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Optics Communications 266 (2006) 512–520 www.elsevier.com/locate/optcom

Group delay in Bragg grating with linear chirp O.V. Belai, E.V. Podivilov, D.A. Shapiro

*

Photonics Laboratory, Institute of Automation and Electrometry, Russian Academy of Sciences, Siberian Branch, 1 Koptjug Avenue, Novosibirsk 630090, Russia Received 29 November 2005; received in revised form 17 May 2006; accepted 17 May 2006

Abstract An analytic solution for Bragg grating with linear chirp in the form of conﬂuent hypergeometric functions is analyzed in the asymptotic limit of long grating. Simple formulas for reﬂection coeﬃcient and group delay are derived. The simpliﬁcation makes it possible to analyze irregularities of the curves and suggest the ways of their suppression. It is shown that the increase in chirp at ﬁxed other parameters decreases the oscillations in the group delay, but gains the oscillations in the reﬂection spectrum. The asymptotic considerations are in good agreement with numerical calculations. 0002 2006 Elsevier B.V. All rights reserved. PACS: 42.81.Wg; 78.66.0002w Keywords: Fiber Bragg grating; Optical ﬁlters; Direct scattering problem; Group delay

1. Introduction Optical ﬁlters based on ﬁber gratings attract particular interest because of their applications in high-speed lightware communications [1], ﬁber lasers [2] and sensors [3]. The Bragg reﬂector is based on periodic modulation of the refractive index along the ﬁber [4,5]. Gratings with the period varied along their length are known as chirped. The theory of linearly chirped grating holds the central place in the ﬁber optics. Chirped grating is of importance because of its applications as a dispersion-correcting or compensating devices [6]. The study of linearly chirped grating is also helpful for approximate solution of more general problem of complex Gaussian modulation [7]. The group delay as a function of wavelength is a linear function with additional oscillations. For applications the problem is to minimize the amplitude of regular oscillations and the ripple resulting from the errors of manufacturing [8]. *

Corresponding author. Tel.: +7 3833309021; fax: +7 3833333863. E-mail address: [email protected] (D.A. Shapiro).

0030-4018/$ - see front matter 0002 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.05.032

The purpose of this work is to present and study a solution to the equations for amplitudes of coupled waves in quasi-sinusoidal grating with quadratic phase modulation. The solution to coupled-wave equations is derived in terms of the conﬂuent hypergeometric functions. Their asymptotic expansion in terms of Euler C-functions makes it possible to obtain relatively simple formulas for reﬂectivity and group delay. The simpliﬁcation enables analysis of irregularities of the curves and suggestions on the ways of their suppression. The paper is organized as follows. The equations for coupled waves in the grating are derived in Section 2. Their analytic solutions are obtained and compared with numerical results in Section 3. The asymptotic behavior is treated in Section 4. Some estimations and qualitative explanations are discussed in Section 5. 2. Equations for slow amplitudes Consider a single-mode ﬁber with the weakly modulated refractive index n + dn(z). Electric ﬁeld E(z) satisﬁes onedimensional Helmholtz equation

O.V. Belai et al. / Optics Communications 266 (2006) 512–520

' 0002 00032 # d2 E 2dnðzÞ dnðzÞ 2 þ þk 1þ E ¼ 0; dz2 n n

k¼

xn ; c

ð1Þ

where z is the coordinate, k is the wavenumber in glass outside the grating (where dn(z) = 0), x, c are the frequency and speed of light. The addition dn(z) to mean refractive index n may be a function with phase and amplitude modulation. A family of analytical solutions for amplitude modulation was obtained in [9]. Below we treat a case of phase modulation dnðzÞ ¼ 2b cos hðzÞ; n

ð2Þ

where h(z) is the phase, constant b is the modulation depth. Since b 0003 1 we neglect the quadratic term in (1). The phase is general quadratic function 2

hðzÞ ¼ az =2 þ jz þ h0 ;

ð3Þ

where j is the frequency of spatial modulation at z = 0, h0 is the constant phase shift. The condition of slow phase variation is 0004 0004 0004dh 0004 0004 0002 j0004 0003 j: ð4Þ 0004 dz 0004 Let us introduce complex amplitudes a, b of waves running in positive and negative directions E ¼ aeikz þ be0002ikz : Keeping only resonant terms and neglecting the parametric resonance of higher orders at the detuning q ¼ k 0002 j=2 0003 k 0 ¼ j=2; b0 ¼ 0002ik 0 be2ikz0002ihðzÞ a;

a(-L)

a(L)

b(-L)

b(L)=0

Fig. 1. The statement of the scattering problem.

(q > 0) the coordinate of the turning point is positive: z0 > 0. Consider the Bragg grating located in interval 0002L 6 z 6 L. The problem of left reﬂection coeﬃcient calculation is illustrated by Fig. 1. Boundary conditions are deﬁned by the scattering problem statement. We set amplitude b at the right end equal to zero bðLÞ ¼ 0

ð8Þ

and get the reﬂection and transmission coeﬃcients r¼

bð0002LÞ ; að0002LÞ

t¼

aðLÞ : að0002LÞ

ð9Þ

The chirp is weak and satisfy (4), then the equations for complex amplitudes are valid when aL 0003 j:

ð10Þ

Note that set (7) is symmetric under transformation a ! 0002a, q ! 0002q, a M b. Then the right reﬂection coeﬃcient can be obtained from the expression for left one by changing signs of parameters a and q. 3. Solution

ð5Þ

we get the equations for coupled waves a0 ¼ ik 0 be00022ikzþihðzÞ b;

513

ð6Þ

where prime denotes z-derivative. Within the limits of resonance approximation (5) we replace k in front of exponents by k0. Set (6) conserves jaj2 0002 jbj2, since the signs in right-hand sides of equations are diﬀerent. The same equations with identical signs conserve the sum of populations jaj2 + jbj2 and describe 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 [10,11]. Finding the derivatives of (6) with respect to z we get complex conjugated second-order equations a00 0002 iaðz 0002 z0 Þa0 0002 k 20 b2 a ¼ 0; b00 þ iaðz 0002 z0 Þb0 0002 k 20 b2 b ¼ 0: ð7Þ Here z0 = (2k 0002 j)/a = 2q/a is the coordinate of resonance point for the wave with wavenumber k. It is the turning point where the wave with given q is reﬂected. The parametric resonance for central wavenumber q = k 0002 j/2 = 0 occurs at z = 0. Let a > 0, then for the red detuning (q < 0) we have z0 < 0, in the opposite case of blue detuning

Eqs. (7) are reduced to the conﬂuent hypergeometric form by the substitution t = ia(z 0002 z0)2/2: 0002 0003 1 0002 t a_ þ iga ¼ 0; t€a þ 2 where dot denotes the derivative with respect to new variable t, g ¼ b2 k 20 =2a is the adiabatic parameter. The equation for second amplitude b is complex conjugated. The general solutions at 0002L < z < L are linear combinations aðzÞ ¼ A1 u1 ðzÞ þ A2 u2 ðzÞ;

bðzÞ ¼ B1 u00041 ðzÞ þ B2 u00042 ðzÞ

ð11Þ

of the Kummer conﬂuent hypergeometric functions [12]: 0002 0003 1 2 u1 ¼ F 0002ig; ; iaðz 0002 z0 Þ =2 ; 2 0002 0003 1 3 0002 ig; ; iaðz 0002 z0 Þ2 =2 ; u2 ¼ ðz 0002 z0 ÞF ð12Þ 2 2 a x aða þ 1Þ x2 þ þ 000500050005; F ða; c; xÞ ¼ 1 þ c 1! cðc þ 1Þ 2! where A1, A2, B1, B2 are constants and the asterisk denotes the complex conjugation. The solution for optical waveguide was obtained in [13]. The solution for two-level system describing nonadiabatic population inversion was obtained in [14].

