Transverse Mode Structure and Output Stability of
the p-Ge Terahertz Laser
J. Niels Hovenier, Ekaterina E. Orlova, Tjeerd O. Klaassen, Associate Member, IEEE, and W. Tom Wenckebach
Abstract—The p-Ge terahertz laser with flat external cavity mirrors exhibits intensity modulations at well-defined nonequidis-tant frequencies in the 8–80-MHz range, resulting from beating of transverse modes. A theoretical description of the mode structure of this cavity is given. Results compare well with experimental data on beat frequencies and far-field beam profiles obtained using a metal mesh out-coupling window. Based on calculations, a cavity with a reduced size back mirror is proposed to suppress higher order transverse modes. Experiments prove that a reduction of the mirror to about 70% of the end-face surface leads to an effective suppression of the intensity modulations. Such a cavity exhibits a strongly improved pulse-to-pulse stability of the output and a much better shape of the micro pulse train under mode-locked operation.
Index Terms—Mode structure, p-Ge terahertz laser.
I. INTRODUCTION
T
HE action of the p-Ge laser is based on the population in-version between the light- and heavy-hole subbands, cre-ated in crossed electric- and magnetic- fields. It pro-vides terahertz emission in the 70–200- m wavelength range [1]. The laser operates at temperatures below about 20 K. The large heating by the high voltage excitation necessitates short pulsed (few micrometers) operation. The cryogenic cooling and the high index of refraction of germanium force the use of very simple cavities in case the broad-band nature of the laser emission has to be retained. Commonly, the two mirrors of the cavity are simply pressed against the end faces of the laser crystal to avoid etalon effects. For coupling radiation out of the cavity, either one mirror contains a small central hole (hole output coupling) or one mirror is smaller than the crystal end face, and light “leaks” out alongside its edges (edge output cou-pling). We have found that, in order to achieve maximum output power and a good beam profile, the use of a metal mesh—cov-ering the total end face—as an output coupler is preferable [2]. Although it imposes some restrictions on the emission band-width, in this case emission across the total end face occurs, enabling the observation of the optical-field distribution in the cavity. An unexpected drawback is found to be the presence of strong fluctuations in the output power, with characteristic fre-quencies in the 8–80 MHz range. These appear to originate fromManuscript received June 21, 2001; revised January 3, 2002. The work of E. E. Orlova was supported in part by INTAS.
J. N. Hovenier, T. O. Klaassen, and W. T. Wenckebach are with the Depart-ment of Applied Physics, Faculty of Applied Sciences, Delft University of Tech-nology, 2600 GA Delft, The Netherlands (e-mail: Niels@hfwork3.tn.tudelft.nl). E. E. Orlova is with the Institute for Physics of Microstructures, Russian Academy of Science, Nizhny Novgorod 603600, Russia.
Publisher Item Identifier S 0018-9197(02)02947-0.
Fig. 1. (a) Time-resolved laser output forB = 0:5 T ( 175 m) for a cavity with large back mirror. Apart from low-frequency intensity variations, a strong 770–MHz modulation due to self mode locking is visible. (b) Fourier spectrum of laser pulse showing intensity modulations at 10 and 50 MHz.
beating of transverse modes in the cavity. Far-field beam pro-file measurements confirm the presence of higher order modes in our cavity. With hole- or edge-output coupling, this effect is much less because, due to the special geometry, a number of transverse modes are not supported in the cavity and/or are nearly not ouput coupled. The aim of this work is to give a con-cise theoretical description of the observed beat frequencies and measured far-field beam profiles. Based on that knowledge, cal-culations will be given to show that the higher order modes can be suppressed by employing a back mirror of reduced size. Fi-nally, the experimental results on pulse shapes, beam profiles, and micro pulse trains under mode-locked operation, employing this transverse mode suppression, will be presented and dis-cussed.
