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Multimode operation of ARROW waveguides
Optics Communications 102 ( 1993 ) 2 1 7 - 2 2 0 North-Holland
OPTICS COMMUNICATIONS
Multimode operation of ARROW waveguides Jacek M. K u b i c a
a, Jerzy Gazecki b a n d Geoffrey K. Reeves c
a Institute of Physics, Warsaw University o f Technology, Koszykowa 75, 00-622 Warsaw, Poland b Department of Communication andElectronicEngineering, RoyalMelbournelnstitute of Technology, GPO Box 2476 V, Melbourne Victoria 3001, Australia c Microelectronics and Materials Technology Centre, Royal Melbourne Institute of Technology, GPO Box 2476 V, Melbourne Victoria 3001, Australia
Received I December 1992;revised manuscript received 19 May 1993
It is shown that antiresonant reflectingoptical waveguidesexhibit multimode operation when a refractive index difference of the order of 10-2 is introduced between the core and the second claddinglayer.
Antiresonant reflecting optical waveguides (ARROW's) offer new design freedom in the field of semiconductor integrated optics. Optical confinement in these waveguides is provided by the high reflectivity of Fabry-P6rot resonators formed in a system of cladding layers [ 1 ]. It has been shown that the ARROW geometry can be useful for many practical devices, such as semiconductor lasers [2], wavelength selective photodetectors [3,4], or remote all-optical switches [5 ]. One of the main advantages of the ARROW guide is its single-mode character even for a large thickness of the core, which is essential for many applications. Single-mode operation of the ARROW waveguide refers to a situation where with a series of leaky waves, only one is well confined to the core and propagates with low loss. Until recently only an ARROW with a very thick core was shown to support more than one low-loss leaky wave [6]. In this paper we show that singlemode operation of the ARROW is strongly affected by the refractive index difference between the core and one of the cladding layers. Figure 1 shows a typical ARROW structure grown on a high-index substrate. The core layer of thickness d~ and low refractive index nl is separated from the substrate by the interference cladding system formed by the higher-index cladding layer (d2, n2) and the second cladding layer (d3, n3), usually of the same refractive index as that of the core. High refractive
index of the substrate results in leaky character of all guided waves. To investigate the modal properties of this system, we have used the standard transfer matrix approach that applies to leaky waves supported by an arbitrary multilayer planar structure [7]. It provides a simple algebraic equation which expresses the model condition and follows from the requirement of self-consistent fields at the interfaces between the guide and its surrounding media 7emil +TcTsml2 4-m21 q- ysm22 = 0 ,
( 1)
where rnpq are elements of the transfer defined as M=HMj J
11 j
cosOj - i y j sin Oj
( - i / y j ) sinq~j~ cosOj ]'
(2)
2 2 ~)j(c,s) = (konjfc,s) _flz)l/2/koz ° (for TE modes), Zo is the impedance of free space, nc and ns are the refractive indices of the cover and substrate, respectively, O j = ~ ( k ~ n 2 _fl2)~/2, ko is the vacuum wavenumber, and fl is the propagation constant, which for leaky waves is written as
fl=koncfr+ ia/2 .
(3)
The value of 4.34ot represents the propagation loss (decibels per unit length). The cladding layers beneath the core are designed to form antiresonant Fabry-P6rot cavities, which
0030-4018/93/$06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
217
Volume 102, number 3,4
OPTICS COMMUNICATIONS
1 October 1993
Re core>
dl
nl
d2 d3
n2 n3
substrate
ns
Fig. 1. Geometry of the ARROW waveguide. ensures low-loss single-mode operation of the waveguide. In general, the antiresonances of a F a b r y - P r r o t cavity are relatively broad with respect to its phase thickness n d . Consequently, the fabrication tolerance of the thicknesses of the cladding layers dj is quite large, as can be seen from the respective characteristics [6]. Following the F a b r y - P r r o t analogy one might conclude that the tolerance of the refractive indices in the A R R O W claddings is also large. On the contrary, according to our analysis the modal properties of the A R R O W guide are very sensitive to the difference between the refractive indices of the core and the second cladding layer. This is illustrated in fig. 2 which shows the a t t e n u a t i o n of the TE polarized waves as a function of the refractive index 1000
. . . .
,
.
.
.
