Energy Level Statistics of Spontaneous Emission Spectra

Firstly, we employed the experimental spontaneous emission spectra of the VCSELs to perform a statistical analysis. With the relation of k_z =2π λ₀ , the subthreshold emission spectrum ^{ρ λ}

( )

can be changed from a function of the emission wavelength λ to a function of the transverse wave number K by using the relation ^{K =}

(

^{2π λ}

)

^2−kz2 . Figures 3.9(a) and 3.9(b) are plots of measured subthreshold emission spectrum ^ρSE

( )

^K for equilateral-triangular and stadium-shaped VCSELs, respectively. We searched all the peak positions in the experimental spectra and recorded these wave numbers as the sequence of eigenvalues

{

^{K K1, 2,…Ki,…}

}

. The spacings ^si =

(

^Ki+¹−^Ki

)

∆^K between adjacent eigenvalues were subsequently obtained by calculating the mean spacing ∆K. We obtained 817 and 548 spacings of eigenmodes for the equilateral-triangular and stadium-shaped VCSELs. Figure 3.10 shows the experimental statistics for the nearest-neighbor

eigenvalue spacing distribution ^{p s}

( )

in the form of a histogram. It can be seen that the statistical results for the equilateral-triangular VCSELs and stadium-shaped VCSELs are in good agreement with a Poisson distribution ^{p s}

( )

^=exp

( )

^{−s and a}

Wigner distribution ^{p s}

( ) (

^{= πs 2 exp}

) (

^{−πs2 4}

)

. The result is completely consistent with the theoretical prediction that the level statistics of energy spectra in integrable (chaotic) systems correspond to Poisson (Wigner) distribution.

= 2 b a

aaaa

^{= = = =}

^{66 66 66 66}

^μμmμμ

a

= 2 b a

aaaa

^{= = = =}

^{66 66 66 66}

^μμmμμ

a

Fig. 3.8 Schematic of the equilateral-triangular and stadium-shaped VCSELs device structure

0 1 2 3 4 5

equilateral-triangular and (b) stadium-shaped VCSELs.

Fig. 3.10 The experimental statistics for the nearest-neighbor eigenvalue spacing distribution ^{p s}

( )

in the form of a histogram for (a) equilateral-triangular and (b) stadium-shaped device. The curves represent (a) Poisson distribution and (b) Wigner distribution.

3.4.2 Extracting POs from Spontaneous Emission Spectra

As discussed in secion 3.3.2, with the experimental data, the Fourier transform of the subthreshold spectrum ^ρɶSE

( )

^L can be numerically calculated. Figure 3.11 (top row) depicts the calculated results for the path-length spectrum ^ρɶSE

( )

^{L 2}

corresponding to the experimental data shown in Fig. 3.9(a). We experimentally found that there was no obvious difference in the path-length spectra from sample to sample for device growth in the same batch. To make a comparison with the quantum-billiard spectrum, we calculated the Fourier transform of the density of states for an equilateral-triangular quantum billiard. The quantized wave numbers in an equilateral-triangular quantum billiard of side a can be analytically given by [26, 27]

( )

^{2 2}

, 4 3

km n = π a m + −n mn for integral values of m and n, with the restriction that

m ≥ 2 n

. The length spectrum was numerically calculated with the expression

( )

1 ² 2 ^,

N N ikm nL

N L n m ne

ρ =

∑ ∑

_{= =} ^{and N =15} . Figure 3.11 (bottom row) depicts the calculated results for the billiard model. Note that the path length of the periodic orbit

( )

^{p q,} can be expressed as ^{L p q}

( )

^{, =a 3 p2+ pq+q2 . If} p and q have common factors, such an orbit categorically corresponds to a recurrence of a simpler one in which the particle undergoes two or more periods. It can be seen that the experimental length spectrum agrees very well with the theoretical spectrum of the billiard model to exhibit a series of sharp peaks at multiples of the lengths of the primitive periodic orbits.