II. MODEBEATING
The Ga-doped Ge laser crystal under investigation has the
shape of a rectangular parallelepiped with mm
mm mm, respectively. The capacitive mesh mirror with a 25- m period and a plain reflective mirror are pressed against the 5 7 crystal end faces. The facets are completely cov-ered by the electrical contacts for the uniform electrical
excita-tion field . The magnetic field is
perpendicular to both and the long axis of the crystal. At the facets, near the mesh out coupler, the 1 10-mm elec-trodes for active mode locking are placed [3]–[5]. A z-cut quartz window in the tail of the cryostat enables the observation of the emitted radiation at room temperature. The time-resolved optical output is registered using a very fast room temperature GaAs–AlGaAs heterostructure detector [6].
To illustrate the impact of the transverse mode-beating ef-fects, in Figs. 1–3 the pulse shapes for a few different situations 0018–9197/02$17.00 © 2002 IEEE
Fig. 2. (a) Typical pulse shape forB = 0:9 T ( 90–110 m) for a cavity with large back mirror. (b) Fourier spectrum showing the presence of a large number of modulation frequencies.
Fig. 3. Start of laser action atB = 0:5 T under active mode locking conditions. The intensity of the micro pulse train is modulated at 25 MHz. are shown for a cavity with a 5 7 mm back mirror. In Fig. 1,
the laser pulse for T ( m) is shown.
Be-sides strong modulations at the cavity roundtrip frequency (770 MHz), due to self mode locking, Fourier components at about 10 and 50 MHz are present. In Fig. 2, a typical pulse shape for
T ( – m) is shown, exhibiting
modula-tions at frequencies of 6, 11, 31, 40, 61, 69, 73, and 770 MHz. Fig. 3 shows the micro pulse train under mode locking con-ditions. The steadily increasing intensity of the micro pulses, separated by the 1.3-ns cavity roundtrip time, is also strongly modulated due to transverse mode-beating. To obtain additional information concerning the transverse modes present, far-field beam patterns were measured with a -mm pyroelectric
detector (with video bandwidth kHz), mounted on
a step-motor driven – stage.
The far-field intensity contours for various values of are given in Fig. 4. as a function of and , where z is the
axial distance from the laser crystal and and .
The and scales represent the far-field divergence angles of the laser beam.
Diffraction causes the beam profile to be elongated along the -direction because . In a low field, the beam pat-tern vaguely shows two intensity maxima with a small sep-aration. Toward higher fields, the two-fold structure becomes more easily distinguishable and exhibits a larger separation. At still larger fields, a four-fold splitting appears. Clearly, these beam profiles indicate the presence of higher order transverse modes and the mode structure depends on field strength. The
internal reflection within the laser crystal [8], [10] as well as Gaussian modes similar to that of an open quasioptic resonator [7], [11] have been considered. The latter only occur when external mirrors are applied to the crystal faces. Both models imply an equidistant frequency spectrum of the transverse modes. The structure of the optical field in a p-Ge laser with an external spherical mirror has been calculated using quasioptical methods, not taking into account the influence of the crystal boundaries [11]. These boundaries, however, do influence the field structure in a resonator with flat external mirrors, as used in the present investigations. The field in the present resonator is described by Maxwell’s equations for a dielectric medium
with boundary conditions for the
facets (ideally conducting surfaces) and and continuous across the facets. We do not account for a possible influence of the active medium on the field distribution.
The field distribution for electric and magnetic waves is given by the electric and magnetic Hertz vectors parallel to z the axis with the complex amplitudes
and :
(1) From boundary conditions on the metallic surfaces and on the mirrors, we obtain the conditions
. The dielectric boundary conditions lead to
, where for even parity waves and
for odd parity waves, .