.
•
,
,
•
difference r/l-r/3, while keeping the core index /71 constant. The numerical parameters correspond to the recently reported GaAs/A1GaAs A R R O W structure operating at the wavelength 2 = 0 . 8 3 ~tm [8]: dl = 4 . 0 p.m, d2=0.14 ~tm, d3=2.0 p.m, no= 1.0, n l = 3.28, n 2 = 3 . 5 8 and ns=3.64. The results show that a slight decrease of n3 below the value of n] leads to a reduction of the attenuation of the TEl A R R O W mode, and also to the occurrence of additional lowloss leaky waves. All these waves are well confined to the core layer, which is shown in fig. 3 for the structure with n ~ - n 3 = 0.01. Therefore, we can consider them as higher-order A R R O W modes and the whole structure as a multimode A R R O W waveguide. The attenuation of the A R R O W modes decreases with decreasing n3. This is easy to explain, when one refers to the zigzag wave model and takes into ac-
100
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i . . . . 0010
i . . . . 0015
i . . . . 0 020
TE2
0 025 1 O0
RefracLive index difference nl-n3 Fig. 2. Attenuation of TE modes as a function of the refractive index difference between the core and the second cladding layer nl - n3, with a constant value of n~= 3.28. Other parameters are: d~=4.0 gm, d2=0.14 gm, ds=2.0 gm, no= 1.0, n2=3.58 and n, = 3.64. The wavelength2 = 0.83 gm. 218
-soo
- 1 25
m
ooo
3oo
~oo
9oo
Depth (#m) Fig. 3. Electric field profiles of the TEl and TE2 ARROW modes in the waveguidestructure with n~- ns = 0.01. Other parameters are as in fig. 2.
Volume 102, number 3,4
OPTICS COMMUNICATIONS
1 October 1993
1000
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0 0001 325 . . . . 0,000
i . . . . . . . . 0005 0010
i . . . . 0015
0 00001
J . . . . 0020
Fig. 4. Dispersion of TE modes as a function of the refractive index difference between the core and the second cladding layer n~- n3, with constant value ofn~ = 3.28. Other parameters are as in fig. 2.
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~
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,
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,
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250
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Fig. 6. Attenuation of TE modes as a function of the thickness of the second cladding layer d3 in the ARROW structure with no= 1.0, nl = 3.28, n2= 3.58, n3= 3.27, n,= 3.64, dl =4.0 p.m, and d2 = 0.14 p.m (solid lines). The dashed lines correspond to the case when d2=0, i.e. when the structure forms a conventional waveguide on a high-index substrate with a buffer layer. Note, that in the latter case the numbering starts with TEo, whereas in the ARROW the TE0 symbol corresponds to the bound mode guided by the high-index cladding layer, which is not presented in this figure. The wavelength 2 = 0.83 p.m.
o 0032 1 25
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;.~
' ....
050
Thickness of second cladding layer d3 (#m)
RefracLiveindexdifferencent-n3
g~
....
O.O0 0.025
1 O0
0.0025
00018
00011
, 130
, 100
,
, , , 1 70 190
,
~ 210
,
~ 230
,
, 250
,
, , i , i 2.70 2.90 310
"-~
0 50
r~
0 25 0 O0
0
, 3.30
R e f r a c L i v e i n d e x of t h e c o r e n l Fig. 5. Critical value of the refractive index difference between the core and the second cladding layer for the fundamental ARROW mode (at which FTIR occurs), as a function of the refractive index of the core nl. d~ = 4.0 p.m and ,t=0.83 gm. c o u n t that the angle o f refraction in the first c l a d d i n g layer 02 is very close to the critical angle o f the interface b e t w e e n the c l a d d i n g layers Oc,=sin-l(n3/ t/2 ), (for n l = n3:02 < 0c~). In this case, a slight change o f the 02 or 0~r leads to a d r a m a t i c change in the reflectivity o f the interface and c o n s e q u e n t l y o f the w h o l e resonator. In particular, the value 0cr m a y be reduced by decreasing the i n d e x n3. T h i s does not affect the effective i n d i c e s o f the A R R O W m o d e s , as one can see in the respective regions o f the dispersion characteristics (fig. 4 ) . It m e a n s , that the propagation angles in all layers, i n c l u d i n g the angle 02, remain unchanged. Thus, with decreasing n3, 0~r tends to a fixed v a l u e o f 02 resulting in the increase o f the
~,~ O ©
rEo
-0
25
-0
50
-0
75
-1
O0
-1
25
nc - 1 . 0
-3
O0
nl - 3 . 2 8
0 O0
na - 3 . 2 7
300
600
nt - 3 . 6 4
900
Depth (#m) Fig. 7. Electric field profiles of the TEo and TEl modes in a conventional waveguide formed by the structure of fig. 3 with d2 = 0.