Figure 3.12 (top row) depicts the calculated result for the Fourier-transformed spectrum ^ρɶSE

( )

^{L 2} of the experimental data shown in Fig. 3.9(b). To make a comparison with the quantum-billiard spectrum, we employed the so called expansion method [28, Appendix A] which we have mentioned in section 2.3.1 to calculate the theoretical eigenvalue density for the stadium billiard with the same geometry. The numerical result of the quantum-billiard model is shown in Fig. 3.12 (bottom row). It is found that the positions of the experimental peaks for the short-range periodic orbits,

( )

^{L a <3.0}, agree well with the theoretical analysis. The short-range periodic orbits

are associated with the scar modes that are numerically found to be rather insensitive to the geometry imperfection. For the long-range length distribution, the experimental spectrum comes close to the theoretical one to exhibit the complicated oscillations without conspicuous peaks. Numerical results indicate that the detailed structure in the long-range length distribution is more or less changed by the tiny perturbation, even though the salient feature of the complicated oscillations is quite similar.

Therefore it is somewhat problematic to make a more quantitative comparison between the experimental and theoretical peaks for the long-range periodic orbits.

Nevertheless, it is judiciously confirmed that the subthreshold emission spectra of the VCSELs with classically chaotic shape can manifest the path length distributions to be in good agreement with the characteristics of the quantum-billiard model.

Fig. 3.11 Fourier transformed spectrum ^ρɶSE

( )

^{L 2}. The experimental and numerical results are displayed as mirror images.

Fig. 3.12 Fourier transformed spectrum ^ρɶSE

( )

^{L 2}. The experimental and numerical results are displayed as mirror images.

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Schuhmann, and T. Weiland, “Wave dynamical chaos in a superconducting three-dimensional Sinai billiard,” Phys. Rev. Lett. 79, 1026 (1997).

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Schmit, “Experimental versus numerical eigenvalues of a Bunimovich stadium billiard: A comparison,” Phys. Rev. E 60, 2851 (1999).

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[8] M. L. Du, and J. B. Delos, “Effect of closed classical orbits on quantum spectra:

ionization of atoms in a magnetic field,” Phys. Rev. Lett. 58, 1731 (1987).

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Schmit, and E. Bogomolny, “Inferring periodic orbits from spectra of simply shaped microlasers,” Phys. Rev. A 76, 023830 (2007).

[10] E. Bogomolny, R. Dubertrand, and C. Schmit, “Trace formula for dielectric cavities: General properties,” Phys. Rev. A 78, 056202 (2008).

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Schäfer, “Experimental test of a trace formula for two-dimensional dielectric resonators,” Phys. Rev. E 81, 066215 (2010).

[12] E. Bogomolny, N. Djellali, R. Dubertrand, I. Gozhyk, M. Lebental, C. Schmit, C.

Ulysse, and J. Zyss, “Trace formula for dielectric cavities. II. Regular, pseudointegrable, and chaotic examples,” Phys. Rev. E 83, 036208 (2011).

[13] T. Gensty, K. Becker, I. Fischer, W. Elsäßer, C. Degen, P. Debernardi, and G. P.

Bava, “Wave chaos in real-world vertical-cavity surface-emitting lasers,” Phys.

Rev. Lett. 94, 233901 (2005).

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Thierry-Mieg, “Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity,” Phys. Rev. Lett. 81, 1110 (1998).

[17] A. M. García‐García, “Finite-size corrections to the blackbody radiation laws,”

Phys. Rev. A 78, 023806 (2008).

[18] H. Odashima, M. Tachikawa, and K. Takehiro, “Mode selective the‐ rmal radiation from a microparticle,” Phys. Rev. A. 80, 041806 (2009).

[19] M. I. Nathan, A. B. Fowler, and G. Burns, “Oscillations in GaAs spontaneous emission in Fabry-Perot cavities,” Phys. Rev. Lett. 11, 152 (1963).

