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The grating coupler: Comparison of theoretical and experimental results
Volume 13, number 1
January 1975
OPTICS COMMUNICATIONS
THE GRATING COUPLER: COMPARISON OF THEORETICAL AND EXPERIMENTAL RESULTS * A. JACQUES and D.B. OSTROWSKY Thomson-CSF, B.P. no. 10, 91401 Orsay, France Received 18 October 1974
Previous theoretical treatments of the grating coupler have been extended to include a realistic case (lossy, two-layer waveguide) which we have studied experimentally. Both microscopic structural parameters (thicknesses, complex indices, modulation depth, etc.) and macroscopic coupling parameters (coupling efficiencies, lengths, etc.) have been measured. Better agreement between theoretical predictions and experimental results is obtained for the coupling efficiencies than for the coupling lengths and we explain why this is to be expected.
1. Introduction
2. Theoretical considerations
Since the appearance of the original paper [l] demonstrating the use of grating couplers for coupling a laser beam into and out of thin-film waveguides, a considerable number of papers devoted to this problem [2- 171 has been published. Most of these papers have been primarily theoretical in nature. Confrontation of theoretical with experimental results has been rather sketchy and limited usually to comparing the calculated and observed values of maximum coupling efficiency. In this paper we describe our work aimed at providing a more comprehensive theoretical/ experimental confrontation. In the theoretical part of this paper (sect. 2) we ex-. tend the previous theories to the case which we have examined experimentally: a four level waveguide with an important absorptive loss in one of the layers. In the experimental part (sect. 3) we describe the method used to determine simultaneously the optimal coupling length and the maximum coupling efficiency. Good agreement is found for the coupling efficiency. Less satisfactory agreement is obtained for the optimum coupling length and we explain why this is to be expected, and why the coupling length, rather than the maximum efficiency, is actually a more sensitive experimental test of the value of a given theory.
The grating coupler problem can be divided into two complementary problems. One which we shall call the macroscopic problem concerns the exchange of energy between the various interacting optical modes. This problem can be solved, using the well-known coupled mode formalism, without any knowledge of the actual values of the various coupling coefficients. The second problem which we shall call the microscopic problem consists of determining the values of the coupling coefficients for the problem in question. In the following two subsections we outline our approach to these two problems.
* This work was partially supported by the D.R.M.E. 74
2. I. Macroscopic considerations The structure
we are considering
is shown
in fig. 1.
If a plane wave of amplitude A,-,is incident at an angle 0 permitting resonance coupling to the mode m of the guide structure, the amplitude Cm of this mode will evolve according to
d Cm(~1 ___ = K&o - (ffm+ Tm)Cm(X) dx 7
where Km = coupling coefficient for mode m, 0~~ = recoupling coefficient from mode m, and 7m = absorption coefficient for mode m. The’ solution of (1) assuming C,(O) = 0 is:
January 1975
OPTICS COMMUNICATIONS
Volume 13, number 1 Incident plane wave
Z
k
0
no i 1.515
Y
X
Fig. 2. Typical wavefunctions (2,0), b) a superficial mode
n3<
n,<
We want to predict the optimal efficiencies and coupling lengths of various gratings. To do so, we will have to relate the coefficients K,, am and 7, to the particular guide, grating, and optical electric Geld configurations associated with a given coupler. This leads us to the next subsection.
nlc n2
Fig. 1. Four-level structure.
C,(x) = 4)
(y
m
:“,m U -
exp [-(~m+~,)xlI.(2)
2.2. Microscopic considerations
This solution enables us to determine the behavior of the coupling efficiency. The coupling efficiency at the point L is defined as the ratio of the guided power at that point (I C,(L) 12) to the total power incident between 0 and L. The maximum efficiency we can attain is given by: 1K,12
(3)
qmax = 0e81 cLnr + 7m *y
where w = 2 77c/Ao, c = velocity of light, ho = vacuum wavelength k, = 2n/Xo, p. = vacuum permeability, 0 = angle of incidence. If we consider the case of a reverse coupler (i.e. one in which the incident beam enters via the substrate and couples to the guided wave via the minus first order of diffraction) we find a maximum coupling efficiency r)(Lo) = 0.81 (1 +7&,)-l
.
