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Family of exact solutions for reflection spectrum of Bragg grating
Optics Communications 215 (2003) 295–301 www.elsevier.com/locate/optcom
Family of exact solutions for reflection spectrum of Bragg grating D.A. Shapiro * Institute of Automation and Electrometry, Siberian Branch, Russian Academy of Sciences, 1 Koptjug Ave., Novosibirsk 630090, Russia Received 12 May 2002; accepted 22 November 2002
Abstract Two-parametric family of exactly solvable profiles of quasi-sinusoidal Bragg gratings is found. A simple formula is derived for their reflection coefficient as a function of wavelength. The detailed analysis of the reflection spectrum width is presented. The solution is helpful as a starting point of optical filter synthesis. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 42.81.Wg; 78.66.)w Keywords: Fiber Bragg grating; Optical filters; Direct scattering problem
1. Introduction Optical filters based on fiber grating have attracted particular interest nowadays because of their applications in high-speed lightware communications [1]. The distributed Bragg reflector consists of periodically spaced elements with varying refractive index of the waveguide material. The array parameters are chosen to strongly reflect light in a narrow band of wavelength. To determine the array parameters a designer needs solving the inverse problem of light scattering, i.e., reconstructing the grating from reflectometric data.
*
Tel.: +7-3832-344021; fax: +7-3832-333863. E-mail address: [email protected].
The problem can be reduced to the GelÕfand– Levitan–Marchenko (GLM) integral equation [2]. The direct problem of calculating a reflection spectrum from given grating profile is well understood, then most of the present literature on this subject analyzes methods of inverse problem solutions or grating synthesis. Song and Shin [3] rederived the GLM equation and its solution when the reflection coefficient was a rational function of the wavenumber. Paladian [4] found exact solutions for the rational functions: for a single resonance mode and an array with nonreciprocal reflection properties. Brinkmeyer [5] proposed an iteration algorithm for inverse problem without GLM. It simplifies mathematics, but badly converges for grating with high reflection. Iteration method for GLM developed by Peral et al. [6]. Recent notable advance centers around the fast
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296
D.A. Shapiro / Optics Communications 215 (2003) 295–301
algorithm of synthesis based on casuality argument. Paladian [7] demonstrated the scheme of calculation for continuous functions. Skaar et al. [8] described the discrete method of ‘‘layer-peeling’’, compared it with continuous analog, and noticed that this technique is known in quantum mechanics, geophysics and electronics (transmission line design). Nevertheless, a question of interest remains in the direct problem. Not many exact solutions are known. They are the rectangular barrier, hyperbolic secant, and its truncated analogs [9]. Finding new exactly solvable profiles is important if only for test of algorithms. The explicit solutions are also helpful for conceptual understanding of the scattering. The aim of the present paper is to find the analytical solutions and extend the family of exactly solvable profiles. The paper is organized as follows. Section 2 describes the substitution suggested by Bambini and Berman [10] for probability amplitudes of a two-level system. The Zakharov–Shabat equations are reduced to Gauss hypergeometric form and two-parametric family of exactly solvable profiles is found. The spectral profile of reflection coefficient is derived in Section 3 in terms of elementary functions and plotted at different values of parameters. The width of reflection spectrum is estimated in Section 4. Section 5 summarizes the qualitative results.
Fig. 1. Quasi-sinusoidal addition to the refractive index dn as a function of coordinate z. Thick curves show the envelope.
where aðzÞ is the envelope shown in Fig. 1, hðzÞ is the phase deviation. The grating is quasi-sinusoidal, when aðzÞ; hðzÞ are slow functions. Let us introduce amplitudes of waves running in positive and negative directions dE ¼ ik a1 eikz þ a2 e ikz : E ¼ a1 eikz þ a2 e ikz ; dz Keeping only resonant terms and neglecting the parametric resonance of higher orders at q ¼ k j=2 k, we get da1 da2 ¼ ikpðzÞa2 e 2iqz ; ¼ ikp ðzÞa1 e2iqz ; ð3Þ dz dz where pðzÞ ¼ aðzÞeihðzÞ =4. Denoting the dimensionless coordinate as f ¼ kz and detuning as n ¼ q=k, we have
2. Hypergeometric equation
a01 ¼ ipðfÞa2 e 2inf ;
ð4Þ
Consider a single-mode fiber with the refractive index n þ dnðzÞ; dn n. Steady-state electric field EðzÞ satisfies one-dimensional Helmholtz equation d2 E 2dnðzÞ xn 2 ; ð1Þ þ k 1 þ E ¼ 0; k ¼ dz2 n c
a02 ¼ ip ðfÞa1 e2inf ;
ð5Þ
where z is coordinate, k is the wavenumber at z ! 1, where dnðzÞ ! 0, x; c are the frequency and speed of light. The addition to refractive index is quasi-sinusoidal function 2dnðzÞ ¼ aðzÞ cos ½ jz þ hðzÞ 1; n
ð2Þ
where the prime is the derivative with respect to f. 2 2 The set conserves ja1 j ja2 j , since the signs at ip and ip are different. The same equations with identical signs conserve the sum of ‘‘populations’’ jai j2 and describes the amplitudes of probability in two-state quantum system. The exact solutions in this case are of importance in quantum optics, then they are studied in details (see [11] and references therein). Calculating the derivative of (4) and substituting a2 ; a02 from Eqs. (4) and (5) we get one secondorder equation
D.A. Shapiro / Optics Communications 215 (2003) 295–301
p0 0 a001 þ 2in a1 jpj2 a1 ¼ 0: p
297
ð6Þ
For simplicity let us consider the envelope without phase modulation, i.e., with h ¼ 0 and real p. When h 6¼ 0 is a constant it also does not enter into Eq. (6). Let us factorize pðfÞ ¼ bP into b and normalized envelope P ðfÞ. Parameter b is the constant of coupling between forward and backward waves. The area under P ðfÞ is assumed equal p. The linear transformation of the independent variable [10] Z f t¼ qðf0 ÞP ðf0 Þ df0 ð7Þ
Fig. 2. Profiles of solvable envelopes at l ¼ 1: k ¼ 0 (solid line); )0.5 (dotted); 1 (dashed).
