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Chromatic dispersion in arrow waveguides
Volume 76, number 3,4
OPTICS COMMUNICATIONS
1 May 1990
CHROMATIC DISPERSION IN ARROW WAVEGUIDES
M.S. WARTAK ~, J. CHROSTOWSKI, Wenyan JIANG 2 Dlvtston of Electrwal Engtneenng, Natzonal Research Councd, Ottawa, Ontario KIA OR6, Canada
and B.A. SYRETT Department of Electromcs, Carleton Umverstty, Ottawa, Ontario K I S 5B6, Canada Received 5 December 1989
Compound chromatic dispersion has been calculated for ARROW wavegmdes. It shows anomalous behavlour when compared w~th regular asymmetric wavegmdes
There has been a growing interest in antiresonant reflecting optical waveguldes (ARROW), both in passive [1-4] and active [5] devices where optical confinement is achieved on the basis of antiresonant Fabry-Perot reflection. These grades have unique properties when compared to conventional structures based on total internal reflection. They offer low loss, polarization and wavelength selectivity as well as ease of fabrication. The refractive index and size of the core layer allow flexibility in coupling to an optical fibre with minimal loss. The fact that ARROW guides can be implemented on semiconductor substrate, makes them ideal for integration with electronic devices on a single chip. As future optical elements will probably be used in superfast applications, the most severe limitations will arise from the group velocity dispersion ( G V D ) also known as chromatic dispersion, of the picosecond and subplcosecond pulses propagating along those elements. In fact, on the time scale of 10-100 fs a significant dispersion can occur over distances as short as one centimeter. It is a purpose of this note
to analyse the chromatic dispersion of the ARROW guides. ARROW guides have been recently analysed [6] using the equivalent transmission-line and transverse resonance method. For the optimum values of parameters the dispersion relation for the TE mode in the ARROW guide is ~
1+
22 N x / ~2 2
tan ( k d x / ~ - N~fr) = 0
(1) where nc is the core refractive index, d is the core thickness (fig. 1 ) and the complex propagation constant fl xs related to the complex effective index N~ff by f l = k N ~ f r = k n e f f + lOt/2 .
(2)
St 02 SI3N4 SlO 2
Department of Electrical Engineenng, Unxverslty of Ottawa, Ottawa, Ontario Kl N 6N4, Canada. z Present address Canadian Marconi Company, Montreal, Canada
Sl s u b s t r a t e Fig 1. ARROW wavegmde structure
217
Volume 76, n u m b e r 3,4
OPTICS COMMUNICATIONS
Here k=2rc/2 is the w a v e n u m b e r in v a c u u m with wavelength 2, whereas c~ (cm - ~ ) represents the propagation loss. F o r the structure under consideration, losses are very small and we can assume them to be zero C h r o m a t i c disperston o f the guide is defined as D = dud2,
(3)
where the transit time or the group delay per unit length associated with a m o d e propagating through the m e d i u m is given by r= (l/c) dfl/dk.
(4)
Here c is the speed o f hght in a vacuum. Using the dispersion relation ( 1 ), one finds
d/3 dk
1A(knc) (d/dk) (knc)+kB fl A+B
(5
4= 1 + d ~ / f l Z - k 2 sec 2 ( d , , / ~ n ~ _ f 1 2 ) ,
(6
B = (k2n 2 --
(7
where
J~2)/(j~2-k2)
For typical values o f p a r a m e t e r s [ 6 ], the ratio A / B ~s of the o r d e r I0 2 - 10 3. This allows us to approximate eq. (5) as
d f l / d k ~ (k/fl) n c M ( 2 ) ,
(8)
where M ( 2 ) = nc - 2 dnc/d2 .
