cladding layer. Although the light-coupling mechanisms in a waveguide-based LPG and a fiber LPG are basically the same, a waveguide-based LPG exhibits a much wider range of optical characteristics because of the additional degrees of freedom available in the design of optical waveguides. We therefore expect a wide range of applications with waveguide-based LPGs, especially in the construction of integrated-optic devices. 2. Analysis The waveguide structure is shown in Fig. 1, which consists of a thick substrate of refractive index ns, a guiding film of refractive index nf and thickness df , a cladding layer of refractive index ncl and thickness dcl, and an external medium of refractive index nex that extends to infinity, where nf ncl ns, nex. We assume that the waveguide supports only the fundamental TE0 and TM0 mode with ncl N0 nf , where N0 is the mode index, and an LPG with period is embedded in the guiding film. The LPG allows light coupling from the fundamental mode to the cladding TEm and TMm modes whose mode indices Nmm 1, 2, 3, . . . are smaller than ncl, i.e., ns Nm ncl. The cladding layer plays a key role in the present study. Without the cladding layer, the structure reduces to a three-layer slab waveguide. The LPG in a three-layer slab waveguide can at best couple light to the radiation or substrate modes, but the efficiency will be low because of the small field overlap between the guided mode and the radiation mode. Even though light coupling can still take place, no distinct resonance wavelengths will be seen because the radiation modes cover a continuum of mode indices. For the LPG to function, therefore, it is essential to The authors are with the Department of Electronic Engineering, Optoelectronics Research Centre, City University of Hong Kong, Tat Chee Avenue, Hong Kong, China. V. Rastogi’s email address is vipul.rastogi@cityu.edu.hk. Received 4 January 2002; revised manuscript received 26 July 2002. 0003-693502306351-05$15.000 © 2002 Optical Society of America 20 October 2002 Vol. 41, No. 30 APPLIED OPTICS 6351
n(x) where I=Bo-Bm-2T/A represents the phase mismatch;K =(koAno2/8co)denotes the coupling coefficient with c the speed of light in vacuum and wo the permeability;and n foEoE dx is the overlap integral that measures the spatial overlap between nf the guided and the cladding mode fields in the guid- ing film region.Equations(3)and(4)can be solved ncl- analytically,and the variation of the power in the guided mode with the propagation distance is given as ns K Pa(2)=A(2)P=P1-sinyz (5) where y2=k2+r2/4 and Po =A(0)2.Using Eq. (5),we can study the variation of the transmitted nex- power with the wavelength for given waveguide and grating parameters.In general,maximum light Fig.1.Refractive-index profile of a planar waveguide with a clad- coupling takes place at wavelengths that correspond ding layer,where a long-period grating lies in the region 0<x<d. to I=0,which are called the resonance wavelengths 入0 create a set of discrete cladding modes by introduc- 入o=(No-Nm)A, (6) tion of a cladding layer. Our analysis follows the coupled-mode theory,in where No=Bo/ko and Nm Bm/ko (m =1,2,3,...) which the total field in the waveguide is expressed as are evaluated at Ao Equation (6)is referred to as a superposition of the guided and the cladding mode the phase-matching condition of the grating. fields.21 Here we consider only the TE modes and express the total field as 3.Numerical Results and Discussion We first study the relationship between the reso- 1 = [A(2)E()expli(t-Boz)] nance wavelengths and the period of the grating. The following waveguide parameters are used:ns= 1.5,nr=1.52,ng=1.51,nex=1.0(air),dr=2.0μm +B(z)Em(x)exp[i(t-Bmz)]+cc,(1) da =30.0 um,grating length L 2.5 cm,and index where Eo(x)and E(x)(real functions)are the power- modulation Ano2/2n 2 x 10-4.In our calcula- tions,the index modulation is assumed to confine in normalized fields of the guided and cladding modes, respectively,and can be obtained if one solves the the guiding film only.Any additional index modu- eigenvalue equations of the four-layer slab lation in other regions will affect only the coupling waveguide22;A(z)and B(z)are the corresponding efficiency without changing the qualitative nature of z-dependent amplitude coefficients,and Bo and Bm the results.LPGs of this type could be