1
The envelope P ðfÞ is even function with maximum at f ¼ 0 only at k ¼ 0. In general case maximum
reduces Eq. (6) to € a1 þ
q0 þ 2inq b2 _ a1 ¼ 0; a 1 q2 P q2
ð8Þ
where the weight function qðzÞ is to be determined below, a dot denotes the derivative with respect to new variable t, 0 < t < 1, while f varies from 1 to þ1. Comparing (8) with the Gauss hypergeometric equation €þ w
c ða þ b þ 1Þt ab w ¼ 0; w_ tð1 tÞ tð1 tÞ
we get two conditions b2 ab : ¼ tð1 tÞ q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Its solution is the weight function qðtÞ ¼ tð1 tÞ and relation ab ¼ b2 . Changing parameters c 1=2 ¼ 2il; a þ b ¼ 2ik, where k; l are real, we obtain two-parametric family of exactly solvable envelopes pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tð1 tÞ tl P ðtÞ ¼ ; fðtÞ ¼ ln : ð9Þ kþl kt þ l ð1 tÞ
q0 þ 2inq c ða þ b þ 1Þt ; ¼ q2 P tð1 tÞ
The parameters must satisfy conditions l > 0; kþ l > 0. At k ¼ 0 set (9) gives the explicit profile with the shape of hyperbolic secant 1 P ðfÞ ¼ : 2l coshðf=2lÞ
ð10Þ
There are two other cases of explicit but more complicated profiles k ¼ l and k ¼ l=2.
1 Pmax ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 lðk þ lÞ
ð11Þ
is located at t0 ¼ l=ðk þ 2lÞ, as shown in Fig. 2. Since t ¼ 1=2 corresponds to z ¼ 0, the maximum point lies at positive z at k < 0 or at negative z at k > 0. At k < 0 the right front is steeper, at k < 0 the left is. From (7) follows that the area under curve P ðfÞ is really independent of k; l: Z 1 Z 1 dt ¼ p: ð12Þ P ðfÞ df ¼ qðtÞ 1 0 Thus, parameter l gives the characteristics length of envelope at k ¼ 0. For arbitrary k >ffi l the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi length is of the order of l ¼ 2p lðk þ lÞ, as follows from (11) and (12). The two-parametric family of exactly solvable envelopes is the same as for two-level atom [10], whereas the solution of the hypergeometric Eq. (8) is different due to Ô Õ sign in the third term.
3. Reflection spectrum The solution can be expressed in terms of the Gaussian hypergeometric function a1 ðtÞ ¼ c1 F ða; b; c; tÞ þ c2 t1 c F ða þ 1 c; b þ 1 c; 2 c; tÞ;
ð13Þ
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D.A. Shapiro / Optics Communications 215 (2003) 295–301
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a ¼ i nk b2 þ n2 k2 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b ¼ i nk þ b2 þ n2 k2 ;
ð14Þ
1 c ¼ þ 2iln: 2 Let us consider the incident wave e ifn running from þ1 to the left, the boundary condition at z ¼ 1 is a1 ðt ¼ 0Þ ¼ 0. Therefore, c1 ¼ 0, c2 is arbitrary, since the right reflection coefficient, the ratio of amplitudes of falling and reflecting waves, is independent of c2 : 1 a1 ðt ¼ 1Þ d 2inf ¼ ibe lim qðtÞ ln a1 ðtÞ ; r¼ t!1 a2 ðt ¼ 1Þ dt R ¼ jrj2 :
where a; b; c are defined by (14). With help of relation CðzÞCð1 zÞ ¼ p=sinpz [12] the expression can be simplified qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosh 2p b2 þ k2 n2 cosh 2pkn qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : RðnÞ ¼ cosh 2p b2 þ k2 n2 þ cosh 2pðk þ 2lÞn ð15Þ The reflection coefficient as a function of n (the wavelength in dimensionless variables) is plotted in Fig. 3 at fixed l ¼ 1; k ¼ 0 and different b. The profile is triangular at low b and trapezoidal at high b. The profile for fixed b ¼ 1; k ¼ 0 and varying l is shown in Fig. 4. At l 1 the shape approaches a rectangular function. The next Fig. 5 shows the case of different k and fixed b ¼ l ¼ 1: The filter becomes narrower at higher k.
