Dynamical manipulation of quadratic non-linearity photonic crystal gap solitons through thermooptic induced index modulations

Dynamical manipulation of quadratic non-linearity

photonic crystal gap solitons trough thermooptic induced

index modulations

Frederico C. Moreiraa and Solange B. Cavalcantib

aUniversidade de Lisboa, Av. Prof. Gama Pinto, Lisbon, Portugal;bUniversidade Federal de Alagoas, Av. Lourival Melo Mota, Maceio, Brazil

ABSTRACT

We investigate optical spatial solitons in a one-dimensional photonic crystal composed of two materialspossessing quadratic nonlinearity subject to an array of microheaters. The periodic temperature gradientinduces a variation in the refraction index due to the thermooptic effect. In contrast to the photoniccrystal, one can easily change parameters like period and modulation depth of the induced index modu-lation. In addition to that one can easily move the modulation in time and introduce defects. We foundthat is possible to steer solitons, change its shape, split it in two or more lower power solitons and movethem independently.

Keywords: solitons, thermooptic effect, photonic crystals, nonlinear optics, quadratic solitons

1. INTRODUCTION

During the last decades a great deal of attention was focused on the study of electromagnetic waves inphotonic band gap (PBG) materials. Periodic nonlinear systems host many useful phenomena, which arestrongly enhanced when the wave frequencies border a photonic band edge. Then the wave group velocityvanishes and the efficiency of nonlinear processes is significantly increased in comparison with that ofa uniform material. One of the most remarkable phenomena observed in this kind of situation is theformation of gap solitons.1 Another interesting feature associated with such structures is the possibilityof mode coupling, and in particular the second-harmonic generation (SHG)2 as well as more sophisticatedfrequency conversion processes, based on the quasi phase matching (QPM)3 achieved artificially by meansof a structural periodicity built into the medium, among which we mention simultaneous second and thirdharmonic generation4 and fractional frequency conversion.5

In this work we study Gap Solitons (GS) in a one-dimensional Photonic Band Gap material composedof materials possessing quadratic nonlinearity. A way to obtain such a periodic arrangement is in the formof the so called Photonic Crystal (PC), where two or more dielectric materials with different dielectricconstants are periodically layered. But PCs are not flexible in the sense that once they’re manufacturedcharacteristic properties as period and dielectric function cannot be changed.

New possibilities arise when a superlattice is used, i.e. when the underlining periodic structures isadditionally smoothly modulated. On one hand appearing of minibands allow for creation of localizedstructures in the frequency domain, not supported by the underlining nonlinear periodic structure and onthe other hand, one can achieve desired phase-matching conditions in an a priori defined spatial domains.This last statement becomes clear if one interprets smooth modulations of the lattice parameters in thesmooth modulation of the underlining band-gap structure.

To make the imposed additional smooth periodic modulation to be most efficient in controlling thewave propagation, one should use a flexible way of modulating the parameters. To this end we recallthat a dielectric function of any material depends on the temperature, i.e. by changing temperatureof a given structure one can introduce additional variations into the dielectric function distribution inseveral ways, such as introducing localized, non periodic or even periodic variations over the length ofthe structure.6–8 Additional flexibility reached in this way is due to possibility of the parameter variation

Further author information: (Send correspondence to Frederico Moreira)Frederico Moreira.: E-mail: [email protected], Telephone: 55 82 9619 6248Solange Cavalcanti.: E-mail: [email protected], Telephone: 55 82 3214 1427

1

not only in space but also in time, by varying the temperature. We mention that temperature gradientswere already successfully used to induce modulations of the dielectric function necessary for observationof optical Bloch oscillations,6 designing switches for communications9 and optical multiplexers.10

In the present paper we consider GSs in a one-dimensional multilayer stack composed of two χ(2)-materials and subjected to additional periodic modulations of the dielectric function induced by thethermo-optic effect.

We use the inherent flexibility of design of the Thermooptic Index Modulation (TIM) over a PC topropose a way to steer initially standing solutions in space. We also found a way to introduce periodicoscillations of GSs by changing the amplitude and/or period of the TIM. We also propose a switch torelease the energy of a GS in the form of propagating waves.