For can be
approx-imated by , with a relative error
. Here, for
electric waves and and for
magnetic waves. (It should be noted that the mode of the metallic waveguide does not fulfill the conditions of the dielectric boundary of the actual resonator.) For , the mode frequencies are given by
(2)
with and
Here, and are the free-space wavelength and velocity of light, respectively. Longitudinal modes are separated by the
(a) (b)
(c) (d)
Fig. 4. Theoretical and experimental far-field beam profiles for cavity with large back mirror. (a) Theoretical beam profile for a mixture of the (1, 0) and the (1, 1) modes at = 175 m. a Experimental beam profile; B = 0:50 T. (b) Theoretical beam profile for the (2, 0) mode at = 100 m. b Experimental beam profile;B = 0:77 T. (c) Theoretical beam profile for the (2, 2) mode at = 100 m. c Experimental beam profile; B = 1:32 T. (d) Experimental near field profile forB = 0:90 T.
roundtrip frequency. The dependence of the frequency differ-ences for the first few transverse modes on the free-space wave-length is presented in Fig. 5.
The distribution of the optical field in the Fraunhofer region
(Fresnel numbers ) for the lower order
trans-verse modes is obtained using Fresnel integrals, assuming that the field amplitude changes much faster in the transverse direc-tion than in the longitudinal direcdirec-tion and . The field on the crystal facet resulting from each mode is a superposi-tion of two (for modes) or four travelling waves with . The influence of the mesh period on
the distribution of the field at the facet is not taken into account, since its period is much less than the radiation wavelength. The far-field distribution for the modes is then given by (3), shown at the bottom of the next page, where
and
The two/four traveling waves of each mode lead to two/four in-tensity maximums in the far-field region, shifted from the
lon-Fig. 5. Calculated wavelength dependence of frequency differences between transverse modes(L; M) for cavity with large back mirror.
gitudinal axis by , corresponding to
the zero phase difference for the radiation coming from the dif-ferent parts of the sample facet. The widths of the maxima are
.
Because the finite width of the maxima is independent of mode number, the lowest order mode does not show an ob-servable splitting of the maxima.
As the far-field intensity distribution is the sum of the inten-sities of the fields produced by the different modes present, the maximum beam divergence is determined by the highest order transverse mode present
B. Comparison With Experiment
Using the data depicted in Fig. 5, one may try to identify the experimentally observed beat frequencies. For the low-field
case— T and m—the frequency differences
between the (1, 1) mode and the (1, 0) and (2, 1) modes, respec-tively, may be attributed to the 10- and 50-MHz beats.
In high fields— T and m—the
situa-tion is more difficult as laser excitasitua-tion occurs simultaneously over a broad wavelength range [2], [6]. For a proper identifica-tion, combined data on frequencies and emission wavelength are needed. However, a few frequencies seem to be evident:
– MHz, – MHz, and
– MHz.
In the high field case, the number of transverse modes is larger than at low fields. It is noted that the observed beat frequen-cies do not form an equidistant spectrum, which proves that the Gaussian mode structure proposed in [7] for the p-Ge laser with flat mirrors is incorrect.
tense modes. For the low field situation the profile calculated for a mixture of the (1, 0) and the (1, 1) modes (Fig. 4(a)) has a clear resemblance with Fig. 4(a)’ concerning the posi-tion of the maxima. The experimental beam profile exhibits a larger transverse extension due to the presence of higher order transverse modes. The dominant intensity of low-order modes, deduced from the other experimental data, appears to be con-firmed. In Fig. 4(b), the shape of the (2, 0) mode with the two intensity maxima is given, that compares well with the exper-imental shape in Fig. 4(b)’ for intermediate field strength. Fi-nally, the (2, 2) mode pattern in Fig. 4(c) exhibits the four-fold maximum structure resembling the pattern in Fig. 4(c)’.
Thus the observed far-field profiles can be attributed to com-binations of wave guide modes. To explain the actual distribu-tion of the intensity between the modes and the asymmetry of the observed beam patterns, the influence of the gain distribu-tion in the active medium on the radiadistribu-tion field structure should be taken into account.
IV. MODESTRUCTURE OFRESONATORWITHSMALL BACKMIRROR
A. Theory
To reduce the instability of the laser output due to mode beating, and improve the beam quality, we have investigated the possibility of increasing the losses for the higher order transverse modes by reducing the size of the back mirror. Due to cross-relaxation processes, the suppression of modes in the p-Ge laser leads to the redistribution of spectral power density to other—high quality—modes [9]. The total radiation intensity need not be reduced much with the increase of losses for some of the transverse modes, provided that the losses for axial modes are not changed substantially.