Replacement of l; k by their linear combinations m ¼ l þ k=2;
g ¼ k=2; 0 < jgj < m
makes the profile (9) invariant with respect to substitution t ! 1 t; g ! g: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tð1 tÞ tmþg P ðtÞ ¼ ; f ¼ ln : g þ m 2gt ð1 tÞm g After the transformation only the sign of coordinate changes: P ! P ; f ! f. Then one can get the left reflection coefficient as well by changing the sign of g at fixed m. Below we study the intensity reflection coefficient R that is the same for both directions. The limit can be calculated using the formula for analytic continuation of the hypergeometric function [12]
Fig. 3. Reflection coefficient as a function of detuning n at k ¼ 0; l ¼ 1: from the top down b ¼ 2; 1:5; 1; 0:5.
F ða; b; c; tÞ ¼ A1 F ða; b; a þ b c þ 1; 1 tÞ þ A2 ð1 tÞ
c a b
F ðc a; c b; c a b þ 1; 1 tÞ; A1 ¼
CðcÞCðc a bÞ ; Cðc aÞCðc bÞ
A2 ¼
CðcÞCða þ b cÞ : CðaÞCðbÞ
The intensity reflection coefficient takes the form of ratio of Euler C-functions Cða c þ 1ÞCðb c þ 1Þ 2 ; R ¼ b2 Cð1 aÞCð1 bÞ
Fig. 4. The same as in Fig. 3 at k ¼ 0; b ¼ 1 and different l: from the top down l ¼ 1; 2; 3; 4.
D.A. Shapiro / Optics Communications 215 (2003) 295–301
At N ¼ 2 we have 1 ln 2 þ cosh 2pb w1=2 ¼ 4pl pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ð1 þ cosh 2pbÞð3 þ cosh 2pbÞ :
299
ð18Þ
For small reflectance ðpb 1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 w1=N ln½2N 1 þ N ðN 1Þ ; 4pl in particular, Fig. 5. The same as in Fig. 3 at l ¼ 1; b ¼ 1: from the top down k ¼ 0:75; 0:5; 1; 3; 10.
The expression (15) can be simplified in some particular cases. At n ¼ 0 the saturation in the center is described by formula: Rð0Þ ¼ tanh2 pb:
ð16Þ
At k ¼ 0 for symmetric envelope Eq. (15) yields RðnÞ ¼
cosh 2pb 1 : cosh 2pb þ cosh 4pln
ð17Þ
Its asymptotics at n ! 1 is exponent RðnÞ expð 4pjnjÞ, that corresponds to tails of the spectrum in semi-logarithmic coordinates. At b ! 0 Eq. (17) reduces into 2 pb RðnÞ ¼ : cosh 2pln
w1=2 ¼
0:742 : 2pl
ð19Þ
For high reflectance ðpb 1Þ w1=N
b 1 þ lnðN 1Þ; 2l 4pl
particularly, w1=2 ¼ b=2l: The width w1=2 is shown in Fig. 6 as a function of b at several values of l. The numerically calculated width coincides with (18). Along with w1=2 the effective width w is shown, calculating from the alternative definition Z 1 1 w ¼ RðnÞ dn: Rð0Þ 0 The value w is the width of equivalent rectangle, that has the same height and area. Calculating the integral we have w ¼
b cothpb: 2l
ð20Þ
It is the square of the Fourier-transform of the envelope. 4. Filter bandwidth The half-width at 1=N maximum, w1=N , can be found from the equation Rðw1=N Þ ¼
1 Rð0Þ: N
For symmetric envelope k ¼ 0 the width is 1 w1=N ¼ ln ðN 1Þ cosh 2pb þ N 4pl qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ððN 1Þ cosh 2pb þ N Þ2 1 :
Fig. 6. Half-width w1=2 at half maximum of the spectral profile vs. b at k ¼ 0 and different l: from the top down l ¼ 0:5; 1; 2; 4. Dotted lines indicate approximation (20).
