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The analytic solutions (AS) for the spectral properties of short- and long-period waveguide gratings with the effects of discrete phase shift (PS), Gaussian-apodization (GA) and superstructure are presented in this paper, which are derived from the Fourier mode coupling (FMC) theory proposed recently. The spectral properties include the reflectivity of short-period gratings, and the transmission of long-period gratings. The calculated spectra based on the analytic solutions are achieved and compared with measured cases and that on the transfer matrix (TM) method, in the case of changing grating parameters. The AS-based calculation requires the average time of several milliseconds at common PC, and the AS-based efficiency is improved up to ~6700 times the TM-based one. The comparisons have confirmed that the FMC-based analytic solutions are suitable for the real-time and accurate analyses of some non-uniform waveguide gratings.
©2011 Optical Society of America
1. Introduction
Waveguide grating has been developed into a critical component with wavelength selectivity, which can be induced in slab or planar waveguide, or optical fiber. According to coupling directions, waveguide gratings are classified to be two categories: short- (i.e., Bragg) and long-period gratings [1, 2]. Most gratings for practical applications are non-uniform (NU) gratings [2, 3], such as phase-shift (PS), Gaussian-apodized (GA), superstructure and linearly chirped gratings etc., since they can shape spectra. GA waveguide gratings with short- or long-period are used extensively for optical communication and sensing, especially for wavelength-division-multiplexed systems [4, 5], as they can reduce the undesirable sidelobes prevalent in uniform-grating spectra, and shape spectra extremely sharp. Superstructure Bragg grating (SBG, namely sampled Bragg grating) has a special grating structure described as a series of equally spaced Bragg grating (BG) segments [6]. An SBG can couple a propagating core mode simultaneously to the backward core mode and the co-directional cladding modes, like Bragg grating and long-period grating, respectively. Thus it will result in multi-wavelength Bragg reflection and attenuation band in transmission spectra. SBG has been widely applied to comb-like reflector, multi-wavelength laser, dispersion compensation and refractive index sensing etc.. PS grating has a discrete phase shift at some position, and leads to an ultra-narrowband transmission related to the position and the magnitude of the shifted phase [7]. It can be used for all-optical switching, optical coder/decoder and ultra-narrowband filter etc..
The analysis of spectral property is very important for the design, fabrication and application of a waveguide grating or its mask. The approaches to analyzing NU-gratings are usually numerical techniques, such as transfer matrix (TM) [8, 9] (including Rouard’s method [10]), direct Fourier transform [11] (FT, including fractional FT [12]), numerical integration (including Runge-Kutta alg
orithm [13]), the numerical implementations of Bloch wave [14] and electromagnetic scattering [15, 16] etc.. The TM method is most often preferred, but is rarely the fastest, since the whole grating is divided into many uniform pieces and then described by multiplying a number of 2 × 2 matrices. The numerical integration to solve the differential coupling equations of a waveguide grating is time-consuming. The methods based on Bloch wave and electromagnetic scattering are less employed, as they are substantially more complicated and also time-consuming. The Runge-Kutta method is generally inferior to the TM method in precision and efficiency. In the direct FT method, the Fourier transform result of index perturbation in any waveguide grating is directly taken for the approximation of spectral property, which amounts to neglecting the nonlinear term in coupling equation, and thus is adapted only for weak-coupling gratings.
Waveguide gratings can be inscribed by using ultraviolet irradiation, CO2 laser radiation, electric-arc discharges and mechanical pressure. The writing processes that use the point-to-point technique can produce a complex pattern in the refractive index modulation, since the writing parameters can be changed after each point. This performance can be exploited easily to explore NU gratings [17, 18]. For timely changing or modifying the writing parameters, especially for the real-time control in the procedure of fabrication, the design and analysis of NU-grating need the real-time and accurate calculations of the spectral properties such as reflectivity and transmission. But for modeling complex NU-gratings, the numerical methods aforementioned possess low efficiency and remain to miss providing real-time calculations, and also are restricted by accumulative computation errors. Therefore it is worthy to look for an excellent approach to fast and accurately analyzing the spectral properties of NU-gratings.
