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Anomalous modal spectrum of radiation of DFB and DBR lasers
Volume 60, number 1,2
OPTICS COMMUNICATIONS
15 October 1986
A N O M A L O U S MODAL SPECTRUM OF RADIATION OF DFB AND DBR LASERS A.A. S P I K H A L ' S K I I General Physics Institute, Academy of Sciences of the USSR, Moscow 117942, USSR Received 27 August 1985; revised manuscript received 25 June 1986
The modal spectrum of waveguide DFB and DBR lasers is examined theoretically. It is found that the modal spectrum of the laser output can differ greatly from the spectrum inside the laser cavity. Numerical results are presented for examples of DFB and DBR lasers.
1. Introduction
Lasers with distributed feedback (DFB) or distributed Bragg reflectors ( D B R ) are interesting and perspective devices of quantum electronics. They are among the most important elements o f integrated optics [ 1-4]. These integrated optical lasers are often performed on the base of multimode waveguides; thus, different waveguide modes may appear in laser radiation. For different applications o f the lasers, especially in optical communications, it is of importance to stabilize the modal spectrum o f the radiation and to maintain the modal purity. The modal purity and stability are necessary for a laser to be employed in integrated optical multiplex systems, optical heterodyne transmission systems, and many other waveguide applications. Distortions of the modal spectrum are now considered mainly as a consequence o f imperfections, random variations o f waveguide parameters, and similar deviations o f the laser structure from the perfect one. However, our analysis has shown that even the perfect DFB and DBR structures can destroy the purity of modal spectrum and, moreover, lead to unusual spectra of waveguide modes in laser radiation. Consider a corrugated optical waveguide used as the DFB or DBR laser (fig. la or lb). Suppose that a waveguide mode (with the mode index m) is excited by a pumping o f this laser. The ordinary condition on the grating period A, the mode wavelength in vacuum 2,,, and the mode propagation constant b,,, is 0 030-401/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
PUMPING
FILM ~ . . . .
(a)
~" ~
SLI&STRRT~
PUMPINCr
(b) Fig.l. Schemes of the waveguide laser with distributed Bragg reflectors (a) and distributed feedback (b). bm(2,n) =Tr r/A,
(1)
where r is the Bragg order o f the distributed feedback. Here we are interested in the conditions, by which the correlation (1) is true for the mode m~, whereas the main radiation of the laser is the mode rn2; this process is called here the process with anomal modal spectrum.
2. Grounds of the anomal modal spectrum The physical grounds of the anomal modal spectrum are closely connected with the process of wave diffraction by a restricted grating. The latter process 23
Volume 60, number 1,2
OPTICS COMMUNICATIONS
the m o d e power flux propagates through the DBRs ( T',...... a n d 7'~ ....... are the right and left transmission coefficients ). The rest of the m o d e power flux is scattered into all others guided and leakage modes of the considered waveguide. This scattering is illustrated in fig. 2c as an appearence of the guided m o d e m (the diffraction efficiency of the m o d e mL into tile m o d e m2 is R', ...... or RI ....... for the right or left mirror). The vacuum wavelengths o f the modes m~ and m, arc the same. the propagation constants t~.... a n d h ..... are different, hut only the constant /7.... was supposed to satisfy the Bragg condition ( 1 ). the mirrors o f the considered laser p r o v i d e weak retlectance for the m o d e m> Hence. though the laser caxity was considered to be adjusted for generation of the m-mode, the circulation of the mode m~ in the cavity and the wave scattering in the D B R s providc all possible guided and leakage modes in the laser output (fig. 2d).