fabricated in Ge-doped silica waveguides or polymer waveguides are the corresponding propagation constants at opti- cal frequency @The total field Y satisfies the fol- by laser writing.LPGs by corrugations with con- lowing scalar wave equation: ventional etching techniques are also possible for a wide range of glass and polymer materials.The in- a2型a2型 dex modulation we choose here is typical of ultravio- +2+bn2()+△nx,2=0, .x2 (2) let laser written gratings in doped silica.The results are shown in Fig.2(a),where the curves cor- where An2(x,z)=Ano2 sin(2m/A)z is the sinusoidal respond to couplings from the TEo guided mode to the index perturbation in the z-direction that represents different cladding modes.The curves in Fig.2(a) the grating and Ano2 is the amplitude of the pertur- can be termed as the phase-matching curves,as they bation;ko=2T/A is the free-space wavenumber with are obtained from the phase-matching condition,Eq. A the free-space wavelength.Substituting from (6).The phase-matching curves help us to choose a Eq.(1)into Eq.(2)and using the slowly varying en- grating period to filter out a certain wavelength from velope approximation,we arrive at the following two the transmission spectrum of the waveguide.As coupled-mode equations: shown in Fig.2(a),the number of cladding modes available for light coupling decreases as the grating dA period increases.The phase-matching curve can KBexp(irz), (3) turn backward at a particular value of grating period, dz and,as a result,admit two different resonance wave- lengths.The existence of double resonance wave- dB =-KAexp(-ilz). (4) lengths for a particular cladding mode can be dz understood from the fact that the phase-matching 6352 APPLIED OPTICS/Vol.41,No.30/20 October 2002
create a set of discrete cladding modes by introduction of a cladding layer. Our analysis follows the coupled-mode theory, in which the total field in the waveguide is expressed as a superposition of the guided and the cladding mode fields.21 Here we consider only the TE modes and express the total field as 1 2 A z E0 xexp i t 0 z B z Em xexp i t m z cc, (1) where E0x and Emx real functions are the powernormalized fields of the guided and cladding modes, respectively, and can be obtained if one solves the eigenvalue equations of the four-layer slab waveguide22; Az and Bz are the corresponding z-dependent amplitude coefficients, and 0 and m are the corresponding propagation constants at optical frequency . The total field satisfies the following scalar wave equation: 2 x2 2 z2 k0 2 n2 x n2 x, z 0, (2) where n2 x, zn0 2 sin2z is the sinusoidal index perturbation in the z-direction that represents the grating and n0 2 is the amplitude of the perturbation; k0 2 is the free-space wavenumber with the free-space wavelength. Substituting from Eq. 1 into Eq. 2 and using the slowly varying envelope approximation, we arrive at the following two coupled-mode equations: dA dz Bexpiz, (3) dB dz Aexp iz, (4) where 0 m 2 represents the phase mismatch; k0n0 2 8c0 denotes the coupling coefficient with c the speed of light in vacuum and 0 the permeability; and 0 df E0Emdx is the overlap integral that measures the spatial overlap between the guided and the cladding mode fields in the guiding film region. Equations 3 and 4 can be solved analytically, and the variation of the power in the guided mode with the propagation distance is given as PA z A z 2 P01 2 2 sin2 z , (5) where 2 2 2 4 and P0 A0 2 . Using Eq. 5, we can study the variation of the transmitted power with the wavelength for given waveguide and grating parameters. In general, maximum light coupling takes place at wavelengths that correspond to 0, which are called the resonance wavelengths 0: 0 N0 Nm, (6) where N0 0k0 and Nm mk0 m 1, 2, 3, . . . are evaluated at 0. Equation 6 is referred to as the phase-matching condition of the grating. 