One way available for implementing the real-time calculations is to get their analytic (namely closed-form) solutions, since the analytic solution (AS) of spectral property may straightforwardly describe the effect of parameter changes on the spectral responses, and can provide accurate and fast calculation. However in traditional viewpoint, it is quite difficult to obtain analytic solution for the spectral property of complex NU-gratings, as the cross- and self-coupling coefficients in differential coupled-mode equations are complex and variable along propagating axis. Fourier mode coupling (FMC) theory was proposed recently for the analyses of spectral responses of short- and long-period waveguide gratings, which is fit for uniform, non-uniform and other arbitrary gratings. According to the FMC theory, we try to derive and confirm the general and analytic solutions for the reflectivities of short-period NU-gratings, and for the transmissions of long-period ones. These NU-gratings are with the effects of phase shift or Gaussian-apodization or superstructure. The analytic solutions can be applied to the real-time and accurate analyses for these NU waveguide gratings, which are the fastest as the well-known coupled-mode theory for uniform gratings.
2. FMC theory for short- and long-period gratings
The presences of index perturbations within short- and long-period waveguide gratings will couple a propagating core mode to the backward core mode and the forward cladding modes, respectively. The mode couplings in short- and long-period waveguide gratings are illustrated in Fig. 1(a) and (b) , respectively.
Fig. 1 Schematic diagram of the mode couplings in (a) short- and (b) long-period waveguide gratings.
We assume that an induced index perturbation is created uniformly across waveguide and nonexistent outside the waveguide. According to the Fourier mode coupling theory [19, 20], the general coupling equation for short- and long-period gratings can be constructed to be
where z is propagating axis, Δn(z) is the profile of index perturbation along axis z, m and s are the orders of propagating mode and coupled mode, respectively, Bm(z) and Bs(z) are the amplitude coefficients of propagating and coupled modes, respectively, βm and βs are the propagation constants of propagating and coupled modes, respectively; sign “±” is replaced with “+” for Bragg grating, and with “-” for long-period one; ks is the coupling coefficient between the propagating and coupled modes, defined aswhere A is cross-section area, A∞ is whole cross-section area, r and ϕ are the axes of cylindrical coordinate system, n0 is the refractive index of waveguide, ω is angular frequency, ε0 is the permittivity in vacuum, Em and Es are the electric fields of the propagating and coupled modes, respectively. The overlap integral in Eq. (2) must be numerically calculated for fiber gratings, because of the Bessel-function solutions for the electric fields of optic fiberThe integration of Eq. (1) is performed along the propagating axis z within the limits of the start and end positions of waveguide gratings, which directly leads to
where L is grating length. The boundary conditions are Bm(0) = 1, Bs(L) = 0 for Bragg grating, and Bm(0) = 1, Bs(0) = 0 for long-period grating. According to the boundary conditions and the law of flux conservation [21], the relations between the amplitude coefficients Bm(z) and Bs(z) can be described by the following Eqs. (4) and (5) for short- and long-period gratings, respectivelyWe define the Fourier transform (FT) of the index perturbation Δn(z) within any waveguide grating, as that
where γs(νs) and ηs(νs) represent the real and imaginary components, respectively, H and φ are the modulus and phase, respectively, of the Fourier transform result; νs = (nm + δn0 ± ns)/λ, where λ is the wavelength, δn0 is the effective-index change caused by the index perturbation, nm and ns are the effective index of propagating and coupled modes, respectively. Equation (6) can be implemented by discrete Fourier transform suitable for all types of waveguide gratings, also by analytic solution if there exists closed-form Fourier integration suitable for some non-uniform gratings. Note that ns and ks are related to the order s of coupled mode in a short- or long-period grating.By considering two-mode coupling and the boundary conditions, and substituting Eqs. (6) and (4) or (5) into Eq. (3), we can obtain Bm(z) and Bs(z), and then exactly get the general solutions for the power reflectivity R of a short-period grating, and for the power bar-transmission T of a long-period grating, given by
Here the R and T are power reflection and transmission coefficients, respectively. Some waveguide gratings, such as Moiré, superstructure, raised-cosine- and Gaussian-apodized gratings etc., can be described by modulated sine or cosine functions. In their Fourier transform results from Eq. (6), the modulus H out of resonant band is close to 0, and the phase φ within the resonant band is an approximate constant of about π/2 or -π/2 with small deviation, which lead to γs ≈0 and |ηs|≈H. Thus the phase φ has a quite weak influence on the profiles of the reflective and transmissive spectra, and then R and T can be approximately reduced to Eqs. (9) and (10), respectively
From Eq. (6) and the general solutions R and T, we can derive the analytic solutions for the spectral properties of some non-uniform waveguide gratings, such as superstructure Bragg grating, Gaussian-apodized and phase-shift gratings with short- or long-period. For the derivations and calculations of the following analytic solutions, we will employ Eqs. (7) and (8), or Eqs. (9) and (10).