(a)
m
]
•~,..~..,_.._~-
le)
3. Theoretical model for the analysis .,
*',?×e Fig.2. Light scattering b~¢a restricted grating. ( a ) The conversion of a singular propagation direction into a continuous spectrum of propagation directions (in the gap between the lens and the grating). The source and the screen are in the focal plane. (b)-(d) A cross-section of the DBR laser. The sequence of excitation of the mode m2 in the laser output. Solid arrows are the m~-mode. light ones the m2-mode. (e) The DBF laser considered as a DBR one with a pair of active reflectors. transforms the single propagation direction o f the initial wave into a continuous spectrum of propagation directions o f the afler-diffractional waves (see fig. 2a). The same situation occurs in waveguide optics: a single waveguide m o d e excites all possible waveguide and leakage m o d e s through diffraction by a grating having limited dimensions. Apply these preliminary remarks to a D B R laser. Consider a D B R laser (fig. 2b), the cavity o f which is adjusted to generate a waveguide m o d e with the m o d e index m~. Suppose that the m o d e m~ has been already excited in the laser. Let the m w m o d e propagating in the positive x-direction (as fig. 2b). The right D B R reflects a considerable part o f the m o d e power flux ( R ~,...... is the reflection coefficient for the right reflector, R I ....... for the left one). Some part o f 24
15 October 1986
Here we shall outline a theoretical model which is helpful to evaluate the problem in question. Our model is based on the fact that o r d i n a r y efficiencies o f the scattering o f the lasing m o d e into another one are small enough. Low levels of the scattering p e r m i t to consider the problem as a two-step one: (1) taking into account ordinary descriptions of the DFB and DBR lasers (e.g., Kogelnik and Shank [ 5 ], Chinn and Kelley [6], etc.), one can obtain the lasing m o d e distribution along the laser cavity: ( 2 ) using the obtained m o d e distribution, one can estimate the scattering o f the lasing waveguide m o d e into other guided and leaky modes. Since the first step is well-known, consider more detailed the second. Let us have already o b t a i n e d the mL-mode distribution in the laser cavity (fig. 2b), and P .... be the m w m o d e power flux at the beginning o f the right DBR ( x = x 2 ) . Thus, the m , - m o d e output through the right m i r r o r will be P',,,,=P,,,, T',, ...... .
21
The o u t p u t through the left m i r r o r will be P',,,,=p,,,,
7'~,, ...... (R',, ...... /R', .......
)'~.
3)
Volume 60, number 1,2
OPTICS COMMUNICATIONS
15 October 1986
R~f
where the last multiplier presents the ratio of the power fluxes falling to the left ( x = x ~ ) and right (x = x2) mirrors. The outputs of the me-mode through the right and left DBR will be
10o2 4z
(5)
where we have placed in square brackets the part of the m l-mode converted into the me-mode by the left and the right mirror (in eq. (4) and eq. (5)); Q,,: is the factor of amplification of the me-mode after the single crossing of the active region of the laser. Eqs. (4), (5) are written at the condition of low levels of the scattering R,%,,,,,_<
R~....... <
R ~,....... <
R ~,..... <
M3 M~ M~
P. (~,)
P~,,2 = [P .... R t........ ( R r...... IR I. . . . . . .] I/2"J-,,,2 I T r .... Q .... (4) P~,,, = (P,,, R~, ...... ) T t,...... Q .....
M~~2
Nj
i~ ux~ (a)
o.o¢/.
(b) o
~
2
3
~ mm
fO0 X
(6) J
which was mentioned above.
"1ram
4. Numerical results (c) Relations ( 2 - 5 ) permit to estimate the relative presence of the mode mJ in the laser output Mi=(P,,,,+P,,,,
(7)
P,,~+P,,~).
Expressions for R~(!,).,, and --,,,m,Tm)have been already presented in ref. [ 7] as R and T; to calculate (for example) Rr,,,,,,,, one must place parameters of the right DBR in expressions [7] and calculate the efficieny of reflection of the mode mi into the mode mj. It should be pointed out that the reflection coeffiecients R r(~) (as well as the transmission coefficients) may oscillate with variations of the lasing wavelength (fig. 3a). Here we are interested in maximum values of the anomal modal spectrum. Therefore, the next numerical results will use _~ instead R (see the difference shown in fig. 3a) in calculations n f R m ) with m, ~ m,. In terms [7 ],/~ is, approximately, ~"
nit/Hi\
v ~
1~= I (Ao/Ap) [(Co~t)~1 + t h ( l m ( t k l ) ) ) ]
2 I,
-
.
lnlln/
(8)
see also ref. [ 8 ] for more details. A short note must be added with respect to DFB lasers. The upper formalism is acceptable to a DFB laser as if the latter is a DBR with a pair of active
nrn
Fig.3. (a) The qualitative dependences of R(2) and /~(2). (b) Values of Mj versus the length of the Bragg mirror. Parameters of each DBR are: grating period 0.143 Hm, grating depth 0.3 Hm, corrugation from the air side, refractive indices of the film and substrate 3.6 and 3.4, light wavelength in vacuum 1 pm. j is the mode index. (c) Values M , for the DFB laser with different grating lengths, l, and depths, hg. The DFB structure is with the corrugation from the air side, refractive indices of the film and substrate are 3.59 and 3.141, the film thickness is 0.7/tm, the grating period is 0.115/~m, the Bragg condition is refered to TE 2mode of the waveguide considered.
