Application of discrete layer peeling algorithm for design of selective photonic devices obeying the constraints of the fabrication process

Marcin Wielichowski*a, Sergiusz Patelaa

aFaculty of Microsystems Electronics and Photonics, Wrocław University of Technology, Wrocław, Poland

ABSTRACT

Results of the Discrete Layer Peeling (DLP) inverse scattering algorithm do not directly correspond to Bragg grating's technological parameters. We propose a method of transforming a stack of discrete complex reflectors resulting from DLP into physical and geometrical parameters of a Bragg grating suitable for fabrication in planar technology. Particularly, the method keeps lengths of grating sections (teeth and grooves) above a required technological minimum. While the stack of complex reflectors is a natural output DLP, it can also be computed from a space distribution of the complex coupling coefficient determined in other ways.

Keywords: Optical couplers, gratings, waveguides, optical switches.

1. INTRODUCTION The apodization of Bragg gratings etched on top of planar waveguides poses specific problems not present in gratings manufactured via photosensitivity. The most straightforward way is varying the etch depth. Although possible, it is hard to assure reproducibility. Various approaches based on binary etch have been proposed. They are: varying the duty cycle [1], varying the period [2], grating sampling [3], introduction of sections without grating [4], curving the waveguide under uniform grating [5], varying the transversal length of grating tooth [6], low period gratings [7].

Most of the above methods, except for those from [3] and [7], do not enable the realization of arbitrarily low values of coupling coefficient while maintaining a minimum length of grating tooth or groove required by technological constraints.

We give a systematic analysis of a method based on varying both the duty cycle and grating period, capable of transforming complicated distributions of the complex coupling coefficient into physical and geometrical parameters of a binary etched grating that meets the technological constraint for tooth or grove minimal length. The method directly takes a stack of discrete complex reflectors as input. The stack can be directly generated by the DLP algorithm or computed from spatial distribution of the coupling coefficient obtained in some other way.

Section 2 derives the method. Section 3 shows an exemplary numerical result of the method. Sections 4 and 5 include some further discussion and conclusions, respectively.

2. ANALYSIS We will consider a stack of M equally spaced discrete complex reflectors as in Fig. 1. Distances between reflectors equal ∆ and 1ρ , 2ρ , …, Mρ are complex reflection coefficients. Values of the coefficients are independent of the light wavelength. A reflector together with the free-propagation space that immediately follows that reflector will be called a stack unit cell.

We will be replacing each stack unit cell with a planar unit cell depicted in Fig. 2 i.e. with a fragment of the planar Bragg grating. Each planar unit cell consists of N rectangular teeth and N grooves having the effective refraction indices hn

and ln , respectively. Note that the last (N -th) groove in planar unit cell has a different length than all 1−N preceding groves.

*marcin.wielichowski@pwr.wroc.pl; phone +48 71 320 3223; fax +48 71 328 3504; www.wemif.pwr.wroc.pl/photonicsgroup

Silicon Photonics and Photonic Integrated Circuits, edited by Giancarlo C. Righini, Seppo K. Honkanen, Lorenzo Pavesi,Laurent Vivien, Proc. of SPIE Vol. 6996, 69961P, (2008) · 0277-786X/08/$18 · doi: 10.1117/12.785812

Proc. of SPIE Vol. 6996 69961P-12008 SPIE Digital Library -- Subscriber Archive Copy

29CK I

3

IU!4

COII29CK I

II

Fig. 1 Stack unit cells.

v1

u1

z

planar unit cell j-1 planar unit cell j planar unit cell j+1

nh

nl

neff

dh dl dpdhdh dl

v1

u1

Fig. 2 Planar unit cells.

We will now calculate the complex reflection coefficient of a planar unit cell using the Transfer Matrix Method (TMM) [10]. Below are given the transfer matrices for the tooth-groove interfaces

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡=

111

111

lh

lh

lhlh

hl

hl

hlhl TT

ρρ

τρρ

τ (1)

and for the free-propagation distances within teeth and grooves

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡=

−− li

li

lhi

hi

h ee

Te

eT φ

φ

φ

φ

00

00

. (2)

All symbols are defined below

( ) ( )( ) ( )lhlhlhhl

lhllhlhhhl

nnnnnnnnnn

+−=−=+=+=

//2/2

ρρττ

(3)

λπφφ/2===

kdkndkn lllhhh (4)

where λ is the wavelength of light.

Transfer matrix for entire planar unit cell is an N -th power of a product of the transfer matrices (1)-(2)

( )Nllhhhl TTTTT = . (5)Matrix T connects the complex amplitudes of incident and reflected waves (see Fig. 2) in the following way

Proc. of SPIE Vol. 6996 69961P-2

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

2

2

1

1

vu

Tvu

. (6)

Note that in (5) we have ignored for simplicity the length pd and just replaced it with ld . This can be done since pd does not affect neither amplitude nor phase of reflection.