2. THERMOOPTIC MODULATION OVER A DIELECTRIC STACK

We consider a TE electromagnetic wave E = (E, 0, 0) with frequency ω1 propagating along z-directionperpendicular to a periodically placed dielectric layers of two alternating materials with period ℓ, atleast one of which is a χ(2)-medium. So we have the relative dielectric function ǫ0 (z) = ǫ0 (z + ℓ) andquadratic nonlinearity χ(2) (z) = χ(2) (z + ℓ).

The stack is in contact with a surface that heats it with a periodic temperature profile alongz−direction (this can be done using microheaters 11). Due to the thermooptical effect the total relativedielectric function after the system reached thermal equilibrium is given has an additional contributionof a function of the local temperature (see e.g.12), and one can write it as a Taylor’s series

ǫ (z, ω) = ǫ0 (z, ω) + η2f (z) g (ηz) . (1)

The local thermooptic coefficient is represented by the periodic function f (z + ℓ) = f (z). The temper-ature modulation of the structure is given by g (ηz). The dimensionless parameter η is

η2 ∼ ∆T

⟨

∂ǫ

∂T

⟩

, (2)

a function of the thermooptic coefficient function average 〈∂ǫ/∂T 〉 and amplitude of temperature modu-lation ∆T . The dielectric function at the average temperature is given by ǫ0 (z, ω). Since the thermoopticcoefficient of most common nonlinear materials is very small (of the order 10−6/K). The parameter η issmall enough to one make a perturbative approach even in a range of temperature gradient of hundredsof Kelvin. So we focus on the case η ≪ 1. We consider the function g(ηz) to have a period d, defined as

d ∼ ℓ

η. (3)

As an experimentally feasible example we consider a periodic stack of period ℓ = 6.28µm composed oflayers of AgGaS2 and AlAs with lengths 5.34µm and 0.94µm respectively. At the temperature rangewe are dealing with, both materials are transparent for a fundamental field (FF) with frequency ofω0 = (2πc)/(5.01µm) and the corresponding second harmonic (SH) angular frequency 2ω0. The speedof light is given by c. For simplicity we define a general periodic step-function in the interval ℓ as

F (z, Fa, Fb, a, b) =

Fa when |z| < a/2,Fb when a/2 < |z| < b+ a/2,

(4)

Here a + b = ℓ. We have for the AgGaS-AlAs grating ǫ0(z, ω1) = F (z, 5.49, 8.12, a, b), ǫ0(z, ω2) =F (z, 5.56, 8.23, a, b) and f(z) = 10−4F (z, 7.4, 1.4, a, b)/(Kη2).

These parameters refer to a ∆T = 100K about an average temperature of 297K one has the amplitudeof the thermooptic modulation of the dielectric function to be η2 = 10−2. From Eq. (3) we obtain aperiod of the thermooptic pertubation to be d = 10ℓ = 62.8µm.

Following the previous considerations we introduce the one-dimensional Maxwell equation for a lin-early polarized electromagnetic field

−c2∂2Ex

∂z2+

∂2Dx

∂t2= 4π

∂2PNL

∂t2, (5)

2

0 0.5 10.16

0.17

0.18

0.19

0.2

0.21

0.22

0.23

0.24

0.25(a)

kℓ/π

ω

0 0.5 1 1.5 2−1

−0.5

0

0.5

1(b)

φ1

,2

0 50 100 150 2000

2

4

6

8

(c)

z/ℓ

ǫ

0 1 2

6

8

10

Figure 1. (a)The FF(line) and SH(dashed) bands of the PC are calculated for the average temperature of 297K.(b)Bloch functions calculated at the the band-edges of the FF(lines) and SH(dashed). (c) The large-scale thermoopticmodulation. Small graph shows the fast-scale fixed PC.

and consider the electric field in the x direction as a superposition of two waves with frequencies ω0 and2ω0

Ex (z, t) = E1 (z, t) e−iω0t + E2 (z, t) e

−2iω0t + c.c. (6)

The displacement field is given by the constitutive relation

Dx (z, t) =

∫ ∞

−∞

ǫ (z, ω)Ex (z, ω) e−iωtdω. (7)

In the case of weak quadratic nonlinearity we have the nonlinear polarization components in the quasi-monochromatic approximation

PNL = ηχ(2)0 (z)

(

E1E2e−iω0t + E2

1e−2iω0t

)

+ c.c. (8)

Here χ(2) (z) = ηχ(2)0 (z) where χ0 is adimensional function of order one.