300
Its asymptotics is 1=2pl; pb 1; w b=2l; pb 1:
D.A. Shapiro / Optics Communications 215 (2003) 295–301
ð21Þ
Therefore, the width increases with b and decreases with l in agreement with Figs. 3 and 4. The dotted curve deviates from the solid one at small b, since the reflection spectrum in this limit is far from rectangular, then the half-width w1=2 is not equal to w . The difference between the effective width and HWHM is up to 25% at small b. To be certain one should compare the first line of (21) with (19). In general case k 6¼ 0 the rough approximation can be found from (20) by replacement l ! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi lðk þ lÞ: b coth pb w pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð22Þ 2 lðk þ lÞ Fig. 7 displays the numerically calculated halfwidth at half maximum as a function of k together with approximation (22). The dotted lines show that even though the approximation is crude, it qualitatively reproduces the dependence. Only at small b the approximation becomes poor in limits k ! l; þ1: To outline the domain of applicability of the approximation let us find the poles of reflection coefficient (15) on the complex plane n: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 iðN þ Þðk þ 2lÞ 4b2 lðk þ lÞ ðN þ 12Þ k2 2 n ; ¼ n 4lðk þ lÞ N ¼ 0; 1; 2; . . .
The pattern is symmetric with respect to real and imaginary axes. At b ¼ 0 poles for all n are placed in the imaginary axis. At growing b poles appear with nonzero real part. The first four poles n 0 ; n 1 with Im n 6¼ 0 occur at b2 ¼
k2 : 16lðk þ lÞ
ð24Þ
At much greater b there is a finite number of offaxis poles are aligned along the ellipse 2 2 Re n Im n þ ¼ 1: A B Its small semi-axis is A2 ¼ b2 =4lðk þ lÞ, aspect ratio is B=A ¼ 1 þ 2l=k. For symmetric profile k ! 0 the ellipse stretches along the imaginary axis and turns into a pair of vertical lines n N ¼ ðb þ iðN þ 12ÞÞ=2l. They merge at b ! 0, when all the poles lie in the imaginary axis. The effective width is determined by a nearest to the real axis pole, i.e., the small semi-axis, hence b w A ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 lðk þ lÞ in agreement with (22) at b 1. Fig. 8 illustrates condition (24). The curve restricts minimum and maximum values of k=l, between which the approximation is valid at given b.
ð23Þ
Fig. 7. The width w1=2 vs. k at l ¼ 1: b ¼ 0:3 (solid line), 1 (long dashes), 3 (short dashes). Dotted lines indicate approximation (22).
Fig. 8. The boundary (24) of the applicability region of approximation (22). Inset shows an example of the poles on complex n-plane (b ¼ 1; l ¼ 5; k ¼ 1:5).
D.A. Shapiro / Optics Communications 215 (2003) 295–301
301
5. Conclusions
Acknowledgements
For the qualitative treatment let us express the spectral width in terms of grating experimental parameters: the number of periods M l=2p and depth e ¼ maxz ð2dn=nÞ of modulation of the refractive index. For considered case the effective pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi number is M lðk þ lÞ. The distributed Bragg reflector must be several times longer to provide the smooth transition to unperturbed refractive index and avoid Gibbs oscillations outside the filter band. The coupling constant b 2eM is the product of the depth and length. The relative spectral width w dk=k grows up with b and decreases with l; k. As follows from (16), the grating is of small reflectance at pb 1. In this case the reflection can be described by the first Born approximation of the quantum-mechanical scattering theory. Then the amplitude reflection coefficient rðkÞ is close to the Fouriertransform of the profile pðzÞ. From (21) we get the relative width dk=k 1=M in accordance with the uncertainty relation. For high-reflectance case the Born expansion is not valid, still theory of parametric resonance is appropriate. If the detuning n exceeds the depth of modulation e, the parametric resonance of the lowest order vanishes. From (21) we get dk=k e in agreement with the prescription. Thus, for synthesis of narrow-band and high-reflectance Bragg filter a designer must choose the small depth and large number of strokes M e 1 1.
Author is grateful to S.A. Babin and E.V. Podivilov for fruitful discussions. The work was supported by the UTAR Scientific Inc., Vancouver. The support of Russian Ministry of Industry, Science and Technology (program ‘‘Physics of quantum and nonlinear processes’’) and Russian Foundation for Basic Research (Grants # 00-0217973, 00-15-96808) is also acknowledged.