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The relations between constants can be obtained from set (6) near resonance point z = z0 where a = A1 + A2(z 0002 z0) + O((z 0002 z0)2), b = B1 + B2(z 0002 z0) + O((z 0002 z0)2):

The left reﬂection and transmission coeﬃcients (9) can be expressed in terms of conﬂuent hypergeometric functions

A2 2 ¼ ik 0 beih0 0002iaz0 =2 ; B1

r¼

B2 2 ¼ 0002ik 0 be0002ih0 þiaz0 =2 : A1

ð13Þ

The right boundary condition (8) yields the ratio of coeﬃcients A1 and A2 A1 1 B2 q¼ ¼ 2 2 A 2 k 0 b B1 0005 0006 F ig; 12 ; 0002iaðL 0002 z0 Þ2 =2 0005 0006: ¼0002 b2 k 20 ðL 0002 z0 ÞF 12 þ ig; 32 ; 0002iaðL 0002 z0 Þ2 =2

ð14Þ

2

e0002ih0 þiaz0 =2 u00041 ð0002LÞ þ b2 k 20 qu00042 ð0002LÞ ; qu1 ð0002LÞ þ u2 ð0002LÞ ik 0 b qu1 ðLÞ þ u2 ðLÞ t¼ ; qu1 ð0002LÞ þ u2 ð0002LÞ

ð15Þ

where functions u1,2 are deﬁned by (12) and q is deﬁned by (14). The reﬂection spectrum, i.e., the reﬂectivity R = jrj2 as a function of detuning q, is shown in Fig. 2(a). The central frequency of the spectrum comes to resonance at z = 0, in the middle of grating. The central part of the spectrum has a ﬂat top at high adiabatic parameter, as the upper

1

Reflectivity

0.8

0.6

0.4

0.2

(a) 0 -300

-200

-100

0

q, cm-1

100

200

300

0.7

0.68

Reflectivity

0.66

0.64

0.62

0.6

(b) 0.58

0

10

20

30

q, cm-1

40

50

60

Fig. 2. (a) Reﬂection spectrum R(q) at a = 600 cm00022, L = 0.5 cm, k0 = 6 · 104 cm00021, from the top down b = b0, b0/2, b0/4, where b0 = 0.67 · 1000023. (b) A part of the lower curve b = b0/4, crosses denote the numerical result.

O.V. Belai et al. / Optics Communications 266 (2006) 512–520

curve shows, and the maximal reﬂectivity is close to 1. The width of central part is proportional to the length L. The reﬂectivity is relatively high if the turning point z0 lies inside the grating jz0j < L. This inequality gives the bandwidth jqj < aL/2. There is no parametric resonance at higher detuning, when jz0j > L, and then the reﬂectivity is small. Fig. 3(a) shows how the bandwidth grows up with the chirp parameter a at ﬁxed modulation depth b. The adiabatic parameter decreases with a, then the reﬂectivity in the center decreases from curve to curve. The spectrum was recalculated numerically from Helmholtz equation (1) by the T-matrix approach. The number of points per period of spatial modulation was ﬁxed at N = 32, then the step varied along the grating. The numer-

515

ical spectrum for n = 1.5 at the same parameters as analytical expression is shown in Fig. 2(b) by crosses. The numerical results are very close to analytical. The deviation of T-matrix solution from the Kummer formulas might be caused by three reasons: (i) failure of the condition (4) for the phase variation of the grating, (ii) departure of the light frequency from the resonance, (iii) inﬂuence of the quadratic term (dn/n)2 in Helmholtz equation controlled by parameter b. Since dimensionless parameters controlling the validity of coupled-wave approximation are small in Fig. 2(b): b 0006 1000023, aL/j 0006 5 · 1000023, the deviation is negligible. At higher parameter a the deviation of the solutions to coupled-wave equations from that to Helmholtz equation increases, but not dramatically, as shown in

1 0.9 0.8

Reflectivity

0.7 0.6 0.5 0.4 0.3 0.2 0.1

(a)

0 -600

-400

-200

0

q, cm-1

200

400

600

0.79 0.78 0.77

Reflectivity

0.76 0.75 0.74 0.73 0.72 0.71 0.7

(b) 0

50

100

q, cm

-1

150

200

Fig. 3. (a) Reﬂection spectrum from the top down at L = 0.5 cm, b = 0.33 · 1000023, k0 = 6 · 104 cm00021 and a = a0, 2a0, 3a0, where a0 = 600 cm00022. (b) A part of the lower curve a = 3a0, crosses denote the numerical result.

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for small b. A fragment of group delay characteristics is shown in Fig. 4(b) along with numerical calculations. Dots obtained from numerical calculation are very close to the curve given by analytical formula. It is diﬃcult to analyze the solution in its general form. In particular the cumbersome expression for group delay, the derivative of (15) with respect to the detuning, is not presented here. We simplify expressions using the asymptotics of Kummer functions in Section 4.

Fig. 3(b). The main reason of the deviation is resonance approximation (5). Comparing Figs. 2(b) and 3(b) we see that the latter involves higher detuning, then the deviation becomes greater. The group delay found from analytical solution (15) is plotted in Fig. 4(a) at the same parameters as the reﬂection spectrum in Fig. 2. The deviation of curves from the linear dependence, the group delay ripple, manifests itself as oscillations with variable frequency. The frequency grows up towards the blue edge of the spectrum in agreement with results from [5,15]. For the negative chirp (or when the incident light enters from the right) the frequency grows up towards the red edge of spectrum. The maximum deviation from the averaged slope decreases with decreasing modulation depth b. Meanwhile, the ripple in reﬂectivity increases

4. Asymptotics The asymptotic expressions for the reﬂection coeﬃcient can be obtained from (15) in two cases. The ﬁrst case is the resonance condition at the left end, namely, detuning

200 180 160

Group delay, ps

140 120 100 80 60 40 20

(a)

0 -20

-100

-50

0

q, cm-1

50

100

120

Group delay, ps

100

80

60

40

20

(b) 0 -90

-80

-70

-60

-50

-40

-30

-20

-10

0

q, cm-1 Fig. 4. (a) Group delay (ps) as a function of detuning q (cm00021) at a = 600 cm00022, L = 0.5 cm, k0 = 6 · 104 cm00021, from the top down b = b0, b0/2, b0/4, where b0 = 0.67 · 1000023. (b) Comparing with numerical calculation denoted by dots.