[20] M. L. Du, F. H. Wang, Y. P. Jin, Y. S. Zhou, X. H. Wang, and B. Y. Gu,

“ Oscillations in the spontaneous emission rate of atoms in a dielectric slab:

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Chapter 4

Analogous Experiments for Quantum Billiard Wave

Functions from Lasing

Modes of VCSELs

It is confirmed that near the lasing threshold, the lasing modes of VCSELs can be used to investigate fundamental physics such as pattern formation and wave functions in quantum billiards. In nonlinear optics, pattern formation is a significant phenomenon which results from the interaction between electromagnetic field and nonlinear media [1-4]. In other words, the forming of patterns are determined by the unstable or instability of the optical systems. The structures of lasing modes are usually simple since the Fresnel number in small-area VCSELs is small. In contrast, broad-area VCSELs whose Fresnel number is large can exhibit lasing modes with complicated spatial structure and lead to various kinds of patterns. Ten years ago, Hegarty has reported various forms of patterns in broad-area oxide-confined VCSELs [5, 6].

In quantum worlds, the behaviors of particles were described by waves, and hence it is an important task to obtain wave functions of the systems. It is usually hard to directly get wave functions from experiments because of the difficulty of measurements in microscopic scale. Fortunately, analogy between different physical systems enables us to extract the information of wave functions other wave systems.

As mentioned in Sec. 1.3, due to the short cavity length, the governing equation of electric field in VCSELs reduced to quasi-two-dimensional Helmholtz equation which is completely equivalent to 2D time-independent Schrödinger equation. That is to say, wave functions of 2D quantum billiards are able to be experimentally visualized in the transverse patterns of VCSELs. The intriguing discovery opens a new avenue to understand behaviors of waves in mesoscopic systems.

In recent years, several reports have been proposed to discuss characteristics of lasing modes of VCSELs. It has been experimentally found that vector polarization patterns in square-shaped VCSELs can show the structure of vector vortex lattice [7].

Superscarred mode which is the wave pattern associated with stable periodic orbit can be observed in VCSELs with integrable cavity such as square and equilateral triangle [8, 9]. It is interesting that chaotic wave patterns can also be seen in both square-shaped and equilateral triangular VCSELs due to the existence of perturbation in systems [10]. Diffraction in time effect that is the famous quantum transient phenomena in matter waves has also been verified to be observed with free-space

propagation of lasing modes emitted from VCSELs [11].

Most reports mentioned above mainly concentrated on VCSELs with integrable cavity. For VCSELs with chaotic cavity, we just know that they are possible to generate two distinct types of wave patterns: chaotic wave patterns and scarred mode.

The investigation on the feature of lasing modes in chaotic shaped VCSELs had not ever been explicated thoroughly.

Since the far-field pattern is the Fourier transform of the near-field pattern in the limit of paraxial approximation, the momentum-space wavefunctions of 2D quantum billiards can be analogously observed with the high-order lasing modes of VCSELs [9]. Generation of the higher-order transverse modes can provide more interesting perspectives for the exploration of the quantum–classical connection. However, during the free-space propagation of the higher-order lasing modes, non-paraxial contributions to the total wavevector k may significantly influence the far-field patterns. Therefore, it is essentially important to develop an appropriate correcting method for extracting the momentum-space wavefunctions from the experimental far-field patterns with the substantial non-paraxial contribution.

In this chapter, the above mentioned significant issues about lasing modes of VCSELs under threshold operation were further investigated. Firstly, we present the typical near field patterns of VCSELs with square and equilateral triangular aperture and investigate how these lasing modes correspond to the coordinate-space wave functions of a 2D quantum billiard with the same shape. We will then systematically explore characteristics of typical lasing patterns in two popular chaotic shaped VCSELs: ripple square shaped VCSELs and stadium-shaped VCSELs. Moreover, we discuss the correspondence between far-field transverse patterns of VCSELs and the momentum-space wave functions of a 2D quantum billiard. A reliable method is developed for extracting the momentum-space wavefunctions from the experimental far-field patterns with the substantial non-paraxial contribution.