mirrors (the interface of these DBRs should be supposed in maximum of the lasing mode distribution, see fig. 2e). The mentioned expressions remain valid. For a numerical testing of the anomal modal spectrum in DBR laser, we have chosen the structure of fig. l a, each mirror of which is similar to one considered by Stoll and Yariv [9]. We have suggested a near threshold regime so that we could isolate the results from pumping levels and accept Q,,, ~ 1. In fig. 3b we have presented the modal spectrum in the laser output versus length of the gratings used as DBRs. The modal spectrum of the DFB laser has been calculated for the laser of Shank et al. [ 10]. Results 25
Volume 60, number 1,2
OPTICS COMMUNICATIONS
are shown in fig. 3c, they have been obtained at the near threshold condition. The modal spectrum is presented versus grating depth.
5. Discussion The above numerical illustrations are referred to 'perfect' laser structures, i.e. with no distortions of the waveguide or the distributed reflectors. Once more we like to emphasize that even perfect DBRs and DFB provide all waveguide modes in the laser radiation, which is the consequence of the linite length of the distributed feedback. It is obvious that a similar excitation of the waveguide mode spectrum appears in two-dimentional waveguide lasers (some
o
............
Li,;ijk (a)
.......
i. c,A'M
15 October 1986
of them are sketched in fig. 4). It is of interest that the mode scattering in the latter laser structures can turn the polarization of the laser output. This fact is illustrated in fig. 4 by the following way: suppose the laser cavities (in fig. 4) to be adjusted to lase TEmodes; oblique angles of incidence lead to the TETM mode conversion in the grating regions [ 11 ]: a high quality of the cavity for the TE-modes, and remarkable efficiency of the TE-TM scattering can provide the laser output having mainly tile TM polarization.
6. Conclusion Surface light scattering by restricted waveguide gratings can provide the anomal modal spectrum of radiation of the DFB and DBR lasers. It has both advantages and disadvantages. Disadvantages: (1) the discussed phenomenon provides the modal noises even for highly fabricated Bragg reflectors or distributed feedback: (2) the Bragg condition ( 1 ) cannot be used for forecasting the output modal spectrum (being useful in the determination of the lasing mode). Advantages: ( 1 ) the alteration of the modal spectrum or polarization of the output radiation may have practical applications: the anomal part of laser output spectrum may have a convient use in designing of laser operation control and management.
(b)
References
L~
(c)
Fig.4. Different schemes of lasers (in plane). Solid arrows show the waveguide mode which is stimulated by a pumping, light arrows are the modes excited by diffractional scattering; polarizations of the lasing mode and scattered ones may be orthogonal.
26
[ 1] H. Kogelnik, IEEE Trans. on Microwave Fheory and Techniques MTT 23 (1975) 2, [ 2 ] J.-J. (?lair, Introduction a l'optique int6grOe ( Masson, Paris ) [3] K. Utaka, K. Kobayashi and Y. Susematsu, IEEE J. Quant, Electron. QE-17 (1981) 651. [4] R. Wolski. P. Szczepanski, W. Wojdak and W. Wolinski, Optica Applicata 13 (1983) 431. [5] H. Kogelnik and C.V. Shank, J. Appl. Phys. 43 (1972) 2327 [ 6 ] S.R. Chinn and P.L. Kelley, Optics Comm. l 0 ( I t)74 ) 123. [7] A.A. Spikhal'skii, Optics Comm. 57 (1986) 375. [ 8 ] A.A. Spikhal'skii, Sov. J. QuanI. Electron. 14 ( 1984 ) o I 'L [9] H. Stoll and A. Yariv, Optics Comm. 8 (1973) 5. [ 10] C.V. Shank, R.V. Schmidt and B.I. Miller, Appt. Phys. t_ett. 25 (1974) 200. [ 11 ] M.T. Wlodarzyk and S.R. Seshadri, J. Opt. Soc. AnL ~\ 2 (1985) 171.