We use the method of multiple scales13 in order to get equations for the envelopes E1,2. The naturalparameter of the system is η. We expand the spatial coordinate as z =

∑∞j=0 η

jzj and the time coordinate

as t =∑∞

j=0 ηjtj .Where all variables zj and tj are regarded as independent from each other. The final

envelope equations in adimensional form are

i∂u1

∂τ+ ρ

∂2u1

∂ζ2− V1(ζ)u1 + 2u1u2 = 0, (9)

i∂u2

∂τ+ σ

∂2u2

∂ζ2− 2V2(ζ)u2 + u2

1 = 0. (10)

In above the following definitions apply

t =2

ω0τ, z =

√

ω0

ω′′1

ζ, ρ =ω′′1

|ω′′1 |, σ =

ω′′2

ω′′1

, V1 (ζ) = −f1g

(

√

ω′′1/ω0ζ

)

, V2 (ζ) = −f2g

(

√

ω′′1/ω0ζ

)

.

Here ω′′j (j = 1, 2) are the curvatures of the the band diagram on the respective frequencies jω0 and

the average coefficients fj and χ are given by

3

fj =1

ℓ

∫ ℓ

0

ϕ(j)νj (z)ϕ

(j)νj

(z)f(z0)dz0,

χ =4π

ℓ

∫ ℓ

0

ϕ(2)ν2

(z)χ(2)0 (z)

(

ϕ(1)ν1

(z))2

dz.

Where the Bloch functions ϕ(j)νj

(z) are solutions of the linear equations at the νj-th band edge

−c2d2ϕ(j)

νj

dz2− (jω0)

2ǫ (z, jω0)ϕ

(j)νj

= 0. (11)

Here we consider only solutions of (11) at the center of band diagram. The electric field can bereconstructed from (9) as

E1 (z, t) = ηϕ(1)ν1

(z)u1√2χ

+ η2

∞∑

ν 6=ν1

cν,ν1ϕ(1)ν (z)

√

ω0

ω′′1

1√2χ

∂u1

∂z,

E2 (z, t) = ηϕ(2)ν2

(z)

u2

(

η√

ω′′

1

ω0z, η2t

)

χ+ η2

∞∑

ν 6=ν2

cν,ν2ϕ(2)ν (z)

√

ω0

ω′′1

1√2χ

∂u1

∂z.

Where

ci,j =1

ℓ

(

∫ ℓ

0

ϕ(j)νj

(z)∂

∂zϕ(j)

νj(z)dz

)

2c2

(ωl − jω0).

Here we are going to consider only localized solutions where ρ = 1 and σ > 0, that can support gapsolitons. We write a stationary solution as

uj (ζ, τ) = wj (z) e−ijΩτ . (12)

Where jΩ/2 are the detunings from the solutions from the frequencies ωj. one can find that the envelopefunctions wj are real valued functions. Furthemore, if the modulation Vj is an even function,w2 is aneven function and w1 can be even or odd. subject to14

−d2w1

dζ2+ (V1(ζ) − Ω)w1 − 2w1w2 = 0, (13a)

−σd2w2

dζ2+ 2 (V2(ζ)− 2Ω)w2 − w2

1 = 0. (13b)

3. THE TOTAL GAP

Before we start the study of GS of Eqs.(13) we investigate the solutions of the linear equations

−ρd2A1

dζ2+ (V1(ζ)− Ω)A1 = 0, (14a)

−σd2A2

dζ2+ (V2(ζ) − 2Ω)A2 = 0, (14b)

as they will prove to be of fundamental importance on the study of the asymptotic properties and onthe required conditions to the existence of a gap soliton.