3. FMC-based analytic solutions
3.1 For short- and long-period GA-gratings
Both the short- and long-period GA-gratings have the similar expression of index perturbation ΔnG(z), such that
where α is Gaussian-apodized coefficient, Λ and δn are the nominal period and amplitude of the perturbation, respectively. In Eq. (11), the value of the period Λ of short-period GA-grating is different from that of long-period grating. Equations (11) and (6) allow us to obtain the closed-form γs = 0, and ηs aswhere σ = νs−1/Λ. Then Eqs. (7) and (8), together with Eq. (12), let us get the analytic solutions for the reflectivity RG of GA Bragg grating, and for the transmission TG of long-period GA grating (LP-GAG), given by Eqs. (13) and (14), respectively where σΒ = (nm + δn0 + ns)/λ−1/Λ and σF = (nm + δn0-ns)/λ−1/Λ are the detuning parameters of counter- and co-directional couplings, respectively. Note that parameters ns and ks in short-period gratings are different from those in long-period gratings.3.2 For superstructure Bragg grating
In fact, SBG is a periodically modulated Bragg grating, and its index perturbation ΔnSB(z) as shown in Fig. 2 can be modeled as
where P is segment length (i.e., modulation period), q is the ordinal number of a segment, N is the number of segments; Λ, a, δn and δ0 are the period, length, AC and DC components, respectively, of the perturbation in each Bragg grating segment.From Eqs. (6) and (15), we can derive the analytic modulus HSB of the Fourier transform result of the SBG index perturbation, written to be
where the parameters i, δ and σ are integral, δ = δn and σ = σb = (nm + ns + δ0)/λ−1/Λ, respectively, for backward coupling, and are i = 1, δ = δ0 and σ = σsc = (nm-ns + aδ0/P)/λ, respectively, for co-directional coupling. By substituting Eq. (16) into Eq. (9), the analytic solution of the power reflectivity RSB of the SBG takes the form asIn SBG, its index perturbation will couple one core mode simultaneously to several co-directional cladding modes if only their phases are matching, and the bar-transmission is only related to the residual light exclusive of the reflection and the cross-transmissions. Therefore the combination of Eqs. (10), (16) and (17) leads to the whole bar-transmission TSB of an SBG, approximately described by
Equation (17) indicates that the reflective peak wavelength λp(g) and the interval Δ between two adjacent reflection peaks are governed by the following Eqs. (19) and (20), respectively
where g is the serial number of the reflection peak, λ0 = 2(nm + δ0)Λ is the wavelength of the central reflection peak labeled by the serial number “0”.3.3 For short- and long-period phase-shift gratings
Like analogous GA gratings, both the short- and the long-period PS-gratings possess the similar form of index perturbation ΔnP(z) expressed by
where Λ is the period of the index perturbation, ϕ and l are the magnitude and position of a discrete phase shift, respectively. Incorporating Eq. (21) into Eq. (6), we can get the modulus square H2 of the Fourier transform result of the index perturbation, given byBy substituting Eq. (22) into Eqs. (9) and (10), we can also obtain the analytic solutions for the reflectivity Rp of short-period PS grating, and for the transmission Tp of long-period one, which seem complex. If a discrete phase shift occurs at the center of a waveguide grating, Rp and Tp will be simplified to be the following Eqs. (23) and (24), respectively
where σΒ and σF are the detuning parameters as defined previously for GA-gratings.4. Simulations and comparisons
We now apply the analytic solutions abovementioned to the calculations of spectral properties, which include the reflectivities of short-period gratings, and the transmissions of long-period ones. These gratings are with the effects of Gaussian-apodization or superstructure, phase shift. The TM method is most often chosen, and suitable for the spectral analyses of various non-uniform gratings. The AS-based calculations will be compared with the TM-based ones. The comparisons can confirm the accuracy and efficiency of the AS-based analyses.