reflectivity o f the interface b e t w e e n the c l a d d i n g layers a n d c o n s e q u e n t l y , to the reduction o f the attenu a t i o n o f the A R R O W m o d e s . S i m i l a r results are expected w h e n the refractive i n d e x o f the core nl is c h a n g e d w h i l e keeping n3 at a constant value. In this case the angle 02 increases w i t h increasing n 1 according to the a p p r o x i m a t e f o r m u l a
02=sin-ill(n2_ 4d2a='~"21 J j
(4)
219
Volume 102, number 3,4
OPTICS COMMUNICATIONS
and tends towards the critical angle 0¢r that is kept constant. It is i m p o r t a n t to note that the total internal reflection, which should occur at 02=0cr is d i s t u r b e d by the presence o f the high-index substrate, a n d thus frustrated total internal reflection ( F T I R ) can be expected. The reflectivity o f the interface between the cladding layers is continuous at the transition point to the region o f F T I R [9 ], hence this transition cannot be recognized in fig. 2. The value o f the refractive index difference n ~ - n 3 at which F T I R occurs for the f u n d a m e n t a l A R R O W m o d e can be evaluated using the following a p p r o x i m a t e formulas Antclr ) = n, -- ( n ~ - 2 2 / 4 d ~ )
'/2 ,
A n ~2) = ( n ] + 2 2 / 4 d ~ ) 1 / 2 - n3
(5)
(6)
where A n ~ ) corresponds to the case when n3 is varA~(2) when n~ is ied while keeping n~ constant, a n a• z~',cr varied while keeping n3 constant. Figure 5 shows a plot o f eq. ( 5 ) as a function o f the core refractive index//1 with 2 = 0 . 8 3 ~tm and dl = 4 . 0 ~tm. As can be seen the value o f A n ~ ) is o f the order o f 10 -3 over the range o f n~ considered. F o r our structure with n l = 3.28, the value o f An ~r~) is about 0.0016. Hence, we can conclude that the m u l t i m o d e operation which appears in fig. 2 for larger values o f n l - / / 3 , involves F T I R at the interface between the cladding layers. The fields o f the A R R O W m o d e s which undergo F T I R at the interface between the cladding layers are evanescent in the second cladding layer. They cannot form a standing wave in this layer and consequently the layer does not behave as a resonator but as a k i n d o f a buffer. This is clearly seen in fig. 6 which shows the a t t e n u a t i o n o f the TE m o d e s as a function o f the thickness d3 for the structure with n l - n 3 = 0 . 0 1 . The attenuation o f the TE~ a n d TE2 A R R O W m o d e s decreases continuously with increasing d3. This b e h a v i o r is typical for m o d e s supported by a conventional waveguide grown on a highindex substrate where a buffer layer is i n c o r p o r a t e d to m i n i m i z e r a d i a t i o n losses into the substrate. In fig. 6 we provide a comparative characteristic o f such a waveguide which is f o r m e d by the A R R O W structure w i t h / / ~ - / / 3 = 0 . 0 1 and d 2 = 0 . It is evident, that the m u l t i m o d e A R R O W m a y be regarded as a conventional waveguide on a high-index substrate with
220
1 October 1993
an additional antiresonant high-index cladding layer. As can be seen in fig. 6 the presence o f this layer resuits in a reduction o f the attenuation o f the lowesto r d e r leaky waves, regarded as A R R O W modes, leading to a better separation between them and the higher-order modes. As predicted by Duguay et al. [ 1 ] the fields o f the A R R O W modes are also better confined to the core layer than their counterparts in the conventional structure (fig. 7). In conclusion, we have shown that the m o d a l properties o f the antiresonant reflecting optical waveguides are very sensitive to the refractive index difference between the core a n d the second cladding layer. We have shown, that a very small refractive index difference ( o f the order o f 10 -2 ) results in m u l t i m o d e operation o f the A R R O W . This is o f particular i m p o r t a n c e in the fabrication o f A R R O W ' s i n t e n d e d for single-mode applications, as the tolerance for refractive indices is then very small. On the other hand, high sensitivity to the refractive index changes makes the A R R O W geometry applicable to m a n y devices, including optical switches and modulators. This work was supported by the Australian Bilateral Science and Technology ( D I T A C ) Program and ATERB grants, a n d in part by the Polish Council o f Scientific Research ( K B N ) .