The exponentiation in (5) will be performed using the Sylvester’s theorem [11]. For a unimodular matrix we have

( )( ) ( )( )

( )( )

( ) ( )( )⎥⎦⎤

⎢⎣

⎡Ω−−Ω

ΩΩ

Ω−−Ω×

Ω=⎥

⎦

⎤⎢⎣

⎡1sinsin

sinsin

1sinsinsin

1NND

NBNCNNA

DCBA N

(7)

where

( ) ( )DA +=Ω21cos . (8)

After performing the exponentiation in (5) and then calculating the complex reflection coefficient r of entire planar unit cell using the formula [9]

2221 /TTr = (9)where 21T and 22T are the submatrices of T , we arrive at a fairly complicated expression

( ) ( )( )( )( ) ( ) ( )( )pNpNq

pNqr 111

12

cos1sincossincossin

−−

−

−−=λ (10)

where

( ) ( ) ( ) ( )( )lh

lhlhlh nn

nnp2

sinsincoscos22 +

−=φφφφ (11)

( ) ( ) lihi

lh

lhlihi

lh

lh ennnn

ennnn

q φφφφ −−−

⎟⎟⎠

⎞⎜⎜⎝

⎛ +−+

+=

41

4

22

1 (12)

( )( )lihilihi

lh

lh eennnn

q φφφφ −−− −−

=4

22

2 . (13)

Let us now assume that hd and ld obey the following conditions

BragghlBraggll

BragghBragghh

dddddd

,,

,,

ηηη

+=−=

(14)

where 1,0∈η , lη will be calculated later and Bragghd , and Braggld , meet the Bragg condition

( ) ( )00

0,0,

/22/2/

λπππ

===

knkdnkd lBragglhBraggh (15)

where 0λ is the light wavelength of the centre of grating's reflection peak.

Figure Fig. 3 shows the modulus of r in function of η for several sample values of lη treated as parameter. For fixed

values of N , hn and ln , modulus of the maximum reflection coefficient maxr is independent of lη and it can be calculated as

Proc. of SPIE Vol. 6996 69961P-3

0max ==

ηrr . (16)

00.20.40.60.81

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

ηab

s(pl

anar

uni

t cel

l ref

lect

ion

coef

ficie

nt)

0 20 40 60 80 1000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

dh [nm]

ηl=1

ηl=0.91211

ηl=0.86248

Fig. 3 Modulus of planar unit cell reflection coefficient r for several values of parameter ηl. The refrective indexes assumed

are nh=3.3087, nl=3.3023 and N=46.

Explicitly, maxr has the form

⎟⎟⎠

⎞⎜⎜⎝

⎛ += −

lh

lh

nnnn

Nr2

coshtanh22

1max (17)

what, for 1max <<r , can be approximated as

( )( )lh

lh

nnnn

Nr+−

≈ 2max . (18)

The formula above can be interpreted as Fresnel's reflection coefficient at tooth-groove boundary multiplied by N2 (i.e. N2 reflections). Results of (17) and (18) for exemplary values of hn an ln are shown in Fig. 4.

Zeros of ( )ηr we will find by analyzing the argument of the sin function in the numerator of (10). For lh nn ≈ , i.e.

under conditions of planar grating, ( ) ( ) 12/22 ≈+ lhlh nnnn so p from (11) takes the form

( )lhp φφ += cos . (19)We can then write

( ) πφφ nN lh =+ (20)where n is natural.

Proc. of SPIE Vol. 6996 69961P-4

50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1

planar unit cell teeth count

plan

ar u

nit c

ell

refle

ctio

n co

effic

ient

Fig. 4 Planar unit cell reflection coefficient r in function of teeth count calculated using formula (17) (solid line) and (18)

(dashed line). The pairs of lines from botom to top are calculated for increasing difference between nh and nl. The values used are: nh=3.3087, nl=3.3023; nh=3.3187, nl=3.2923 and nh=3.3287, nl=3.2823.

After substituting (4) and (14) into (20), we find

( )( )llh

h

nnNnNn

ηη

−−

+=21 . (21)

Substituting n = 1−N in (21) and using (14) we find a length min,hd closest to Bragghd , , for which r is zero

( )⎟⎟⎠

⎞⎜⎜⎝

⎛−

−=llh

h

hh nnN

nnk

dη

π 212 0

min, (22)

As stated at the beginning of this section, stack unit sections are to be substituted with planar unit sections. We are now ready to give precise criteria of such a substitution. First, using (17) or (18) we find maxr such that

( ),...,,max 321max ρρρ≥r (23)and using (22) we find lη such that

min,min, technolh dd ≥ (24)where min,technold means a minimum technologically realizable length of hn section. Note that, according to (14), for

lη >0, all ln sections will have lengths higher than Bragghd , so we assume their technological realizability. Second, by

numerically solving (10) for η (having substituted (14) to (10)) we find M planar unit cells such that

MMrrr ρρρ === ...,,, 2211 . (25)Condition (25) defines the amplitude properties of planar unit sections. For defining the phase properties, we will use a general result from [8]