4.1 Determined Factor for Transverse Order of Lasing Modes

In section 1.3, it has been discussed that VCSELs are quasi-two dimensional systems and their transverse fields obey 2D Helmholtz equation which is given by

( )

2 2

, 0

t kt E x y

∇ +  =

 

 



(4.1)

with k_t ² = k² −k_z ²

, where ^k=

(

^{k k kx, y, z}

)

is emission wave vector determined by active layer of quantum well and k is longitudinal wave vector related to the _z thickness of active layer. Quantum wells are semiconductor materials that properties are sensitive to operating temperature. When the device temperature varies, both longitudinal wave vector and gain profile of quantum well will shift due to variations in refractive index and band gap energy. The changes in refractive index with temperature are always smaller than that of band gap energy and thus the shift of gain peak with temperature is greater than the shift in longitudinal wave vector.

Accordingly, transverse wave vector of k is able be detuned to quite different values _t with varying temperature. In general, band gap energy of quantum well increases with decreasing temperature and therefore it is feasible to obtain lasing modes with large transverse wave vector of k (i.e. short wavelength) at low temperature. In addition, _t the value of transverse wave vector is often described by frequency detuning which is given by Ω = −ω ω_c , where ω =ck n is emission angular frequency and

c ck nz

ω = is longitudinal angular frequency.

We pick one of broad-area VCSELs to measure the influence of temperature on lasing wavelength. Figure 4.1 shows temperature dependence of emission wavelength (red star) as well as longitudinal wavelength (black star) for a stadium-shaped VCSEL with area of 60 30 m× µ ². The linear fitting curves for these data were also depicted.

As our expected, both cases red shift with increasing temperature and linear fitting curves for emission wavelength have larger slope compared to longitudinal wavelength. Lasing modes with large transverse wave vector (i.e. short wavelength)

will be generated at low temperature. high-order lasing modes.

In summary, operating temperature and aperture size are vital factors for determining transverse order of lasing modes for VCSELs.

220 230 240 250 260 270 280 290 300 802

220 230 240 250 260 270 280 290 300 802

Fig. 4.1 Temperature dependence of emission wavelength (red star) and longitudinal wavelength (black star) for a stadium-shaped VCSEL with area of 60 30 m× µ ²

4.2 Experimental Setup for Near-Field Pattern Measurement

Figure 4.2 depicts the schematic view of experimental setup for Near-Field pattern measurement. The VCSELs were placed in a cryogenic system with a temperature stability of 0.1 K in the range of 80–300 K. A power supply providing current with a precision of 0.01 mA was utilized to drive the VCSEL. The near-field patterns were measured by a charge-coupled device (CCD) camera (Coherent, Beam-Code) with an objective lens (Mitsutoyo, numerical aperture 0.9).

VCSEL

Objective Lens

CCD Camera

Cryogenic System

DC Power Supplier VCSEL

Objective Lens

CCD Camera

Cryogenic System

DC Power Supplier

Fig. 4.2 Schematic view of experimental setup for Near-Field Pattern Measurement

4.3 Typical Lasing Modes in Integrabe Shaped VCSEL

In this section, typical lasing modes in square-shaped and equilateral-Triangular VCSEL with varied temperature are studied. Various types of superscarred wave patterns can be seen in both devices. We also investigated temperature dependence of near-field patterns for square-shaped VCSELs with different aperture size for validating the discussion in previous section.

4.3.1 Square-Shaped VCSEL

The morphology of near-field patterns of VCSELs can be affected by operating temperature and injected current. Here we operate a broad-area square-shape oxide-confined VCSEL at different device temperature to explore typical lasing patterns under threshold condition. The size of the oxide aperture was 40 40 m× µ ² and the emission wavelength was designed to be around 800 nm.

Figure 4.3 shows schematic of the laser device structure, temperature dependence of threshold current, and experimental near-field patterns that are characteristically observed at different device temperatures. The lasing patterns are generally robust and reproducibly observed under the same experimental circumstances. The lasing pattern shown in Fig. 4.3(a) is obtained at the operating temperature of 295 K. It can be seen that due to the imperfection of the system, the pattern exhibit complex low order structure rather than regular arrangements. The lasing state at the operating temperature of 280 K is found to dramatically change to a superscarred mode that related to classical periodic orbits with ( , )p q =(1, 2), as seen in Fig. 4.3(b). When the operating temperature decreases to 250 K, the lasing pattern shown in Fig. 4.3(c) exhibits a stripe structure. When the operating temperature further decreases to 230 K, the near-field pattern shown in Fig. 4.3(d) presents superscarred mode with multi-diamond structure. For the operating temperature below 220 K, the experimental pattern shown in Fig. 4.3(e) corresponds to another