The linear potentials satisfy Vj(ζ) = Vj(ζ+d). They can induce bands gaps. We define the individual

gap intervals as ∆(j)νj = [Ω

(j)νj ,Ω

(j)νj+1 ]. Where Ω

(j)νj is the νj-th edge (νj = 0, 1, 2...) of the corresponding

4

−1 0 1−1

0

1

Ω(1)2

=1.5264

−0.5 0 0.5−0.5

0

0.5

Ω(2)2

=0.8809

−1 0 1−2

0

2

Ω(1)1

=0.3919

−1 0 1−1

0

1

Ω(2)1

=−0.4912

−0.5 0 0.50

0.5

1

Ω(1)0

=−0.1571

ζ/d−0.5 0 0.50

0.5

1

Ω(2)0

=−0.5119

ζ/d0 0.5−1

−0.5

0

0.5

1

1.5

2

k1/(2π),k

2/(2π)

Ω

Ω(1)2

Ω(2)1

Ω(1,2)1,2

Ω(1)2

Ω(1)0

Ω(2)0

Ω(2,2)3,4

Ω(1)1

Ω(2)2

Ω(2)4

Ω(2,2)3,4

Ω(1,2)1,2

Ω(2)3

Ω(2,1)1,0

Figure 2. Left panel show the wave vectors k1(lines) and k2(dashed lines) for the structure defined by V1 =α1 cos (2πζ/d) and V2 = α2 cos (2πζ/d). Where α1 = −1.14 and α2 = −2.42 The total gap regions are representedby gray bars. Right panels show several periodic band-edge solutions.

gap. A total gap is defined as a intersection of the band gaps of Eqs.(3), ∆(1)ν1 ∩∆(2)

ν2 , ∆(L,U)νL,νU = [Ω

(L)νL ,Ω

(U)νU ].

Where the lower edge L = 1, 2 and upper edge U = 1, 2 can have any combination.

The Eqs.(3) has two linearly independent Floquet solutions15

Aj,±(ζ) = φj(±ζ) exp [± (ikj + µj) ζ]. (15)

Here φj,±(ζ) are periodic functions and µj and kj are real constants.

We consider only kj = 0 solutions. Such solutions are real valued and, on a band edge, one can provethat Aj,±(ζ) = φj(±ζ).

4. LOCALIZED MODES AND IT’S ASYMPTOTIC BEHAVIOUR

Let us consider the asymptotic behavior of the function w1 as |ζ| → ∞. In this limit one has for localizedsolutions wj → 0. Since for large ζ one always have w1w2 → 0, the FF field must decay as the solutionof the Eq.(14a)

This equation has decaying (growing solutions) only if Ω belongs to a gap of (14a) (here we notconsider the cases where Ω coincides with an edge of the gap). Then it follows that for large values of ζ

w1 → C1φ1(−ζ)e−µ1ζ (16)

describes the asymptotic behavior of w1. In equation above C1 is a constant. Let us now turn toEq.(13b).Notice that it can be considered as a non homogeneous linear equation of the unknown w2

satisfying the conditions dw2/dζ(0) = 0 and w2(±∞) = 0

w2(ζ) = − 1

σW

∫ ∞

−∞

G(ζ, ζ′)w21(ζ

′)dζ′. (17)

Where the Green’s function is given by

G(ζ, ζ′) =

φ+(ζ)φ−(ζ′)e−µ2(ζ

′−ζ) when ζ < ζ′,

φ+(ζ′)φ−(ζ)e

−µ2(ζ−ζ′) when ζ > ζ′(18)

Where for simplicity we have written φ±(ζ) = φ2,±(ζ). In Eq.(17) the Wronskian is given by W =A+(A−)ζ − A−(A+)ζ . Now we study the asymptotic behaviour of Eq.(17). Let’s consider the limit

5

−10 −5 0 5 10−0.5

0

0.5

1Ω = 0.6

−10 −5 0 5 10−0.4

−0.2

0

0.2

0.4

ζ /d

Figure 3. Solid lines represents the functions wj . The linear solutions C1A1 and C+A−(|ζ|) are represented by

dashed lines. The figure correspond to a localized mode with Ω = 0.6 and µ = 0.2566. See how both wj are veryclose to their linear asymptotic solutions. The parameters are σ = 0.5, α1 = −1.21 and α2 = −2.42.