3. Numerical Results and Discussion We first study the relationship between the resonance wavelengths and the period of the grating. The following waveguide parameters are used: ns 1.5, nf 1.52, ncl 1.51, nex 1.0 air, df 2.0 m, dcl 30.0 m, grating length L 2.5 cm, and index modulation n0 2 2nf 2 104 . In our calculations, the index modulation is assumed to confine in the guiding film only. Any additional index modulation in other regions will affect only the coupling efficiency without changing the qualitative nature of the results. LPGs of this type could be fabricated in Ge-doped silica waveguides or polymer waveguides by laser writing. LPGs by corrugations with conventional etching techniques are also possible for a wide range of glass and polymer materials. The index modulation we choose here is typical of ultraviolet laser written gratings in doped silica. The results are shown in Fig. 2a, where the curves correspond to couplings from the TE0 guided mode to the different cladding modes. The curves in Fig. 2a can be termed as the phase-matching curves, as they are obtained from the phase-matching condition, Eq. 6. The phase-matching curves help us to choose a grating period to filter out a certain wavelength from the transmission spectrum of the waveguide. As shown in Fig. 2a, the number of cladding modes available for light coupling decreases as the grating period increases. The phase-matching curve can turn backward at a particular value of grating period, and, as a result, admit two different resonance wavelengths. The existence of double resonance wavelengths for a particular cladding mode can be understood from the fact that the phase-matching Fig. 1. Refractive-index profile of a planar waveguide with a cladding layer, where a long-period grating lies in the region 0 x df . 6352 APPLIED OPTICS Vol. 41, No. 30 20 October 2002
2.4 △B(m 2 TE E 0.0425 2.0 de =4.5 um 18 TE2 0.0415 TE Λ=1544m (a) 1.6 0.0405 1.4 0.0395 .2 1.3 1.4 1.5 1.6 1.7 1.8 1.0 200 300 400500600 700 800 900 0.035 A (um) (a) da=5.5μm 0.034 1.0 0.033 (b) 0.8 TE 0.032 Λ=1944m 0.03 "a/(Tva 0.6 1.2 1.3 1.41.5 1.6 1.71.8 1.9 2.0 2.1 TE2 0.0238 0.4 0.0234 da =7.5 um 0.2 TE 0.0230 (c) 0.0 1.0 1.2 1.4 1.6 1.8 2.0 2.2 0.0226 A=278 um 入(m) (b) Fig.2.(a)Phase-matching curves for a long-period grating in a 0.0222 .4 1.5 1.6 1.71.81.92.02.1 planar waveguide with n,=1.5,n=1.52,na 1.51,nx=1.0, 入(um) d,=2.0 um,and d=30.0 um.The dashed vertical line marks Fig.3.Variation of the phase mismatch AB between the TEo and the grating period A 290 um.(b)Transmission spectrum of a TE,modes as a function of wavelength for (a)d=4.5 um,(b)d= grating that is 2.5 cm long and has a period of 290 um,showing double resonance wavelengths for the coupling to the TE mode. 5.5 um,and (c)da=7.5 um.The dashed straight lines show the values of 2/A and their points of intersection with the curves give the resonance wavelengths. condition Eq.(6)is a nonlinear equation in wave- length and thus admits multiple roots.In our case, phase mismatch between the TEo guided mode and for example,a grating period of 290 um gives two TE1 cladding mode,△p=βo-B,as a function of resonance wavelengths for the TE mode,but only wavelength for three different values of cladding one resonance wavelength for each of the TEs,TE2 thickness.Figure 3(a)shows the case de=4.5 um, and TE modes.The dual-resonance phenomenon which gives a well-defined single resonance wave- has also been observed in a long-period fiber grating length at 1.55 um with a grating period of 154 um. but for a very high-order cladding mode (e.g.,the When we increase the cladding thickness to 5.5 um, LPo1s mode).23 The relatively thin cladding layer of the curve becomes flat over a wide range of wave- our waveguide allows dual resonance to take place for lengths,as shown in Fig.3(b).In this case,a 1.6- a low-order cladding mode. cm-long grating with a period of 194 um gives a The transmission spectrum of the waveguide with 345-nm-wide rejection band,which is shown in Fig.4. A =290 um is presented in Fig.2(b),where two A further increase in the cladding thickness can pro- well-separated resonance wavelengths for the TE duce a U-shape curve,resulting in double resonance mode are clearly shown.It can be seen from Fig. wavelengths with an appropriate choice of the grat- 2(b)that the bandwidth of the rejection band in- ing period.As shown in Fig.3(c),in the case da creases with the resonance wavelength,which is con- 7.5 um,a grating period of 278 um gives two reso- sistent with the fact that the bandwidth is nance wavelengths at 1.56 um and 2.0 um. proportional to the square of the resonance wave- We next investigate how the changes in the clad- length.5 The strength of the rejection band in- ding parameters affect the transmission spectrum of creases with the order of the