superscarred mode that concentrates on classical periodic orbits with ( , )p q =(1,1). Experimental results reveal that when the operating temperature decreases, the near-field patterns of square-shaped VCSEL tend to generate superscar modes rather than eigenfunctions of square planar waveguides. This is because in real world, due to the existence of perturbation, the observed modes are composed of eigenstates.

Figures 4.4(a)–4.4(c) illustrate the calculated patterns corresponding to Fig.

4.3(b)–4.3(e) by using representation of coherent states in quantum billiards. The excellent agreement between the experimental and numerical patterns confirms that the near-field patterns of VCSELs can be analogously regarded as coordinate-space wave functions of a 2D quantum billiard.

a a a

a = = = = 40404040μμμμm

300 K

(a) ^(b)

280 K

^(c)

250 K

230 K

(d) ^(e)

210 K

210 220 230 240 250 260 270 280 290 300 0

10 20 30 40 50

ΙΙΙΙth (mA)

T em perature (K )

a a a

a = = = = 40404040μμμμm

300 K

(a) ^(b)

280 K

^(c)

250 K

230 K

(d) ^(e)

210 K

210 220 230 240 250 260 270 280 290 300 0

10 20 30 40 50

ΙΙΙΙth (mA)

T em perature (K )

Fig. 4.3 Intensity patterns of transverse near-field patterns at temperatures of (a) 300K (room temperature), (b) 280 K, (c) 250 K, (d) 230 K, and (e) 210 K.

(a) (b) (c)

(a) (b) (c)

Fig. 4.4 The calculated patterns by using representation of quantum coherent states in square billiards corresponding to Figures 4.3(b)–4.3(e), respectively.

4.3.2 Comparison of Typical Lasing Modes in Square-Shaped VCSELs with Different Aperture Size

It has been discussed in section 4.1 that aperture size of VCSELs is also a crucial factor for determining the transverse order of lasing modes. In this section, square-shaped broad-area VCSELs with different aperture size were designed to observe this phenomenon. The emission wavelengths of both devices are approximately around 800 nm.

Figures 4.5–4.7 show schematic of device structure, temperature dependence of threshold current and experimental near-field patterns for square-shaped VCSELs with area of 20 20 m× µ ², 30 30 m× µ ², and 40 40 m× µ ². For the VCSEL with aperture size of 20 20 m× µ ², due to relatively large transverse mode spacing, typical wave patterns show irregular and low-order structures and almost unchanged as the operating temperature decreases from 300 K to 180 K. In contrast, For the VCSELs with area of 30 30 m× µ ² and 40 40 m× µ ², with varying operating temperature, kinds of superscarred modes can be generated. The same type of superscarred mode in the two devices presents distinct detailed structure and transverse order owing to the diversity of their intrinsic properties and aperture size. Nevertheless, on the whole, the appearances of lasing modes with varied temperature have a specific regularity. That is, wave patterns with stripe structure are feasible to be generated at high operating temperature. On the other hand, superscarred mode with diamond shape can often be obtained at low operating temperature.

1 8 0 2 0 0 2 2 0 2 4 0 2 6 0 2 8 0 3 0 0 0

1 0 2 0 3 0

ΙΙΙΙ th (mA)

T e m p e r a t u r e ( K )

a a a a =20=20μ=20=20μμmμ

300 K

(a) ^(b)

290 K

^(c)

280 K

^(d)

270 K

260 K

(e) ^(f)

250 K

^(g)

240 K

^(h)

230 K

220 K

(i) ^(j)

210 K

^(k)

200 K

^(l)

180 K

1 8 0 2 0 0 2 2 0 2 4 0 2 6 0 2 8 0 3 0 0

0 1 0 2 0 3 0

ΙΙΙΙ th (mA)

T e m p e r a t u r e ( K )

a a a a =20=20μ=20=20μμmμ

300 K

(a) ^(b)

290 K

^(c)

280 K

^(d)

270 K

260 K

(e) ^(f)

250 K

^(g)

240 K

^(h)

230 K

220 K

(i) ^(j)

210 K

^(k)

200 K

^(l)

180 K

Fig. 4.5 Schematic of device structure, temperature dependence of threshold current and experimental near-field patterns for square-shaped VCSELs with area of

20 20 m× µ 2.

