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VOLUME 89, NUMBER 8
PHYSICAL REVIEW LETTERS
19 AUGUST 2002
Modal Interference and Dynamical Instability in a Solid-State Slice Laser
with Asymmetric End-Pumping
Kenju Otsuka, Jing-Yuan Ko, Tsong-Shin Lim, and Hironori Makino
Department of Human and Information Science, Tokai University, 1117 Kitakaname, Hiratsuka, Kanagawa 259-1292, Japan
(Received 18 March 2002; published 5 August 2002)
We observed complicated emission patterns consisting of different transverse modes and associated
intensity pulsations at beat frequencies between pairs of transverse eigenmodes in a solid-state thin-slice
Fabry-Perot laser with asymmetric end-pumping. The dependence of transverse patterns and pulsation
frequencies on pump power has been demonstrated. The interference among nonorthogonal transverse
eigenmodes, which are formed in a deformed Fabry-Perot microcavity possessing an asymmetric, gradient
refractive-index potential for optical waves, is proposed for explaining observed instabilities. Intensity
modulations have been remarkably reproduced by numerical simulations of model equations.
083903-1
0031-9007=02=89(8)=083903(4)$20.00
general, in which polarization dynamics is adiabatically
eliminated. The observed intensity modulation at much
higher frequencies than the intrinsic relaxation oscillation
frequency and a resonantly excited chaotic pulsation have
been reproduced by numerical simulations of a proposed
laser equation including interference effect of nonorthogonal transverse lasing fields.
The experimental setup is shown in Fig. 1(a). A 7-mmsquare c-plate Nd-direct compound LiNdP4 O12 (LNP)
laser with a 0.3-mm-thick plane-parallel Fabry-Perot resonator configuration was attached to a Cu heat sink that had
a 5-mm-diameter hole in the center. The present thin-slice
laser cavity possesses an extremely large Fresnel number
(a)
LD (808 nm)
LNP
WM
PbS
MC
y
SFPI
z
x
OL (10×M)
(b)
20
1.0
0.9
10
0.8
0.7
0
0.6
-10
-20
-20
-10
0
x (µm)
10
20
BS
Relative Temperature Rise (a.u.)
Formations of microcavity laser modes such as whispering gallery modes [1], scarred modes [2], and related
chaotic waves [3,4] have attracted much attention in recent
years for understanding wave formations in microstructure
resonators surrounded by curved hard walls. In these microdisk or microcylinder lasers without end mirrors for
feedback, laser modes are formed through light reflections
at the surrounded hard wall. On the other hand, conventional thin-slice Fabry-Perot microcavity lasers having end
mirrors with symmetric end-pumping, such as verticalcavity surface-emitting laser diodes (VCSEL’s), have revisited in a different context for understanding spatiotemporal dynamics of transverse modes in Fabry-Perot
microcavities. Most recently, Mulet and Balle studied
transverse mode dynamics of VCSEL’s numerically.
They showed that stable transverse patterns consisting of
different orthogonal linearly polarized modesLPml (m 1; 2; . . . and l 0; 1; . . . ), which are derived by solving
the eigenvalue problems assuming the parabolic transverse
refractive-index distribution due to the injection current,
are formed in the stationary state [5]. The mode profile has
m 1 zeros in the radial direction, whereas 2l zeros are
in the angular direction. Then, an intriguing question
arises: Does any generic dynamic behavior appear when
the ‘‘asymmetric’’ end-pumping is applied to thin-slice
Fabry-Perot lasers?
We report here on the formation of a variety of transverse mode patterns, which differ from conventional orthogonal Hermite-Gaussian modes in Fabry-Perot resonators, in a solid-state thin-slice laser with laser-diode
asymmetric end-pumping. Intensity pulsations at beat frequencies between coexisting transverse eigenmodes have
been found. The modal beats between nonorthogonal transverse eigenmodes, which are formed in a thin-slice FabryPerot cavity possessing an asymmetric transverse optical
confinement effect, are proposed to result in intensity
pulsation through the interaction of electric fields with
the atomic system. While, in usual symmetrical pumping
no dynamical instability takes place in class B lasers, in
PACS numbers: 42.55.Rz, 42.55.Sa, 42.55.Xi, 42.60.Mi
y (µm)
DOI: 10.1103/PhysRevLett.89.083903
PD
DO
SA
(c)
mirror mirror
mirror mirror
y
x
x
z
y
z
FIG. 1 (color online). (a) Experimental configuration of a LDpumped LNP slice laser with sheetlike pumping. LD: laser
diode; OL: microscope objective lens; LNP: LiNdP4 O12 laser;
BS: beam splitter; WM: multiwavelength meter; MC: monochrometer; PD: photodetector; DO: digital oscilloscope; SA: rf
spectrum analyzer; SFPI: scanning Fabry-Perot interferometer;
PbS: PbS photoimage tube. (b) Calculated relative temperature
distribution in two dimensions. Thermal conductivity of LNP,
K 0:032 J=s cm K, was assumed. (c) Thermally induced
refractive-index distributions within the crystal.
