The function of 2D wave-billiard wall in the VCSEL device comes from the fact that the large index discontinuity between the oxide layer and surrounding semiconductor leads to a total internal reflection of a wave incident upon the boundary. As shown in Fig. 3.1-1 (a), the separability of the wave function in the VCSEL device enables the wave vectors to be decomposed into kz and kt, where kz is the wave-vector component along the direction of vertical emission and kt is the transverse wave-vector component. The vertical dimension of the cavity is designed to have a large kz component and a relatively small transverse component kt, generally kt < 0.12 kz. The angle between the photon-velocity vector and the normal vector of the boundary surface, tan1(kz kt), can be calculated to be greater than 1.45 rad.
On the other hand, the critical angle for the total reflection is given by )
(
sin1 nox nGaAs , , where is the effective refractive index of the oxide layer and is the effective refractive index of the semiconductor cavity. With 1.5 and 3.5, it can be confirmed that the angle between the photon-velocity vector and the normal vector of the boundary surface is certainly greater than the critical angle for the total reflection, as illustrated in Fig. 3.1-1 (b). As a consequence, the lateral oxide boundaries can be modeled as rigid walls and the losses through the wall boundaries are extremely low.
nox
Under the circumstance of paraxial optics, kt << kz, the longitudinal field is significantly small in comparison with the transverse field. Therefore, the electric field can be approximated to have only transverse components and no longitudinal component, i.e. so-called quasi-TEM waves. After separating the z component in the wave equation, we are left with a two-dimensional Helmholtz equation:
2t kt2
x,y 0, where means th2t lacian operator operating on the coordinates in the transverse plane ande Lap
x,y is a scalar wave function that describes the transverse distribution of the laser mode. As a result, the transverse eigenfunctions of the oxide-confined VCSEL device are equivalent to the eigenfunctions of the 2D Schrodinger equation with hard wall boundaries of the same geometry.
(a)
Fig.3.1-1. (a) The schematic diagrams for vertical-cavity surface-emitting laser. he separability of the wave function in the VCSEL device enables the wave vectors to be decomposed into kz and kt. (b) The illustration of a wave a wave incident upon the current-guiding oxide boundary would undergo total internal reflection for kt kz.
noxide=1.5 kt
nGaAs=3.5 k
kz Plane of Incidence
k
kz
kt
nGaAs=3.5 noxide=1.5
3.2 Experimental Setup
The experimental setup is schematically depicted in Fig. 3.2-1 and Fig, 3.2-2 present the detailed photos of the equipments. The VCSEL devices are mounted on a cooper holder (Fig. 3.2-2 (a)) with good thermal conductance and placed in the cryogenic system (Janis, VPF-100, Fig. 3.2-2 (b)) that is operating with liquid nitrogen. The temperature is controlled by a temperature controller (Neocera, LTC-11) with a temperature stability of 0.1 K at the range of 80-300 K. The VCSEL is driven by a DC power supplier (KEITHLEY 2400) with a precision of 0.005 mA.
The near-field patterns were re-imaged into a CCD camera (Coherent, Beam-Code) with a microscope objective lens (Mitsutoyo, M Plan Apo, NA=0.9, Fig.3.2-2 (c)). The objective lens is placed in a tube that is connected with the cryogenic system and can be tilted by the seven valves (Fig.3.2-2 (d)). A polarizer was used to obtain polarization-resolved near field patterns. The spectral information of the laser output was measured by a Fourier optical spectrum analyzer (ADVANTEST Q8347) with a Michelson interferometer.
Fig.3.2-1. The schematic diagrams for the experimental setup.
VCSEL
Objective Lens
Beam Splitter
CCD Camera DC Power Supplier
Cryogenic System
Optical Spectrum Analyzer Polarizer
VCSEL
Objective Lens
Beam Splitter
CCD Camera DC Power Supplier
Polarizer
Cryogenic System
Optical Spectrum Analyzer
(b) (a)
(c) (b)
Fig.3.2-2. (a) The VCSEL mounted on the copper holder. (b) Side view of the cryogenic system. (c) The objective lens with NA=0.9 (d) Face view of the cryogenic system.