References [ 1] M.A. Duguay, Y. Kokubun, T.L. Koch and L. Pfeiffer, Appl. Phys. Lett. 49 (1986) 13. [2] T.L. Koch, E.G. Burkhardt, F.G. Storz, T.J. Bridges and T. Sizer II, IEEE J. Quantum Electron. QE-23 (1987) 889. [3] T.L. Koch, P.J. Corvini, W.T. Tsang, U. Koren and B.I. Miller, Appl. Phys. Lett. 51 (1987) 1060. [4]T. Baba, Y. Kokubun and H. Watanabe, J. Lightwave Technol. 8 (1990) 99. [ 5 ] U. Trutschel, M. Mann, F. Lederer, C. W~ichter and A.D. Boardman, Appl. Phys. Lett. 59 ( 1991 ) 1940. [6] J.M. Kubica, D. Unamchandani and B. Culshaw, Optics Comm. 78 (1990) 133. [7] J. Chilwell and I. Hodgkinson, J. Opt. Soc. Am. A1 (1984) 742. [8] A. Malag, J.M. Kubica, J. Gazecki, M.W. Austin and G.K. Reeves, Proc. 17th Australian Conf. on Optical Fibre Technology (ACOFT-17), 1992, Hobart, Tasmania, p. 102. [ 9 ] H. Anders, Thin films in optics (The Focal Press, London, 1967) p. 44.
OPTICS COMMUNICATIONS
Multimode operation of ARROW waveguides Jacek M. K u b i c a
a, Jerzy Gazecki b a n d Geoffrey K. Reeves c
a Institute of Physics, Warsaw University o f Technology, Koszykowa 75, 00-622 Warsaw, Poland b Department of Communication andElectronicEngineering, RoyalMelbournelnstitute of Technology, GPO Box 2476 V, Melbourne Victoria 3001, Australia c Microelectronics and Materials Technology Centre, Royal Melbourne Institute of Technology, GPO Box 2476 V, Melbourne Victoria 3001, Australia
Received I December 1992;revised manuscript received 19 May 1993
It is shown that antiresonant reflectingoptical waveguidesexhibit multimode operation when a refractive index difference of the order of 10-2 is introduced between the core and the second claddinglayer.
Antiresonant reflecting optical waveguides (ARROW's) offer new design freedom in the field of semiconductor integrated optics. Optical confinement in these waveguides is provided by the high reflectivity of Fabry-P6rot resonators formed in a system of cladding layers [ 1 ]. It has been shown that the ARROW geometry can be useful for many practical devices, such as semiconductor lasers [2], wavelength selective photodetectors [3,4], or remote all-optical switches [5 ]. One of the main advantages of the ARROW guide is its single-mode character even for a large thickness of the core, which is essential for many applications. Single-mode operation of the ARROW waveguide refers to a situation where with a series of leaky waves, only one is well confined to the core and propagates with low loss. Until recently only an ARROW with a very thick core was shown to support more than one low-loss leaky wave [6]. In this paper we show that singlemode operation of the ARROW is strongly affected by the refractive index difference between the core and one of the cladding layers. Figure 1 shows a typical ARROW structure grown on a high-index substrate. The core layer of thickness d~ and low refractive index nl is separated from the substrate by the interference cladding system formed by the higher-index cladding layer (d2, n2) and the second cladding layer (d3, n3), usually of the same refractive index as that of the core. High refractive
index of the substrate results in leaky character of all guided waves. To investigate the modal properties of this system, we have used the standard transfer matrix approach that applies to leaky waves supported by an arbitrary multilayer planar structure [7]. It provides a simple algebraic equation which expresses the model condition and follows from the requirement of self-consistent fields at the interfaces between the guide and its surrounding media 7emil +TcTsml2 4-m21 q- ysm22 = 0 ,
( 1)
where rnpq are elements of the transfer defined as M=HMj J
11 j
cosOj - i y j sin Oj
( - i / y j ) sinq~j~ cosOj ]'
(2)
2 2 ~)j(c,s) = (konjfc,s) _flz)l/2/koz ° (for TE modes), Zo is the impedance of free space, nc and ns are the refractive indices of the cover and substrate, respectively, O j = ~ ( k ~ n 2 _fl2)~/2, ko is the vacuum wavenumber, and fl is the propagation constant, which for leaky waves is written as
fl=koncfr+ ia/2 .
