Relation between Statistical Models

The validity and equivalence between the density distributions

P

_CS( v

I

; ) and )

; ( g

I

P

_ imply that the two parameters v and g are related. The relationship between v and g according to the experimental results is marked with blue dots in Fig. 5.2.3. We employ an empirical form of v 1 exp[ 0.08 g^0.85] to express the relationship between v and g, as depicted with a solid line in Fig. 5.2.3. The empirical expression indicates the two properties: one is v

 1

as

g

 to indicate no WL effects and the other is v_c0.06 with g1 to signify the SL threshold. In other words, the statistical properties for the WL and SL effects can be manifested with the the chi-square distributions with the parameters in the region of

c 1

v  v and 0 v v_c, respectively. Taking the familiar parameter g as a standard of scaling, the careful mapping of g and v of the two models helps to clarify the regime of different extent of localization with the new parameter v.

Fig. 5.2.3 Blue dots: The relation between v and g according to the experimental data.

Red line: Empirical form for the relationship between v and g.

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Chapter 6

Summary and Future Work

6.1 Summary

In section 2.1.1, we have theoretically derived the eigenstates of the coupled isotropic HO, which reveals the continuous transformation from the HG to the LG states. In section 2.2.2, we have performed the analogous optical experiment to systematically reconstruct the transformational relation between the HG and LG modes with a cylindrical-lens mode converter. In section 2.1.2, we have verified that the spatial morphologies of the Lissajous states can be continuously transformed into the trochoidal states with spatial morphologies corresponding to the trochoidal curves by converting the HG components into the corresponding LG modes. In section 2.2.3, we have further exploited the optical Lissajous modes and a /2 cylindrical lens mode converter to perform the spatial transformation in analogous way and to generate the intermediate optical modes between the optical Lissajous and trochoidal modes.

Experimental realization confirmed a notable method to generate the spatial coherent states with various orbital morphologies. The present method is expected to be constructive for investigating the spatial transformation of optical coherent waves. In section 2.1.3, we have systematically investigated the quantum signatures of the eigenstates corresponding to the coupled commensurate HO with SU(2) coupling interactions. The eigenstates are shown to be concentrated on the multiple periodic orbits that transform from the multiple Lissajous orbits to the multiple trochoidal orbits. In section 2.2.4, we have explored the analogous observation of the laser

transverse modes from large-Fresnel-number degenerate cavities via varying pumping size. It has been experimentally verified that the 3D coherent lasing waves corresponding to the quantum states with multiple Lissajous orbits can be systematically generated by enlarging the pumping spot size. We also employ the propagating property of the lasing modes to manifest the role of the phase factor introduced by the SU(2) coupling interactions. Moreover, we apply the cylindrical-lens mode converter to confirm the transformational relation between the multiple Lissajous orbits and the multiple trochoidal orbits. Section 2.3 is the further extension of section 2.1 and 2.2. We develop a novel method of creating optical vortex array by the conversion of a standing-wave Laguerre-Gaussian (LG) mode.

Theoretically, by employing the transformational relation, the standing-wave LG mode is verified to be transformed from a pair of crisscrossed Hermite-Gaussian (HG) modes, embedded with optical vortex array, consists of a TEMn,m mode and a TEMm,n

mode. Due to close correspondence between the transformational relation and the mode conversion of astigmatic lenses, we successfully generate the optical vortex array by transforming a standing-wave LG mode into the crisscrossed HG modes via a

/2 cylindrical lens mode converter. The investigation may provide useful insight in the study of the vortex light beam and its further applications.

In chapter 3, we have exploited the Bessel’s integral to analytically manifest the ray-wave correspondence between high-order Bessel beams and geometric modes in circular billiards. We also experimentally demonstrated that the Bessel-related geometric modes can be strikingly generated by utilizing a large-aperture cylindrical waveguide with controlling the extent of the incident angle. Moreover, we demonstrated that the free-space propagation of the output beam emerging from the cylindrical waveguide could be used to investigate the transient dynamics of the geometric modes. We believe that the present investigation can provide an important

insight into quantum physics and wave optics.

In chapter 4, we have theoretically demonstrated the mode transformation of the high-order standing-wave LG0,l modes when undergoing the phase-matching SHG.

The SHG for the standing-wave LG0,l modes has been verified to cause the formation of the centrally focused beams which propagate with their transverse intensity profiles changed. The theoretical analysis reveals that the revolution of the centrally focused beam along the longitudinal axis results from the interference of a set of traveling-wave LGp,0 and a standing-wave LG0,2l modes according to different Gouy phase shift. Furthermore, we have employed a diode-pumped solid-state laser with intracavity SHG to carry out the experiment. By controlling the spot sizes of the doughnut-shaped pump profiles, we have effectively generated the high-order standing-wave LG0,l modes of varying orders for frequency-doubling. The experimental results of the second-harmonic waves are shown to be in good agreement with the theoretical analysis. Our studies might provide some useful insights into the wave functions for the nonlinear conversion.

In chapter 5 we have experimentally generated the optical patterns from the conical SHG process to investigate the disordered wave functions with different extents of WL from extended to pre-localized states. It has been numerically confirmed that the statistical characteristics of experimental disordered wave functions can be explained very well with the RV-NLS model. Furthermore, we have found that the fractional chi-square distributions are nearly equivalent to the distributions of the RV-NLS model. With this result, the concept of the fractional degrees of freedom can be used to manifest the extent of localization for the disordered wave functions. It is believed that the present work can bring more insight into the localization phenomena of diverse disordered systems.