Diffraction by the edges of the reduced mirror
strongly influences the mode structure of the resonator. If the mirror is substantially smaller than the size of the crystal end face, the mirror dimensions determine the characteristic transverse dimension of the radiation field. Then the effect of the crystal boundaries may be neglected and the open resonator model can be used for the analysis of the mode
Fig. 6. Calculated dependence of frequency differences between transverse modes(L; M) for a cavity with small back mirror as a function of back mirror size for = 175 m.
structure. The edges of the large mesh output-coupling mirror only slightly influence the field distribution; this mirror can, thus, be regarded as infinite. According to Huygens’ principle, the transverse distribution of the field in such a resonator is the same as in a resonator with double the length, and terminated
by two mirrors of size .
Vainstein [12] has given an approximate analytical solution for the open resonator problem. Provided that
and
, and , the field
distribu-tion at the mirror can be approximated by
(4)
where and are given by
(5)
( for ‘ ’ and for ‘-’ in the
corresponding parts of (4)).
are the and components of the electric field for the
and modes, respectively.
To justify the use of this approximation for the present cavity, the field intensity at the position of the crystal boundaries has been calculated using the Fresnel integral method. It is found that for the lower order modes (1, 1) and (1, 2), the intensities at the crystal boundaries are below 4% of the maximum for
.
Frequencies of the modes are obtained from (2) using the values of and from (5). The dependence of the frequency differences on the relative mirror size for some lower order
modes, for m, are shown in Fig. 6.
The diffraction losses in the resonator, determined by the imaginary part of the frequency, are presented in Fig. 7. As the small-signal gain for this laser is of the order of 0.01 cm , for
a mirror size in the range the lowest
Fig. 7. Diffraction losses of modes(L; M) for = 175 m as a function of back mirror size.
Fig. 8. (a) Pulse shape forB = 0:5 T ( 175 m) for cavity with back mirror sizea =a = b =b 0:84. Apart from 770-MHz oscillations, only a very weak beating in the tail of the laser pulse is observed. (b) Fourier spectrum of laser pulse showing the 54-MHz transverse-mode beating component. order mode (1, 1) is expected to exhibit a positive gain, whereas the higher modes will have a negative gain value.
The far-field pattern of the
open resonator modes is
(6) where
B. Comparison With Experiment
For the experimental verification, a 4.2 5.8-mm back
mirror has been chosen, i.e., . At low fields,
only a very weak mode beating—rather late in the pulse—is observed (see Fig. 8). The beat frequency of 54 MHz is close to that of the calculated difference between the (1, 2) and (1, 1) modes, as shown in Fig. 6. The pulse to pulse stability has improved considerably compared to that with a large back mirror. The delay between start of the electrical excitation pulse and start of observable laser action has increased slightly, indicating some decrease of the small-signal gain. In Fig. 9(a ), the experimental far-field beam pattern is seen to show a close
Fig. 9. (a) Theoretical far-field profile for the (1, 1) mode in a cavity with small back mirror at = 175 m. (a ) Experimental far-field beam profile at
B = 0:5 T for cavity with small back mirror.
Fig. 10. Start of laser action atB = 0:5 T under mode locking conditions for a cavity with small back mirror. A smooth increase of micro-pulse intensity is observed.
agreement with the calculated one in Fig. 9(a) for the (1, 1) mode. In general, very strong self-mode locking is observed. Also, under active mode-locking conditions, the stability has improved as can be seen from the very regular pulse train in Fig. 10. It is evident that the use of a smaller mirror results in a very efficient suppression of the higher order cavity modes.