cladding mode,which is a given grating.We assume a grating period of 388 due to a larger overlap integral with a higher-order um and a grating length of 1.8 cm and consider only cladding mode. the coupling to the TE mode.In Fig.5,we show The transmission spectrum of the grating depends that the resonance wavelength of the grating is strongly on the fashion in which the phase mismatch shifted from 1.55 to 1.24 um by changing the cladding between the interacting modes varies with the wave- thickness de from 10 to 20 um,while keeping the length.The cladding parameters can be used to ma- refractive index of the cladding constant(nc=1.51). nipulate this phase mismatch to obtain a desired It can be seen that as the cladding thickness in- spectrum.To illustrate this,we plot in Fig.3 the creases the strength of the grating decreases and the 20 October 2002/Vol.41,No.30/APPLIED OPTICS 6353
condition Eq. 6 is a nonlinear equation in wavelength and thus admits multiple roots. In our case, for example, a grating period of 290 m gives two resonance wavelengths for the TE4 mode, but only one resonance wavelength for each of the TE3, TE2 and TE1 modes. The dual-resonance phenomenon has also been observed in a long-period fiber grating but for a very high-order cladding mode e.g., the LP015 mode.23 The relatively thin cladding layer of our waveguide allows dual resonance to take place for a low-order cladding mode. The transmission spectrum of the waveguide with 290 m is presented in Fig. 2b, where two well-separated resonance wavelengths for the TE4 mode are clearly shown. It can be seen from Fig. 2b that the bandwidth of the rejection band increases with the resonance wavelength, which is consistent with the fact that the bandwidth is proportional to the square of the resonance wavelength.5 The strength of the rejection band increases with the order of the cladding mode, which is due to a larger overlap integral with a higher-order cladding mode. The transmission spectrum of the grating depends strongly on the fashion in which the phase mismatch between the interacting modes varies with the wavelength. The cladding parameters can be used to manipulate this phase mismatch to obtain a desired spectrum. To illustrate this, we plot in Fig. 3 the phase mismatch between the TE0 guided mode and TE1 cladding mode, 0 1, as a function of wavelength for three different values of cladding thickness. Figure 3a shows the case dcl 4.5 m, which gives a well-defined single resonance wavelength at 1.55 m with a grating period of 154 m. When we increase the cladding thickness to 5.5 m, the curve becomes flat over a wide range of wavelengths, as shown in Fig. 3b. In this case, a 1.6- cm-long grating with a period of 194 m gives a 345-nm-wide rejection band, which is shown in Fig. 4. A further increase in the cladding thickness can produce a U-shape curve, resulting in double resonance wavelengths with an appropriate choice of the grating period. As shown in Fig. 3c, in the case dcl 7.5 m, a grating period of 278 m gives two resonance wavelengths at 1.56 m and 2.0 m. We next investigate how the changes in the cladding parameters affect the transmission spectrum of a given grating. We assume a grating period of 388 m and a grating length of 1.8 cm and consider only the coupling to the TE1 mode. In Fig. 5, we show that the resonance wavelength of the grating is shifted from 1.55 to 1.24 m by changing the cladding thickness dcl from 10 to 20 m, while keeping the refractive index of the cladding constant ncl 1.51. It can be seen that as the cladding thickness increases the strength of the grating decreases and the Fig. 2. a Phase-matching curves for a long-period grating in a planar waveguide with ns 1.5, nf 1.52, ncl 1.51, nex 1.0, df 2.0 m, and dcl 30.0 m. The dashed vertical line marks the grating period 290 m. b Transmission spectrum of a grating that is 2.5 cm long and has a period of 290 m, showing double resonance wavelengths for the coupling to the TE4 mode. Fig. 3. Variation of the phase mismatch between the TE0 and TE1 modes as a function of wavelength for a dcl 4.5 m, b dcl 5.5 m, and c dcl 7.5 m. The dashed straight lines show the values of 2 and their points of intersection with the curves give the resonance wavelengths. 20 October 2002 Vol. 41, No. 30 APPLIED OPTICS 6353