(4)
The maximum efficiency always occurs for an optimum coupling length, Lo, given by LOS 1.2S(t~,+7,)-~
.
qPm(z) for: a) a grand mode
(01, (k,W, = 20; k,W, = 4.5).
(5)
If the absorption coefficient, 7,) equals zero, eqs. (3), (4) and (5) reduce to the results obtained by other authors who -treated the lossless case.
To derive the actual values of the various coupling coefficients, we have used a first-order perturbation theory similar to that described by several authors [6,14]. In such a theory the perturbation is expressed in terms of overlap integrals involving - the Fourier expansion describing the grating, - the incident electric field mode functions, - the guided electric field mode functions. Since the main difference between our work and that of the previous authors lies in the form of the electric field mode functions involved in the fourlevel structure which we consider, we shall only discuss the form of the interacting modes (the reader is referred to the literature [6] for details of the remainder of the calculation). We must consider two ensembles of interacting modes: 1) Modes of the continuum (incident field). These modes, whose electric field (assumed to be polarized in the y direction) is of the form (PC&z) exp (iox), have a continuous spectrum. That is to say, the constant of propagation /3 can assume all values in the interval [0, n&J. This ensemble includes: a) air modes: 0 < f3 < n3k0, and b) substrate modes: n3ko
< p < noko.
75
OPTICS COMMUNICATIONS
Volume 13, number 1
_____“,,_-_RFgionll
“,___--Rrglo”l 1515
--___-“2
1.586
!6!1
Fig. 3. Normalized thickness of layer 2 (k,Wa) versus the effective index of refraction &&, of the first few discrete modes of propagation (n, = 1; k, W, = 20).
January 1975
tion for our structure. We have plotted the normalized thickness of layer 2 (koW2) versus the effective index of refraction, /3,/ko, of the first few discrete modes of propagation. The two indices give the number of zero crossing of the electric field amplitude in the layers 1 and 2 respectively. The numerical values which are applicable to the experiments we shall discuss were: no - 1.515, nl = 1.566, n2 = 1.618, n3 = 1; koW1 = 20. We note that the curves are continuous at the transition between region II (superficial modes) and region I (grand modes). Using fig. 3 we can determine the fl of the guided mode and hence the form of its electric field distribution. Since we know the form of the incident field and the grating, we can compute the various coupling coefficients. The absorption coefficient 7,,, is given by:
(6)
2) The discrete spectra modes (guided field). The
electric fields are of the form cp,(z) exp (ipmx). The &,r are the eigenvahres of an eigenmode equation and have discrete values lying in the interval [no/co, n2ko]. For our case nl n2, which we shall not consider here, one simply permutes the indices 1,2 and the modes exchange roles). In fig. 2 we show typical wave functions cpfor each of the discrete type of modes. In fig. 3 we give the results of a computer solution of the eigenmode equa-
where the complex index of layer j is written as $ = nj + i ni and Ph signifies the fraction of the energy of mode m propagating in layer j. Using the techniques just discussed we have calculated the various coefficients for the guide/grating structure of fig. 1 and, via eqs. (3) and (S), the optimal efficiency and coupling length for the various modes of the guide. In table 1, we have compared our results to those obtained by Petit and coworkers who used numerical methods to obtain an exact solution [7-g]. We note that there is a good agreement be-
Table 1 ml,ml
m
PQ(%) n;
rml+,,
Pmlk, a
0,o
0
22.6
0 10-a
l,o
1
26.5
0 1O-3
2,0
2
26.8
0 10-s
Theoretical/experimental
76
0 1.1
1.564
confrontation.