3.3 Typical Lasing Modes of Square VCSEL
The SEM image and the optical microscope image of the square VCSEL device used in this work are shown in Fig. 3.3-1 and Fig. 3.3-2, respectively. The bright region in Fig. 3.3-2 displays the spontaneous emission to manifest the details on the square boundary. The edge length of the aperture is measured to be about 40 μm.
The control parameters of this experiment are the device temperature and pumping currents. Since the lasing patterns will become multi modes, we only focus on the lasing modes at near-threshold currents. The temperature dependence of threshold current is shown in Fig. 3.3-3 (a) and Fig. 3.3-3 (b)-(d) depicts the experimental near-field patterns of one of our square VCSEL devices at temperatures as indicated.
These patterns are robust; they remain unchanged for the durations of at least 1.0mA, and can be reproduced under the same experimental circumstances. The lasing pattern shown in Fig. 3.3-3(b) is obtained at room temperature. The lasing patterns of VCSEL are typically multi-mode emission because of the thermal fluctuations.
The lasing state at the operating temperature of 285K becomes a bouncing-ball mode, as seen in Fig. 3.3-3(c). When the operating temperature further decreases to 250K, the near-field pattern dramatically changes to a multi-diamond pattern hat is a superscar mode associated with several POs, as shown in Fig. 3.3-3(d). For the operating temperature below 230K, the experimental pattern shown in Fig. 3.3-3(e) corresponds to another superscar mode that is localized on a single PO. The behaviors of each VCSEL devices are different but their characteristics are generally the same. In conclusion, the temperatures at which the bouncing-ball or superscar modes appear are not all the same for all devices, but it can be sure that the bouncing-ball mode appears at a higher temperature than that for the superscar mode.
The multi-POs superscar mode does not always exist, but the temperature at which it appears is higher than the temperature for a single-PO superscar mode if it ppears.
Most devices lase with single-PO superscar mode at low temperatures.
These lasing patterns can be analogously interpreted by the quantum-billiard model.
The eigenstates of square billiard are given by (2.1.1)
1 2
We find that the bouncing-ball mode shown in Fig. 3.3-3(c) is not merely an eigenstate but more like a linear combination of two eigenstates
43,11( , )sin(0.6 )x y 42,16( , ) cos(0.6 )x y
(3.3.2)
The experimental result and theoretical simulation are depicted in Fig. 3.3-4 (a) and (a’), respectively, for convenient comparison. Such kind of bouncing-ball modes are prevalent in square VCSEL; we show some other two cases in Fig. 3.3-4 (b) and (c).
The corresponding mathematical expressions of Fig. 3.3-4 (b’) and (c’) are
40,11( , )sin(0.35 )x y 39,14( , ) cos(0.35 )x y
The other representative lasing pattern is the superscar mode that localized on the diamond-shaped PO. As discussed in chapter 2, the superscar can be expressed as
, , 1
It can be found that the superscar mode shown in Fig. 3.3-3 (e) can be interpreted by . We can compare the experimental and theoretical patterns from Fig.
3.3-5(a) and Fig. 3.3-5(a’). The low-temperature lasing modes of most square VCSELs are dominated by the superscar modes. Fig. 3.3-5 (b) and (c) display two other similar superscar modes observed from the other devices. The corresponding theoretical results are also shown in Fig. 3.3-5 (b’) and (c’) for comparisons.
1,1,0.57
We can also reconstruct the multi-POs superscar mode by superposing SU(2) coherent states. Based on thorough numerical analysis, the experimental multi-POs superscar modes can be found to be well reconstructed by
1,1,0.25 1,1,0.57 1,1,0.8
32,20 ( , ) 0.7 32,30 ( , ) 0.9 32,20 ( , )
C x y C x y C x (3.3.6)
Fig. 3.3-6 (a) and (a’) depict the experimental and theoretical results. Although the multi-POs mode is not as prevalent as the bouncing-ball and single-POs superscar mode, it also commonly appears in the transition regime between the two popular modes. Fig. 3.3-6 (b) and (c) show the other two paradigmatic multi-POs superscar modes. The two typical multi-POs modes can be reconstructed by
1,1,0.28 1,1,0.64
and are shown in Fig. 3.3-6 (b’) and (c’), respectively. As we have demonstrated, the transverse mode of square VCSELs can be well reconstructed by quantum-billiard wave functions.