OPTICS COMMUNICATIONS
15 October 1986
A N O M A L O U S MODAL SPECTRUM OF RADIATION OF DFB AND DBR LASERS A.A. S P I K H A L ' S K I I General Physics Institute, Academy of Sciences of the USSR, Moscow 117942, USSR Received 27 August 1985; revised manuscript received 25 June 1986
The modal spectrum of waveguide DFB and DBR lasers is examined theoretically. It is found that the modal spectrum of the laser output can differ greatly from the spectrum inside the laser cavity. Numerical results are presented for examples of DFB and DBR lasers.
1. Introduction
Lasers with distributed feedback (DFB) or distributed Bragg reflectors ( D B R ) are interesting and perspective devices of quantum electronics. They are among the most important elements o f integrated optics [ 1-4]. These integrated optical lasers are often performed on the base of multimode waveguides; thus, different waveguide modes may appear in laser radiation. For different applications o f the lasers, especially in optical communications, it is of importance to stabilize the modal spectrum o f the radiation and to maintain the modal purity. The modal purity and stability are necessary for a laser to be employed in integrated optical multiplex systems, optical heterodyne transmission systems, and many other waveguide applications. Distortions of the modal spectrum are now considered mainly as a consequence o f imperfections, random variations o f waveguide parameters, and similar deviations o f the laser structure from the perfect one. However, our analysis has shown that even the perfect DFB and DBR structures can destroy the purity of modal spectrum and, moreover, lead to unusual spectra of waveguide modes in laser radiation. Consider a corrugated optical waveguide used as the DFB or DBR laser (fig. la or lb). Suppose that a waveguide mode (with the mode index m) is excited by a pumping o f this laser. The ordinary condition on the grating period A, the mode wavelength in vacuum 2,,, and the mode propagation constant b,,, is 0 030-401/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
PUMPING
FILM ~ . . . .
(a)
~" ~
SLI&STRRT~
PUMPINCr
(b) Fig.l. Schemes of the waveguide laser with distributed Bragg reflectors (a) and distributed feedback (b). bm(2,n) =Tr r/A,
(1)
where r is the Bragg order o f the distributed feedback. Here we are interested in the conditions, by which the correlation (1) is true for the mode m~, whereas the main radiation of the laser is the mode rn2; this process is called here the process with anomal modal spectrum.
2. Grounds of the anomal modal spectrum The physical grounds of the anomal modal spectrum are closely connected with the process of wave diffraction by a restricted grating. The latter process 23
Volume 60, number 1,2
OPTICS COMMUNICATIONS
the m o d e power flux propagates through the DBRs ( T',...... a n d 7'~ ....... are the right and left transmission coefficients ). The rest of the m o d e power flux is scattered into all others guided and leakage modes of the considered waveguide. This scattering is illustrated in fig. 2c as an appearence of the guided m o d e m (the diffraction efficiency of the m o d e mL into tile m o d e m2 is R', ...... or RI ....... for the right or left mirror). The vacuum wavelengths o f the modes m~ and m, arc the same. the propagation constants t~.... a n d h ..... are different, hut only the constant /7.... was supposed to satisfy the Bragg condition ( 1 ). the mirrors o f the considered laser p r o v i d e weak retlectance for the m o d e m> Hence. though the laser caxity was considered to be adjusted for generation of the m-mode, the circulation of the mode m~ in the cavity and the wave scattering in the D B R s providc all possible guided and leakage modes in the laser output (fig. 2d).