1.563 *0.002 1.556
1.553
0 0.5
C
1.567
0 0.6
&Jhl) b
‘So f 0.004 1.535
1.533
1.527 to.006
17
a
b
645
602
420
284
537
533
393
388
460
453
379
309
c
1270 *420
512 285
207 *120
a) Exact theory (Petit and coworkers). b, Perturbation
(%I
a
b
28.7
22.4
14.9
10.6
29.6
26.5
15.9
16.8
29.5
26.8
17.6
18.3
C
17 f 1.5
15.2 *25
*l;
theory. ‘) Experimental results.
Volume 13, number 1
Incident
OPTICS COMMUNICATIONS
January 1975
depends on the microscopic grating parameters (modulation, grating form) in the same way. Errors in these parameters will therefore tend to annul one another. On the other hand, the coupling length [(es. (S)] does not depend on a ratio of terms and is, therefore, far more sensitive to errors in the microscopic parameters. For example, for our case, a variation of the grating modulation depth of 100 A leads to a 10% change in the coupling length and to only 0.5% change in coupling efficiency. However, even if we take into account the uncertainty in grating modulation depth (? 200 A) and in rri (+ 5 X lo+) we are unable to explain the theoretical/experimental discrepancies observed in the optimum coupling lengths. We must conclude, therefore, that while we can predict coupling efficiencies we are not yet able to predict coupling lengths.
beam
Acknowledgements Fig. 4. Experimental set-up.
tween the two groups of theoretical values. In the following section we discuss the experiment used to measure the actual values of coupling efficiencies and lengths.
We wish to thank Mme C. Moronvalle for her assistance on the experimental aspects of this work. References [II M.L. Dakss, L. Kuhn, P.F:Heidrich and B.A. Scott, Appl. Phys. Lett. 16 (1970) 523.
3. Experimental results and conclusion This experiment is shown schematically in fig. 4. A uniform plane wave (6328 A) is incident at the appropriate coupling angle, on the guide/grating structure. The interaction length is defined by a variable slit placed before the structure. Light coupled into the guide leaves via the prism P and is measured by the detector D. The coupling efficiency is defined as the ratio of the power detected at D to the power incident over the coupling length. By varying the slit width, the coupling efficiency as a function of coupling length is determined for each of the modes of the guide. The maximum efficiencies and optimum coupling lengths measured for the three modes of the guide described by fig. 1 are given in table 1. We note that, while a good agreement between theory and experiment is obtained for the maximum efficiencies, the results are quite bad for the optimum coupling length. This can be understood if we consider the form of eqs. (3) and (5). Eq. (3) for the maximum efficiency, is a ratio of the coefficients /Km I2 and o,,, , each of which
VI M. Kogelnik and T. Sosnowski, Bell. Syst. Tech. J. 49 (1970) 1602. [31 T. Tamir and H.L. Bert& J. Opt. Sot. Am. 61 (1972) 1397. 141 J.H. Harris, R.K. Wirm and D.G. Dalgoutte, Appl. Opt. 11(1972) 2234. 151 L.L. Hope, Opt. Commun. 5 (1972) 179. [61 K. Ogawa, W.S.C. Chang, B.L. Sopri and F.J. Rosenbaum, IEEE J. Quantum Electron, QE 9 (1973) 29. [71 M. Nevierre, R. Petit and Mi.J. Cadilhac, Opt. Commun. 9 (1973) 113. PI M. Nevierre, P. Vincent, R. Petit and M.J. Cadilhac, Opt. Commun. 9 (1973) 48. PI M. Nevierre, P. Vincent, R. Petit and M.J. Cadilhac, Opt. Commun. 9 (1973) 240. [lOI D.B. Ostrowsky, A. Jacques, Appl. Phys. Lett. 18 (1971) 556. ill1 D.G.Dalgoutte, Opt. Commun. 8 (1973) 124. 1121 P.K. Tien, Appl. Opt. 10 (1971) 2395. D31 P.K. Tien, R. Ulrich and R.J. Martin, Appl. Phys. Lett. 14 (1969) 291. iI41 D. Marcuse, Bell. Syst. Tech. J. 48 (1969) 3187. [W A. Yariv, J. Quantum Electronics, QE 9 (1973) 919. 1161 R. Ulrlch, J. Opt. Sot. Amer. 61 (1971) 1467. 1171 S.T. Peng, H.L. Bertoni and’T. Tamir, Opt. Commun. 10 (1974) 91.