O.V. Belai et al. / Optics Communications 266 (2006) 512–520

q = 0002aL/2 for which z0 = 0002L. In this case it follows from (12) that u1(0002L) = 1, u2(0002L) = 0, and then from (15) r0007

e0002ih0 þiaL ikbq

2 =2

2 =2

00070002

e0002ih0 þiaL pﬃﬃﬃﬃ ig

2

R ¼ jrj 0007 tanh pg:

Cð1=2 0002 igÞ ; Cð0002igÞ

ð16Þ ð17Þ

The other case is when the resonance point z0 being far from both ends inside the grating: jqj < aL/2 and a(z0 ± L)2/2 1. The asymptotic expression of the conﬂuent hypergeometric functions [12] at jarg xj < p 0002 ip0002 0003a CðcÞ e CðcÞ x a0002c ex ; F ða; c; xÞ 0007 þ Cðc 0002 aÞ x CðaÞ þ1; Im x > 0; 0002¼ ð18Þ 00021; Im x < 0 allows one to simplify expression (15). The reﬂection coeﬃcient can be written using (12) ' 0002 00032 #00022ig pﬃﬃﬃﬃﬃ i/0002ip=4þ2iq2 =a0002ih a 2q 0 r 0007 0002 R0 e Lþ 2 a 0006 qﬃﬃﬃﬃ0005 2ig00021=2 0002iwþ 2ig00021=2 0002iw0002 1 þ eþip=40002i/ Rg0 wþ e þ w0002 e 0006; pﬃﬃﬃﬃﬃﬃﬃﬃ0005 00022ig00021=2 iw þi/0002ip=4 2ig00021=2 0002iw0002 0002i/þip=4 1 þ gR0 wþ e þ þ w0002 e ð19Þ 2

where w±(q) = a(L ± 2q/a) /2, / = arg[C(ig)C(1/2 + ig)], R0 = 1 0002 e00024pg and we omit terms of the order of 1/aL2 0003 1. The enumerator and denominator of the fraction in the second line of (19) are close to 1, if w± 1. Then the formula for reﬂectivity becomes simple R 0007 R0 ¼ 1 0002 e00024pg :

ð20Þ

Both curves (17) and (20) are shown in Fig. 5. One can see their saturation, moreover, when the turning point is z0 = 0002L, the saturation occurs later than when the turning point is far from both ends. The group delay obtained from (19) is

Fig. 5. The reﬂectivity R as a function of adiabatic parameter g at z0 = 0002L (solid line) and at z0 = 0 (dashed).

517

dargr n dIm ln r ¼ dx c dq sﬃﬃﬃﬃﬃﬃﬃﬃ ( n 4q 2g 0007 00022 ½ð1 þ R0 Þ cos c a R0 a

f ¼

ð/ 0002 p=4 þ wþ 0002 2g ln wþ Þ

)

þ ð1 0002 R0 Þ cosð/ 0002 p=4 þ w0002 0002 2g ln w0002 Þ ;

ð21Þ

where we neglect terms of the order of 1/aL2 0003 1. Expression (21) involves three terms. The ﬁrst (in the ﬁrst line) gives the averaged slope. It is a linear function within the bandwidth. Its slope depends on parameter a. At 1 0002 R0 0003 1 the second term (the second line) gives the ripple, chirped oscillations. The frequency of these oscillations is double distance from left end of the grating to the reﬂection point z0. Their frequency w0þ ðqÞ ¼ 2L þ 4q=a grows up towards the blue edge of the spectrum. When reﬂectivity R0 becomes smaller, the last term (the third line in Eq. (21)) proportional to T0 = 1 0002 R0 comes into eﬀect. It gives the additional oscillations with variable frequency w00002 ðqÞ ¼ 2L 0002 4q=a that grows up towards the red edge of the spectrum. It is precisely the sum of two chirped oscillations with signiﬁcantly diﬀerent frequencies that the left part of the lower curve in Fig. 4(a) displays. Magniﬁed view of the corresponding fragment is shown in Fig. 4(b). If we change the sign of chirp parameter a, then functions w± switch their roles: w+ M w0002. Therefore at high reﬂectivity 1 0002 R0 0003 1 the spatial frequency of leading oscillations w0þ ¼ 2L þ 4q=a decreases towards the shorter wavelengths. The amplitude of oscillations in group delay (21) increases when R0 tends to unity, while that in the spectrum decreases. Formula for the reﬂection inside the bandwidth can be obtained from (19) with the accuracy to the next order of transparency T0 = 1 0002 R0 pﬃﬃﬃﬃﬃﬃﬃﬃ R 0007 R0 þ 2 gR0 ð1 0002 R0 Þ ' cosð/ 0002 p=4 þ wþ 0002 2g ln wþ Þ pﬃﬃﬃﬃﬃﬃﬃﬃ a=2ðL þ 2q=aÞ # cosð/ 0002 p=4 þ w0002 0002 2g ln w0002 Þ pﬃﬃﬃﬃﬃﬃﬃﬃ þ : ð22Þ a=2ðL 0002 2q=aÞ At high reﬂectivity R0 ! 1 oscillations in (22) are suppressed. The ﬁrst term in square brackets describes oscillations with frequency 2L + 4q/a, their amplitude gains towards the red edge of the spectrum. The second term corresponds to oscillations with frequency 2L 0002 4q/a. Then amplitude grows towards the blue edge. Both approximate formulas (22) and (21) for oscillations are plotted in Figs. 6 and 7, respectively. As ﬁgures illustrate, the asymptotic expressions nearly coincide with exact Kummer solutions. The departure of the simple formula from the Kummer solution (left edge in Fig. 6 and both edges in Fig. 7) occurs at the limit of applicability of the asymptotic expansion.

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O.V. Belai et al. / Optics Communications 266 (2006) 512–520 110 100 90

Group delay, ps

80 70 60 50 40 30 20 10 0

-60

-40

-20

0

q, cm-1

20

40

60

Fig. 6. The group delay calculated according to asymptotic formula (21) at a = 600 cm00022, L = 0.5 cm, k0 = 6 · 104 cm00021, b = 0.67 · 1000023. Dots denote the Kummer solution.

0.72

0.7

Reflectivity

0.68

0.66

0.64

0.62

0.6

0.58

0.56

-60

-40

-20

0

-1

20

40

60

q, cm

Fig. 7. Reﬂection spectrum calculated by asymptotic formula (22) at a = 600 cm00021, b = 0.17 · 1000023, k0 = 6 · 104 cm00021. Dots denote the Kummer solution.