1.0 1.56 54 o 1.52 0.6 1.48 46 0.4 1.4 1.42 0.2 de 5.5 um Λ=194μm 1.40 1.0 1.1 1.2 1.3 1.4 1.5 L=1.6 cm (a) 1.3 1.4 1.5 1.61.7 1.81.9 2.0 1.8 入(um) 1.7 Fig.4.The transmission spectrum of the grating corresponding to Fig.3(b),showing a rejection band as wide as 345 nm. 1.6 1.5 resonance wavelength becomes less sensitive to the cladding thickness.In Fig.6,we show the variation 1.4 of the transmission spectrum of the grating with the 1.3 refractive index of the cladding nc while keeping the cladding thickness constant (d=10 um).The res- 1210 1.1 1.2 1.3 14 nex (b) Fig.7.Variation of the resonance wavelength(for the coupling to the TE,mode)as a function of the refractive index of the external 1-PA(L)/Po medium for (a)the case of a single-resonance wavelength and (b) the case of double-resonance wavelengths. 68 0.6 0.4 0.2 onance wavelength is shifted from 1.82 to 1.18 um by 18 changing nd from 1.509 to 1.512,whereas the 12 strength of the grating is not much affected.With polymeric material such as PMMA (polymethyl- 14 methacrylate)as the cladding material,whose refrac- 1.6 dei (um)16 1.5 tive index has a typical temperature sensitivity of 18 1.4 -1.1 x 10-4/C (Ref.24),a change of temperature 3 入(um) by 1C can shift the resonance wavelength by as 20 1.2 much as 23 nm,which suggests potential applica- Fig.5.Variation of the normalized coupled power (to the TE tions as wavelength-tunable filters and switches us- mode)with d,showing the effect of the cladding thickness on the ing the thermal-optic effect transmission spectrum. We next study how the external refractive index nex affects the transmission spectrum of the grating. In the first example,we assume na=1.51,da=10 1-PA(L)/Po um,A =388 um,and L 1.8 cm and consider the 8 coupling to the TE mode.The variation of the res- onance wavelength with the external index is shown in Fig.7(a).The results are similar to those ob- tained for a long-period fiber grating.17.18 We can 1.509 achieve a shift of~100 nm by changing the refractive index of the external region from 1.0 to 1.5,which is 1.510 comparable with that obtained in a long period fiber grating.17.18 In the next example,we assume ne 1.512,da=6.5μm,A=262.5μm,andL=1.2cm ncl 1.511 so that double resonance wavelengths are present for the TE mode.As shown in Fig.7(b),a change in the refractive index of the external medium can shift the 1.512 11,11.21.31.41.51.61.71.81.9 入(um) two resonance wavelengths in opposite directions. Fig.6.Variation of the normalized coupled power(to the TE, The separation between the two resonant wave- mode)with na,showing the effect of the refractive index of the lengths could be exploited for the measurement of the cladding on the transmission spectrum. refractive index of the external medium.A similar 6354 APPLIED OPTICS/Vol.41,No.30/20 October 2002
resonance wavelength becomes less sensitive to the cladding thickness. In Fig. 6, we show the variation of the transmission spectrum of the grating with the refractive index of the cladding ncl while keeping the cladding thickness constant dcl 10 m. The resonance wavelength is shifted from 1.82 to 1.18 m by changing ncl from 1.509 to 1.512, whereas the strength of the grating is not much affected. With polymeric material such as PMMA polymethylmethacrylate as the cladding material, whose refractive index has a typical temperature sensitivity of 1.1 104 °C Ref. 24, a change of temperature by 1 °C can shift the resonance wavelength by as much as 23 nm, which suggests potential applications as wavelength-tunable filters and switches using the thermal-optic effect. We next study how the external refractive index nex affects the transmission spectrum of the grating. In the first example, we assume ncl 1.51, dcl 10 m, 388 m, and L 1.8 cm and consider the coupling to the TE1 mode. The variation of the resonance wavelength with the external index is shown in Fig. 7a. The results are similar to those obtained for a long-period fiber grating.17,18 We can achieve a shift of 100 nm by changing the refractive index of the external region from 1.0 to 1.5, which is comparable with that obtained in a long period fiber grating.17,18 In the next example, we assume ncl 1.512, dcl 6.5 m, 262.5 m, and L 1.2 cm, so that double resonance wavelengths are present for the TE1 mode. As shown in Fig. 7b, a change in the refractive index of the external medium can