77
January 1975
OPTICS COMMUNICATIONS
THE GRATING COUPLER: COMPARISON OF THEORETICAL AND EXPERIMENTAL RESULTS * A. JACQUES and D.B. OSTROWSKY Thomson-CSF, B.P. no. 10, 91401 Orsay, France Received 18 October 1974
Previous theoretical treatments of the grating coupler have been extended to include a realistic case (lossy, two-layer waveguide) which we have studied experimentally. Both microscopic structural parameters (thicknesses, complex indices, modulation depth, etc.) and macroscopic coupling parameters (coupling efficiencies, lengths, etc.) have been measured. Better agreement between theoretical predictions and experimental results is obtained for the coupling efficiencies than for the coupling lengths and we explain why this is to be expected.
1. Introduction
2. Theoretical considerations
Since the appearance of the original paper [l] demonstrating the use of grating couplers for coupling a laser beam into and out of thin-film waveguides, a considerable number of papers devoted to this problem [2- 171 has been published. Most of these papers have been primarily theoretical in nature. Confrontation of theoretical with experimental results has been rather sketchy and limited usually to comparing the calculated and observed values of maximum coupling efficiency. In this paper we describe our work aimed at providing a more comprehensive theoretical/ experimental confrontation. In the theoretical part of this paper (sect. 2) we ex-. tend the previous theories to the case which we have examined experimentally: a four level waveguide with an important absorptive loss in one of the layers. In the experimental part (sect. 3) we describe the method used to determine simultaneously the optimal coupling length and the maximum coupling efficiency. Good agreement is found for the coupling efficiency. Less satisfactory agreement is obtained for the optimum coupling length and we explain why this is to be expected, and why the coupling length, rather than the maximum efficiency, is actually a more sensitive experimental test of the value of a given theory.
The grating coupler problem can be divided into two complementary problems. One which we shall call the macroscopic problem concerns the exchange of energy between the various interacting optical modes. This problem can be solved, using the well-known coupled mode formalism, without any knowledge of the actual values of the various coupling coefficients. The second problem which we shall call the microscopic problem consists of determining the values of the coupling coefficients for the problem in question. In the following two subsections we outline our approach to these two problems.
* This work was partially supported by the D.R.M.E. 74
2. I. Macroscopic considerations The structure
we are considering
is shown
in fig. 1.
If a plane wave of amplitude A,-,is incident at an angle 0 permitting resonance coupling to the mode m of the guide structure, the amplitude Cm of this mode will evolve according to
d Cm(~1 ___ = K&o - (ffm+ Tm)Cm(X) dx 7
where Km = coupling coefficient for mode m, 0~~ = recoupling coefficient from mode m, and 7m = absorption coefficient for mode m. The’ solution of (1) assuming C,(O) = 0 is:
January 1975
OPTICS COMMUNICATIONS
Volume 13, number 1 Incident plane wave
Z
k
0
no i 1.515
Y
X
Fig. 2. Typical wavefunctions (2,0), b) a superficial mode
n3<
n,<
We want to predict the optimal efficiencies and coupling lengths of various gratings. To do so, we will have to relate the coefficients K,, am and 7, to the particular guide, grating, and optical electric Geld configurations associated with a given coupler. This leads us to the next subsection.
nlc n2
Fig. 1. Four-level structure.
C,(x) = 4)
(y
m
:“,m U -
exp [-(~m+~,)xlI.(2)
2.2. Microscopic considerations
This solution enables us to determine the behavior of the coupling efficiency. The coupling efficiency at the point L is defined as the ratio of the guided power at that point (I C,(L) 12) to the total power incident between 0 and L. The maximum efficiency we can attain is given by: 1K,12
(3)
qmax = 0e81 cLnr + 7m *y
where w = 2 77c/Ao, c = velocity of light, ho = vacuum wavelength k, = 2n/Xo, p. = vacuum permeability, 0 = angle of incidence. If we consider the case of a reverse coupler (i.e. one in which the incident beam enters via the substrate and couples to the guided wave via the minus first order of diffraction) we find a maximum coupling efficiency r)(Lo) = 0.81 (1 +7&,)-l
.