l Ma~ 1990
as a funcnon of wavelength The results are shown m fig 2. We have plotted chromatic dispersion for a regular a s y m m e t r i c slab guide with thickness t, single m o d e fiber [7] and, for comparison, the material's curve Refractive index o f the core, n,,, has been m o d e l l e d using the Sellmeier equation wtth p a r a m eters given by Fleming [8] It is clear that the ARR O W wavegmde shows a very interesting b e h a v l o u r when c o m p a r e d with a regular asymmetric guide [9 ] It has large positive values o f D for all considered wavelengths. The total waveguide and material d~sperslon for A R R O W and a s y m m e t r i c slab guides are shown m fig 3. F o r an a s y m m e t r i c guide the crossover wavelength is shifted with respect to the material's dispersion by a small a m o u n t toward larger values, whilst the cross-over wavelength for the ARR O W guide shows a large shift towards the smaller wavelengths. Due to large difference m chromatic dispersion between A R R O W and conventional a s y m m e t r i c wavegmde one can consider the posslblht~ o f buildmg an lnterferometric sensor using a standard rib wavegulde for one arm and an A R R O W waveguide for the other. The sensor should exhibit zero geometrical offset but the optical offset should change by several fringes as the source laser wavelength is chirped. Such a change would m i m i c the effect trad m o n a l l y achieved through the use o f an active phase
(9) 60
For silica the funcUon M ( 2 ) takes the values between 1.46 and 1.47 for the wavelengths within 0 9 and 2.0 #m. Using the above approximanons, chromatic dispersion can be written as
D=Dm + D w ,
10)
11 )
C nef
n,fr
.
12)
neff,,/
Solving the dispersion equation ( 1 ) a n d using the above formulas, one can plot c h r o m a t i c dispersion 218
o
I
2
J 13
-
I 20 7
- ~ 09
11 lambda
1 ( n ~ - 2 dnc/d~) 2 1 - ~ 2c
i
,
E3
n~ d2n~ d~ 2 ,
and guide dispersion
D~--
I
O..
where the material dispersion is Dm= -
C
1
--~ 15
' 17
9
[ tim ]
Fag 2 C o m p a r i s o n of c h r o m a t i c dispersion versus wavelength for different systems Curve I A R R O W (core thickness d e = 4 gin, core refractive Index no= 1 45) Curve 2 m a t e r i a l ' s dispersion. C u r v e 3 a s y m m e t r i c slab grade (film thickness t = 4 jam film refractive Index nf= 1 45, substrate index n s = I 44) Curve 4 single-mode fiber, from ref [ 7 ]
Volume 76, number 3,4
OPTICS COMMUNICATIONS
80-]
E
40.
t-
0
.£
/J
//
t~ -80
N e t w o r k s A r c h i t e c t u r e s P r o j e c t a n d by the o p e r a t i n g g r a n t o f N a t u r a l Sciences & E n g i n e e n n g R e s e a r c h C o u n c i l o f C a n a d a . U s e f u l discussions w~th M. O ' S u l l i v a n a n d J. B e r n a r d are a c k n o w l e d g e d .
j
f
References
/
~Q.-40
8
I
1 May 1990
12
14
16
18
lambda [pm] F~g. 3. Total chromatic d~spers~on versus wavelength for &fferent systems. The values of parameters as m fig 2 Curve 1. ARROW grade dispersion Curve 2' material's &sperslon Curve 3: asymmetric guide dispersion.
shifter m o n e a r m o f the i n t e r f e r o m e t e r . Th~s w o r k was partially s u p p o r t e d by the T e l e c o m munication Research Institute of Ontario-Photonic
[ 1] M A Duguay, Y Kokubun and T L. Koch, Appl. Phys Lett 49 (1986) 13. [2] T.L. Koch, E.G Burkhardt, F.G. Storz, T J Bridges and T Slzer, IEEE J Quantum Electron 23 (1987) 889 [ 3 ] Y. Baba, Y Kokuban, T Sasalo and K. Iga, J. Ltghtwave Tech 16 (1988) 1440 [ 4 ] T Baba and Y Kokubun, IEEE Photomcs Technology Letters l (1989) 232 [ 5 ] D. Botez, L J. Mawst, G Peterson and T J. Roth, Appl Phys Lett 54 (1989) 2183 [6 ] W Jlang, J Chrostowskt and M Fontame, Optics Comm. 72 (1989) 180. [7] A. Suglmura, K. Dalkoku, N. Imoto and T Mlya, IEEE J. Quantum Electron. 16 (1980) 215 [8] J W. Fleming, Electron. Len 14 (1978) 326 [ 9 ] A Dlenes, Y Peng and A Knoesen, Appl Optics 28 (1989) 12.