shift the two resonance wavelengths in opposite directions. The separation between the two resonant wavelengths could be exploited for the measurement of the refractive index of the external medium. A similar Fig. 4. The transmission spectrum of the grating corresponding to Fig. 3b, showing a rejection band as wide as 345 nm. Fig. 5. Variation of the normalized coupled power to the TE1 mode with dcl, showing the effect of the cladding thickness on the transmission spectrum. Fig. 6. Variation of the normalized coupled power to the TE1 mode with ncl, showing the effect of the refractive index of the cladding on the transmission spectrum. Fig. 7. Variation of the resonance wavelength for the coupling to the TE1 mode as a function of the refractive index of the external medium for a the case of a single-resonance wavelength and b the case of double-resonance wavelengths. 6354 APPLIED OPTICS Vol. 41, No. 30 20 October 2002
phenomenon has been observed with a long-period 7.A.A.Abramov,B.J.Eggleton,J.A.Rogers,R.P.Espindola,A. fiber grating for a higher-order cladding mode.25 Hale,R.S.Windeler,and T.A.Strasser,"Electrically tunable In our study,we have considered only the TE efficient broad-band fiber filter."IEEE Photon.Technol.Lett. modes of the planar waveguide to demonstrate the 11.445-447(1999). general properties of an LPG in a waveguide.The 8.D.M.Costantini,C.A.P.Muller,S.A.Vasiliev,H.G.Lim- analysis for the TM modes is obvious and should berger,and R.P.Salathe,"Tunable loss filter based on metal- coated long-period fiber grating,"IEEE Photon.Technol.Lett. produce similar results.LPGs can also be formed in 11,1458-1560(1999). channel waveguides,and it is always possible to de- 9.O.Deparis,R.Kiyan,O.Pottiez,M.Blondel,I.G.Korolev,S.A. sign polarization-insensitive LPGs with zero- Vasiliev,and E.M.Dianov,"Bandpass filters based on pi- birefringence waveguide structures.26 shifted long-period fiber gratings for actively mode-locked er- bium fiber lasers,"Opt.Lett.26,1293-1241 (2001). 4. Conclusion 10.M.Das and K.Thyagarajan,"Wavelength-division multiplex- We have analyzed the transmission characteristics of ing isolation filter using concatenated chirped long period grat- an LPG in a planar waveguide with a cladding layer. ings,”0pt.Commun.197,67-71(2001). Our numerical results show that the cladding param- 11.K.S.Chiang,Y.Liu,M.N.Ng,and S.Li,"Coupling between eters of the waveguide have significant effects on the two parallel long-period fibre gratings,"Electron.Lett.36, 1408-1409(2000). transmission spectrum of the LPG.Unlike a fiber 12.D.B.Stegall and T.Erdogan,"Dispersion control with use of whose dimensions and materials are standardized,a long-period fiber gratings,"J.Opt.Soc.Am.A 17,304-312 waveguide can be fabricated into many shapes with (2000). many different materials.The control of the clad- 13.M.Das and K.Thyagarajan,"Dispersion compensation in ding parameters can therefore provide great flexibil- transmission using uniform long period fiber gratings,"Opt. ity in the control of the transmission spectrum of the Commun.190.159-163(2001). LPG.Furthermore,we can envision many new ap- 14.V.Bhatia and A.M.Vengsarkar,"Optical fiber long-period plications with waveguide-based LPGs by exploita- grating sensors,"Opt.Lett.21,692-694(1996). tion of the numerous material systems available for 15.V.Bhatia,D.Campbell,R.O.Claus,and A.M.Vengsarkar. making waveguides.Active waveguide devices "Simultaneous strain and temperature measurement with based on LPGs using electro-optic and thermal-optic long-period gratings,"Opt.Lett.22,648-650(1997). materials are possible.Thin-film biochemical and 16.V.Grubsky and J.Feinberg,"Long-period fiber gratings with environmental sensors based on LPGs also look at- variable coupling for real-time sensing applications,"Opt. Lett.25,203-205(2000). tractive.The possibility of making various kinds of 17.H.J.Patrick,A.D.Kersey,and F.Bucholtz,"Analysis of the devices using a multilayer overlay in the cladding is response of long-period fiber gratings to the external index of another advantage offered by planar LPGs.We be- refraction,"J.Lightwave Technol.16,1606-1612(1998). lieve that LPG in waveguide offers a promising ap- 18.K.S.Chiang,Y.Liu,M.N.Ng,and X.Dong,"Analysis of proach to the design of a wide range of integrated- etched long-period fibre grating and its response to external optic devices and sensors. refractive index,"Electron.Lett.36,966-967 (2000). 