ζ → ∞. First we rewrite Eq.(17) as

−σWw2 = A+(ζ)

∫ ∞

ζ

A− (ζ′)w21 (ζ

′) dζ′+

+A−(ζ)

∫ ζ

−∞

A+(ζ′)w2

1 (ζ′) dζ′. (19)

Where we have written A±(ζ) = A2,±(ζ). For ζ ≫ 0 the right integral on Eq.(19) approaches

C+ =

∫ +∞

−∞

A+(ζ′)w2

1 (ζ′) dζ′. (20)

If the parameter µ defined byµ = 2µ1 − µ2, (21)

satisfies µ > 0 then the integral C+ is obviously finite(The integrand ultimately decays as a exponentialfunction).We can rewrite in this case Eq.(19) as

−σWw2 = A+(ζ)

∫ ∞

ζ

A− (ζ′)w21 (ζ

′) dζ′−

−A−(ζ)

∫ ∞

ζ

A+ (ζ′)w21 (ζ

′) dζ′−

− C+A−(ζ). (22)

Since by hypothesys 2µ1 > µ2, ζ ≫ 0 we have w2 ≈ − C+

σWA−(ζ). Both FF and SH decay as free Floquet

solutions as is shown in Fig.3.

If µ < 0 w2, the SH is driven by the −w21 term in Eq.(13b) and only Eq.(19) can be used.

5. NUMERICAL STUDY OF LOCALIZED MODES

We use a shooting method for both C1 and C± parameters in equations Eq.(19) or Eq.(22) in conjunctionwith Eq.(16) to determine initial conditions at ζ0 to satisfy the conditions w′

1(0) = w′2(0) = 0 or w1(0) =

w′2(0) = 0. Since the solution of both solutions are related by the constant C1 (it’s present in both

asymptotic formulas for w1 and w2) one reduces the region of the parameters space by guessing valuesof C± close the their values when one substitute w1(ζ) = C1A1(|ζ|) in their respective formulas. So infact the possible parameter region of C± is bounded by the choice of C1. In our implementation we usethe shooting method in two steps. We use a shooting algorithm for C1 in order to satisfy w′

1(0) = 0.

6

But in each iteration of the algorithm we make a shooting of the parameter C± requiring w′2(0) = 0. So

when one finds a suitable C1, the two conditions are satisfied and the program stops. The integrationof Eqs.(13) is done alternating two steps. First one integrate (13a) (wee use Runge-Kutta method)assuming a constant w2(ζ) to find w1(ζ −∆ζ). Then we use a linear interpolation of w1 in the interval∆ζ in Eq.(13b) to find w2(ζ − ∆ζ) and so on. In order to improve accuracy one can also include arelaxation method by reintroducing the w2(ζ − ∆ζ) in Eq.(13a) and iterate until a given numericaltolerance is assured in each step. For the numerical solutions we use a specific form for the linear

0 0.5 1 1.50

2

4

6

8

10

12

14

16

18

20

Ω

P

Figure 4. Figure shows the total power of the ζ0 = 0 branch(line), and the ζ0 = d/2 branch(dotted-dashed line)as functions of Ω. The parameters are σ = 0.5, α1 = −1.21 and α2 = −2.42.

potential, V1(ζ) = α1 cos (2πζ), and V2 (ζ) = α2 cos (2ζ/ d). Where α1 = −1.14 and α2 = −2.42. We alsohave the parameter σ = 0.5. All parameters are calculated for the already described GaAs2−AlAs stack,only the cos (2πζ) shape of the potentials is arbitrary, but we think it’s a useful representation for thecase of a smooth periodic temperature gradient, as is the case. One can easily see that Vj(ζ − d/2) arealso even functions. So we can have branches that can be centered at ζ0 = 0 or ζ0 = d/2. We have infact found one branch at each symmetry axis of the chosen potentials. We see in Fig.5 that the GS ofthe two fundamental branchs increase monotonically in power from 0 on the lower edge of the total gap

in the interval ∆(1,2)1,2 = [0.392, 0.62] and diverges on the right edge of the gap. This is a general property

of this type of total gap, as one can see in Fig.2, the total gap is limited by the FF on the left and by

the SH on the right. We found two fundamental branches detaching from the left edge of the gap ∆(1,2)1,2 .