4.1 Short- and long-period GA-gratings
A Bragg GA-grating induced in a single mode fiber has the index perturbation described by the parameters: Λ = 0.5245μm, L = 9mm, δn = 2.5 × 10−4 and α = 20. The Bragg GA-grating couples propagating core mode LP01 to the counter-propagating core mode LP01 with the effective index and the coupling coefficient of 1.4775 and 2530.6πN/s, respectively. Figure 3 plots the calculated reflectivities of the GA-grating. These reflective spectra are calculated in the case of changing the length (a), the amplitude (b) and the Gaussian-apodized coefficient (c), under otherwise identical conditions, according to Eq. (13) (solid lines labeled 1, 2 and 3) and the TM method with the section number of 150 (dotted lines labeled 1', 2' and 3′). The labels 1', 2' and 3′ are corresponding to the labels 1, 2 and 3, respectively. In Fig. 3(a), the lines labeled 1, 2 and 3 represent the cases with L = 9, 6 and 3mm, respectively. In Fig. 3(b), the lines labeled 1, 2 and 3 represent the cases with δn = 2.5 × 10−4, 1.5 × 10−4 and 0.7 × 10−4, respectively. In Fig. 3 (c), the lines labeled 1, 2 and 3 represent the cases with α = 12, 42 and 80, respectively.
Fig. 3 Calculated reflectivities of a Bragg GA-grating, according to Eq. (13) (solid lines labeled 1,2 and 3) and the TM method (dotted lines labeled 1', 2' and 3′), in the case of changing length (a), amplitude (b) and Gaussian-apodized coefficient (c).
A long-period GA-grating induced in a single mode fiber is with the following parameters: Λ = 400μm, nm = 1.4654, ns = 1.4619, L = 90mm, α = 20 and δn = 4 × 10−4. The long-period GA-grating couples the core mode LP01 to the co-propagating cladding mode with the coupling coefficient of 115.64πN/s. Figure 4 illustrates the calculated bar-transmissions of the long-period grating. The bar-transmissions in the case of changing the grating length are based on Eq. (14) (solid lines labeled 1, 2 and 3) and the TM method with the section number of 100 (dotted lines labeled 1', 2' and 3′). The labels 1', 2' and 3′ are also corresponding to the labels 1, 2 and 3, respectively. The lines labeled 1, 2 and 3 represent the cases with L = 90, 50 and 30mm, respectively.
Fig. 4 Calculated bar-transmissions of a long-period GA-grating, according to Eq. (14) (solid lines labeled 1,2 and 3) and the TM method (dotted lines labeled 1', 2' and 3′).
Figure 3 and Fig. 4 show that the Eq. (13)-based reflection band is just slightly narrower than the TM-based one, and that the Eq. (14)-based rejection band is slightly broader than the TM-based one, under the condition of strong coupling. From the two figures, it is evident yet that the calculated spectra based on the analytic solutions Eqs. (13) and (14) are still in excellent agreement with those on the TM method, in aspects of profile, resonant wavelength, sidelobe restraint, reflection and loss peaks etc.. Thus the analytic solutions Eqs. (13) and (14) are fit for the spectral analyses of short- and long-period GA-gratings, respectively.
4.2 Superstructure Bragg grating
To verify the accuracy of AS-based calculations, we employ the analytic solutions to calculate the spectral properties of practical SBGs, and compare the calculated spectra with the measured spectra and the TM-based ones. A 16-mm-long SBG was induced in a single-mode fiber with the effective index nm = 1.4484 of core mode. The practical SBG is described by the following parameters: Λ = 0.5275μm, P = 550μm, δn = 2 × 10−4, δ0 = 7.45 × 10−4, a = 110μm, N = 29 for co-directional coupling. According to Eq. (17) and the TM method, the reflectivities of the SBG are calculated and then demonstrated in Fig. 5(a) and (b) , respectively. Figure 5(c) illustrates the measured reflectivity of the practical SBG, which is adapted from Ref [22]. Figure 5(a), (b) and (c) show that this SBG causes many reflection peaks exclusive of the 5th left peak. Using Eqs. (19) and (20), we can calculate the central wavelength λ0 = 1528.85nm and the peak-wavelength interval Δ = 1.4663nm.