19.S.Khaliq,S.W.James,and R.P.Tatam,"Fiber-optic liquid- The work was supported by a grant from the Re- level sensor using a long-period grating,"Opt.Lett.26,1224- search Grants Council of the Hong Kong Special Ad- 1226(2001). ministrative Region,China [Project No.CityU 1160/ 20.B.H.Lee,Y.Liu,S.B.Lee,S.S.Choi,and J.N.Jang, 01E1. "Displacements of the resonant peaks of a long-period fiber grating induced by a change of ambient refractive index,"Opt. References Lett.22,1769-1771(1997). 1.A.M.Vengsarkar,J.R.Pedrazzani,J.B.Judkins,P.J.La- 21.A.Yariv,Optical Electronics in Modern Communications (Ox- maire,N.S.Bergano,and C.R.Davidson,"Long-period fiber- ford U.Press,New York,1997). grating-based gain equalizers,"Opt.Lett.21,336-338(1996). 22.M.J.Adams,An Introduction to Optical Waveguides (Wiley, 2.P.F.Wysocki,J.B.Judkins,R.P.Espindola,M.Andrejco,and New York,1981)Chap.2,pp.75-77. A.M.Vengsarkar,"Broad-band erbium-doped fiber amplifier 23.X.W.Shu,X.M.Zhu,Q.L.Wang,S.Jiang,W.Shi,Z.J. flattened beyond 40 nm using long-period grating filter,"IEEE Huang,and D.X.Huang,"Dual resonant peaks of LPois clad- Photon.Technol.Lett.9,1343-1345(1997). ding mode in long-period gratings,"Electron.Lett.35,649- 3.J.R.Qian and H.F.Chen,"Gain flattening fibre filters using 651(1999). phase-shifted long period fibre gratings,"Electron.Lett.34, 24.R.S.Moshrefzadeh,M.D.Radcliffe,T.C.Lee,and S.K 1132-1133(1998). Mohpatra,"Temperature dependence of index of refraction of 4.M.K.Pandit,K.S.Chiang,Z.H.Chen,and S.P.Li,"Tunable polymeric waveguides,"J.Lightwave Technol.10,420-425 long-period fiber gratings for EDFA gain and ASE equaliza- (1992). tion,"Microwave Opt.Technol.Lett.25,181-184(2000). 25.X.W.Shu,X.M.Zhu,S.Jiang,W.Shi,and D.X.Huang,"High 5.A.M.Vengsarkar,P.J.Lemaire,J.B.Judkins,V.Bhatia,T sensitivity of dual resonant peaks of long-period fiber grating Erdogan,and J.E.Sipe,"Long-period fiber gratings as band- to surrounding refractive index changes,"Electron.Lett.35, rejection filters,"J.Lightwave Technol.14,58-65(1996). 1580-1581(1999) 6.B.H.Lee and J.Nishii,"Notch filters based on cascaded mul- 26.W.P.Wong and K.S.Chiang,"Design of polarization- tiple long-period fibre gratings,"Electron.Lett.34,1872-1873 insensitive Bragg gratings in zero-birefringence ridge (1998). waveguides,"IEEE J.Quantum Elect.37,1138-1145(2001). 20 October 2002/Vol.41,No.30/APPLIED OPTICS 6355
phenomenon has been observed with a long-period fiber grating for a higher-order cladding mode.25 In our study, we have considered only the TE modes of the planar waveguide to demonstrate the general properties of an LPG in a waveguide. The analysis for the TM modes is obvious and should produce similar results. LPGs can also be formed in channel waveguides, and it is always possible to design polarization-insensitive LPGs with zerobirefringence waveguide structures.26 4. Conclusion We have analyzed the transmission characteristics of an LPG in a planar waveguide with a cladding layer. Our numerical results show that the cladding parameters of the waveguide have significant effects on the transmission spectrum of the LPG. Unlike a fiber whose dimensions and materials are standardized, a waveguide can be fabricated into many shapes with many different materials. The control of the cladding parameters can therefore provide great flexibility in the control of the transmission spectrum of the LPG. Furthermore, we can envision many new applications with waveguide-based LPGs by exploitation of the numerous material systems available for making waveguides. Active waveguide devices based on LPGs using electro-optic and thermal-optic materials are possible. Thin-film biochemical and environmental sensors based on LPGs also look attractive. The possibility of making various kinds of devices using a multilayer overlay in the cladding is another advantage offered by planar LPGs. We believe that LPG in waveguide offers a promising approach to the design of a wide range of integratedoptic devices and sensors. The work was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China Project No. CityU 1160 01E . References 1. A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, P. J. Lamaire, N. S. 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