(4)
The maximum efficiency always occurs for an optimum coupling length, Lo, given by LOS 1.2S(t~,+7,)-~
.
qPm(z) for: a) a grand mode
(01, (k,W, = 20; k,W, = 4.5).
(5)
If the absorption coefficient, 7,) equals zero, eqs. (3), (4) and (5) reduce to the results obtained by other authors who -treated the lossless case.
To derive the actual values of the various coupling coefficients, we have used a first-order perturbation theory similar to that described by several authors [6,14]. In such a theory the perturbation is expressed in terms of overlap integrals involving - the Fourier expansion describing the grating, - the incident electric field mode functions, - the guided electric field mode functions. Since the main difference between our work and that of the previous authors lies in the form of the electric field mode functions involved in the fourlevel structure which we consider, we shall only discuss the form of the interacting modes (the reader is referred to the literature [6] for details of the remainder of the calculation). We must consider two ensembles of interacting modes: 1) Modes of the continuum (incident field). These modes, whose electric field (assumed to be polarized in the y direction) is of the form (PC&z) exp (iox), have a continuous spectrum. That is to say, the constant of propagation /3 can assume all values in the interval [0, n&J. This ensemble includes: a) air modes: 0 < f3 < n3k0, and b) substrate modes: n3ko
< p < noko.
75
OPTICS COMMUNICATIONS
Volume 13, number 1
_____“,,_-_RFgionll
“,___--Rrglo”l 1515
--___-“2
1.586
!6!1
Fig. 3. Normalized thickness of layer 2 (k,Wa) versus the effective index of refraction &&, of the first few discrete modes of propagation (n, = 1; k, W, = 20).
January 1975
tion for our structure. We have plotted the normalized thickness of layer 2 (koW2) versus the effective index of refraction, /3,/ko, of the first few discrete modes of propagation. The two indices give the number of zero crossing of the electric field amplitude in the layers 1 and 2 respectively. The numerical values which are applicable to the experiments we shall discuss were: no - 1.515, nl = 1.566, n2 = 1.618, n3 = 1; koW1 = 20. We note that the curves are continuous at the transition between region II (superficial modes) and region I (grand modes). Using fig. 3 we can determine the fl of the guided mode and hence the form of its electric field distribution. Since we know the form of the incident field and the grating, we can compute the various coupling coefficients. The absorption coefficient 7,,, is given by:
(6)
2) The discrete spectra modes (guided field). The
electric fields are of the form cp,(z) exp (ipmx). The &,r are the eigenvahres of an eigenmode equation and have discrete values lying in the interval [no/co, n2ko]. For our case nl
where the complex index of layer j is written as $ = nj + i ni and Ph signifies the fraction of the energy of mode m propagating in layer j. Using the techniques just discussed we have calculated the various coefficients for the guide/grating structure of fig. 1 and, via eqs. (3) and (S), the optimal efficiency and coupling length for the various modes of the guide. In table 1, we have compared our results to those obtained by Petit and coworkers who used numerical methods to obtain an exact solution [7-g]. We note that there is a good agreement be-
Table 1 ml,ml
m
PQ(%) n;
rml+,,
Pmlk, a
0,o
0
22.6
0 10-a
l,o
1
26.5
0 1O-3
2,0
2
26.8
0 10-s
Theoretical/experimental
76
0 1.1
1.564
confrontation.
1.563 *0.002 1.556
1.553
0 0.5
C
1.567
0 0.6
&Jhl) b
‘So f 0.004 1.535
1.533
1.527 to.006
17
a
b
645
602
420
284
537
533
393
388
460
453
379
309
c
1270 *420
512 285
207 *120
a) Exact theory (Petit and coworkers). b, Perturbation
(%I
a
b
28.7
22.4
14.9
10.6
29.6
26.5
15.9
16.8
29.5
26.8
17.6
18.3
C
17 f 1.5
15.2 *25
*l;
theory. ‘) Experimental results.