Cathode Anode
Submount
Cathode Anode
Submount
Fig. 3.3-1. The SEM image of square VCSEL device
Fig. 3.3-2. Optical microscope image view from the aperture of the VCSEL.
The bright region display the spontaneous emission to manifest the details on the square boundary.
20 30
220 240 260 280 300
Threshold Current (mA)
Temperature (K)
(d)
(e) (c)
(b)
(a)
Fig. 3.3-3. (a) The temperature dependence of the threshold current and the lasing modes observed at temperatures of (b) 295K (room temperature) (c) 285K (d) 250K (e) 230K.
(a’) (a)
(c’) (c)
(b’) (b)
Fig. 3.3-4. (a)-(c) The bouncing ball modes observed in different square VCSEL devices. (a’)-(c’) The theoretical explanations of (a)-(c), which are expressed by Eq. (3.3.2)-(3.3.4), respectively.
(a’) (a)
(b’) (b)
(c’) (c)
Fig. 3.3-5. (a)-(c) Various superscar modes observed in different square VCSEL devices. (a’)-(c’) Theoretical interpretation of (a)-(c) by SU(2) coherent states C1,1,0.5736,10 ( , )x y , C38,61,1,0.46( , )x y , and C40,251,1,0.8( , )x y respectively.
Fig. 3.3-6. (a)-(c) Various multi-POs superscar modes observed in different square VCSEL devices. (a’)-(c’) Theoretical patterns of (a)-(c) given by Eq. (3.3.6)-(3.3.8), respectively.
(b) (b’)
(c) (c’) (a’) (a)
3.4 Chaotic Wave Function in Rippled-Square VCSEL
Recently, Li et al. reported interesting quantum chaotic phenomena in ripple billiard [LRW02]. In last section we have confirmed that transverse mode of VCSEL is equivalent to the wave function of quantum billiard. Hence, we can analogously observe the quantum chaotic wave function and experimentally investigate the statistical properties of the chaotic wave functions. Although the statistical properties of chaotic wave function have been theoretically well studied [MK88], the experimental wave functions are interfered in the measuring processes.
It is known that microwave cavities have been used to obtain the statistics of chaotic wave state [KKS95, ŠHK+97, SHS04]. However, the statistical properties of the chaotic wave functions emitted from VCSELs have never been studied.
Fig. 3.4-1 shows the pattern of the spontaneous emission that manifests the details on the ripple boundary. The forms for the bottom and top walls of the ripple are approximately expressed as
0.044 1 exp 13.5 1 for the bottom wall ripple are described with the same functional form. The size of the oxide aperture is 45×45μm
a
2. Figures 3.4-2 (a) and (b) show the near-threshold lasing patterns of the rippled VCSEL at temperatures of T260K and T220K, respectively. It can be seen that the two patterns exhibit similar morphology as a chaotic wave function as shown in Ref. [OGH87]. We can validate that the two patterns are chaotic wave
functions or not by testing its statistical properties. Since the intensity patterns do not contain sufficient information, the reconstruction of the wave functions is practically useful for studying the statistical properties of the chaotic modes.