Fig. 5 Calculated and measured reflectivities of an SBG. (a) and (b) are based on Eq. (17) and the TM method, respectively. (c) is the measured one adapted from Ref [22].
There is also an example of the transmission of practical SBG. The SBG was induced in a single mode fiber by using a 10-mm-long uniform phase mask, a 30-mm-long amplitude mask with period Λ = 500μm and duty ratio = 0.5. During the writing process, the amplitude mask was placed on top of the phase mask. The SBG with the perturbation of δn = 1.4 × 10−4 and δ0 = 3.59 × 10−4, reflects the light around central reflection wavelength of 1561nm, and couples the core mode simultaneously to three co-propagating cladding modes with the coupling coefficients of 153.1πΝ/s, 218.3πΝ/s and 254πN/s, respectively. The co-directional couplings result in the losses around 1530nm, 1566nm and 1632nm, corresponding to three values nm-ns = 2.881 × 10−3, 2.953 × 10−3 and 3.085 × 10−3, respectively, as determined by our simulations. The Eq. (18)-based and the measured bar-transmissions of the SBG are shown in Fig. 6(a) and (b) , respectively, where (b) is adapted from Ref [23].
Fig. 6 Calculated and measured bar-transmissions of an SBG. (a) is the calculated one based on Eq. (18). (b) is the measured one adapted from Ref [23].
By comparing the profiles, central wavelengths and peak intervals of the reflective spectra in Fig. 5, we can find that the Eq. (17)-based reflectivity of the SBG is in good agreement with the measured and the TM-based ones, whereas the TM-based one has a little discrepancy relative to the measured case. Figure 6 exhibits that the Eq. (18)-based transmission of another SBG is quite close to its measured one, in aspects of profile, loss peak and 3dB bandwidth. The comparisons demonstrate that the analytic solutions Eqs. (17) and (18) can be used for the accurate analyses of SBG spectra.
4.3 Short- and long-period phase-shift gratings
Now the spectral properties of PS gratings are simulated according to the TM method and the analytic solutions Eqs. (23) and (24). A short-period phase-shift grating, namely phase-shifted Bragg grating (PSBG), is analyzed by using Eq. (23). The PSBG is with the sinusoidal index perturbation described by the parameters below: Λ = 0.28μm, δn = 2 × 10−4, nm = 1.4773, L = 2.1mm and l = L/2. Figure 7(a), (b), (c) and (d) illustrate the calculated reflectivities of the PSBG with the shifted-phase ϕ = 0, π/2, π and 3π/2, respectively, which are based on the analytic solution Eq. (23) (solid lines) and the TM method (dotted lines).
Fig. 7 Calculated reflectivities of a PSBG with ϕ = 0 (a), π/2 (b), π (c) and 3π/2 (d), according to the analytic solution Eq. (23) (solid lines) and the TM method (dotted lines).
A long-period phase-shift grating (LP-PSG) is also analyzed by exploiting the analytic solution Eq. (24). The LP-PSG has the sinusoidal index perturbation described by the following parameters: Λ = 500μm, δn = 4 × 10−4, L = 40mm, l = L/2, nm = 1.4773 and ns = 1.4746. Figure 8(a), (b), (c) and (d) demonstrate the calculated bar-transmissions of the LP-PSG with the shifted phase ϕ = 0, π/2, π and 3π/2, respectively, which are derived from the analytic solution Eq. (24) (solid lines) and the TM method (dotted lines).
Fig. 8 Calculated bar-transmissions of a LP-PSG with shifted phase ϕ = 0 (a), π/2 (b), π (c) and 3π/2 (d), according to Eq. (24) (solid lines) and the TM method (dotted lines).