The turning point should be located far from the ends of grating, i.e., jL ± 2q/aj Leﬀ = (2p/jaj)1/2. The dependence on parameters a, b in Figs. 2 and 3 can also be explained by the asymptotic expressions. At ﬁxed chirp parameter a the adiabatic parameter g = (k0b)2/2a in (20) decreases with decreasing the modulation depth b. Then reﬂectivity R0 at q = 0 is relatively small and oscillations with amplitude 1 0002 R0 in the spectrum become noticeable. At ﬁxed b, on the contrary, the adiabatic parameter decreases with increasing a. It is the reason of

the most evident oscillations in the spectrum corresponding to the higher chirp parameter a. 5. Discussion The reﬂectivity is maximal at k = k0 = j/2 = p/K, where K is the period of modulation in the middle of the grating, at z = 0. The spatial frequency of modulation h 0 (z) = az + j depends on coordinate z. Then at some distance from the center the wave with k = k0 comes out from the

O.V. Belai et al. / Optics Communications 266 (2006) 512–520

resonance. The dephasing occurs when h = az2/2 0006 p, i.e., at distance z = Leﬀ 0006 (2p/a)1/2. The eﬀective number of strokes along length Leﬀ should be large Meﬀ = Leﬀ/ K = (2p/a)1/2K0002 1 1. Moreover, to provide the high reﬂectivity it should satisfy the stricter limitation of dense grating Meﬀb J 1. From here we get a condition for adiabatic parameter b2 k 20 J 1=4p: 2a The bandwidth of the reﬂection spectrum is aL, as shown in Section 3. The fronts of spectrum are determined by the eﬀective length Leﬀ. When point z0 = 2q/a is placed outside the grating at distance 0006 Leﬀ from the end, the reﬂection almost vanishes. The width of fronts is dq = aLeﬀ = 1/Leﬀ. The fronts are steep while Leﬀ 0003 L, i.e., in the limit of long grating. Phase modulation h(z) provides the parametric resonance condition for diﬀerent wavelengths. The shorter waves meet their resonance at longer distance z0 = 2q/a, and then the group delay of blue light is more than that of red one, Fig. 4. The linear dependence of the average group delay (21) upon the detuning has also simple explanation. The delay s = f/vgr is deﬁned by double distance from starting point to the resonance for given wavenumber f 0007 2z0 = 4(k 0002 k0)/a. Here vgr is the group velocity of light. If the chirp a is negative, then the slope of delay characteristics becomes negative. The ripple outside the reﬂection spectrum bandwidth, Figs. 2 and 3, with period p/L are the Gibbs oscillations originated by steep boundaries, i.e., reﬂection from the grating ends. The aperiodic oscillations inside the bandwidth arise from the triple-mirror cavity with moving middle mirror, Fig. 8. The wave reﬂected to the left from turning point z0 could reﬂect back to the right from the left end of the grating. Then the cavity appears between z = 0002L and z0; its eﬀective length is l = z0 + L = 2q/ a + L. It results in oscillations with period p/(L + 2q/a). The cavity with variable ‘‘mirror’’ is longer for blue spectrum and shorter for red, then the frequency of oscillations increases with q, as mentioned in paper [15]. At R0 ! 1 these oscillations are suppressed in the reﬂection spectrum, but remain in the group delay characteristics. If the reﬂectivity is not close to 1, the additional oscillations come into eﬀect due to the ‘‘right’’ cavity with variable ‘‘left mirror’’. Their period p/(L 0002 2q/a) on the contrary is longer for red spectrum. Here, oscillations g¼

I -L

II z0 = 2q/α

+L

Fig. 8. Conﬁguration of compound cavity: left ‘‘mirror’’ z = 0002L is the left end of grating, right ‘‘mirror’’ z = +L is the right end. Middle variable ‘‘mirror’’, the turning point z = z0, is located at diﬀerent positions depending on the wavelength. Then the ripple frequencies are determined by the variable lengths of sub-cavities I and II.

519

are suppressed at R0 ! 1 both in reﬂection spectrum and group delay characteristics. 6. Conclusions Thus, the analysis of the reﬂection spectrum and group delay of linearly chirped grating becomes simple if the turning point z0 = 2q/a is far from both ends of the grating compared to the eﬀective length Leﬀ = (2p/a)1/2. Formulas for reﬂectivity demonstrate the irregular oscillations in the reﬂection spectrum when the adiabatic parameter is not large. The oscillations are aperiodic and their amplitude slowly increases from the center of spectrum. The nature of the oscillations is reﬂection in compound cavity with a mobile middle ‘‘mirror’’. There are two terms in asymptotic expression. The ﬁrst has a period p/(L + z0) (round trip in the left sub-cavity), the second – p/(L 0002 z0) (round trip in the right sub-cavity). The oscillations in group delay characteristics have the same origin. The diﬀerence is that the right sub-cavity takes a negligible part in forming the oscillations of group delay characteristics at R0 ! 1. The amplitude of oscillations is suppressed at high chirp parameter a even at ﬁxed reﬂectivity. The conservation of high reﬂectivity with increasing a requires increasing parameter b. In order to suppress both oscillations one must choose as high the modulation depth as possible, but the limitation exists in ﬁber Bragg grating manufacturing. The alternative method to diminish the unwanted echo might be to provide the signal dephasing by apodization, i.e., smoothing the grating proﬁle [5]. Acknowledgments Authors are grateful to S.A. Babin for fruitful discussions. The work is partially supported by the CRDF Grant RUP1-1505-NO-05 and the Government support program of the leading research schools (NSh-7214.2006.2). References [1] G.A. Thomas, D.A. Ackerman, P.R. Prucnal, S.L. Cooper, Physics Today 53 (9) (2000) 30. [2] M.J.F. Digonnet (Ed.), Rare-Earth-Doped Fiber Lasers and Ampliﬁers, Marcel Dekker Inc., New York, 2001. [3] E. Udd (Ed.), Fiber Optics Sensors: An Introduction for Engineers and Scientists, Wiley, New York, 1991. [4] A. Othonos, K. Kalli, Fiber Bragg Gratings: Fundamentals and Applications in Telecommunications and Sensing, Artech House, Norwood, MA, 1999. [5] R. Kashyap, Fiber Bragg Gratings, Academic Press, New York, 1999. [6] F. Ouellette, Opt. Lett. 12 (10) (1987) 847. [7] J.T. Sheridan, A.G. Larkin, Opt. Commun. 236 (1–3) (2004) 87. [8] M. Sumetsky, B.J. Eggleton, J. Opt. Fiber Commun. Rep. 2 (2005) 256. [9] D.A. Shapiro, Opt. Commun. 215 (4–6) (2003) 295. [10] L. Allen, J.H. Eberly, Optical Resonance and Two-Level Atoms, Dover, New York, 1986. [11] L. Carmel, A. Mann, Phys. Rev. A 61 (5) (2000) 052113.

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[12] H. Bateman, A. ErdelyiHigher Transcendental Functions, vol. 1, McGrow-Hill, New York, 1953. [13] M. Matsuhara, K.O. Hill, A. Watanabe, JOSA 65 (7) (1975) 804. [14] P. Horwitz, Appl. Phys. Lett. 26 (6) (1975) 306.

[15] S. Bonino, M. Norgia, E. Riccardi, in: Proc. IOOC-ECOC’97, Edinburgh, 22–25 September 1997, IEE Conf. Pub. # 448., vol. 3, 1997, p. 194.