We first demonstrate the logic of the method for reconstructing the wave function before going on. As demonstrated in Fig. 3.4-3, consider an unknown function
( )x
which is the stationary wave function of the system. Experimentally, one can only observed its intensity distribution | ( ) | x 2. How can we obtain ( ) x from the experimental result | ( ) | x 2? Firstly, we have to find the square root of | ( ) | x 2, i. e. | ( ) | x . Then we can see that | ( ) | x is formed by many nodal domains which are separated by the nodes. In | ( ) | x these nodal domains are all positive but they actually may be negative in ( ) x and the signs of the neighboring two nodal domains are different. We set the nodal domains with minus signs to be zero and obtain the positive part of ( ) x denoted as p( )x . Finally, ( ) x can be obtained by
( ) 2x p( ) | ( ) |x x
(3.4.2)
Extending to 2D case, the patterns shown in Fig. 3.4-2 (a) and (b) are the experimentally observed intensity distributions and we are going to obtain the corresponding wave function. In order to reconstruct the wave functions, we need to deduce the field point matrix (xi,yj)from the experimental intensity point matrix , where the indices (i, j) denote the pixel positions of the CCD camera and the total pixel number of the experimental data is 200×200. Since the nodal lines separate the positive and negative domains of the wave function, a so-called positive wave distribution zero for the domains with the opposite sign [SHS04].
Figures 3.4-4(a) and (b) depict the patterns of |p(xi,yj)| for two chaotic modes shown in Figures 3.4-2 (a) and (b), respectively. With the positive wave distribution |p(xi,yj)| , the experimental wave function (xi,yj) can be determined by
( , ) 2 |x yi j p( , ) | | ( , )x yi j x yi j |
. (3.4.3)
Since the experimental wave functions are too coarse to explore the statistical properties completely, the eigenfunction expansion technique is utilized to find analytical expressions for (xi,yj). With the eigenstates of 2D square billiards as a basis, the experimental chaotic wave function can be expressed as
1 2 in the x and y direction, respectively, and denote the expansion coefficients.
Even though some other bases can be chosen for the expansion, the simple analytical form of the eigenstates of 2D square billiards leads to the calculation to be extremely straightforward. The orthogonality relation leads to be
a n1 numerically calculated by a summation:
1 2
Figures 3.4-5(a) and (b) show the intensity plots of | corresponding to Figures 3.4-2 (a) and 3.4-2(b), respectively. These ring areas signify the random directional distribution of transverse wave vectors k, since corresponds to the weighting in k-space. The distribution of in Figure 3.4-5(a) has a mean radius
1,2
With the expansion coefficients the experimental wave functions can be reconstruct by inserting into Eq. (3.4.4). The reconstructed intensity patterns for Fig. 3.4-2 (a) and (b) are displayed in Fig. 3.4-6 (a) and (b), respectively.
We can see that it is very successful in reconstructing the experimentally observed chaotic modes. The statistical properties of the two chaotic modes can now be precisely studied. As discussed in Sec. 2.3, one obtains chaotic wave function in the form of a Gaussian distribution for amplitude,
1,2
|Cn n |
1,2 |
|Cn n
2 2
( ) 1 exp( )
2 2
P
(3.4.8)
, where σ is the standard deviation given by 1/ A with A denotes the area of the billiard. Fig. 3.4-7 (a) and (b) depict the amplitude distributions of the wave functions shown in Fig. 3.4-8 (a) and (b), respectively, with the fitting curves described by Eq. (3.4.8). In addition to Gaussian amplitude distribution, the intensity distribution of a chaotic wave function is shown to be Porter-Thomas distribution
( ) 1 exp( )
2 2 P I I
I
K
. (3.4.9)
Figs. 3.4-9 (a) and (b) illustrate the intensity distributions of the reconstructed patterns Figs. 3.4-7 (a) and (b), respectively.
It can be seen that there are slight variations between our statistical results and theoretical predictions, especially for the case of T 260 . This phenomena may be caused by the thermal fluctuation that results in a broadening of the deviation (r) of the nearly degenerate modes in experiment.
a
0
x x a
0 y ya
Fig. 3.4-1. Experimental pattern of the spontaneous emission to manifest the details on the ripple boundary.
(a) (b)
Fig. 3.4-2. Near-threshold lasing patterns of the rippled VCSEL at temperatures of (a)T260K and (b)T220K.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.
Fig. 3.4-3. (a) An unknown wave function (b) The intensity distribution (c) Square Root of intensity distribution (d) Positive part of the wave function (e) Demonstration of 2p( ) | ( )x x | (f) The result of 2p( ) | ( )x x |.