In the high field region, a reduction of the number of beat frequencies is observed, using an intermediate mirror size
(4.5 6.5 mm ). However, with the 4.2 5.8 mm back
mirror, no laser action occurs at all, even though earlier exper-iments showed that, with a large back mirror, the small-signal gain in high fields is larger than in low fields. The reason for the absence of high field stimulated emission with a small mirror must be directly related to the different mode structure observed for the two field regions. Whereas in low fields, the (1, 1) mode is dominant, in high fields, the higher order modes are dominant; possibly the (1, 1) mode is totally absent. With a strong reduction of the size of the back mirror and the subsequent destruction of the gain of higher order transverse modes, apparently no mode with positive gain remains.
is, thus, probably related to the influence of the properties of the active medium on the radiation distribution in the crystal. Such phenomena have not been investigated so far.
V. CONCLUSION
The transverse mode structure of the p-Ge terahertz laser with a cavity consisting of a large back mirror and a metal mesh output-coupling mirror has been studied. The experimental re-sults on beat frequencies and far-field beam profiles are well described by a simple model for transverse waveguide modes.
Decreasing the size of the back mirror leads to an effective suppression of higher order modes, resulting in much better pulse shapes and pulse trains under mode locking conditions. The theoretical description for this case closely fits the experi-mental data.
The peculiar field dependence of the mode structure found in this specific laser crystal is not yet understood.
ACKNOWLEDGMENT
The authors wish to thank F. H. Groen for suggesting wave-guide modes to be the origin of the beats in the output inten-sity of the laser. This work is part of the of the European TMR Network “Inter European Terahertz Action (INTERACT)” Re-search Program.
REFERENCES
[1] Optical Quantum Electron., vol. 23, 1991. Special issue on Far Infrared Semiconductor Lasers.
[2] T. O. Klaassen, J. N. Hovenier, W. Th. Wenckebach, A. V. Muravjov, S. G. Pavlov, and V. N. Shastin, “The pulsed and mode lockedp-Ge THz laser: Wavelength dependent properties,” in Proc. EOS/SPIE Int. Symp.:
Conf. Terahertz Spectroscopy & Applications, vol. 3828, SPIE, Munich,
Germany, June 1999, pp. 58–67.
[3] J. N. Hovenier, R. C. Strijbos, W. Th. Wenckebach, A. V. Muravjov, S. G. Pavlov, and V. N. Shastin, “Active mode locking of ap-Ge hot hole laser,” Appl. Phys. Lett., vol. 71, pp. 443–445, July 1997.
[4] J. N. Hovenier, T. O. Klaassen, W. Th. Wenckebach, A. V. Muravjov, S. G. Pavlov, and V. N. Shastin, “Gain of the mode lockedp-Ge laser in the low field region,” Appl. Phys. Lett., vol. 72, pp. 1140–1142, Mar. 1998. [5] J. N. Hovenier, M. C. Diez, T. O. Klaassen, W. Th. Wenckebach, A. V. Muravjov, S. G. Pavlov, and V. N. Shastin, “Thep-Ge Terahertz laser: Properties under pulsed and mode locked operation,” IEEE Trans.
Mi-crowave Theory Tech., vol. 48, pp. 670–676, 2000.
[6] S. Winnerl, W. Seiwerth, E. Schomberg, J. Grenzer, K. F. Renk, C. J. G. Langerak, A. F. G. van der Meer, D. G. Pavel’ev, Yu. Koschurinov, A. A. Ignatov, B. Melzer, V. Ustinov, S. Ivanov, and P. S. Kop’ev, “Ultrafast detection and autocorrelation of picosecond THz radiation pulses with a GaAs/AlAs superlattice,” Appl. Phys. Lett., vol. 73, pp. 2983–2985, 1998.
[7] A. V. Bespalov, “Temporal and mode structure of the interband p-germa-nium laser emission,” Appl. Phys. Lett., vol. 66, pp. 2703–2705, 1995.
[8] M. Helm, K. Unterainer, E. Gornik, and E. E. Haller, “New results on stimulated emission from p-germanium in crossed fields,” in Proc. 5th
Int. Conf. Hot Carriers in Semiconductors, Boston, MA, July 20–24,
1987.