Group delay in Bragg grating with linear chirp O.V. Belai, E.V. Podivilov, D.A. Shapiro

*

Photonics Laboratory, Institute of Automation and Electrometry, Russian Academy of Sciences, Siberian Branch, 1 Koptjug Avenue, Novosibirsk 630090, Russia Received 29 November 2005; received in revised form 17 May 2006; accepted 17 May 2006

Abstract An analytic solution for Bragg grating with linear chirp in the form of conﬂuent hypergeometric functions is analyzed in the asymptotic limit of long grating. Simple formulas for reﬂection coeﬃcient and group delay are derived. The simpliﬁcation makes it possible to analyze irregularities of the curves and suggest the ways of their suppression. It is shown that the increase in chirp at ﬁxed other parameters decreases the oscillations in the group delay, but gains the oscillations in the reﬂection spectrum. The asymptotic considerations are in good agreement with numerical calculations. 0002 2006 Elsevier B.V. All rights reserved. PACS: 42.81.Wg; 78.66.0002w Keywords: Fiber Bragg grating; Optical ﬁlters; Direct scattering problem; Group delay

1. Introduction Optical ﬁlters based on ﬁber gratings attract particular interest because of their applications in high-speed lightware communications [1], ﬁber lasers [2] and sensors [3]. The Bragg reﬂector is based on periodic modulation of the refractive index along the ﬁber [4,5]. Gratings with the period varied along their length are known as chirped. The theory of linearly chirped grating holds the central place in the ﬁber optics. Chirped grating is of importance because of its applications as a dispersion-correcting or compensating devices [6]. The study of linearly chirped grating is also helpful for approximate solution of more general problem of complex Gaussian modulation [7]. The group delay as a function of wavelength is a linear function with additional oscillations. For applications the problem is to minimize the amplitude of regular oscillations and the ripple resulting from the errors of manufacturing [8]. *

Corresponding author. Tel.: +7 3833309021; fax: +7 3833333863. E-mail address: [email protected] (D.A. Shapiro).

0030-4018/$ - see front matter 0002 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.05.032

The purpose of this work is to present and study a solution to the equations for amplitudes of coupled waves in quasi-sinusoidal grating with quadratic phase modulation. The solution to coupled-wave equations is derived in terms of the conﬂuent hypergeometric functions. Their asymptotic expansion in terms of Euler C-functions makes it possible to obtain relatively simple formulas for reﬂectivity and group delay. The simpliﬁcation enables analysis of irregularities of the curves and suggestions on the ways of their suppression. The paper is organized as follows. The equations for coupled waves in the grating are derived in Section 2. Their analytic solutions are obtained and compared with numerical results in Section 3. The asymptotic behavior is treated in Section 4. Some estimations and qualitative explanations are discussed in Section 5. 2. Equations for slow amplitudes Consider a single-mode ﬁber with the weakly modulated refractive index n + dn(z). Electric ﬁeld E(z) satisﬁes onedimensional Helmholtz equation

O.V. Belai et al. / Optics Communications 266 (2006) 512–520

' 0002 00032 # d2 E 2dnðzÞ dnðzÞ 2 þ þk 1þ E ¼ 0; dz2 n n

k¼

xn ; c

ð1Þ

where z is the coordinate, k is the wavenumber in glass outside the grating (where dn(z) = 0), x, c are the frequency and speed of light. The addition dn(z) to mean refractive index n may be a function with phase and amplitude modulation. A family of analytical solutions for amplitude modulation was obtained in [9]. Below we treat a case of phase modulation dnðzÞ ¼ 2b cos hðzÞ; n

ð2Þ

where h(z) is the phase, constant b is the modulation depth. Since b 0003 1 we neglect the quadratic term in (1). The phase is general quadratic function 2

hðzÞ ¼ az =2 þ jz þ h0 ;

ð3Þ

where j is the frequency of spatial modulation at z = 0, h0 is the constant phase shift. The condition of slow phase variation is 0004 0004 0004dh 0004 0004 0002 j0004 0003 j: ð4Þ 0004 dz 0004 Let us introduce complex amplitudes a, b of waves running in positive and negative directions E ¼ aeikz þ be0002ikz : Keeping only resonant terms and neglecting the parametric resonance of higher orders at the detuning q ¼ k 0002 j=2 0003 k 0 ¼ j=2; b0 ¼ 0002ik 0 be2ikz0002ihðzÞ a;

a(-L)

a(L)

b(-L)

b(L)=0

Fig. 1. The statement of the scattering problem.

(q > 0) the coordinate of the turning point is positive: z0 > 0. Consider the Bragg grating located in interval 0002L 6 z 6 L. The problem of left reﬂection coeﬃcient calculation is illustrated by Fig. 1. Boundary conditions are deﬁned by the scattering problem statement. We set amplitude b at the right end equal to zero bðLÞ ¼ 0

ð8Þ

and get the reﬂection and transmission coeﬃcients r¼

bð0002LÞ ; að0002LÞ

t¼

aðLÞ : að0002LÞ

ð9Þ

The chirp is weak and satisfy (4), then the equations for complex amplitudes are valid when aL 0003 j:

ð10Þ

Note that set (7) is symmetric under transformation a ! 0002a, q ! 0002q, a M b. Then the right reﬂection coeﬃcient can be obtained from the expression for left one by changing signs of parameters a and q. 3. Solution

ð5Þ

we get the equations for coupled waves a0 ¼ ik 0 be00022ikzþihðzÞ b;

513

ð6Þ

where prime denotes z-derivative. Within the limits of resonance approximation (5) we replace k in front of exponents by k0. Set (6) conserves jaj2 0002 jbj2, since the signs in right-hand sides of equations are diﬀerent. The same equations with identical signs conserve the sum of populations jaj2 + jbj2 and describe 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 [10,11]. Finding the derivatives of (6) with respect to z we get complex conjugated second-order equations a00 0002 iaðz 0002 z0 Þa0 0002 k 20 b2 a ¼ 0; b00 þ iaðz 0002 z0 Þb0 0002 k 20 b2 b ¼ 0: ð7Þ Here z0 = (2k 0002 j)/a = 2q/a is the coordinate of resonance point for the wave with wavenumber k. It is the turning point where the wave with given q is reﬂected. The parametric resonance for central wavenumber q = k 0002 j/2 = 0 occurs at z = 0. Let a > 0, then for the red detuning (q < 0) we have z0 < 0, in the opposite case of blue detuning

Eqs. (7) are reduced to the conﬂuent hypergeometric form by the substitution t = ia(z 0002 z0)2/2: 0002 0003 1 0002 t a_ þ iga ¼ 0; t€a þ 2 where dot denotes the derivative with respect to new variable t, g ¼ b2 k 20 =2a is the adiabatic parameter. The equation for second amplitude b is complex conjugated. The general solutions at 0002L < z < L are linear combinations aðzÞ ¼ A1 u1 ðzÞ þ A2 u2 ðzÞ;

bðzÞ ¼ B1 u00041 ðzÞ þ B2 u00042 ðzÞ

ð11Þ

of the Kummer conﬂuent hypergeometric functions [12]: 0002 0003 1 2 u1 ¼ F 0002ig; ; iaðz 0002 z0 Þ =2 ; 2 0002 0003 1 3 0002 ig; ; iaðz 0002 z0 Þ2 =2 ; u2 ¼ ðz 0002 z0 ÞF ð12Þ 2 2 a x aða þ 1Þ x2 þ þ 000500050005; F ða; c; xÞ ¼ 1 þ c 1! cðc þ 1Þ 2! where A1, A2, B1, B2 are constants and the asterisk denotes the complex conjugation. The solution for optical waveguide was obtained in [13]. The solution for two-level system describing nonadiabatic population inversion was obtained in [14].