 2002 The American Physical Society
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VOLUME 89, NUMBER 8
PHYSICAL REVIEW LETTERS
of 1:6 105 , which ensures plane-wave approximation for
light propagation along the 0.3-mm laser crystal. In short,
lasing eigenmodes can be expressed by the product of
transverse optical-field pattern and longitudinal standingwave pattern which is determined by the round-trip condition [5]. An end surface was coated to be transmissive at
the laser diode (LD) pump wavelength of 808 nm (85%
transmission) and highly reflective (99:9%) at the lasing
wavelength around 1050 nm. The other surface was coated
to be 1% transmissive at the lasing wavelength. The thickness of the LD active layer along the vertical direction y
was 1 m and the emitting width along the horizontal
direction x was 100 m, where the LD pump light was
linearly polarized along the x axis. We collimated the LD
pump beam and the resultant collimated beam was directly
focused on the laser crystal surface by using a microscope
objective lens of 10M magnification. The mode profile of
the focused LD beam on the input surface of the crystal
was rectangular (20 m 2 m), yielding the aspect ratio of 10. The uniform intensity distribution in the near
field and the absence of multiple lobes in the far field were
confirmed by an infrared PbS photoimage tube followed by
a monitor with a beam profiler. The absorption length for
the LD light, that is the length at which the pump beam
intensity drops to 1=e, was about 100 m. The pump light
was strongly absorbed near the input mirror; however, the
longitudinal temperature distribution tends to be uniform
in the equilibrium due to the thermal diffusion and the heat
dissipation from the pumped surface of the crystal to the
air. As a result, a strongly asymmetric temperature distribution is thought to be created in the x-y plane. An example
of two-dimensional temperature distributions calculated by
the boundary-element method is shown in Fig. 1(b), where
the dark rectangle corresponds to the focused pump beam
region. The resultant refractive-index distribution due to
the thermally induced refractive-index change are depicted
in Fig. 1(c). In short, a strongly deformed thin-slice FabryPerot cavity possessing an asymmetric graded refractiveindex distribution is formed by sheetlike LD pumping. The
thermally induced potential for optical waves possessing a
strongly asymmetric transverse optical confinement effect
loses its lateral symmetry which ensures usual HermiteGaussian cavity modes [6]. The deformed cavity structure
and corresponding mode pattern are expected to change
depending on the pump power, i.e., temperature rise.
In LNP lasers possessing anisotropic spectroscopic
properties, the stimulated emission cross section along
the b axis is the largest, and the linearly polarized emission
along the b axis occurs independently of the pump condition, e.g., polarization direction of the pump beam. Let us
first show examples of lasing patterns emitted from the
LNP slice laser with sheetlike end-pumping. Near-field and
far-field lasing patterns were measured by the infrared PbS
photoimage tube. The laser exhibited a single longitudinalmode family at first, then two and three longitudinal-mode
families exhibiting similar transverse patterns appeared
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19 AUGUST 2002
with increasing pump power. Far-field patterns of the
single modal output at 1048 nm are shown in Fig. 2(a)
for increasing pump powers where the LNP b axis was set
parallel along the x axis. The transverse pattern is found to
make successive structural changes with increasing pump
power. The structural difference between near-field pattern
and far-field patterns was not seen. According to the symmetry of the thermally induced deformed cavity, the global
structure of observed transverse patterns was symmetric
with respect to the xb and the ya axes. In the low pumppower region, thermally induced optical confinement effect is weak and usual Hermite-Gaussian modes were
observed. However, as the pump power was increased,
these modes became unstable due to the increased strongly
asymmetric transverse optical confinement effect, and
complicated transverse patterns appeared.
We have tested LNP lasers with different thicknesses,
0.3, 0.5, 1, and 3 mm. In the case of 1-mm and 3-mm LNP
lasers, transverse pattern formations shown above were not
observed and even lasing itself did not occur in the highpump power region. This may result from the fact that the
two-dimensional asymmetric optical confinement effect
along the longitudinal direction was weak and the diffraction loss was increased because of much longer cavity
length than the absorption length.
Next, let us focus on optical spectra and dynamical
behaviors of the present laser. Optical spectra were measured by a scanning Fabry-Perot interferometer (Burleigh
FIG. 2. (a) Far-field pattern change at 1048 nm with increasing
pump power in the sheetlike pumping scheme. (b) Far-field
pattern (right) and optical spectrum (left) at 1048 nm in the
symmetric circular pumping scheme (pump power: 140 mW). In
both cases, the crystal b axis was parallel to the x axis.