219
OPTICS COMMUNICATIONS
1 May 1990
CHROMATIC DISPERSION IN ARROW WAVEGUIDES
M.S. WARTAK ~, J. CHROSTOWSKI, Wenyan JIANG 2 Dlvtston of Electrwal Engtneenng, Natzonal Research Councd, Ottawa, Ontario KIA OR6, Canada
and B.A. SYRETT Department of Electromcs, Carleton Umverstty, Ottawa, Ontario K I S 5B6, Canada Received 5 December 1989
Compound chromatic dispersion has been calculated for ARROW wavegmdes. It shows anomalous behavlour when compared w~th regular asymmetric wavegmdes
There has been a growing interest in antiresonant reflecting optical waveguldes (ARROW), both in passive [1-4] and active [5] devices where optical confinement is achieved on the basis of antiresonant Fabry-Perot reflection. These grades have unique properties when compared to conventional structures based on total internal reflection. They offer low loss, polarization and wavelength selectivity as well as ease of fabrication. The refractive index and size of the core layer allow flexibility in coupling to an optical fibre with minimal loss. The fact that ARROW guides can be implemented on semiconductor substrate, makes them ideal for integration with electronic devices on a single chip. As future optical elements will probably be used in superfast applications, the most severe limitations will arise from the group velocity dispersion ( G V D ) also known as chromatic dispersion, of the picosecond and subplcosecond pulses propagating along those elements. In fact, on the time scale of 10-100 fs a significant dispersion can occur over distances as short as one centimeter. It is a purpose of this note
to analyse the chromatic dispersion of the ARROW guides. ARROW guides have been recently analysed [6] using the equivalent transmission-line and transverse resonance method. For the optimum values of parameters the dispersion relation for the TE mode in the ARROW guide is ~
1+
22 N x / ~2 2
tan ( k d x / ~ - N~fr) = 0
(1) where nc is the core refractive index, d is the core thickness (fig. 1 ) and the complex propagation constant fl xs related to the complex effective index N~ff by f l = k N ~ f r = k n e f f + lOt/2 .
(2)
St 02 SI3N4 SlO 2
Department of Electrical Engineenng, Unxverslty of Ottawa, Ottawa, Ontario Kl N 6N4, Canada. z Present address Canadian Marconi Company, Montreal, Canada
Sl s u b s t r a t e Fig 1. ARROW wavegmde structure
217
Volume 76, n u m b e r 3,4
OPTICS COMMUNICATIONS
Here k=2rc/2 is the w a v e n u m b e r in v a c u u m with wavelength 2, whereas c~ (cm - ~ ) represents the propagation loss. F o r the structure under consideration, losses are very small and we can assume them to be zero C h r o m a t i c disperston o f the guide is defined as D = dud2,
(3)
where the transit time or the group delay per unit length associated with a m o d e propagating through the m e d i u m is given by r= (l/c) dfl/dk.
(4)
Here c is the speed o f hght in a vacuum. Using the dispersion relation ( 1 ), one finds
d/3 dk
1A(knc) (d/dk) (knc)+kB fl A+B
(5
4= 1 + d ~ / f l Z - k 2 sec 2 ( d , , / ~ n ~ _ f 1 2 ) ,
(6
B = (k2n 2 --
(7
where
J~2)/(j~2-k2)
For typical values o f p a r a m e t e r s [ 6 ], the ratio A / B ~s of the o r d e r I0 2 - 10 3. This allows us to approximate eq. (5) as
d f l / d k ~ (k/fl) n c M ( 2 ) ,
(8)
where M ( 2 ) = nc - 2 dnc/d2 .