References [1] G.A. Thomas, D.A. Ackerman, P.R. Prucnal, S.L. Cooper, Phys. Today (9) (2000) 30. [2] V.E. Zakharov, A.B. Shabat, Zh. Eksp. Teor. Fiz. 61 (1) (1971) 118. [3] G.H. Song, S.Y. Shin, J. Opt. Soc. Am. 2 (11) (1985) 1905. [4] L. Poladian, Phys. Rev. E 54 (3) (1996) 2963. [5] E. Brinkmeyer, Opt. Lett. 20 (8) (1995) 810. [6] E. Peral, J. Capmany, J. Marti, IEEE J. Quantum Electron. 32 (12) (1996) 2078. [7] L. Poladian, Opt. Lett. 25 (11) (2000) 787, Errata: Opt. Lett. 25 (18) (2000) 1400. [8] J. Skaar, L. Wang, T. Erdogan, IEEE J. Quantum Electron. 37 (20) (2001) 165. [9] G.L. Lamb Jr., Elements of Soliton Theory, Wiley, New York, Toronto, 1980. [10] A. Bambini, P.R. Berman, Phys. Rev. A 23 (5) (1981) 2496. [11] L. Carmel, A. Mann, Phys. Rev. A 61 (5) (2000) 052113. [12] H. Bateman, A. Erdelyi, Hihger Transcendental Functions, vol. 1, McGraw-Hill, New York, Toronto, London, 1953.
Family of exact solutions for reflection spectrum of Bragg grating D.A. Shapiro * Institute of Automation and Electrometry, Siberian Branch, Russian Academy of Sciences, 1 Koptjug Ave., Novosibirsk 630090, Russia Received 12 May 2002; accepted 22 November 2002
Abstract Two-parametric family of exactly solvable profiles of quasi-sinusoidal Bragg gratings is found. A simple formula is derived for their reflection coefficient as a function of wavelength. The detailed analysis of the reflection spectrum width is presented. The solution is helpful as a starting point of optical filter synthesis. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 42.81.Wg; 78.66.)w Keywords: Fiber Bragg grating; Optical filters; Direct scattering problem
1. Introduction Optical filters based on fiber grating have attracted particular interest nowadays because of their applications in high-speed lightware communications [1]. The distributed Bragg reflector consists of periodically spaced elements with varying refractive index of the waveguide material. The array parameters are chosen to strongly reflect light in a narrow band of wavelength. To determine the array parameters a designer needs solving the inverse problem of light scattering, i.e., reconstructing the grating from reflectometric data.
*
Tel.: +7-3832-344021; fax: +7-3832-333863. E-mail address: [email protected].
The problem can be reduced to the GelÕfand– Levitan–Marchenko (GLM) integral equation [2]. The direct problem of calculating a reflection spectrum from given grating profile is well understood, then most of the present literature on this subject analyzes methods of inverse problem solutions or grating synthesis. Song and Shin [3] rederived the GLM equation and its solution when the reflection coefficient was a rational function of the wavenumber. Paladian [4] found exact solutions for the rational functions: for a single resonance mode and an array with nonreciprocal reflection properties. Brinkmeyer [5] proposed an iteration algorithm for inverse problem without GLM. It simplifies mathematics, but badly converges for grating with high reflection. Iteration method for GLM developed by Peral et al. [6]. Recent notable advance centers around the fast
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 2 2 3 3 - 2
296
D.A. Shapiro / Optics Communications 215 (2003) 295–301
algorithm of synthesis based on casuality argument. Paladian [7] demonstrated the scheme of calculation for continuous functions. Skaar et al. [8] described the discrete method of ‘‘layer-peeling’’, compared it with continuous analog, and noticed that this technique is known in quantum mechanics, geophysics and electronics (transmission line design). Nevertheless, a question of interest remains in the direct problem. Not many exact solutions are known. They are the rectangular barrier, hyperbolic secant, and its truncated analogs [9]. Finding new exactly solvable profiles is important if only for test of algorithms. The explicit solutions are also helpful for conceptual understanding of the scattering. The aim of the present paper is to find the analytical solutions and extend the family of exactly solvable profiles. The paper is organized as follows. Section 2 describes the substitution suggested by Bambini and Berman [10] for probability amplitudes of a two-level system. The Zakharov–Shabat equations are reduced to Gauss hypergeometric form and two-parametric family of exactly solvable profiles is found. The spectral profile of reflection coefficient is derived in Section 3 in terms of elementary functions and plotted at different values of parameters. The width of reflection spectrum is estimated in Section 4. Section 5 summarizes the qualitative results.
Fig. 1. Quasi-sinusoidal addition to the refractive index dn as a function of coordinate z. Thick curves show the envelope.