(3)
The value of 4.34ot represents the propagation loss (decibels per unit length). The cladding layers beneath the core are designed to form antiresonant Fabry-P6rot cavities, which
0030-4018/93/$06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
217
Volume 102, number 3,4
OPTICS COMMUNICATIONS
1 October 1993
Re core>
dl
nl
d2 d3
n2 n3
substrate
ns
Fig. 1. Geometry of the ARROW waveguide. ensures low-loss single-mode operation of the waveguide. In general, the antiresonances of a F a b r y - P r r o t cavity are relatively broad with respect to its phase thickness n d . Consequently, the fabrication tolerance of the thicknesses of the cladding layers dj is quite large, as can be seen from the respective characteristics [6]. Following the F a b r y - P r r o t analogy one might conclude that the tolerance of the refractive indices in the A R R O W claddings is also large. On the contrary, according to our analysis the modal properties of the A R R O W guide are very sensitive to the difference between the refractive indices of the core and the second cladding layer. This is illustrated in fig. 2 which shows the a t t e n u a t i o n of the TE polarized waves as a function of the refractive index 1000
. . . .
,
.
.
.
.
•
,
,
•
difference r/l-r/3, while keeping the core index /71 constant. The numerical parameters correspond to the recently reported GaAs/A1GaAs A R R O W structure operating at the wavelength 2 = 0 . 8 3 ~tm [8]: dl = 4 . 0 p.m, d2=0.14 ~tm, d3=2.0 p.m, no= 1.0, n l = 3.28, n 2 = 3 . 5 8 and ns=3.64. The results show that a slight decrease of n3 below the value of n] leads to a reduction of the attenuation of the TEl A R R O W mode, and also to the occurrence of additional lowloss leaky waves. All these waves are well confined to the core layer, which is shown in fig. 3 for the structure with n ~ - n 3 = 0.01. Therefore, we can consider them as higher-order A R R O W modes and the whole structure as a multimode A R R O W waveguide. The attenuation of the A R R O W modes decreases with decreasing n3. This is easy to explain, when one refers to the zigzag wave model and takes into ac-
100
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~ (..)
TE(
1o 1 oo
~ ~
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i
--0 7 5
i
0 0001 . . . . 0000
, . . . . 0 005
i . . . . 0010
i . . . . 0015
i . . . . 0 020
TE2
0 025 1 O0
RefracLive index difference nl-n3 Fig. 2. Attenuation of TE modes as a function of the refractive index difference between the core and the second cladding layer nl - n3, with a constant value of n~= 3.28. Other parameters are: d~=4.0 gm, d2=0.14 gm, ds=2.0 gm, no= 1.0, n2=3.58 and n, = 3.64. The wavelength2 = 0.83 gm. 218
-soo
- 1 25
m
ooo
3oo
~oo
9oo
Depth (#m) Fig. 3. Electric field profiles of the TEl and TE2 ARROW modes in the waveguidestructure with n~- ns = 0.01. Other parameters are as in fig. 2.
Volume 102, number 3,4
OPTICS COMMUNICATIONS
1 October 1993
1000
"c
3 28
'
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'
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~
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001 0001
TEs
0 0001 325 . . . . 0,000
i . . . . . . . . 0005 0010
i . . . . 0015
0 00001
J . . . . 0020
Fig. 4. Dispersion of TE modes as a function of the refractive index difference between the core and the second cladding layer n~- n3, with constant value ofn~ = 3.28. Other parameters are as in fig. 2.