Fig. 3.4-4. (a) and (b) The intensity plots of the positive wave functions (b)
| ) , (
| p xi yj for experimental results shown in Figs. 3.4-2 (a) and (b), respectively.
(b) (a)
(a) (b)
n1
n2
n1
n2
Fig. 3.4-5. (a) and (b) Distribution of the coefficients obtained by Eq.
(3.4.6) for experimental results shown in Figs. 3.4-1 (a) and (b).
|Cm n, |
Fig. 3.4-6. (a) and (b) The reconstructed patterns with the eigenfunction expansion method for experimental results.
(b) (a)
(a)
(b)
Fig. 3.4-7. (a) and (b) Tthe amplitude distributions of the wave functions shown in Fig. 3.4-6 (a) and (b), respectively.
-4 -3 -2 -1 0 1 2 3 4
0.0 0.1 0.2 0.3 0.4 0.5
P
(
)
Histogram Fitting Curve
-4 -3 -2 -1 0 1 2 3 4
0.0 0.1 0.2 0.3 0.4
0.5 H istogram
Fitting C urve
P
(
)
(a)
(b)
Fig. 3.4-8. (a) and (b) The intensity distributions of the patterns shown in Fig.
3.4-6 (a) and (b), respectively.
0 2 4 6 8 10 12 14
10-4 10-3 10-2 10-1 100
P(I)
I
Histogram Fitting Curve
0 2 4 6 8 10 12 14
10-4 10-3 10-2 10-1 100
Histogram Fitting Curve
P(I)
I
3.5 Typical Lasing Modes in Equilateral-Triangular VCSEL
Equilateral triangular billiard is a special polygonal billiard, which is classically nonseparable but integrable system [DR02]. The experimental observation of the lasing modes in equilateral triangular VCSEL may provide useful information for the microdisk lasers experiments with equilateral triangular resonators [CKL+00, HGL00, HGYL01, LCG+04, YAK+07, ] and electron transport phenomena in equilateral triangular quantum dots [CLO+97].
Fig. 3.5-1(a) and (b) show the optical microscope image of the device operated with an electric current under threshold current at room temperature. The bright region indicates the equilateral triangular pattern of spontaneous emission, which can be more clearly visualized in CCD camera as shown in Fig. 3.5-1(c). The edge length of the oxide aperture was measured to be approximately 66.8 μm.
Similarly, we only focus on the lasing patterns at near threshold current. Fig.
3.5-2 shows the temperature dependence of the threshold current in the range from 300K to 120K. Figures 3.5-3(a)-(i) depict the experimental near-field patterns that are characteristically observed at different device temperatures. It is found that the lasing patterns are generally robust and reproducibly observed under the same experimental circumstances. The lasing pattern shown in Fig. 3.5-3(a) is obtained at the operating temperature of 295K and the optical spectrum indicates it to be a multi-mode emission. The lasing state at the operating temperature of 275K is found to dramatically change to a superscar mode that is similar to Fabry-Pérot modes impinging on lateral sides vertically [MMN64], as seen in Fig. 3.5-3(b). When the operating temperature decreases to 195K, the lasing pattern shown in Fig. 3.5-3(e) exhibits a honeycomb structure. As discussed later, the honeycomb morphology corresponds to the pattern of an eigenstate. When the operating temperature further decreases to 175K, the near-field pattern shown in Fig. 3.5-3(f) behaves like a chaotic wave state that can be described as a random superposition of plane waves [OGH87].
For the operating temperature below 135K, the experimental pattern shown in Fig.
3.5-3(i) corresponds to another superscar mode that is related to a geometrical PO [DR02]. This superscar mode is found to be unchanged when the temperature decreases from 135K to 80K. Intriguingly, the lasing pattern displays the transition and coexistence of the chaotic and superscar modes at the other operating temperature.
As shown in Fig. 3.5-3(c), the lasing mode at 255K is a superscar mode like Fig.
