WSEAS TRANSACTIONS on COMMUNICATIONS

Farzin Emami, Amir H. Jafari

Simulation and Applications of Nonlinear Fiber Bragg Gratings FARZIN EMAMI and AMIR H. JAFARI Optoelectronic Research Center, Department of Electrical and Electronic Engineering, Shiraz University of Technology Airport Boulevard, Shiraz IRAN [email protected]

[email protected] http://www.sutech.ac.ir

Abstract: - We have simulated the behaviors of nonlinear fiber Bragg gratings (FBGs). The generalized nonlinear equations governing these structures are solved by a method which uses a Fourier series procedure and a simple iterative method. All of the nonlinear effects are considered. Bragg soliton generation in intrinsic media and birefringence effects in FBGs are studied. We found that the first order dispersion can causes time shifting in the input pulse peak. It is shown that FBGs are proper for optical switching too. They can use in filters, nonlinear fiber optical applications, soliton propagations etc. Key-Words: - Nonlinear Fiber Bragg gratings, Modulation intensity, Solitons, Birefringence, Optical switches, Simulation division multiplexing (WDM) systems [3] and optical multiplexers-demultiplexers with an optical circulator. Propagation of solitons is another application of FBGs. Nonlinear pulse propagation and compression, have been also reported in short period FBGs [8]-[10]. Due to the existence of a stop band in the transmission spectrum of FBGs, known as a photonic band gap (PBG), the nonlinear pulse propagation has many applications (such as an optical switch) in them. For a medium with some nonlinearity, propagation of the waves is possible even though its frequency lies within the stop band such as Bragg solitons in the case of Kerr nonlinearity. Description of pulse propagation is based on a set of non-integrable nonlinear coupled (and maybe non-usual such as found in [11]) partial differential equations. They have solitary wave solutions. In this paper, at first we review a general theory of FBGs. Then, in Section 2, the soliton propagation is discussed in the FBGs and the results of pulse propagation in these structures will be shown. In Section 3, the grating solitons are introduced. Bragg soliton generations are considered in Section 4. Important phenomenon called birefringence in fibers will be the next section contents. Section 6 belongs to a brief review of instabilities in the steady state response of the propagated pulse due to

1 Introduction Application of a special type of the periodic structures called fiber Bragg gratings (FBGs) are introduced for about three decades [1], [2]. They have many applications in the optical communications such as optical filters, couplers, reflectors, dispersion comparators, etc [3], [4]. FBGs are consisting of a periodic modulation of the medium refractive index, along the core of the fiber [5], which has a short length that reflects particular wavelengths of light and transmits all others. In other words, they can be used as an inline optical filter to reject specific frequencies (frequency selector mirrors). Generally, the refractive index variations can be uniform or apodized [1], but usual structures for FBGs are: -

uniform index gratings; apodized structures; chirped grating, and phase-shifted gratings.

An attractive aspect of FBGs is the narrow bandwidth of them which can be used in an optical fiber [6], as notch filters. The large dispersive behaviors of these structures make them good devices for linear dispersion comparators, optical add/drop multiplexers (OADM) [7] in wavelength

ISSN: 1109-2742

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WSEAS TRANSACTIONS on COMMUNICATIONS

Farzin Emami, Amir H. Jafari

modulation instability parameter. FBG applications are subject of Section 7 and finally, conclusions are presented in Section 8.

i

∂A f ∂z

+

i ∂A f + δA f + κAb v g ∂t

(

)

+ γ | A f | 2 +2 | Ab | 2 A f = 0

2 FBG Formulation

i

A simple grating used in a FBG is shown in Fig.1

∂Ab i ∂Ab + δAb + κA f + ∂z v g ∂t

(

)

+ γ | A f | 2 +2 | Ab | 2 Ab = 0 (5) Where δ is the detuning factor, v g is the pulse group

velocity, κ is the coupling coefficient and γ expresses the amount of the medium nonlinearity due to self-phase modulation (SPM) and cross-phase modulation (XPM) which defines as γ =

the nonlinear coefficient of the fiber medium. When a pulse propagates in a fiber, two cases may be considered; the linear case with γ = 0 , and the nonlinear case with γ ≠ 0 . In the linear case with a CW operation of pulse the above equations will be:

Fig. 1. A simple grating

The waves propagated in this medium id diffracted. For the incident angle θi the reflected angle θr we have [1]:

sin(θi ) − sin(θr ) = mλ = nΛ where Λ is the grating period and λ

n2ω0 ; so Aeff is the effective area and n 2 is cAeff

∂A f

+ δA f + κAb = 0 ∂z ∂A i b + δAb + κA f = 0 ∂z

i

(1) is the

n medium wavelength with the medium index of n . For the incident and the diffracted wave numbers, ki and kd, there is a relation with the grating wave number k g = 2 π Λ , as:

One solution of (6) can be written as:

k i − k g = mk g

Ab ( z ) = B 2 exp( −i q z ) + r A1 exp( i q z )

A f ( z ) = A1 exp(i q z ) + r B2 exp(−i q z )

(2)

where q = ± γ 2 − κ 2 and r = −

κ

(7) .