where aðzÞ is the envelope shown in Fig. 1, hðzÞ is the phase deviation. The grating is quasi-sinusoidal, when aðzÞ; hðzÞ are slow functions. Let us introduce amplitudes of waves running in positive and negative directions dE ¼ ik a1 eikz þ a2 e ikz : E ¼ a1 eikz þ a2 e ikz ; dz Keeping only resonant terms and neglecting the parametric resonance of higher orders at q ¼ k j=2 k, we get da1 da2 ¼ ikpðzÞa2 e 2iqz ; ¼ ikp ðzÞa1 e2iqz ; ð3Þ dz dz where pðzÞ ¼ aðzÞeihðzÞ =4. Denoting the dimensionless coordinate as f ¼ kz and detuning as n ¼ q=k, we have
2. Hypergeometric equation
a01 ¼ ipðfÞa2 e 2inf ;
ð4Þ
Consider a single-mode fiber with the refractive index n þ dnðzÞ; dn n. Steady-state electric field EðzÞ satisfies one-dimensional Helmholtz equation d2 E 2dnðzÞ xn 2 ; ð1Þ þ k 1 þ E ¼ 0; k ¼ dz2 n c
a02 ¼ ip ðfÞa1 e2inf ;
ð5Þ
where z is coordinate, k is the wavenumber at z ! 1, where dnðzÞ ! 0, x; c are the frequency and speed of light. The addition to refractive index is quasi-sinusoidal function 2dnðzÞ ¼ aðzÞ cos ½ jz þ hðzÞ 1; n
ð2Þ
where the prime is the derivative with respect to f. 2 2 The set conserves ja1 j ja2 j , since the signs at ip and ip are different. The same equations with identical signs conserve the sum of ‘‘populations’’ jai j2 and describes the amplitudes of probability in two-state quantum system. The exact solutions in this case are of importance in quantum optics, then they are studied in details (see [11] and references therein). Calculating the derivative of (4) and substituting a2 ; a02 from Eqs. (4) and (5) we get one secondorder equation
D.A. Shapiro / Optics Communications 215 (2003) 295–301
p0 0 a001 þ 2in a1 jpj2 a1 ¼ 0: p
297
ð6Þ
For simplicity let us consider the envelope without phase modulation, i.e., with h ¼ 0 and real p. When h 6¼ 0 is a constant it also does not enter into Eq. (6). Let us factorize pðfÞ ¼ bP into b and normalized envelope P ðfÞ. Parameter b is the constant of coupling between forward and backward waves. The area under P ðfÞ is assumed equal p. The linear transformation of the independent variable [10] Z f t¼ qðf0 ÞP ðf0 Þ df0 ð7Þ
Fig. 2. Profiles of solvable envelopes at l ¼ 1: k ¼ 0 (solid line); )0.5 (dotted); 1 (dashed).
1
The envelope P ðfÞ is even function with maximum at f ¼ 0 only at k ¼ 0. In general case maximum
reduces Eq. (6) to € a1 þ
q0 þ 2inq b2 _ a1 ¼ 0; a 1 q2 P q2
ð8Þ
where the weight function qðzÞ is to be determined below, a dot denotes the derivative with respect to new variable t, 0 < t < 1, while f varies from 1 to þ1. Comparing (8) with the Gauss hypergeometric equation €þ w
c ða þ b þ 1Þt ab w ¼ 0; w_ tð1 tÞ tð1 tÞ
we get two conditions b2 ab : ¼ tð1 tÞ q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Its solution is the weight function qðtÞ ¼ tð1 tÞ and relation ab ¼ b2 . Changing parameters c 1=2 ¼ 2il; a þ b ¼ 2ik, where k; l are real, we obtain two-parametric family of exactly solvable envelopes pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tð1 tÞ tl P ðtÞ ¼ ; fðtÞ ¼ ln : ð9Þ kþl kt þ l ð1 tÞ
q0 þ 2inq c ða þ b þ 1Þt ; ¼ q2 P tð1 tÞ
The parameters must satisfy conditions l > 0; kþ l > 0. At k ¼ 0 set (9) gives the explicit profile with the shape of hyperbolic secant 1 P ðfÞ ¼ : 2l coshðf=2lÞ
ð10Þ
There are two other cases of explicit but more complicated profiles k ¼ l and k ¼ l=2.
1 Pmax ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 lðk þ lÞ
ð11Þ
is located at t0 ¼ l=ðk þ 2lÞ, as shown in Fig. 2. Since t ¼ 1=2 corresponds to z ¼ 0, the maximum point lies at positive z at k < 0 or at negative z at k > 0. At k < 0 the right front is steeper, at k < 0 the left is. From (7) follows that the area under curve P ðfÞ is really independent of k; l: Z 1 Z 1 dt ¼ p: ð12Þ P ðfÞ df ¼ qðtÞ 1 0 Thus, parameter l gives the characteristics length of envelope at k ¼ 0. For arbitrary k >ffi l the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi length is of the order of l ¼ 2p lðk þ lÞ, as follows from (11) and (12). The two-parametric family of exactly solvable envelopes is the same as for two-level atom [10], whereas the solution of the hypergeometric Eq. (8) is different due to Ô Õ sign in the third term.