( )jhljjjjjp rrk ,11, 2221 φγρρφ −∠+∠−∆+∠−∠= ++ (26)

Proc. of SPIE Vol. 6996 69961P-5

where j means the unit cell number, hlφ is a phase shift induced by all sections preceding the pd section within the

given planar unit cell, and pφ is the phase shift induced by the pd section. Rewriting (26) for our problem, we have

( ) )( lhjjjjp NNkrr φφγρρφ 121

11 −+−∆+∠+∠−∠−∠= ++ . (27)

Having the value of pφ , pd is calculated as

( ) min,

min,

0

0

,,

/2/

technolp

technolp

hp

hpp dd

ddnk

nkd

<≥

⎩⎨⎧

+=

πφφ

. (28)

Finally, one more condition is needed to assure identical spectral widths of both reflection spectra, i.e. of the reflectors stack and the planar Bragg grating. We will calculate the maximum local coupling coefficients in the stack and in the planar grating. For the stack we have [8]

( )( ),...,,maxtanh1321

1max, ρρρκ −

∆=DLP (29)

and for the grating

( ) ( )max1

1,,,max tanh

11 r

ddNNd jpjljh

−

−+−+=κ . (30)

Neglecting the varying lengths of individual planar unit cells, we will iteratively look for values of hn , ln and N such that

maxmax, κκ ≈DLP . (31)

3. NUMERICAL EXAMPLE As an example result of the procedure described above we will show a transformation of a DLP-generated reflection coefficient from [12] to the parameters of a Bragg grating etched on top of a strip-loaded planar waveguide.

50 100 150 2000

0.02

0.04

0.06

0.08

0.1

complex reflector number

abs(

com

plex

refle

ctor

refle

ctio

n co

effic

ient

)

50 100 150 200-1

-0.5

0

0.5

1

complex reflector number

arg(

com

plex

refle

ctor

refle

ctio

n co

effic

ient

) / π

Fig. 5 Modulus (abs) and angle (arg) of complex reflection coefficients of consecutive complex reflectors ρ1,ρ2,...,ρM.

Proc. of SPIE Vol. 6996 69961P-6

The assumed parameter values are: 3.3087=hn , 3.3023=ln , mµλ 55.1= . Number of grating periods in each

planar unit cell is 46=N , the minimum technologically realizable grating tooth length is assumed to be nmdtechnol 80min, = , thus the coefficient 86248.0=lη (see Fig. 3). The DLP-generated distribution of reflection

coefficient along the stack of complex reflectors is shown in Fig. 5. Both modulus and angle of each complex reflection coefficient is given. Number of complex reflectors is 220 and length of the resulting planar grating is about mµ2200 (see further figures).

0 500 1000 1500 200080

90

100

110

120

130

140

150

z [µm]

d h, dl [n

m]

0 500 1000 1500 200050

100

150

200

250

300

350

z [µm]

d p [nm

]

Fig. 6 Lengths dh of higher refractive index sections (lower line) and lengths dl of lower refractive index sections (upper

line). Lengths dp of phase adjusting sections.

In Fig. 6 there are shown the calculated distributions along the planar grating of the lengths of grating teeth ( hd ),

grooves ( ld ) and phase-adjusting sections ( pd ).

Fig. 7 shows the distribution of grating pitch ( lh dd + ) and grating duty cycle ( ( )lhh ddd +/ ).

0 500 1000 1500 2000229

230

231

232

233

234

z [µm]

grat

ing

pitc

h [n

m]

0 500 1000 1500 200035

40

45

50

z [µm]

grat

ing

duty

cyc

le [%

]

Fig. 7 Grating pitch and grating duty cycle.

Proc. of SPIE Vol. 6996 69961P-7

4. DISCUSSION

As can be seen in (14), the parameter lη determines how strong the modulation of grating period is. Particularly, for

lη =0 only modulation of duty cycle takes place.

The planar grating is from both sides surrounded with waveguide of the effective refractive index hn . Thus the planar

unit cell number 1 lacks the first groove-teeth interface, whereas cell number N has an additional teeth-grove interface. These facts have been neglected in derivations from section II but must be taken into consideration in the calculation of reflection spectra of the resulting frating.

No formula for etch depth has been given. This is because no specific type of planar waveguide has been assumed. For a given type of planar waveguide, the required etch depth and, in fact, all other parameters of the waveguide will be defined by the values of hn and ln .

It is possible to swap the roles of teeth and grooves in the method presented. This might be needed in case the minimum-length constraint is imposed on grating grooves, not teeth.

The calculation of the complex reflector stack from a given coupling coefficient distribution is described in [9].

5. CONCLUSIONS We have proposed a method allowing the transformation of a stack of complex reflectors or, equivalently, a distribution of complex coupling coefficient into physical and geometrical parameters, i.e. effective refractive indices and sections lengths, respectively, of a planar Bragg grating. Values of the resulting geometrical parameters meet the technological constraint.

6. ACKNOWLEDGMENT This work is supported by Polish State Committee for Scientific Research during 2006 – 2009 under Grant 350-730 W12

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