As Ω approaches the right edge ∆(1,2)1,2 , µ1 remains finite but µ2 → 0. The term

C+ = − 1

σW

∫ ∞

−∞

A+ (ζ′)w21 (ζ

′) dz′ (23)

is obviously finite even for an non-decaying w2, provided w1 → C1A1 as is always the case. One can alsosee that in fact the first harmonic power P1, with

Pj = j

∫ ∞

−∞

w2j (ζ)dζ, (24)

remains bounded even on the gap edge. So the FF can be localized on a continuous wave SH background

that of course must approach a infinite P2 for any finite C+ as µ2 → 0. The gap Ω(2,2)3,4 does not have P

decaying to zero at any edge. That’s because µ2 = 0 at both edges. But one has a finite w1 against ainfinite power SH ”background” on the upper edge.

6. THERMOOPTIC GRATING MANIPULATION

In this section we investigate some new effects and control possibilities when one can vary the thermoopticgrating parameters over time. In this work we approach, due to space limitations to two types of

7

−1 −0.5 0 0.5 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

ζ/d

V1

−10 0 10−1

−0.5

0

0.5

1

ζ0=d/2

−10 0 10−0.5

0

0.5

1

ζ0=0

−10 0 10−0.4

−0.2

0

0.2

0.4

ζ0=0

−10 0 10−0.4

−0.2

0

0.2

0.4

ζ0=d/2

ζ0=0

ζ0=d/2

Ω=0.5 Ω=1.3

Ω=1.3Ω=0.5

Figure 5. Left panel shows symmetry the axis of V1. Right panel shows several w1.

Figure 6. Moving GS with ζ0 = 0 and frequency Ω = 0.443 due to a localized defect in the TG. Left panel shows|u1|

2. Left panel shows the thermooptic grating profile.

alterations to the original structure were GSs were previously found, the addition of defects on thestructure and continuous variation of the depth of the gradient of temperature over time. Both are veryeasy to implement just by increasing or diminishing the power of the all the microheaters over time fordepth variation and just changing the power of one or more microheaters to generate a defect. Someeffects one obtain by changing the grating over time are pretty obvious, for example one can ”discharge”the energy acumulated in a GS by just switching off the grating, in the absence of the total gap theenergy in FF and SH is free to propagate in this case. Other effects are nontrivial and we just presenthere an phenomenological approach due the inherent complexity of the present system. Another possiblyuse of this dynamical grating is to move a given GS in space. We found two ways of achieve this goal.The first one is to move the all thermooptic grating in respect to the fixed PC, one way to do this isto create a moving pattern in the fixed microheaters. This is shown in Fig.6. The advantage of thismethod is that if one have more than one GS standing in the PC one can move both at the same velocity.Dinamycally changing the depth of potentials over time can generate stable breathing structures as inFig.6.

7. CONCLUSIONS

We studied the analytical properties of a GS with a quadratic nonlinearity in the presence of a periodicpotential. We found that the asymptotic behaviour of the SH can have two distinct possibilities, todecay as a free wave or as a w2

1 driven solution. We found that the fundamental branch in order toexist in a given gap must satisfy two conditions: at least one edge of the total gap is a edge of the FFpotential and that edge must have an associated Bloch function of even symmetry. A numerical shootingscheme was developed in order to obtain GS of any frequency inside a given gap. It was discussed the

8

Figure 7. Breathing mode for a initial GS with ζ0 = 0 and frequency Ω = 0.393 in a TG modulated as αj(τ ) =α0j(1 + τ/τ0) for τ < τ0, α(τ ) = 2 for τ > τ0. We have defined τ0 = 300. Note the long time oscillations inamplitude. Fourier analysis of the function shows that ΩF = 0.2767.

Figure 8. |u1|2 of a GS with frequency Ω = 0.393 in a TG modulated as Vj(ζ, τ ) = Vj(ζ + 6d sin (2πτ/200)).

behaviour of the fundamental branch in several types of total gaps, each type have distinct properties.The study of a linear stability model showed that instabilities in the studied case are mostly generatedby complex eigenvalues and by internal modes that can unstabilize the GS due to non-zero wavevectorsof internal modes. We suggested the use of thermooptic gratings to introduce a dynamic potential thatcan be changed at will. We found some uses such as a switch to release the optical energy by turning offthe thermooptic grating. We can move GS by introducing defects in the periodic structure or by movingthe grating itself. At last we showed that changing the depth of the potentials one can obtain stable”breathing” strucures.

Acknowledgments

The authors wish to thank Dr. Vladimir Konotop and Dr. Fatkhulla Abdullaev from University ofLisbon for advice.

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