l Ma~ 1990
as a funcnon of wavelength The results are shown m fig 2. We have plotted chromatic dispersion for a regular a s y m m e t r i c slab guide with thickness t, single m o d e fiber [7] and, for comparison, the material's curve Refractive index o f the core, n,,, has been m o d e l l e d using the Sellmeier equation wtth p a r a m eters given by Fleming [8] It is clear that the ARR O W wavegmde shows a very interesting b e h a v l o u r when c o m p a r e d with a regular asymmetric guide [9 ] It has large positive values o f D for all considered wavelengths. The total waveguide and material d~sperslon for A R R O W and a s y m m e t r i c slab guides are shown m fig 3. F o r an a s y m m e t r i c guide the crossover wavelength is shifted with respect to the material's dispersion by a small a m o u n t toward larger values, whilst the cross-over wavelength for the ARR O W guide shows a large shift towards the smaller wavelengths. Due to large difference m chromatic dispersion between A R R O W and conventional a s y m m e t r i c wavegmde one can consider the posslblht~ o f buildmg an lnterferometric sensor using a standard rib wavegulde for one arm and an A R R O W waveguide for the other. The sensor should exhibit zero geometrical offset but the optical offset should change by several fringes as the source laser wavelength is chirped. Such a change would m i m i c the effect trad m o n a l l y achieved through the use o f an active phase
(9) 60
For silica the funcUon M ( 2 ) takes the values between 1.46 and 1.47 for the wavelengths within 0 9 and 2.0 #m. Using the above approximanons, chromatic dispersion can be written as
D=Dm + D w ,
10)
11 )
C nef
n,fr
.
12)
neff,,/
Solving the dispersion equation ( 1 ) a n d using the above formulas, one can plot c h r o m a t i c dispersion 218
o
I
2
J 13
-
I 20 7
- ~ 09
11 lambda
1 ( n ~ - 2 dnc/d~) 2 1 - ~ 2c
i
,
E3
n~ d2n~ d~ 2 ,
and guide dispersion
D~--
I
O..
where the material dispersion is Dm= -
C
1
--~ 15
' 17
9
[ tim ]
Fag 2 C o m p a r i s o n of c h r o m a t i c dispersion versus wavelength for different systems Curve I A R R O W (core thickness d e = 4 gin, core refractive Index no= 1 45) Curve 2 m a t e r i a l ' s dispersion. C u r v e 3 a s y m m e t r i c slab grade (film thickness t = 4 jam film refractive Index nf= 1 45, substrate index n s = I 44) Curve 4 single-mode fiber, from ref [ 7 ]
Volume 76, number 3,4
OPTICS COMMUNICATIONS
80-]
E
40.
t-
0
.£
/J
//
t~ -80
N e t w o r k s A r c h i t e c t u r e s P r o j e c t a n d by the o p e r a t i n g g r a n t o f N a t u r a l Sciences & E n g i n e e n n g R e s e a r c h C o u n c i l o f C a n a d a . U s e f u l discussions w~th M. O ' S u l l i v a n a n d J. B e r n a r d are a c k n o w l e d g e d .
j
f
References
/
~Q.-40
8
I
1 May 1990
12
14
16
18
lambda [pm] F~g. 3. Total chromatic d~spers~on versus wavelength for &fferent systems. The values of parameters as m fig 2 Curve 1. ARROW grade dispersion Curve 2' material's &sperslon Curve 3: asymmetric guide dispersion.
shifter m o n e a r m o f the i n t e r f e r o m e t e r . Th~s w o r k was partially s u p p o r t e d by the T e l e c o m munication Research Institute of Ontario-Photonic
[ 1] M A Duguay, Y Kokubun and T L. Koch, Appl. Phys Lett 49 (1986) 13. [2] T.L. Koch, E.G Burkhardt, F.G. Storz, T J Bridges and T Slzer, IEEE J Quantum Electron 23 (1987) 889 [ 3 ] Y. Baba, Y Kokuban, T Sasalo and K. Iga, J. Ltghtwave Tech 16 (1988) 1440 [ 4 ] T Baba and Y Kokubun, IEEE Photomcs Technology Letters l (1989) 232 [ 5 ] D. Botez, L J. Mawst, G Peterson and T J. Roth, Appl Phys Lett 54 (1989) 2183 [6 ] W Jlang, J Chrostowskt and M Fontame, Optics Comm. 72 (1989) 180. [7] A. Suglmura, K. Dalkoku, N. Imoto and T Mlya, IEEE J. Quantum Electron. 16 (1980) 215 [8] J W. Fleming, Electron. Len 14 (1978) 326 [ 9 ] A Dlenes, Y Peng and A Knoesen, Appl Optics 28 (1989) 12.
219