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PHYSICAL REVIEW LETTERS
1.0
f1
(b)
4
3
2
5
100MHz
1
0.8
0.6
0
1
1.7
1.5
1.3
0.0
0.5
time (µs)
2
power spectral density
f2
voltage
intensity (arb. unit)
(a)
1.0
10
10
10
0
f2 f1
-6
10
10
50
100
0
5
-6
4 3 2
300
200
100
0
40
2
(b)
60
80
pump power (mW)
1
0
0
1
50
time (µs)
100
-10
0
50 100 150
frequency (MHz)
FIG. 3. Optical spectra, waveforms, and corresponding rf
power spectra for different pump powers of (a) 58 mW and
(b) 80 mW. The scaling of the abscissa for optical spectra is the
same for both pump powers.
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(a)
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0
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19 AUGUST 2002
transverse pattern is very close to the simulated pattern for
VCSEL’s with symmetric current injection [5].
Optical spectra and frequency spacing between eigenmodes were found to change depending on the pump power
and the aspect ratio of the pump beam, which critically
changes depending on the crystal position along the z axis
(i.e., defocusing of the pump beam). This is because the
asymmetry of two-dimensional refractive-index distribution (i.e., optical confinement) changes with these parameters, yielding different transverse eigenmodes. The modalbeat (i.e., pulsation) frequency is plotted as a function of
pump power in Fig. 4(a). It should be pointed out that mode
spacing (i.e., energy difference) decreases at first and then
increases with increasing pump power featuring rich bifurcation structures in the high-pump-power regime. In the
low pump-power region, the usual Hermite-Gaussian mode
appeared without exhibiting beat notes. As the pump power
was increased, the optical confinement became effective
and Hermite-Gaussian modes were deformed, resulting in
modal-interference mediate intensity modulation at decreased beat frequencies (i.e., mode spacings) appeared
above a critical pump power. In the high-pump-power
region, on the other hand, the average mode spacing is
considered to have increased by pronounced asymmetry in
the gradient refractive-index profile with increasing pump
power (i.e., refractive-index gradient). In addition, in the
present system, when the beat frequency coincided with
voltage
SAPLUS ; free-spectral range: 2 GHz; resolution: 6 MHz).
Temporal evolutions were measured by an InGaAs photoreceiver (New Focus 1611; 3-dB bandwidth: 1 GHz) followed by a digital oscilloscope (Tektronix TDS 3052;
bandwidth: 500 MHz) where the ‘‘entire’’ output beam
was focused on the detector. Typical optical spectra, intensity waveforms, and corresponding rf power spectra are
shown Fig. 3. From this figure, several significant features
should be addressed. First, the microcavity laser emission
pattern consists of different transverse eigenmodes having
different eigenfrequencies. Second, particular eigenmode
pairs within many eigenmodes are coupled with each other
and produce intensity pulsations at beat frequencies corresponding to the energy (i.e., eigenfrequency) differences
between eigenmode pairs. In the case of Fig. 3(a), mode
pairs, which are indicated by the bridge sign below the
arrows, induce intensity pulsations at frequency f1 and f2 ,
respectively. A ’’breathing’’ oscillation at the difference
frequency of two pulsations, i.e., f1 f2 f1 , is
clearly seen in Fig. 3(a). As for Fig. 3(b), the system exhibited more complicated pulsation involving many mode
pairs indicated by the bridge sign. Note that pulsation
frequencies are 2 orders of magnitude higher than the
intrinsic relaxation oscillation frequency of about 1 MHz.
These features can never be expected in conventional
Hermite-Gaussian laser modes, in which perfect mode
orthogonality is established, because beat notes between
orthogonal sets vanish for the ‘‘entire’’ beam and transverse mode interaction through modal interference is absent. In the present system, however, many transverse
eigenmodes are excited and some of them with the same
parity may violate mode orthogonality. These mode pairs
can generate beats through the modal-interference effect.
Intensity modulations were not observed in the circular
(i.e., symmetric) LD end-pumping scheme due to mode
orthogonality similarly to [5], although complicated spectra featuring fine structures similar to Fig. 3 appeared in the
high-pump-power region exhibiting strong symmetric optical confinement. A typical example of far-field patterns
and optical spectra is shown in Fig. 2(b). The observed
pulsatio
on frequenc y (MHz)
VOLUME 89, NUMBER 8
FIG. 4. (a) Pulsation frequencies as a function of pump power.