1 6 0 1 8 0 2 0 0 2 2 0 2 4 0 2 6 0 2 8 0 3 0 0 0

1 0 2 0 3 0 4 0

ΙΙΙΙth (mA)

T e m p e r a tu r e (K )

a a a

a = = = = 30303030^μμμμm

300 K

(a) ^(b)

290 K

^(c)

280 K

^(d)

270 K

260 K

(e) ^(f)

250 K

^(g)

240 K

^(h)

230 K

220 K

(i) ^(j)

200 K

^(k)

180 K

^(l)

160 K

1 6 0 1 8 0 2 0 0 2 2 0 2 4 0 2 6 0 2 8 0 3 0 0

0 1 0 2 0 3 0 4 0

ΙΙΙΙth (mA)

T e m p e r a tu r e (K )

a a a

a = = = = 30303030^μμμμm

300 K

(a) ^(b)

290 K

^(c)

280 K

^(d)

270 K

260 K

(e) ^(f)

250 K

^(g)

240 K

^(h)

230 K

220 K

(i) ^(j)

200 K

^(k)

180 K

^(l)

160 K

Fig. 4.6 Schematic diagrams of device structure, temperature dependence of threshold current and experimental near-field patterns for square-shaped VCSELs with area of

30 30 m× µ 2.

1 8 0 2 0 0 2 2 0 2 4 0 2 6 0 2 8 0 3 0 0 0

1 0 2 0 3 0 4 0

ΙΙΙΙth (mA)

T e m p e r a t u r e ( K )

a a a

a = = = = 40404040μμμμm

300 K

(a) ^(b)

290 K

^(c)

280 K

^(d)

270 K

260 K

(e) ^(f)

250 K

^(g)

240 K

^(h)

230 K

220 K

(i) ^(j)

210 K

^(k)

200 K

^(l)

180 K

1 8 0 2 0 0 2 2 0 2 4 0 2 6 0 2 8 0 3 0 0

0 1 0 2 0 3 0 4 0

ΙΙΙΙth (mA)

T e m p e r a t u r e ( K )

a a a

a = = = = 40404040μμμμm

300 K

(a) ^(b)

290 K

^(c)

280 K

^(d)

270 K

260 K

(e) ^(f)

250 K

^(g)

240 K

^(h)

230 K

220 K

(i) ^(j)

210 K

^(k)

200 K

^(l)

180 K

Fig. 4.7 Schematic diagrams of device structure, temperature dependence of threshold current and experimental near-field patterns for square-shaped VCSELs with area of

40 40 m× µ 2.

4.3.3 Equilateral-Triangular VCSEL

In this section, we operate a large-aperture equilateral-triangular oxide-confined VCSEL at different device temperature to explore the morphology of near-field patterns. The size of the oxide aperture was about 66 66 m× µ ² and the emission wavelength was designed to be around 780 nm.

Figure 4.8 depicts threshold current and experimental near-field patterns at different device temperatures. The inset shows schematic of the laser device structure.

As shown in Fig. 4.8(a), the lasing mode presents low-order and irregular wave pattern that is obtained at the operating temperature of 300 K. When the operating temperature decreases to 260 K, superscarred mode that is similar to Fabry–Pérot modes impinging on lateral sides vertically [12] can be excited, as depicted in Fig.

4.8(b). At the operating temperature of 240 K, the near-field pattern shown in Fig.

4.8(c) presents a honeycomb structure. As discussed in section 2.1.2, the honeycomb

4.8(c) presents a honeycomb structure. As discussed in section 2.1.2, the honeycomb

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