BEAM PROPAGATION ANALYSIS OF MODAL PROPERTIES OF OPTICAL NONLINEAR WAVEGUIDES

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BEAM PROPAGATION ANALYSIS OF MODAL PROPERTIES OF OPTICAL NONLINEAR

WAVEGUIDES

Z. Jakubczyk, H. Jerominek, R. Tremblay, C. Delisle

To cite this version:

Z. Jakubczyk, H. Jerominek, R. Tremblay, C. Delisle. BEAM PROPAGATION ANALYSIS OF

MODAL PROPERTIES OF OPTICAL NONLINEAR WAVEGUIDES. Journal de Physique Collo-

ques, 1988, 49 (C2), pp.C2-311-C2-314. �10.1051/jphyscol:1988273�. �jpa-00227690�

BEAM PROPAGATION ANALYSIS OF MODAL PROPERTIES OF OPTICAL NONLINEAR WAVEGUIDES

Z. JAKUBCZYK(~), H . JEROMINEK(~), R. TREMBLAY and C. DELISLE Laboratoire de Recherches en Optique et Laser, DBpartement de Physique. Universite Laval, Sainte-Foy, Quebec. G I K 7P4, Canada

% s d : Nous dkrivons des dthodes pour obtenir l',information nkessaire au calcul des constantes de propagation des modes, des poids modaux et des fonctions propres B W i r d'une solution ndrique de l'bquation daHelmholtz. Des exemples de calculs ndriques pour des guides d'onde nonlin6aires sont presentbs.

Abstract: The methods for extracting the information necessary to compute the propagation constants of the modes, the mode weihts and eigenfunctions from a numerical propagating-beam solution of the Helmholtz equation are described. Examples of numerical calculations for nonlinear waveguides are demonstrated.

The beam propgation method (BPM) is an accurate numerical procedure for describing the propagation of an arbitrary incoming optical field through a given optical system. The method has already been used in nonlinear integrated optics to study mode coupling / I / , soliton emission /2/ and modal stability /3/. The BPM can be baaed on either the full scalar-wave (Helmholtz) equation for the electric field, or on the Fresnel equation that is valid for paraxial field propagation. For realistic sources of illumination in integrated optics structures where the thichness of a waveguiding layer is comparable to the wavelength the solution of the Fresnel equation can break down because of the presence of field components with large angular deviations from the propagation direction. In this paper we adopt the more general, Helmholtz equation formalism. Although the numerical solution generated by the BPM emphasizes the beam properties of the field, it implicitly contains all the information necessary for a complete description of the field in terms of modes. In the series of articles (Ref. /4/ and the references therein) Feit and Fleck, Jr. have introduced the methods which thro& the Fourier analysis of the propagating field with respect to the axial distance z can give such modal properties as the propgation constants B n , the relative mode wights and the mode eigenfunctions. These methods were developed for multimode fibers. In this paper we applied these methods to study the modal properties of nonlinear planar waveguides.

The beam propagation method is based on the assumption that the propagation of a single- frequency component of the light in planar waveguide is governed by the scalar Helmholtz

equation : ,,

,.

^A

here E(w,x,z)exp(i't) is the transverse component of the electric field at angular frequency w, and n(x,:E:Z) is the intensity dependent refractive index. It is convenient to express E ( w , x , z ) as the product of a complex field amplitude ~(w,x,z)and a carrier wave moving in the positive z

-

direction:

E(@,x,z) = e(a,x,z) exp(-ikz) (2)

where k= (nc w)/c, and n, is the refractive index of the waveguide cladding. The BPM requires specification of the field as a function of a transverse coordinate at z = 0. The numerical solution is then developed in terms of the initial field by the successive application of operators corresponding to the free space propagation and phase front alteration by the refractive index profile.