3. Reflection spectrum The solution can be expressed in terms of the Gaussian hypergeometric function a1 ðtÞ ¼ c1 F ða; b; c; tÞ þ c2 t1 c F ða þ 1 c; b þ 1 c; 2 c; tÞ;
ð13Þ
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qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a ¼ i nk b2 þ n2 k2 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b ¼ i nk þ b2 þ n2 k2 ;
ð14Þ
1 c ¼ þ 2iln: 2 Let us consider the incident wave e ifn running from þ1 to the left, the boundary condition at z ¼ 1 is a1 ðt ¼ 0Þ ¼ 0. Therefore, c1 ¼ 0, c2 is arbitrary, since the right reflection coefficient, the ratio of amplitudes of falling and reflecting waves, is independent of c2 : 1 a1 ðt ¼ 1Þ d 2inf ¼ ibe lim qðtÞ ln a1 ðtÞ ; r¼ t!1 a2 ðt ¼ 1Þ dt R ¼ jrj2 :
where a; b; c are defined by (14). With help of relation CðzÞCð1 zÞ ¼ p=sinpz [12] the expression can be simplified qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosh 2p b2 þ k2 n2 cosh 2pkn qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : RðnÞ ¼ cosh 2p b2 þ k2 n2 þ cosh 2pðk þ 2lÞn ð15Þ The reflection coefficient as a function of n (the wavelength in dimensionless variables) is plotted in Fig. 3 at fixed l ¼ 1; k ¼ 0 and different b. The profile is triangular at low b and trapezoidal at high b. The profile for fixed b ¼ 1; k ¼ 0 and varying l is shown in Fig. 4. At l 1 the shape approaches a rectangular function. The next Fig. 5 shows the case of different k and fixed b ¼ l ¼ 1: The filter becomes narrower at higher k.
Replacement of l; k by their linear combinations m ¼ l þ k=2;
g ¼ k=2; 0 < jgj < m
makes the profile (9) invariant with respect to substitution t ! 1 t; g ! g: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tð1 tÞ tmþg P ðtÞ ¼ ; f ¼ ln : g þ m 2gt ð1 tÞm g After the transformation only the sign of coordinate changes: P ! P ; f ! f. Then one can get the left reflection coefficient as well by changing the sign of g at fixed m. Below we study the intensity reflection coefficient R that is the same for both directions. The limit can be calculated using the formula for analytic continuation of the hypergeometric function [12]
Fig. 3. Reflection coefficient as a function of detuning n at k ¼ 0; l ¼ 1: from the top down b ¼ 2; 1:5; 1; 0:5.
F ða; b; c; tÞ ¼ A1 F ða; b; a þ b c þ 1; 1 tÞ þ A2 ð1 tÞ
c a b
F ðc a; c b; c a b þ 1; 1 tÞ; A1 ¼
CðcÞCðc a bÞ ; Cðc aÞCðc bÞ
A2 ¼
CðcÞCða þ b cÞ : CðaÞCðbÞ
The intensity reflection coefficient takes the form of ratio of Euler C-functions Cða c þ 1ÞCðb c þ 1Þ 2 ; R ¼ b2 Cð1 aÞCð1 bÞ
Fig. 4. The same as in Fig. 3 at k ¼ 0; b ¼ 1 and different l: from the top down l ¼ 1; 2; 3; 4.
D.A. Shapiro / Optics Communications 215 (2003) 295–301
At N ¼ 2 we have 1 ln 2 þ cosh 2pb w1=2 ¼ 4pl pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ð1 þ cosh 2pbÞð3 þ cosh 2pbÞ :
299
ð18Þ
For small reflectance ðpb 1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 w1=N ln½2N 1 þ N ðN 1Þ ; 4pl in particular, Fig. 5. The same as in Fig. 3 at l ¼ 1; b ¼ 1: from the top down k ¼ 0:75; 0:5; 1; 3; 10.
The expression (15) can be simplified in some particular cases. At n ¼ 0 the saturation in the center is described by formula: Rð0Þ ¼ tanh2 pb:
ð16Þ
At k ¼ 0 for symmetric envelope Eq. (15) yields RðnÞ ¼
cosh 2pb 1 : cosh 2pb þ cosh 4pln
ð17Þ
Its asymptotics at n ! 1 is exponent RðnÞ expð 4pjnjÞ, that corresponds to tails of the spectrum in semi-logarithmic coordinates. At b ! 0 Eq. (17) reduces into 2 pb RðnÞ ¼ : cosh 2pln
w1=2 ¼
0:742 : 2pl
ð19Þ
For high reflectance ðpb 1Þ w1=N
b 1 þ lnðN 1Þ; 2l 4pl
particularly, w1=2 ¼ b=2l: The width w1=2 is shown in Fig. 6 as a function of b at several values of l. The numerically calculated width coincides with (18). Along with w1=2 the effective width w is shown, calculating from the alternative definition Z 1 1 w ¼ RðnÞ dn: Rð0Þ 0 The value w is the width of equivalent rectangle, that has the same height and area. Calculating the integral we have w ¼
b cothpb: 2l
ð20Þ
It is the square of the Fourier-transform of the envelope. 4. Filter bandwidth The half-width at 1=N maximum, w1=N , can be found from the equation Rðw1=N Þ ¼
1 Rð0Þ: N
For symmetric envelope k ¼ 0 the width is 1 w1=N ¼ ln ðN 1Þ cosh 2pb þ N 4pl qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ððN 1Þ cosh 2pb þ N Þ2 1 :
Fig. 6. Half-width w1=2 at half maximum of the spectral profile vs. b at k ¼ 0 and different l: from the top down l ¼ 0:5; 1; 2; 4. Dotted lines indicate approximation (20).