Solid (open) circles correspond to stable (unstable) pulsation
frequency, where ’’unstable’’ implies that the pulsation waveforms are unstable in time. The lasing threshold is 41 mW ( # ).
(b) Chaotic relaxation oscillation observed when the beatfrequency fQB coincided with the relaxation frequency
fR 1 MHz ( + ).
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PHYSICAL REVIEW LETTERS
the intrinsic relaxation oscillation frequency of the laser, as
indicated by the arrow, chaotic relaxation oscillations were
found to be resonantly excited as shown in Fig. 4(b).
Let us show a numerical result indicating modalinterference mediate intensity modulation. The proposed
modal-interference effect between nonorthogonal fields
~ m x; y and E
~ n x; y is expected to introduce a significant
E
gain (i.e., stimulated emission) modulation
R R at a beat fre~ mE
~ n dxdy quency to the laser in the form of N0 B x y E
c:c:, where B is the stimulated emission coefficient and N0
is the population-inversion density. The model equation
including the interference between different nonorthogonal
transverse eigenmode pairs is given as follows:
dNi =dt w 1 Ni
X
1 2Ni E2i E2j =K;
(1)
ji
dEi =dt Ni Ei gEi Ei1 cos
i;i1
gEi Ei1 cos
i;i1 =dt
!i;i1 Di "i t;
i;i1 ;
i; j 1; 2; 3: (3)
where Ni is the normalized excess population-inversion
density, Ei is the normalized field amplitude averaged
over the beam cross section, i;i1 is the phase difference
between the ith lasing mode and its adjacent mode, w is the
relative pump power normalized against the threshold, is
the cross-saturation parameter, K is the lifetime ratio of
fluorescence lifetime # to photon lifetime #p , g is the
interference parameter, !i;i1 %!i;i1 #p is the normalized lasing frequency difference between the ith lasing
mode and its adjacent mode, and t is scaled to the photon lifetime. The last term in Eq. (3) expresses the phase
noise, where Di is the phase noise strength and "i t is
the Gaussian white noise, which satisfies h"i ti 0
0
and h"i t"j t0 i P %ij2%t t . Numerical intensity
2
waveforms (E i Ei ) indicating modal-beat mediate
pulsation featuring breathing mode and chaotic relaxation
oscillation corresponding to Figs. 3(a) and 4(b) are shown
in Fig. 5. Pulsations have been reproduced numerically
in a wide parameter region similar to experimental
observations.
The existence of relaxation eigenfrequencies and the
related resonances through transverse cross saturations of
population inversions has been reported as a mechanism of
destabilization in a bimode laser, in which the beating term
was included in the population dynamics [7]. However,
effective modulations in a wide beat-frequency region
observed in our experiment were not achieved numerically
with her model presumably because of a large scaling
factor K 1 in the population equation for microchip
solid-state lasers.
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0.24
E2 0.20
0.16
(b)
2
0
4000
8000
E2 1
0
0
(2)
d
(a)
19 AUGUST 2002
20000
40000
normalized time
FIG. 5. (a) A numerical breathing intensity modulation for
two mode pairs. The parameters are w 1:05, 0:667, K !23 0:075, D1 3
2000, g 0:002,
!12 0:08,
106 , and D2 3 106 . (b) A chaotic relaxation oscillation
for one mode pair as !12 coincided with the relaxation oscillation frequency. !12 0:008, where other parameters are the
same as (a).
The basic idea of sheetlike end-pumping of thin-slice
lasers presented in this Letter is generally applicable to
many solid-state lasers and VCSEL’s. The present finding
of self-induced pulsations associated with the field interference between nonorthogonal transverse modes would
provide new insights into wave formations and nonlinear
dynamical behaviors in deformed Fabry-Perot microcavity
lasers. Quantitative theoretical studies on wave formations
in asymmetric gradient refractive-index potentials, based
on the difficult eigenvalue problems and spatiotemporal
laser dynamics, are strongly anticipated.
The authors are indebted to Professor J.-L. Chern at
National Cheng Kung University, Taiwan, for providing
us with the fast photoreceiver which we used in the
experiment.
[1] S. L. McCall et al., Appl. Phys. Lett. 60, 289 (1992).
[2] S.-B. Lee et al., Phys. Rev. Lett. 88, 033903 (2002).
[3] J. U. Nöckel and A. D. Stone, Nature (London) 385, 45
(1997).
[4] C. Gmachl et al., Science 280, 1556 (1998).
[5] J. Mulet and S. Balle, IEEE J. Quantum Electron. 38, 291
(2002).
[6] P. B. Wilkinson et al., Phys. Rev. Lett. 86, 5466 (2001).
[7] V. Zehnlé, Phys. Rev. A 57, 629 (1998).
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