Volume 13, number 1
Incident
OPTICS COMMUNICATIONS
January 1975
depends on the microscopic grating parameters (modulation, grating form) in the same way. Errors in these parameters will therefore tend to annul one another. On the other hand, the coupling length [(es. (S)] does not depend on a ratio of terms and is, therefore, far more sensitive to errors in the microscopic parameters. For example, for our case, a variation of the grating modulation depth of 100 A leads to a 10% change in the coupling length and to only 0.5% change in coupling efficiency. However, even if we take into account the uncertainty in grating modulation depth (? 200 A) and in rri (+ 5 X lo+) we are unable to explain the theoretical/experimental discrepancies observed in the optimum coupling lengths. We must conclude, therefore, that while we can predict coupling efficiencies we are not yet able to predict coupling lengths.
beam
Acknowledgements Fig. 4. Experimental set-up.
tween the two groups of theoretical values. In the following section we discuss the experiment used to measure the actual values of coupling efficiencies and lengths.
We wish to thank Mme C. Moronvalle for her assistance on the experimental aspects of this work. References [II M.L. Dakss, L. Kuhn, P.F:Heidrich and B.A. Scott, Appl. Phys. Lett. 16 (1970) 523.
3. Experimental results and conclusion This experiment is shown schematically in fig. 4. A uniform plane wave (6328 A) is incident at the appropriate coupling angle, on the guide/grating structure. The interaction length is defined by a variable slit placed before the structure. Light coupled into the guide leaves via the prism P and is measured by the detector D. The coupling efficiency is defined as the ratio of the power detected at D to the power incident over the coupling length. By varying the slit width, the coupling efficiency as a function of coupling length is determined for each of the modes of the guide. The maximum efficiencies and optimum coupling lengths measured for the three modes of the guide described by fig. 1 are given in table 1. We note that, while a good agreement between theory and experiment is obtained for the maximum efficiencies, the results are quite bad for the optimum coupling length. This can be understood if we consider the form of eqs. (3) and (5). Eq. (3) for the maximum efficiency, is a ratio of the coefficients /Km I2 and o,,, , each of which
VI M. Kogelnik and T. Sosnowski, Bell. Syst. Tech. J. 49 (1970) 1602. [31 T. Tamir and H.L. Bert& J. Opt. Sot. Am. 61 (1972) 1397. 141 J.H. Harris, R.K. Wirm and D.G. Dalgoutte, Appl. Opt. 11(1972) 2234. 151 L.L. Hope, Opt. Commun. 5 (1972) 179. [61 K. Ogawa, W.S.C. Chang, B.L. Sopri and F.J. Rosenbaum, IEEE J. Quantum Electron, QE 9 (1973) 29. [71 M. Nevierre, R. Petit and Mi.J. Cadilhac, Opt. Commun. 9 (1973) 113. PI M. Nevierre, P. Vincent, R. Petit and M.J. Cadilhac, Opt. Commun. 9 (1973) 48. PI M. Nevierre, P. Vincent, R. Petit and M.J. Cadilhac, Opt. Commun. 9 (1973) 240. [lOI D.B. Ostrowsky, A. Jacques, Appl. Phys. Lett. 18 (1971) 556. ill1 D.G.Dalgoutte, Opt. Commun. 8 (1973) 124. 1121 P.K. Tien, Appl. Opt. 10 (1971) 2395. D31 P.K. Tien, R. Ulrich and R.J. Martin, Appl. Phys. Lett. 14 (1969) 291. iI41 D. Marcuse, Bell. Syst. Tech. J. 48 (1969) 3187. [W A. Yariv, J. Quantum Electronics, QE 9 (1973) 919. 1161 R. Ulrlch, J. Opt. Sot. Amer. 61 (1971) 1467. 1171 S.T. Peng, H.L. Bertoni and’T. Tamir, Opt. Commun. 10 (1974) 91.
77