3.5-3(b) but with a background of random wave. Fig. 3.5-3(d) is a mixing of honey-comb eigenstate and random wave. The transition from chaotic wave function to superscar mode is clearly displayed from Fig. 3.5-3(f)-(i). The lasing patterns of each VCSEL devices are different but their characteristics are generally the same.
The analogy between the electromagnetic wave equation in paraxial approximation and the Schrödinger equation enables us to make a detailed connection between the quantum wave functions and the experimental patterns. As discussed in Sec. 2.2, the quantum eigenstates of the equilateral triangular billiard are given by
( )2
with . The eigenstates are the representation of traveling waves.
The standing-wave wave representation of can be expressed as m
. The experimental honeycomb pattern shown in Fig. 3.5-3 (e) can be numerically confirmed to correspond to the wave intensity of
x y , as depicted in Fig. 3.5-4 (b). Superscar modes that are associated
with classical POs can be analytically expressed with the representation of quantum for the standing wave can be given by
*
, ( , , , , ) , ( , , , , ) , ( , , , , )
Tri Tri Tri
N M N M N M
C x y p q x y p q x y p q . (3.5.3)
Based on thorough numerical analysis, the experimental superscar modes can be found to be well reconstructed with the coherent states of C36, 9Tri ( ,x y ; 1, 0, 0.23 ) and C22, 6Tri ( , ;1, 1, 0.35 )x y . Figures 3.5-5(c) and (d) depict the numerical wave patterns of |C36,9(x, y;1,0,0.23)|2 and |C20, 6 ( ,x y ; 1, 1, 0.35 ) | 2 corresponding to the experimental patterns shown in Fig. 3.5-5 (a) and (b), respectively. The excellent agreemen between the experimental and numerical patterns confirms that the quantum formulism is of great importance in describing distinct branches of physics because of the underlying structural similarity.
Conversely, the present analysis also provides a further indication that laser resonators can be designed to demonstrate the quantum phenomenon in mesoscopic physics.
Although an ideal equilateral triangular billiard is integrable, some experimental patterns reveal the property of quantum chaotic modes, as seen in Fig. 3.5-6(a). To prove this pattern is chaotic, the method used in last section is employed to reconstruct this experimental result. Since the eigenstates of equilateral-triangular form a complete set of basis, the wave function can be spanned by [DB02]
( )
Based on the orthogonality of the eigenstates, the expansion coefficients can be obtained by
, where the integration area S is the entire equilateral-triangle billiard. Similarly, the experimental wave function ( , )x yi j can be found by
2 | p( , ) | | ( , ) |x yi j xi yj (3.5.7)
, where p( , )x yi j shown in Fig. 3.5-6 is the positive wave distribution. With the experimental wave function ( , ) x yi j , the integrals in Eq. (3.5.5) and (3.5.6) can be numerically calculated by summations:
( )
( ) the expansion coefficients and into Eq. (3.5.4), we can obtain the reconstructed wave function as shown in Fig. 3.5-7(b). Besides, it has been discussed that the intensity statistics of the chaotic wave functions obey the Porter-Thomas distribution statistics for the reconstructed wave function, as shown in Fig. 3.5-7(c). The good agreement validates that the wave pattern corresponds to a chaotic wave function.
The origin of stationary chaotic modes is inspected to arise from spontaneous imperfections, such as roughness on boundary or unequal of the three internal angles.
In other words, the spontaneous symmetry breaking may cause the real devices with idealized integrable confinements to exhibit the characteristics of nonintegrable systems. As discussed in Ref. [BU94], although a triangular billiard with internal angles to be slightly different from / 3 is intrinsically chaotic, the wave functions can still be scarred by families of POs. Briefly, tiny symmetry breaking can lead to the emergence of superscar as well as chaotic modes in the almost integrable systems.
In other words, the spontaneous symmetry breaking may cause the real devices with idealized integrable confinements to exhibit the characteristics of nonintegrable systems. As discussed in Ref. [BU94], although a triangular billiard with internal angles to be slightly different from / 3 is intrinsically chaotic, the wave functions can still be scarred by families of POs. Briefly, tiny symmetry breaking can lead to the emergence of superscar as well as chaotic modes in the almost integrable systems.