[9] A. V. Murav’ev, I. M. Nefedov, S. G. Pavlov, and V. N. Shastin, “Tunable narrow band laser that operates on interband transitions of hot holes in germanium,” Quantum Electron., vol. 23, pp. 119–124, 1993. [10] E. Bründermann, H. P. Röser, A. V. Muravjov, S. G. Pavlov, and V. N.
Shastin, “Mode fine structure of thep-Ge intervalence band laser mea-sured by heterodyne mixing spectroscopy with an optically pumped ring gas laser,” Infrared Phys. Technol., vol. 36, pp. 59–69, 1995. [11] S. G. Pavlov, “Optical output of the p-Ge intervalence band laser,” in
Proc. Int. Symp.hhPhysics and Engineering of Millimiter and Submil-limiter Wavesii, vol. 1, Kharkov, Ukraine, 1994, pp. 154–157.
[12] L. A. Vainstein, “Open resonators and open waveguides,” Sov. Radio, p. 476, 1966.
[13] J. N. Hovenier, R. M. de Kleijn, T. O. Klaassen, W. T. Wenckebach, D. R. Chamberlin, E. Bründermann, and E. E. Haller, “Mode locked operation of the copper doped germanium terahertz laser,” Appl. Phys. Lett., vol. 77, pp. 3155–3157, 2000.
J. Niels Hovenier was born in Hilversum, The Netherlands. He received the
degree in advanced electronic engineering from the school of Electronics, Hil-versum, The Netherlands, in 1977.
From 1979 to 1990, he was an Electronic Engineer in the NMR Research Group, Physics Department, University of Leiden, The Netherlands. Since 1991, he has been an Opto-Electronic Engineer with the Semiconductors Physics Group, Physics Department, Delft University of Technology, Delft, The Netherlands. He was involved in far-infrared short-pulse/high-power pilot studies on semiconductors using the Dutch Free Electron Laser FELIX. His current research activities include mode-locked p-Ge terahertz lasers and detection of short (picosecond) terahertz pulses.
Ekaterina E. Orlova received the M.Sc. degree from the Radiophysical Faculty
of Gorky State University, Nizhny Novgorod, Russia, in 1989.
She was a Probationer Researcher in the field of semiconductor terahertz lasers based on inter-subband transitions at the Institute of Applied Physics RAS, Nizhny Novgorod, Russia, during 1990-1992. Since 1992, she has been in-volved in theoretical research of terahertz active media based on impurity transi-tionsin semiconductors and semiconductor structures at the Institute for Physics of Microstructures RAS, Nizhny Novgorod, Russia. She was involved in far-in-frared spectroscopy studies of non-equilibrium populated impurity states in sil-icon using the Dutch Free Electron Laser FELIX.
Tjeerd O. Klaassen (A’95) received the M.Sc. and Ph.D. degrees in physics
from the University of Leiden, The Netherlands, in 1967 and 1973, respectively. As a member of staff in the Physics Department, University of Leiden, he was involved in the study of low-dimentional magnetic systems using magnetic resonance and relaxation techniques. In 1985, he became involved in the field of far-infrared spectroscopy of impurities in semiconductors. Since 1990, he has been with the Department of Applied Physics, Delft University of Tech-nology, Delft, The Netherlands, where he is involved in (non)linear far-infrared spectroscopy in semiconductors and the development of active (p-Ge laser) and passive (antennas, tramission lines) components for terahertz electronics.
W. Tom Wenckebach received the M.Sc. and Ph.D. degrees in physics from the
University of Leiden, Leiden, The Netherlands, in 1967 and 1970, respectively. As a member of staff at the Physics Department, University of Leiden, he worked among others on dynamic nuclear spin polarization, nuclear spin or-dering, ENDOR, and MIONP. Since 1990, he has been a Full Professor in the Department of Applied Physics, Delft University of Technology, Delft, The Netherlands. His current fields of research include semiconductor physics, tera-hertz science, and the development of teratera-hertz sources and imaging techniques.