6.2 Future work

In chapter 4, we have thoroughly investigated the formation of the centrally focused beam with the second-harmonic generation (SHG) of a high-order Laguerre-Gaussian mode. It can be found that the obtained second-harmonic waves characterized by their intricate structures reveal fairly different morphologies to the input fundamental waves. The intriguing observations stimulate our interests in the SHG for the localized coherent waves as discussed in chapter 2. However, the restriction of the conversion efficiency for the SHG of such a high-order laser mode might cause the major difficulty in carrying out the experiment. Once we can overcome the limitation, it could be expected that the second-harmonic coherent waves associated with the input localized modes might display considerably complicated configurations.

Appendix A

Derivation of the Eigenstates to the Harmonic Oscillator with SU(2) Coupling

The Hamiltonian for a two-dimensional (2D) isotropic harmonic oscillator with SU(2) coupling can be given by

0 1 1 2 2 3 3

ˆ ˆ ˆ ˆ ˆ

H



H

 

J

 

J

 

J

. (A.1)

In terms of the dimensionless spatial representation, (A.1) can be rewritten as



^{2 2 2 2}



0 ¹

 

²

 

Try to eliminate the second term with



₁, we employ the following transformation for the operators:

cos sin eigenvalues and eigensates ₁

,

_{2 ˆ}

n n H_ in terms of the spatial representation for

H  can be given by

ˆ

We have already got the solutions of ˆ

H  . Such transformations in Eq. (A.3), and

(A.5) enable us to simplify the question with the well-known eigenstates Hermite-Gaussian states of the uncoupled Hamiltonian ˆ

H  . At the end, we will show

that the eignenstates of the coupled oscillator can be expressed as the superposition of Hermite-Gaussian states. Now let us back to our question in the beginning with the 2D coupled harmonic oscillator—solving the eigensates and eigenvalues of it.

Furthermore, using the ladder operators may help us explicitly figure out the question.

The ladder operators can be given by

 

Equations (A.5) can be rewritten in the form of the ladder operators:

 

Since the ground state 0,0 is invariant under transformation of the coordinate, it is obvious that 0,0 _Hˆ  0,0 _Hˆ. Therefore, the eigenstates ₁

,

_{2 ˆ} required for the energy degeneracy. Furthermore, we use the Binomial series to expand Eq. (A.11):

 

Wigner d-coefficient is given by

     

Similarly, according to Eq. (A.3), the ladder operators can be obtained to be

† †

Substitute Eq. (A.14) into Eq. (A.12), an extra phase term is introduced:

 ^{2 2}

 

ˆ

Eq. (A.15) presents the eigenstates of the coupled harmonic oscillator, where one can replace 2



by  and

2

 by



to correspond to Eq. (2.1.11).

Appendix B

Derivation of the General Expression for a High-order Hermite-Gaussian Mode Transforming through an ABCD System with the Huygens Integral

The general form for the Huygens integral in one transverse dimension in terms of the ABCD matrix can be given by



2 2

  

B.1 Input a Hermite-Gaussian Mode

Consider an input beam of a high-order Hermite-Gaussian (HG) mode in the form



1 1



^{1 12}

where



₁ is the wavelength in the medium where the beam is currently located. The field distribution of the HG mode propagates through a distance L hence can be written as

 ^{12 1 2 22}

intensity distribution u x z_n



2

,

2



². Employing the generating function of the Hermite polynomials, ^{2 2}

 

Calculate the exponent with allocation method and simplify the equation with

0 1

Applying the integral identity ^

^exp 

^{ax dx2}



^{ a}

To eliminate the series in the left hand side (LHS) of Eq. (B.6) and find the expression for the amplitude distribution u x z_n



2

,

2



, we should apply again the generating function of the Hermite polynomials to the right hand side (RHS). See the exponent first,

Furthermore, employ the parameter t derived in Eq. (B.9) and define a new parameter 



for term (2) in Eq. (B.7):

 

^{2 2}

Substitute Eq. (B.9) and Eq. (B.11) into Eq. (B.7), Eq. (B.7) can be rewritten as



2 2



Compare the RHS and LHS term-by-term, the field distribution can be given by



2 2



^{ 1 2}

For a high-order HG mode in the two transverse dimension (x,y), Eq. (B.13) can be modified as

B.2 Input a Rotated Hermite-Gaussian Mode

We first expand the rotated high-order HG modes into a set of HG basis without rotation and find the weighting coefficients:

     

From the generating function in the two transverse dimensions, we have

   

assumed that the input mode is an isotropic, i.e. the beam radius in x and y directions are the same. Substitute Eq. (B.17) into the LHS of Eq. (B.18):

2 2

Apply the generating function to Eq. (B.19):

rcos tsin ² 2 cosr tsin x1 rsin tcos ² 2 sinr tcos y1

   

Similarly, the RHS in Eq. (B.18) can be modified as

 

¹

 

¹

Comparing Eq. (B.20) and Eq. (B.21) term-by-term, we can obtain the relation:

   

Therefore, Eq. (B.20) can be rewritten as

  Since Eq. (B.21) is equivalent to Eq. (B.24) and the length is conserved under rotation, we can compare Eq. (B.21) and Eq (B.24) term-by-term to obtain the field distribution

as follows

Consider the case for the input beam traveling through an ABCD system, it is evident that the output beam can be directly obtained tobe

 

²

   

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