δ +q The relation between δ κ and q κ are plotted in

If the incident light be in the fiber axis direction, using (1) we have:

λ = 2nΛ

(6)

Fig.2; this is the dispersion curve.

(3)

This is the Bragg condition [1]. To study the nature of forward and backward waves we can solve the Helmholtz equation considering the periodic variations of the grating refractive index, δn g ( z ) , in the form of a Fourier series as:

δn g ( z ) =

+∞

∑ δn

n = −∞

n

exp(2 π i n ( z / Λ)) (4)

Defining the forward and backward electric field amplitudes as A f and Ab respectively, one can write the governing FBG equations in a coupled form of:

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Fig. 2. Relation between detuning and q

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Farzin Emami, Amir H. Jafari

When − κ < δ < κ the q-value is purely imaginary, we have a reflected wave from FBG and this is called a photonic band-gap too. For δ > κ the wave

Our simulation is based on a Fourier series expansion combined with a simple Jacobi iterative method (a predictor-corrector method) [12]. As you can see, there is an appreciable decrease in the pulse amplitude during its propagation. For lower detuning the propagation delay increased and this is shown in Fig. 5.

would be passed entirely from the grating. The reflectance spectrum and the transmission factor of FBGs can be derived from (1) using the proper boundary conditions. In z=L (where L is the grating length) after solving these equations the reflection coefficient will be:

rg =

i k sin (qL) q cos(qL) − i δ sin( qL)

(8)

It is plotted in Fig. 3 for different detuning factor.

Fig. 5. Pulse propagation delay versus detuning

For nonlinear dispersion and in CW case we have the following relations for A f and Ab :

A f = u f exp(i q z )

Fig. 3. FBG reflectivity versus the detuning factor

Ab = u b exp(i q z )

Due to dispersion effects in FBGs such as group velocity dispersion (GVD) and their increments especially around the stop band, there is a pulse broadening after the pulse propagation through the FBG. It will be serious near the band gap which has a more delay in pulse propagation. We simulate this broadening for a typical pulse propagated in a FBG and our results is seen in Fig. 4 [12].

If p0 denotes

(9)

the

pulse

power,

two

amplitudes u f and u b are:

uf =

p02 1+ f

2

&

p02 1+ f

ub =

2

f

(10)

ub 2 2 2 & P0 = u f + u b . uf Using (10) the values of δ and q are:

where f ≡

k (1 + f 2 ) 3 γ P0 − 2f 2 2 k (1 − f ) γ P0 1 − f − q=− 2f 2 1+ f

δ =−

2 2

(11)

A plot of detuning versus q is shown in Fig. 6 [1]. Note for f = 1 the group velocity is zero and for

f > 1 there is a negative group velocity which Fig. 4. Simulation of pulse propagation in FBG considering dispersion effects

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means the reflected waves.

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Farzin Emami, Amir H. Jafari

detuning (cm-1)

the grating as forward parts E f and E b [7], [8]:

backward

E = E f ( z , t 0 ) exp( i( β 0 z − ω 0 t )) + E b ( z , t ) exp( − i( β 0 z + ω 0 t )) (12) After some mathematical manipulations and simplifications and using the Helmholtz equation the coupled equations derived as:

i[

∂E f ∂z

detuning (cm-1)

−i[

+

1 ∂E f ] + κ E b exp( −2 i δ z ) + v ∂t γ ( | E f |2 +2 | E b | 2 ) E f = 0

∂Eb 1 ∂Eb ] + κ E f exp( 2 i δ z ) − ∂z v ∂t + γ ( | Eb |2 +2 | E f |2 ) Eb = 0 (13)

c and κ , the coupling coefficient, is where v = n0 defined as πn1 λ0 . The detuning from the Bragg wavelength is δ =

Fig. 6. A plot of detuning versus q for (a) f < 0 , (b) f > 0 ; for f = ±1 we are on stop band exactly

factor γ is

n0 ( ω 0 − ω B ) and the nonlinear c

πn2 . As mentioned, the nonlinear terms n0 λ

contain the effects of SPM and XPM [10]. Operating at the Bragg frequency, the field components E f and Eb satisfy the Klein-Gordon

3 Grating Solitons As said, there are nonlinear effects in the fibers grating usually called Kerr effects, and for pulses propagated outside the band gap there is high dispersion. Kerr effect is the index dependency to the field strength I :

equation [10]:

∂2E f

2 1 ∂ Ef = κ 2 Ef ∂z 2 v 2 ∂t 2 ∂ 2 Eb 1 ∂ 2 Eb − − 2 = κ 2 Eb ∂z 2 v ∂t 2

n = n1 + n ( 2 ) I where n1 is the background index and n ( 2 ) defines the amount of the medium nonlinearity. In gratings, there are solitons which are the consequence of two mentioned effects; the Kerr effect and outside the band gap dispersion which are called the grating solitons [3], [4]. For solitons which have spectrum inside the photonic band gap we have gap solitons. Since the amount of the dispersion in the grating are more than the usual fibers, there is a difference between the solitons in the grating and conventional fibers. In fact, the grating solitons are dens respect to the fiber solitons. Soliton analysis can be done using the coupled mode theory (CMT) [5], [6]. Utilizing the CMT, decompose the inside field of

ISSN: 1109-2742

and

−

(14)

Assume E f = exp(i ( Kz − Ωt )) ;

we

have

a

dispersion equation for the structure such as:

Ω K = (( ) 2 − k 2 ) v K for all the frequencies of ω = ω B + Ω between

the frequency range of ω B −

∆ω

< ω < ωB +

2 is a complex variable, with ∆ω ≡ 2 kc

n0

∆ω 2

,

.

To solve the equation for nonlinear case define:

x = (k / 2) ( z + vt ) , y = (κ / 2) ( z − vt ) ,

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δ and γ U = ( ) 0 .5 E f ; κ κ we will have the following normalized equation: ∂U i + V exp(−2 i δ ( x + y )) + ( | U | 2 +2 | V | 2 ) U = 0 ∂x ∂V −i + U exp(2 i δ ( x + y )) + ( | V | 2 +2 | U | 2 )V = 0 ∂x (15) Neglecting SPM, for δ = 0 the above equations are integrable and we can find their solitary equations using the Thirring model as [13]:

coefficient and photonic band-gap controllability [20]. The refractive index can be written as:

σ=

n 2 ( z ) = 1 + χ 0 + δχ cos( 2 k B z ) + χ ( 3 ) | E p |2 (21) the background χ 0 is and δχ (modulation depth) is

its amplitude variations; they both are frequency dependent. So, depending on the input pulse frequency it is possible to change band-gap.

^

U = A Sec1 / 2 (τ / τ 0 ) exp(iθ )

(16)

5 Birefringence Effects on FBGs

^

V = − µ 2 A Sec1 / 2 (τ / τ 0 ) exp(iΦ )

Birefringence (BRG) effects are usually occurred in all types of FBGs. This effect can divide the peak to two parts at the Bragg wavelength [20]. Since the Bragg wavelength depends on the grating period and the effective index, physical and environmental effects such as temperature, stress and stretching the fiber can change the fiber index as:

(17)

with the following definitions:

τ )] & τ0 τ 1 3µ 2 + 4 µ 4 − 1 tan −1 [ sinh( )] (18) Φ = Ψ0 + 8 4 τ0 2 µ + 4µ + 1 ^ v − ve z 4µ 2 A= 8 , τ = t − , µ4 = 4 ve v + ve µ + 4µ + 1 v µ2 2 2 2 v = κ τ ve = ± & 0 2 (1 + 4κ 2 v 2τ 0 )1 / 2 (1 − µ 4 )2

θ = Ψ0 +

1 µ 2 − 4µ 4 − 3 tan 2 µ8 + µ 4 +1

−1

susceptibility

[ sinh(

∂neff

∂λ B )]dP ∂P ∂P ∂neff ∂λ [ 2λ B , 0 ( ) + 2neff 0 ( B )]dT ∂T ∂T

dλ B = [ 2λ B , 0 (

) + 2neff 0 (

(22)

(19) With the aid of the above equations it is possible to have positive or negative solitons depends on the sign of v − v e . Now, the total waves inside the fiber can be written as:

Where P and T are pressure and temperature respectively. The variations of the effective index cause BRG and this is due to the changes in propagation constants of the guided waves. The value of BRG along the z-axis is:

E =(

B=

κ 1/ 2 ^ ) A sec h 1 / 2 ( τ / τ 0 ) {exp [ i ( Θ + β 0 z )] γ

− µ 2 exp[ i ( Φ − β 0 z )]} exp( −iϖ 0 t ) (20)

| ∆n|| + ∆n ⊥ | n0

(23)

and perpendicular indices [21]. Fig. 7 shows the Bragg wavelength variations for different polarizations. Deriving the refractive index and hence the propagation constant, explain the reflectance spectrum of distributed FBGs. It is found that this spectrum has more than one peak in Bragg wavelength and decomposes perfectly for higher forces. The results are plotted in Fig. 8. Such specification can be used in sensors.

When the nonlinear coefficients n2 and n1 be comparable, the Bragg filter would be transparent [14], [15]. Finally, forward and backward solitons are correspond to µ → 0 and µ → ∞ respectively.

in

We can generate the Bragg solitons [16]-[18] in an electro-magnetically induced transparency medium [19]. Such media have big Kerr coefficient (or a large amount of Re χ 3 where the factor χ denotes the medium susceptibility), low absorption

6 Modulation Instability (MI)

[ ]

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n0

= B0 +

n0 is the initial core index, n|| and n⊥ are the parallel

For µ ≈ 1, v e