(a)
m
]
•~,..~..,_.._~-
le)
3. Theoretical model for the analysis .,
*',?×e Fig.2. Light scattering b~¢a restricted grating. ( a ) The conversion of a singular propagation direction into a continuous spectrum of propagation directions (in the gap between the lens and the grating). The source and the screen are in the focal plane. (b)-(d) A cross-section of the DBR laser. The sequence of excitation of the mode m2 in the laser output. Solid arrows are the m~-mode. light ones the m2-mode. (e) The DBF laser considered as a DBR one with a pair of active reflectors. transforms the single propagation direction o f the initial wave into a continuous spectrum of propagation directions o f the afler-diffractional waves (see fig. 2a). The same situation occurs in waveguide optics: a single waveguide m o d e excites all possible waveguide and leakage m o d e s through diffraction by a grating having limited dimensions. Apply these preliminary remarks to a D B R laser. Consider a D B R laser (fig. 2b), the cavity o f which is adjusted to generate a waveguide m o d e with the m o d e index m~. Suppose that the m o d e m~ has been already excited in the laser. Let the m w m o d e propagating in the positive x-direction (as fig. 2b). The right D B R reflects a considerable part o f the m o d e power flux ( R ~,...... is the reflection coefficient for the right reflector, R I ....... for the left one). Some part o f 24
15 October 1986
Here we shall outline a theoretical model which is helpful to evaluate the problem in question. Our model is based on the fact that o r d i n a r y efficiencies o f the scattering o f the lasing m o d e into another one are small enough. Low levels of the scattering p e r m i t to consider the problem as a two-step one: (1) taking into account ordinary descriptions of the DFB and DBR lasers (e.g., Kogelnik and Shank [ 5 ], Chinn and Kelley [6], etc.), one can obtain the lasing m o d e distribution along the laser cavity: ( 2 ) using the obtained m o d e distribution, one can estimate the scattering o f the lasing waveguide m o d e into other guided and leaky modes. Since the first step is well-known, consider more detailed the second. Let us have already o b t a i n e d the mL-mode distribution in the laser cavity (fig. 2b), and P .... be the m w m o d e power flux at the beginning o f the right DBR ( x = x 2 ) . Thus, the m , - m o d e output through the right m i r r o r will be P',,,,=P,,,, T',, ...... .
21
The o u t p u t through the left m i r r o r will be P',,,,=p,,,,
7'~,, ...... (R',, ...... /R', .......
)'~.
3)
Volume 60, number 1,2
OPTICS COMMUNICATIONS
15 October 1986
R~f
where the last multiplier presents the ratio of the power fluxes falling to the left ( x = x ~ ) and right (x = x2) mirrors. The outputs of the me-mode through the right and left DBR will be
10o2 4z
(5)
where we have placed in square brackets the part of the m l-mode converted into the me-mode by the left and the right mirror (in eq. (4) and eq. (5)); Q,,: is the factor of amplification of the me-mode after the single crossing of the active region of the laser. Eqs. (4), (5) are written at the condition of low levels of the scattering R,%,,,,,_<
R~....... <
R ~,....... <
R ~,..... <
M3 M~ M~
P. (~,)
P~,,2 = [P .... R t........ ( R r...... IR I. . . . . . .] I/2"J-,,,2 I T r .... Q .... (4) P~,,, = (P,,, R~, ...... ) T t,...... Q .....
M~~2
Nj
i~ ux~ (a)
o.o¢/.
(b) o
~
2
3
~ mm
fO0 X
(6) J
which was mentioned above.
"1ram
4. Numerical results (c) Relations ( 2 - 5 ) permit to estimate the relative presence of the mode mJ in the laser output Mi=(P,,,,+P,,,,
(7)
P,,~+P,,~).
Expressions for R~(!,).,, and --,,,m,Tm)have been already presented in ref. [ 7] as R and T; to calculate (for example) Rr,,,,,,,, one must place parameters of the right DBR in expressions [7] and calculate the efficieny of reflection of the mode mi into the mode mj. It should be pointed out that the reflection coeffiecients R r(~) (as well as the transmission coefficients) may oscillate with variations of the lasing wavelength (fig. 3a). Here we are interested in maximum values of the anomal modal spectrum. Therefore, the next numerical results will use _~ instead R (see the difference shown in fig. 3a) in calculations n f R m ) with m, ~ m,. In terms [7 ],/~ is, approximately, ~"
nit/Hi\
v ~
1~= I (Ao/Ap) [(Co~t)~1 + t h ( l m ( t k l ) ) ) ]
2 I,
-
.
lnlln/
(8)
see also ref. [ 8 ] for more details. A short note must be added with respect to DFB lasers. The upper formalism is acceptable to a DFB laser as if the latter is a DBR with a pair of active
nrn
Fig.3. (a) The qualitative dependences of R(2) and /~(2). (b) Values of Mj versus the length of the Bragg mirror. Parameters of each DBR are: grating period 0.143 Hm, grating depth 0.3 Hm, corrugation from the air side, refractive indices of the film and substrate 3.6 and 3.4, light wavelength in vacuum 1 pm. j is the mode index. (c) Values M , for the DFB laser with different grating lengths, l, and depths, hg. The DFB structure is with the corrugation from the air side, refractive indices of the film and substrate are 3.59 and 3.141, the film thickness is 0.7/tm, the grating period is 0.115/~m, the Bragg condition is refered to TE 2mode of the waveguide considered.