("on leave from Institute of Physics. Silesian Technical University. Gliwice. Poland '2)~resent address : National Optics Institute, Sainte-Foy, Qu6bec. GIK 7P4, Canada

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988273

C2-312 JOURNAL DE PHYSIQUE

Three different ~eethods are available for d e M n i n g the mode propgation constants Bn /4/. Two of them also #jive the relative mode weights. To compute the Ra from a numerical solution of Eq.1, it is necessary to calculate the Fourier transform of ~(w,x,z) for a particular transverse position (xp) not at- the origin. If we call this Fourier transform

E(O,X~,~), the spectrril density :~(w,xp,R);2 will display a set of resonant peaks, which can be associated with the gukded modes of the waveguide. The second method requires squaring the complex field amplitude and integrating over the waveguide cross section for each increment in z. The resulting complex function of z is then numerically Fourier transformed, and the heights of the peaks which appear at the position of ^fin of $he resulting spectrum will be proportional to the desired weights, The third method, which is the most accurate, is based on a correlation function formed by multiplying the conjugate of the complex field amplitude at z = 0 and the complex field amplitude for arbitrary z and integrating over the waveguide cross section:

The complex field amplitude can be expressed in terms of the waveguide-mode eigenfmtions as:

A. are determined by the input field E(x,~). If Eq.4 is substituted into the correlation function then after multiplying by Hanning window function and taking the Fourier transform, we obtain the spectrum function:

P(R) = An El(B 2

-

^{Bn) =}

C

^WnEl(R

-

^{Bn) (5)}

n n

where Wn are the mode weights, and $t ^(R-Ra ) is the line shape function /4/. If the calculations of the field are made over a total axial distance Z, the accuracy with which the waveguide propagation constants or effective refractive indices of the modes can be determined is:

It is possible to reduce these uncertainties by lengthening Z, but it is far more efficient to fit the line shape function to calculated values of P(B) in the neighborhood of a local maximum.

Once the eigenvalues m e known, the mode eigenfunctions un (x) can be computed by repeating the numerical propagation procedure and performing numerically the integration:

~ ( x )

=

^const

*

1 ~ 8 ( ~ , ~ ) w(e) exp(iBnz) dz (7)

0

where w(z) is the Hanning window function.

Our basic configuration consists of a thin linear film (0

<

x

<

d) with thickness d = 4 pin

and refractive index nt = 1.597, sandwiched between a linear cladding (x

>

d, nc = 1.573) and a nonlinear =ubstrate (x

<

0) with low-power index l l s o = 1.573 and nonlinear index nz = 10- l a

&/W. This data set is !For a particular structure /5/ consisting of a film of Corning 7059 glass deposited by RF magnetron sputtering onto the surface of a Schott GG495 color glass filter.

Throughout the calculatf~ons we have used the initial field corresponding to a collimated

Gaussian beam: 2 2

~ ( x , o )

=

A e-(X-Xo) lo ( 8 )

where A is used to djust the input beam power flux, xo is the displacement of the central coordinate of the beam and a is the Gaussian spot size. The number of transverse points used to represent the transverrle field was 256 or 512 at the wavelength 2

=

^0.4765 pm. A propagation step from 0.1 X to 0.5 2, was used depending on the stability of the solutions.

In order to obtain the results for the considered waveguide in the case where nonlinear effects can be neglected, we performed the calculations of the correlation function for low power flux of 1 0 - 3 W/mm,. The promation distance was Z

=

^{10240 L.} The results are shown in Fig.1; a)

-

^{for the} bean1 central coordinate xo = 2 p (layer centre) and b)

-

^{for xo = 1.333 p.}

The ordinate is recalculated for effective refractive index. In the first case (Fig.la), because

effective refractive indices of the appropriate modes in the first and the second case are less than 10-6.

Fig.2 shows the evolution of the electric field initially excited with the power flux of a)- 104 W/mm and b)

-

^5.104 W/m. In both cases xo = 1.8 pm and o = 2 pm. In Fig.2a the power flux is still too small to cause nonlinear effects, The propagation until Z = 102401. showed that the preservation of the power flux was better than 99%. In Fig.2b the generation of two spertial solitons is clearly visible. Further propgation of this beam showed generation of two more

solitons, the first at about z = 368A. and the second at about z

=

713 )i. Each of these four solitons carry away a power flux of about 0.5.10Q W/mm. We propagated the beam to Z = 11396 h . After the solitons' emission the power loss of the main beam was less than 1%. The power flux at the end of this distance was 2.83.104 W/mm.