<:1~©~
~
00046
,
00039
,
,
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,
,
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,
,
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, ....
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, ....
, ....
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200
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250
, ....
3 O0
350
,
•
~
Fig. 6. Attenuation of TE modes as a function of the thickness of the second cladding layer d3 in the ARROW structure with no= 1.0, nl = 3.28, n2= 3.58, n3= 3.27, n,= 3.64, dl =4.0 p.m, and d2 = 0.14 p.m (solid lines). The dashed lines correspond to the case when d2=0, i.e. when the structure forms a conventional waveguide on a high-index substrate with a buffer layer. Note, that in the latter case the numbering starts with TEo, whereas in the ARROW the TE0 symbol corresponds to the bound mode guided by the high-index cladding layer, which is not presented in this figure. The wavelength 2 = 0.83 p.m.
o 0032 1 25
©
;.~
' ....
050
Thickness of second cladding layer d3 (#m)
RefracLiveindexdifferencent-n3
g~
....
O.O0 0.025
1 O0
0.0025
00018
00011
, 130
, 100
,
, , , 1 70 190
,
~ 210
,
~ 230
,
, 250
,
, , i , i 2.70 2.90 310
"-~
0 50
r~
0 25 0 O0
0
, 3.30
R e f r a c L i v e i n d e x of t h e c o r e n l Fig. 5. Critical value of the refractive index difference between the core and the second cladding layer for the fundamental ARROW mode (at which FTIR occurs), as a function of the refractive index of the core nl. d~ = 4.0 p.m and ,t=0.83 gm. c o u n t that the angle o f refraction in the first c l a d d i n g layer 02 is very close to the critical angle o f the interface b e t w e e n the c l a d d i n g layers Oc,=sin-l(n3/ t/2 ), (for n l = n3:02 < 0c~). In this case, a slight change o f the 02 or 0~r leads to a d r a m a t i c change in the reflectivity o f the interface and c o n s e q u e n t l y o f the w h o l e resonator. In particular, the value 0cr m a y be reduced by decreasing the i n d e x n3. T h i s does not affect the effective i n d i c e s o f the A R R O W m o d e s , as one can see in the respective regions o f the dispersion characteristics (fig. 4 ) . It m e a n s , that the propagation angles in all layers, i n c l u d i n g the angle 02, remain unchanged. Thus, with decreasing n3, 0~r tends to a fixed v a l u e o f 02 resulting in the increase o f the
~,~ O ©
rEo
-0
25
-0
50
-0
75
-1
O0
-1
25
nc - 1 . 0
-3
O0
nl - 3 . 2 8
0 O0
na - 3 . 2 7
300
600
nt - 3 . 6 4
900
Depth (#m) Fig. 7. Electric field profiles of the TEo and TEl modes in a conventional waveguide formed by the structure of fig. 3 with d2 = 0.
reflectivity o f the interface b e t w e e n the c l a d d i n g layers a n d c o n s e q u e n t l y , to the reduction o f the attenu a t i o n o f the A R R O W m o d e s . S i m i l a r results are expected w h e n the refractive i n d e x o f the core nl is c h a n g e d w h i l e keeping n3 at a constant value. In this case the angle 02 increases w i t h increasing n 1 according to the a p p r o x i m a t e f o r m u l a
02=sin-ill(n2_ 4d2a='~"21 J j
(4)
219
Volume 102, number 3,4
OPTICS COMMUNICATIONS
and tends towards the critical angle 0¢r that is kept constant. It is i m p o r t a n t to note that the total internal reflection, which should occur at 02=0cr is d i s t u r b e d by the presence o f the high-index substrate, a n d thus frustrated total internal reflection ( F T I R ) can be expected. The reflectivity o f the interface between the cladding layers is continuous at the transition point to the region o f F T I R [9 ], hence this transition cannot be recognized in fig. 2. The value o f the refractive index difference n ~ - n 3 at which F T I R occurs for the f u n d a m e n t a l A R R O W m o d e can be evaluated using the following a p p r o x i m a t e formulas Antclr ) = n, -- ( n ~ - 2 2 / 4 d ~ )
'/2 ,
A n ~2) = ( n ] + 2 2 / 4 d ~ ) 1 / 2 - n3
(5)
(6)