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The relations between constants can be obtained from set (6) near resonance point z = z0 where a = A1 + A2(z 0002 z0) + O((z 0002 z0)2), b = B1 + B2(z 0002 z0) + O((z 0002 z0)2):

The left reﬂection and transmission coeﬃcients (9) can be expressed in terms of conﬂuent hypergeometric functions

A2 2 ¼ ik 0 beih0 0002iaz0 =2 ; B1

r¼

B2 2 ¼ 0002ik 0 be0002ih0 þiaz0 =2 : A1

ð13Þ

The right boundary condition (8) yields the ratio of coeﬃcients A1 and A2 A1 1 B2 q¼ ¼ 2 2 A 2 k 0 b B1 0005 0006 F ig; 12 ; 0002iaðL 0002 z0 Þ2 =2 0005 0006: ¼0002 b2 k 20 ðL 0002 z0 ÞF 12 þ ig; 32 ; 0002iaðL 0002 z0 Þ2 =2

ð14Þ

2

e0002ih0 þiaz0 =2 u00041 ð0002LÞ þ b2 k 20 qu00042 ð0002LÞ ; qu1 ð0002LÞ þ u2 ð0002LÞ ik 0 b qu1 ðLÞ þ u2 ðLÞ t¼ ; qu1 ð0002LÞ þ u2 ð0002LÞ

ð15Þ

where functions u1,2 are deﬁned by (12) and q is deﬁned by (14). The reﬂection spectrum, i.e., the reﬂectivity R = jrj2 as a function of detuning q, is shown in Fig. 2(a). The central frequency of the spectrum comes to resonance at z = 0, in the middle of grating. The central part of the spectrum has a ﬂat top at high adiabatic parameter, as the upper

1

Reflectivity

0.8

0.6

0.4

0.2

(a) 0 -300

-200

-100

0

q, cm-1

100

200

300

0.7

0.68

Reflectivity

0.66

0.64

0.62

0.6

(b) 0.58

0

10

20

30

q, cm-1

40

50

60

Fig. 2. (a) Reﬂection spectrum R(q) at a = 600 cm00022, L = 0.5 cm, k0 = 6 · 104 cm00021, from the top down b = b0, b0/2, b0/4, where b0 = 0.67 · 1000023. (b) A part of the lower curve b = b0/4, crosses denote the numerical result.

O.V. Belai et al. / Optics Communications 266 (2006) 512–520

curve shows, and the maximal reﬂectivity is close to 1. The width of central part is proportional to the length L. The reﬂectivity is relatively high if the turning point z0 lies inside the grating jz0j < L. This inequality gives the bandwidth jqj < aL/2. There is no parametric resonance at higher detuning, when jz0j > L, and then the reﬂectivity is small. Fig. 3(a) shows how the bandwidth grows up with the chirp parameter a at ﬁxed modulation depth b. The adiabatic parameter decreases with a, then the reﬂectivity in the center decreases from curve to curve. The spectrum was recalculated numerically from Helmholtz equation (1) by the T-matrix approach. The number of points per period of spatial modulation was ﬁxed at N = 32, then the step varied along the grating. The numer-

515

ical spectrum for n = 1.5 at the same parameters as analytical expression is shown in Fig. 2(b) by crosses. The numerical results are very close to analytical. The deviation of T-matrix solution from the Kummer formulas might be caused by three reasons: (i) failure of the condition (4) for the phase variation of the grating, (ii) departure of the light frequency from the resonance, (iii) inﬂuence of the quadratic term (dn/n)2 in Helmholtz equation controlled by parameter b. Since dimensionless parameters controlling the validity of coupled-wave approximation are small in Fig. 2(b): b 0006 1000023, aL/j 0006 5 · 1000023, the deviation is negligible. At higher parameter a the deviation of the solutions to coupled-wave equations from that to Helmholtz equation increases, but not dramatically, as shown in

1 0.9 0.8

Reflectivity

0.7 0.6 0.5 0.4 0.3 0.2 0.1

(a)

0 -600

-400

-200

0

q, cm-1

200

400

600

0.79 0.78 0.77

Reflectivity

0.76 0.75 0.74 0.73 0.72 0.71 0.7

(b) 0

50

100

q, cm

-1

150

200

Fig. 3. (a) Reﬂection spectrum from the top down at L = 0.5 cm, b = 0.33 · 1000023, k0 = 6 · 104 cm00021 and a = a0, 2a0, 3a0, where a0 = 600 cm00022. (b) A part of the lower curve a = 3a0, crosses denote the numerical result.

516

O.V. Belai et al. / Optics Communications 266 (2006) 512–520

for small b. A fragment of group delay characteristics is shown in Fig. 4(b) along with numerical calculations. Dots obtained from numerical calculation are very close to the curve given by analytical formula. It is diﬃcult to analyze the solution in its general form. In particular the cumbersome expression for group delay, the derivative of (15) with respect to the detuning, is not presented here. We simplify expressions using the asymptotics of Kummer functions in Section 4.

Fig. 3(b). The main reason of the deviation is resonance approximation (5). Comparing Figs. 2(b) and 3(b) we see that the latter involves higher detuning, then the deviation becomes greater. The group delay found from analytical solution (15) is plotted in Fig. 4(a) at the same parameters as the reﬂection spectrum in Fig. 2. The deviation of curves from the linear dependence, the group delay ripple, manifests itself as oscillations with variable frequency. The frequency grows up towards the blue edge of the spectrum in agreement with results from [5,15]. For the negative chirp (or when the incident light enters from the right) the frequency grows up towards the red edge of spectrum. The maximum deviation from the averaged slope decreases with decreasing modulation depth b. Meanwhile, the ripple in reﬂectivity increases

4. Asymptotics The asymptotic expressions for the reﬂection coeﬃcient can be obtained from (15) in two cases. The ﬁrst case is the resonance condition at the left end, namely, detuning

200 180 160

Group delay, ps

140 120 100 80 60 40 20

(a)

0 -20

-100

-50

0

q, cm-1

50

100

120

Group delay, ps

100

80

60

40

20

(b) 0 -90

-80

-70

-60

-50

-40

-30

-20

-10

0

q, cm-1 Fig. 4. (a) Group delay (ps) as a function of detuning q (cm00021) at a = 600 cm00022, L = 0.5 cm, k0 = 6 · 104 cm00021, from the top down b = b0, b0/2, b0/4, where b0 = 0.67 · 1000023. (b) Comparing with numerical calculation denoted by dots.