mirrors (the interface of these DBRs should be supposed in maximum of the lasing mode distribution, see fig. 2e). The mentioned expressions remain valid. For a numerical testing of the anomal modal spectrum in DBR laser, we have chosen the structure of fig. l a, each mirror of which is similar to one considered by Stoll and Yariv [9]. We have suggested a near threshold regime so that we could isolate the results from pumping levels and accept Q,,, ~ 1. In fig. 3b we have presented the modal spectrum in the laser output versus length of the gratings used as DBRs. The modal spectrum of the DFB laser has been calculated for the laser of Shank et al. [ 10]. Results 25
Volume 60, number 1,2
OPTICS COMMUNICATIONS
are shown in fig. 3c, they have been obtained at the near threshold condition. The modal spectrum is presented versus grating depth.
5. Discussion The above numerical illustrations are referred to 'perfect' laser structures, i.e. with no distortions of the waveguide or the distributed reflectors. Once more we like to emphasize that even perfect DBRs and DFB provide all waveguide modes in the laser radiation, which is the consequence of the linite length of the distributed feedback. It is obvious that a similar excitation of the waveguide mode spectrum appears in two-dimentional waveguide lasers (some
o
............
Li,;ijk (a)
.......
i. c,A'M
15 October 1986
of them are sketched in fig. 4). It is of interest that the mode scattering in the latter laser structures can turn the polarization of the laser output. This fact is illustrated in fig. 4 by the following way: suppose the laser cavities (in fig. 4) to be adjusted to lase TEmodes; oblique angles of incidence lead to the TETM mode conversion in the grating regions [ 11 ]: a high quality of the cavity for the TE-modes, and remarkable efficiency of the TE-TM scattering can provide the laser output having mainly tile TM polarization.
6. Conclusion Surface light scattering by restricted waveguide gratings can provide the anomal modal spectrum of radiation of the DFB and DBR lasers. It has both advantages and disadvantages. Disadvantages: (1) the discussed phenomenon provides the modal noises even for highly fabricated Bragg reflectors or distributed feedback: (2) the Bragg condition ( 1 ) cannot be used for forecasting the output modal spectrum (being useful in the determination of the lasing mode). Advantages: ( 1 ) the alteration of the modal spectrum or polarization of the output radiation may have practical applications: the anomal part of laser output spectrum may have a convient use in designing of laser operation control and management.
(b)
References
L~
(c)
Fig.4. Different schemes of lasers (in plane). Solid arrows show the waveguide mode which is stimulated by a pumping, light arrows are the modes excited by diffractional scattering; polarizations of the lasing mode and scattered ones may be orthogonal.
26
[ 1] H. Kogelnik, IEEE Trans. on Microwave Fheory and Techniques MTT 23 (1975) 2, [ 2 ] J.-J. (?lair, Introduction a l'optique int6grOe ( Masson, Paris ) [3] K. Utaka, K. Kobayashi and Y. Susematsu, IEEE J. Quant, Electron. QE-17 (1981) 651. [4] R. Wolski. P. Szczepanski, W. Wojdak and W. Wolinski, Optica Applicata 13 (1983) 431. [5] H. Kogelnik and C.V. Shank, J. Appl. Phys. 43 (1972) 2327 [ 6 ] S.R. Chinn and P.L. Kelley, Optics Comm. l 0 ( I t)74 ) 123. [7] A.A. Spikhal'skii, Optics Comm. 57 (1986) 375. [ 8 ] A.A. Spikhal'skii, Sov. J. QuanI. Electron. 14 ( 1984 ) o I 'L [9] H. Stoll and A. Yariv, Optics Comm. 8 (1973) 5. [ 10] C.V. Shank, R.V. Schmidt and B.I. Miller, Appt. Phys. t_ett. 25 (1974) 200. [ 11 ] M.T. Wlodarzyk and S.R. Seshadri, J. Opt. Soc. AnL ~\ 2 (1985) 171.