where A n ~ ) corresponds to the case when n3 is varA~(2) when n~ is ied while keeping n~ constant, a n a• z~',cr varied while keeping n3 constant. Figure 5 shows a plot o f eq. ( 5 ) as a function o f the core refractive index//1 with 2 = 0 . 8 3 ~tm and dl = 4 . 0 ~tm. As can be seen the value o f A n ~ ) is o f the order o f 10 -3 over the range o f n~ considered. F o r our structure with n l = 3.28, the value o f An ~r~) is about 0.0016. Hence, we can conclude that the m u l t i m o d e operation which appears in fig. 2 for larger values o f n l - / / 3 , involves F T I R at the interface between the cladding layers. The fields o f the A R R O W m o d e s which undergo F T I R at the interface between the cladding layers are evanescent in the second cladding layer. They cannot form a standing wave in this layer and consequently the layer does not behave as a resonator but as a k i n d o f a buffer. This is clearly seen in fig. 6 which shows the a t t e n u a t i o n o f the TE m o d e s as a function o f the thickness d3 for the structure with n l - n 3 = 0 . 0 1 . The attenuation o f the TE~ a n d TE2 A R R O W m o d e s decreases continuously with increasing d3. This b e h a v i o r is typical for m o d e s supported by a conventional waveguide grown on a highindex substrate where a buffer layer is i n c o r p o r a t e d to m i n i m i z e r a d i a t i o n losses into the substrate. In fig. 6 we provide a comparative characteristic o f such a waveguide which is f o r m e d by the A R R O W structure w i t h / / ~ - / / 3 = 0 . 0 1 and d 2 = 0 . It is evident, that the m u l t i m o d e A R R O W m a y be regarded as a conventional waveguide on a high-index substrate with
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an additional antiresonant high-index cladding layer. As can be seen in fig. 6 the presence o f this layer resuits in a reduction o f the attenuation o f the lowesto r d e r leaky waves, regarded as A R R O W modes, leading to a better separation between them and the higher-order modes. As predicted by Duguay et al. [ 1 ] the fields o f the A R R O W modes are also better confined to the core layer than their counterparts in the conventional structure (fig. 7). In conclusion, we have shown that the m o d a l properties o f the antiresonant reflecting optical waveguides are very sensitive to the refractive index difference between the core a n d the second cladding layer. We have shown, that a very small refractive index difference ( o f the order o f 10 -2 ) results in m u l t i m o d e operation o f the A R R O W . This is o f particular i m p o r t a n c e in the fabrication o f A R R O W ' s i n t e n d e d for single-mode applications, as the tolerance for refractive indices is then very small. On the other hand, high sensitivity to the refractive index changes makes the A R R O W geometry applicable to m a n y devices, including optical switches and modulators. This work was supported by the Australian Bilateral Science and Technology ( D I T A C ) Program and ATERB grants, a n d in part by the Polish Council o f Scientific Research ( K B N ) .
References [ 1] M.A. Duguay, Y. Kokubun, T.L. Koch and L. Pfeiffer, Appl. Phys. Lett. 49 (1986) 13. [2] T.L. Koch, E.G. Burkhardt, F.G. Storz, T.J. Bridges and T. Sizer II, IEEE J. Quantum Electron. QE-23 (1987) 889. [3] T.L. Koch, P.J. Corvini, W.T. Tsang, U. Koren and B.I. Miller, Appl. Phys. Lett. 51 (1987) 1060. [4]T. Baba, Y. Kokubun and H. Watanabe, J. Lightwave Technol. 8 (1990) 99. [ 5 ] U. Trutschel, M. Mann, F. Lederer, C. W~ichter and A.D. Boardman, Appl. Phys. Lett. 59 ( 1991 ) 1940. [6] J.M. Kubica, D. Unamchandani and B. Culshaw, Optics Comm. 78 (1990) 133. [7] J. Chilwell and I. Hodgkinson, J. Opt. Soc. Am. A1 (1984) 742. [8] A. Malag, J.M. Kubica, J. Gazecki, M.W. Austin and G.K. Reeves, Proc. 17th Australian Conf. on Optical Fibre Technology (ACOFT-17), 1992, Hobart, Tasmania, p. 102. [ 9 ] H. Anders, Thin films in optics (The Focal Press, London, 1967) p. 44.