O.V. Belai et al. / Optics Communications 266 (2006) 512–520

q = 0002aL/2 for which z0 = 0002L. In this case it follows from (12) that u1(0002L) = 1, u2(0002L) = 0, and then from (15) r0007

e0002ih0 þiaL ikbq

2 =2

2 =2

00070002

e0002ih0 þiaL pﬃﬃﬃﬃ ig

2

R ¼ jrj 0007 tanh pg:

Cð1=2 0002 igÞ ; Cð0002igÞ

ð16Þ ð17Þ

The other case is when the resonance point z0 being far from both ends inside the grating: jqj < aL/2 and a(z0 ± L)2/2 1. The asymptotic expression of the conﬂuent hypergeometric functions [12] at jarg xj < p 0002 ip0002 0003a CðcÞ e CðcÞ x a0002c ex ; F ða; c; xÞ 0007 þ Cðc 0002 aÞ x CðaÞ þ1; Im x > 0; 0002¼ ð18Þ 00021; Im x < 0 allows one to simplify expression (15). The reﬂection coeﬃcient can be written using (12) ' 0002 00032 #00022ig pﬃﬃﬃﬃﬃ i/0002ip=4þ2iq2 =a0002ih a 2q 0 r 0007 0002 R0 e Lþ 2 a 0006 qﬃﬃﬃﬃ0005 2ig00021=2 0002iwþ 2ig00021=2 0002iw0002 1 þ eþip=40002i/ Rg0 wþ e þ w0002 e 0006; pﬃﬃﬃﬃﬃﬃﬃﬃ0005 00022ig00021=2 iw þi/0002ip=4 2ig00021=2 0002iw0002 0002i/þip=4 1 þ gR0 wþ e þ þ w0002 e ð19Þ 2

where w±(q) = a(L ± 2q/a) /2, / = arg[C(ig)C(1/2 + ig)], R0 = 1 0002 e00024pg and we omit terms of the order of 1/aL2 0003 1. The enumerator and denominator of the fraction in the second line of (19) are close to 1, if w± 1. Then the formula for reﬂectivity becomes simple R 0007 R0 ¼ 1 0002 e00024pg :

ð20Þ

Both curves (17) and (20) are shown in Fig. 5. One can see their saturation, moreover, when the turning point is z0 = 0002L, the saturation occurs later than when the turning point is far from both ends. The group delay obtained from (19) is

Fig. 5. The reﬂectivity R as a function of adiabatic parameter g at z0 = 0002L (solid line) and at z0 = 0 (dashed).

517

dargr n dIm ln r ¼ dx c dq sﬃﬃﬃﬃﬃﬃﬃﬃ ( n 4q 2g 0007 00022 ½ð1 þ R0 Þ cos c a R0 a

f ¼

ð/ 0002 p=4 þ wþ 0002 2g ln wþ Þ

)

þ ð1 0002 R0 Þ cosð/ 0002 p=4 þ w0002 0002 2g ln w0002 Þ ;

ð21Þ

where we neglect terms of the order of 1/aL2 0003 1. Expression (21) involves three terms. The ﬁrst (in the ﬁrst line) gives the averaged slope. It is a linear function within the bandwidth. Its slope depends on parameter a. At 1 0002 R0 0003 1 the second term (the second line) gives the ripple, chirped oscillations. The frequency of these oscillations is double distance from left end of the grating to the reﬂection point z0. Their frequency w0þ ðqÞ ¼ 2L þ 4q=a grows up towards the blue edge of the spectrum. When reﬂectivity R0 becomes smaller, the last term (the third line in Eq. (21)) proportional to T0 = 1 0002 R0 comes into eﬀect. It gives the additional oscillations with variable frequency w00002 ðqÞ ¼ 2L 0002 4q=a that grows up towards the red edge of the spectrum. It is precisely the sum of two chirped oscillations with signiﬁcantly diﬀerent frequencies that the left part of the lower curve in Fig. 4(a) displays. Magniﬁed view of the corresponding fragment is shown in Fig. 4(b). If we change the sign of chirp parameter a, then functions w± switch their roles: w+ M w0002. Therefore at high reﬂectivity 1 0002 R0 0003 1 the spatial frequency of leading oscillations w0þ ¼ 2L þ 4q=a decreases towards the shorter wavelengths. The amplitude of oscillations in group delay (21) increases when R0 tends to unity, while that in the spectrum decreases. Formula for the reﬂection inside the bandwidth can be obtained from (19) with the accuracy to the next order of transparency T0 = 1 0002 R0 pﬃﬃﬃﬃﬃﬃﬃﬃ R 0007 R0 þ 2 gR0 ð1 0002 R0 Þ ' cosð/ 0002 p=4 þ wþ 0002 2g ln wþ Þ pﬃﬃﬃﬃﬃﬃﬃﬃ a=2ðL þ 2q=aÞ # cosð/ 0002 p=4 þ w0002 0002 2g ln w0002 Þ pﬃﬃﬃﬃﬃﬃﬃﬃ þ : ð22Þ a=2ðL 0002 2q=aÞ At high reﬂectivity R0 ! 1 oscillations in (22) are suppressed. The ﬁrst term in square brackets describes oscillations with frequency 2L + 4q/a, their amplitude gains towards the red edge of the spectrum. The second term corresponds to oscillations with frequency 2L 0002 4q/a. Then amplitude grows towards the blue edge. Both approximate formulas (22) and (21) for oscillations are plotted in Figs. 6 and 7, respectively. As ﬁgures illustrate, the asymptotic expressions nearly coincide with exact Kummer solutions. The departure of the simple formula from the Kummer solution (left edge in Fig. 6 and both edges in Fig. 7) occurs at the limit of applicability of the asymptotic expansion.

518

O.V. Belai et al. / Optics Communications 266 (2006) 512–520 110 100 90

Group delay, ps

80 70 60 50 40 30 20 10 0

-60

-40

-20

0

q, cm-1

20

40

60

Fig. 6. The group delay calculated according to asymptotic formula (21) at a = 600 cm00022, L = 0.5 cm, k0 = 6 · 104 cm00021, b = 0.67 · 1000023. Dots denote the Kummer solution.

0.72

0.7

Reflectivity

0.68

0.66

0.64

0.62

0.6

0.58

0.56

-60

-40

-20

0

-1

20

40

60

q, cm

Fig. 7. Reﬂection spectrum calculated by asymptotic formula (22) at a = 600 cm00021, b = 0.17 · 1000023, k0 = 6 · 104 cm00021. Dots denote the Kummer solution.