Figure 7 shows that the phase-shift-induced change of the middle pit in the Eq. (23)-based reflectivity of the PSBG is identical to that in the TM-based reflectivity. Figure 8 demonstrate that the phase-shift-induced variation of the middle peak in the Eq. (24)-based transmission of the LP-PSG is also the same as the TM-based transmission. Both Fig. 7 and Fig. 8 indicate that the calculated spectra based on the analytic solutions Eqs. (23) and (24) have no obvious difference from those on the TM method. Thus the analytic solutions are suitable for the spectral analyses of short- and long-period phase-shift gratings.
4.4 Calculation efficiency of the analytic solutions
In respect of calculation efficiency in the cases of enough accuracy and resolution, the AS-based efficiency is much higher than the TM-based one that is related to the number of divided sections. All the calculations aforementioned were achieved in a PC as Founder S360R with the Vista OS and Matlab5.3. In the cases of 2-pm wavelength resolution and 4-nm span, the calculation of the TM-based reflection spectrum requires the average time of about 8.34s and 16.726s for short-period GA-gratings with the section number of 100 and 200, respectively, and about 0.3455s for short-period PS-grating, and about 4.855s for the SBG with 58 sections. The calculation of the AS-based reflection spectrum needs the average time of about 2.485ms for the short-period GA- and PS-gratings, and about 15.6ms for the SBG, which are almost real-time. This implies that the AS-based calculation efficiency is improved up to ~6730 times the TM-based one for the reflectivity of non-uniform Bragg gratings.
In the cases of 150-nm wavelength span and 20-pm resolution, the calculation of the TM-based transmission requires the average time of about 23.342s and 46.118s for the long-period GA-gratings with the section number of 100 and 200, respectively, and about 0.702s for the long-period PS-grating, about 44.413s for the SBG with 60 sections. The calculation of the AS-based transmission needs the average time of about 7.9ms for the long-period GA- and PS-gratings, and about 15.5ms for the SBG, which are also almost real-time. This indicates that the AS-based calculation efficiency is increased up to ~5837 times the TM-based one for the transmission of non-uniform long-period gratings. The efficiencies abovementioned show that the real-time analyses for the spectral properties of some non-uniform waveguide gratings can be precisely achieved on the bases of analytic solutions.
5. Conclusion
In conclusion we have presented the general solutions for the spectral properties of short- and long-period waveguide gratings, which are derived from the Fourier mode coupling theory proposed recently. The spectral properties include the reflectivity of short-period grating, and the transmission of long-period grating. According to the general solutions, we have obtained the analytic solutions for the reflectivity and transmission of short- and long-period gratings with the effects of Gaussian-apodization, superstructure and phase shift. The AS-based calculations of the reflectivities and the transmissions of these non-uniform gratings are demonstrated and compared with the TM-based ones, which have confirmed the accuracy and efficiency of the analytic solutions. The calculations and comparisons aforementioned exhibit that the reflectivity and transmission based on the analytic solutions are more close to the measured cases, and are more accurate than those on the TM method, in the aspects of shape, resonant wavelength, bandwidth, sidelobe and coupling strength etc.
The demonstrations abovementioned also indicate that the superstructure, Gaussian-apodized and phase-shifted gratings with short- or long-period can be fast analyzed in the case of enough accuracy, by using the analytic solutions derived from the FMC theory. This is pretty significant, especially for the real-time control in the design and fabrication of a complex grating and its mask. The analytic solutions presented in this paper have overthrown the traditional viewpoint that is the absence of the analytic solutions for the spectral properties of non-uniform (complex) waveguide gratings. Based on the FMC theory, we can also get the analytic solutions for other non-uniform gratings, such as linearly chirped, raised-cosine-apodized and Moiré waveguide gratings. Therefore the FMC theory is suitable for the fast and accurate analyses of many complex waveguide gratings.
Acknowledgment
This work was supported partially by the Key Laboratory of Optical Fiber Sensing and Communications (UESTC), Ministry of Education, China, under grant Open Fund Project 2010, and also by the Key Laboratory of Optoelectronic Technology and Systems, Ministry of Education, China, under grant Visiting Scholar Fund 2011.
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