I

Fig.2. Evolution of electric field for the initial beam with xo = 1.8 m, o = 2 pm and the power flux: a)

-

104 W / m and b)

-

^{5.104 W/m.}

- _g

_7:

-

_f ^{7 -}

5

6 - 3 6 -

d b s:

e

_o^{s -} _.

G 4 -

Y . -E 4 - .

p 3 :

z

^{2 -} ; ^{2 -}

a ' .

U _{C I 1 .} l '

- C

_CI1 ¹

'

^- _-

-

0 ^{0 -} -0 o -

-1

--. . .

^,

.

^,

. .

^, , - - ^-1

In the two cases described above we have also carried out the calculations of the correlation function. Fig.3 shows the examples of the mode eigenfunctions for the propagation of the field from Fig.2b for a)

-

^{TEI and b)}

-

^'I%. The calculations were performed between z = 1156 X and z

= 11396 A .

I

1

*

*

Table 1. Flux

-

dependent changes C&ff and relative mode weights.

1 572 1 577 1 502 1 587 1 592 1 597 1 602 1 572 1 577 1 582 1 587 1 592 1 597 1 600

"eff "eff

Fig.1. Mode spectra P(att) for power flux 10-3 W/m. Gaussian spot size o = 2 um and Gaussian central coordinate: a)

-

xo = 2 pm and b)

-

xo = 1.333 pm.

1

-

^{Anetf .lo-=} ; 2

-

Power flux of a particular mode in relation to TEo mode.

-

Flux [W/mm]

104 5.104

TE2 TE3

1 7.0 17.0

1 26.0 36.7

TEP TEo

2 1.37.10-1 3.31.10-2

2 4.11.10-3 2.90.10-3

1 37.5 54.8 1

0.9 1.1

TEI

2 2.96.10-3 1.82-10-3 2

1 1

1 4.6 7.2

2 2.94.10-2 1.69-10-2

C2-314 JOURNAL DE PHYSIQUE

The mode spectrum for the case with initial pawer flux 5.104 W/rmn and for a propagation between z = 1156 A and z = 11396 h is shown in Fig.4. Analogous calculations were performed for the initial beam with po*rer flux of 104 W/mm for the distance between z = 0 and z = 10240 1. Changes in the effective refractive indices Alleff for pwer fluxes of 104 and 5.104 W/m in comparison with values for the small power-flux of 10-3 W/mm are shown in Table 1. The line shape function fitting procedure showed that the uncertainties of Alleft calculations were smaller than 10-5.

Table 1 also shows the relative mode weights for all five modes.

Fig.3. Amplitude of mode eigenfunctions for the modes a)

-

^{TEI and b)}

-

^!l75

-

₂

5 '"

d 6 14

. C

-

-

13

i

-

12

-

a'

$ i l

-

10

1.572 1.577 1.582 1.587 1.592 1.597 1.60

"eff

Fig.4. Mode spectrum P(lleff) for the initial Fig.5. Electric field evolution for the power flux 5.104 W/m. xo = 1.8 pm initial power flux 5.104 W / m

a = 2 ~ . (xo = 1.8 pu, o = 2

w)

^{in the}

waveguide with attenuation.

We have performed many calculations concerning waveguides with saturation of the nonlinear effect and waveguides with absorption. As an example, Fig.5 shows the evolution of the field with the same parameters as in Fig.2b but for a waveguiding structure with strong linear absorption in the substrate (imaginary part of the dielectric constant c" = 5010-3). The fast decay of the "soliton-like" waves is clearly visible. Moreover, further propgation showed that there was only one more "soliton-like" wave emission at about z = 373 A.

Our studies have shown the great usefulness of the BPM in the analysis of the modal properties of nonlinear waveguiding structures.

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3

(1987), 1652.

/2/ M.A. Gubbels, E.M. Wright, G.I. Stegemm, and C.T. Seaton, J. Opt. Soc. Am. B,

4

(1987), 1837.

-- -

/3/ L. Leine, C. Wachter, U. Tangbein, F. Lederer, J. Opt. Soc. Am. B,

5

(1988), 547.

/4/ M.D. Feit, and J.A. Fleck Jr., Appl. Opt.,

19

(1980), 3140.

/5/ S. Patela, H. Jeraminek, C. Delisle, and R. Tremblay, J. Appl. Phys.,

60

(1986), 1591.