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Its asymptotics is 1=2pl; pb 1; w b=2l; pb 1:
D.A. Shapiro / Optics Communications 215 (2003) 295–301
ð21Þ
Therefore, the width increases with b and decreases with l in agreement with Figs. 3 and 4. The dotted curve deviates from the solid one at small b, since the reflection spectrum in this limit is far from rectangular, then the half-width w1=2 is not equal to w . The difference between the effective width and HWHM is up to 25% at small b. To be certain one should compare the first line of (21) with (19). In general case k 6¼ 0 the rough approximation can be found from (20) by replacement l ! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi lðk þ lÞ: b coth pb w pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð22Þ 2 lðk þ lÞ Fig. 7 displays the numerically calculated halfwidth at half maximum as a function of k together with approximation (22). The dotted lines show that even though the approximation is crude, it qualitatively reproduces the dependence. Only at small b the approximation becomes poor in limits k ! l; þ1: To outline the domain of applicability of the approximation let us find the poles of reflection coefficient (15) on the complex plane n: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 iðN þ Þðk þ 2lÞ 4b2 lðk þ lÞ ðN þ 12Þ k2 2 n ; ¼ n 4lðk þ lÞ N ¼ 0; 1; 2; . . .
The pattern is symmetric with respect to real and imaginary axes. At b ¼ 0 poles for all n are placed in the imaginary axis. At growing b poles appear with nonzero real part. The first four poles n 0 ; n 1 with Im n 6¼ 0 occur at b2 ¼
k2 : 16lðk þ lÞ
ð24Þ
At much greater b there is a finite number of offaxis poles are aligned along the ellipse 2 2 Re n Im n þ ¼ 1: A B Its small semi-axis is A2 ¼ b2 =4lðk þ lÞ, aspect ratio is B=A ¼ 1 þ 2l=k. For symmetric profile k ! 0 the ellipse stretches along the imaginary axis and turns into a pair of vertical lines n N ¼ ðb þ iðN þ 12ÞÞ=2l. They merge at b ! 0, when all the poles lie in the imaginary axis. The effective width is determined by a nearest to the real axis pole, i.e., the small semi-axis, hence b w A ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 lðk þ lÞ in agreement with (22) at b 1. Fig. 8 illustrates condition (24). The curve restricts minimum and maximum values of k=l, between which the approximation is valid at given b.
ð23Þ
Fig. 7. The width w1=2 vs. k at l ¼ 1: b ¼ 0:3 (solid line), 1 (long dashes), 3 (short dashes). Dotted lines indicate approximation (22).
Fig. 8. The boundary (24) of the applicability region of approximation (22). Inset shows an example of the poles on complex n-plane (b ¼ 1; l ¼ 5; k ¼ 1:5).
D.A. Shapiro / Optics Communications 215 (2003) 295–301
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5. Conclusions
Acknowledgements
For the qualitative treatment let us express the spectral width in terms of grating experimental parameters: the number of periods M l=2p and depth e ¼ maxz ð2dn=nÞ of modulation of the refractive index. For considered case the effective pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi number is M lðk þ lÞ. The distributed Bragg reflector must be several times longer to provide the smooth transition to unperturbed refractive index and avoid Gibbs oscillations outside the filter band. The coupling constant b 2eM is the product of the depth and length. The relative spectral width w dk=k grows up with b and decreases with l; k. As follows from (16), the grating is of small reflectance at pb 1. In this case the reflection can be described by the first Born approximation of the quantum-mechanical scattering theory. Then the amplitude reflection coefficient rðkÞ is close to the Fouriertransform of the profile pðzÞ. From (21) we get the relative width dk=k 1=M in accordance with the uncertainty relation. For high-reflectance case the Born expansion is not valid, still theory of parametric resonance is appropriate. If the detuning n exceeds the depth of modulation e, the parametric resonance of the lowest order vanishes. From (21) we get dk=k e in agreement with the prescription. Thus, for synthesis of narrow-band and high-reflectance Bragg filter a designer must choose the small depth and large number of strokes M e 1 1.
Author is grateful to S.A. Babin and E.V. Podivilov for fruitful discussions. The work was supported by the UTAR Scientific Inc., Vancouver. The support of Russian Ministry of Industry, Science and Technology (program ‘‘Physics of quantum and nonlinear processes’’) and Russian Foundation for Basic Research (Grants # 00-0217973, 00-15-96808) is also acknowledged.
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