The turning point should be located far from the ends of grating, i.e., jL ± 2q/aj Leﬀ = (2p/jaj)1/2. The dependence on parameters a, b in Figs. 2 and 3 can also be explained by the asymptotic expressions. At ﬁxed chirp parameter a the adiabatic parameter g = (k0b)2/2a in (20) decreases with decreasing the modulation depth b. Then reﬂectivity R0 at q = 0 is relatively small and oscillations with amplitude 1 0002 R0 in the spectrum become noticeable. At ﬁxed b, on the contrary, the adiabatic parameter decreases with increasing a. It is the reason of

the most evident oscillations in the spectrum corresponding to the higher chirp parameter a. 5. Discussion The reﬂectivity is maximal at k = k0 = j/2 = p/K, where K is the period of modulation in the middle of the grating, at z = 0. The spatial frequency of modulation h 0 (z) = az + j depends on coordinate z. Then at some distance from the center the wave with k = k0 comes out from the

O.V. Belai et al. / Optics Communications 266 (2006) 512–520

resonance. The dephasing occurs when h = az2/2 0006 p, i.e., at distance z = Leﬀ 0006 (2p/a)1/2. The eﬀective number of strokes along length Leﬀ should be large Meﬀ = Leﬀ/ K = (2p/a)1/2K0002 1 1. Moreover, to provide the high reﬂectivity it should satisfy the stricter limitation of dense grating Meﬀb J 1. From here we get a condition for adiabatic parameter b2 k 20 J 1=4p: 2a The bandwidth of the reﬂection spectrum is aL, as shown in Section 3. The fronts of spectrum are determined by the eﬀective length Leﬀ. When point z0 = 2q/a is placed outside the grating at distance 0006 Leﬀ from the end, the reﬂection almost vanishes. The width of fronts is dq = aLeﬀ = 1/Leﬀ. The fronts are steep while Leﬀ 0003 L, i.e., in the limit of long grating. Phase modulation h(z) provides the parametric resonance condition for diﬀerent wavelengths. The shorter waves meet their resonance at longer distance z0 = 2q/a, and then the group delay of blue light is more than that of red one, Fig. 4. The linear dependence of the average group delay (21) upon the detuning has also simple explanation. The delay s = f/vgr is deﬁned by double distance from starting point to the resonance for given wavenumber f 0007 2z0 = 4(k 0002 k0)/a. Here vgr is the group velocity of light. If the chirp a is negative, then the slope of delay characteristics becomes negative. The ripple outside the reﬂection spectrum bandwidth, Figs. 2 and 3, with period p/L are the Gibbs oscillations originated by steep boundaries, i.e., reﬂection from the grating ends. The aperiodic oscillations inside the bandwidth arise from the triple-mirror cavity with moving middle mirror, Fig. 8. The wave reﬂected to the left from turning point z0 could reﬂect back to the right from the left end of the grating. Then the cavity appears between z = 0002L and z0; its eﬀective length is l = z0 + L = 2q/ a + L. It results in oscillations with period p/(L + 2q/a). The cavity with variable ‘‘mirror’’ is longer for blue spectrum and shorter for red, then the frequency of oscillations increases with q, as mentioned in paper [15]. At R0 ! 1 these oscillations are suppressed in the reﬂection spectrum, but remain in the group delay characteristics. If the reﬂectivity is not close to 1, the additional oscillations come into eﬀect due to the ‘‘right’’ cavity with variable ‘‘left mirror’’. Their period p/(L 0002 2q/a) on the contrary is longer for red spectrum. Here, oscillations g¼

I -L

II z0 = 2q/α

+L

Fig. 8. Conﬁguration of compound cavity: left ‘‘mirror’’ z = 0002L is the left end of grating, right ‘‘mirror’’ z = +L is the right end. Middle variable ‘‘mirror’’, the turning point z = z0, is located at diﬀerent positions depending on the wavelength. Then the ripple frequencies are determined by the variable lengths of sub-cavities I and II.

519

are suppressed at R0 ! 1 both in reﬂection spectrum and group delay characteristics. 6. Conclusions Thus, the analysis of the reﬂection spectrum and group delay of linearly chirped grating becomes simple if the turning point z0 = 2q/a is far from both ends of the grating compared to the eﬀective length Leﬀ = (2p/a)1/2. Formulas for reﬂectivity demonstrate the irregular oscillations in the reﬂection spectrum when the adiabatic parameter is not large. The oscillations are aperiodic and their amplitude slowly increases from the center of spectrum. The nature of the oscillations is reﬂection in compound cavity with a mobile middle ‘‘mirror’’. There are two terms in asymptotic expression. The ﬁrst has a period p/(L + z0) (round trip in the left sub-cavity), the second – p/(L 0002 z0) (round trip in the right sub-cavity). The oscillations in group delay characteristics have the same origin. The diﬀerence is that the right sub-cavity takes a negligible part in forming the oscillations of group delay characteristics at R0 ! 1. The amplitude of oscillations is suppressed at high chirp parameter a even at ﬁxed reﬂectivity. The conservation of high reﬂectivity with increasing a requires increasing parameter b. In order to suppress both oscillations one must choose as high the modulation depth as possible, but the limitation exists in ﬁber Bragg grating manufacturing. The alternative method to diminish the unwanted echo might be to provide the signal dephasing by apodization, i.e., smoothing the grating proﬁle [5]. Acknowledgments Authors are grateful to S.A. Babin for fruitful discussions. The work is partially supported by the CRDF Grant RUP1-1505-NO-05 and the Government support program of the leading research schools (NSh-7214.2006.2). References [1] G.A. Thomas, D.A. Ackerman, P.R. Prucnal, S.L. Cooper, Physics Today 53 (9) (2000) 30. [2] M.J.F. Digonnet (Ed.), Rare-Earth-Doped Fiber Lasers and Ampliﬁers, Marcel Dekker Inc., New York, 2001. [3] E. Udd (Ed.), Fiber Optics Sensors: An Introduction for Engineers and Scientists, Wiley, New York, 1991. [4] A. Othonos, K. Kalli, Fiber Bragg Gratings: Fundamentals and Applications in Telecommunications and Sensing, Artech House, Norwood, MA, 1999. [5] R. Kashyap, Fiber Bragg Gratings, Academic Press, New York, 1999. [6] F. Ouellette, Opt. Lett. 12 (10) (1987) 847. [7] J.T. Sheridan, A.G. Larkin, Opt. Commun. 236 (1–3) (2004) 87. [8] M. Sumetsky, B.J. Eggleton, J. Opt. Fiber Commun. Rep. 2 (2005) 256. [9] D.A. Shapiro, Opt. Commun. 215 (4–6) (2003) 295. [10] L. Allen, J.H. Eberly, Optical Resonance and Two-Level Atoms, Dover, New York, 1986. [11] L. Carmel, A. Mann, Phys. Rev. A 61 (5) (2000) 052113.

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[12] H. Bateman, A. ErdelyiHigher Transcendental Functions, vol. 1, McGrow-Hill, New York, 1953. [13] M. Matsuhara, K.O. Hill, A. Watanabe, JOSA 65 (7) (1975) 804. [14] P. Horwitz, Appl. Phys. Lett. 26 (6) (1975) 306.

[15] S. Bonino, M. Norgia, E. Riccardi, in: Proc. IOOC-ECOC’97, Edinburgh, 22–25 September 1997, IEE Conf. Pub. # 448., vol. 3, 1997, p. 194.

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Accounting Manual Bragg.pdf - Free download Ebook, Handbook, Textbook, User Guide PDF files on the internet quickly and easily. This free download contains some brag tag bracelets to use as incentives for good behavior. You are receiving two choices for printing - a page for each bracelet with 5 on each page, and one of each bracelet on 2 pages. This way, you can color code. The typefaces shall not be used in such a manner that they represent competition to Otto Maurer. Modifying the typefaces is prohibited, even in the event that it is necessary for fulfilling personal design requirements.Embedding of the typefaces into electronic documents like PDF is permitted in a secured 'Preview and Print' mode.

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