The factors transver-confinement structure of oxide-aperture and dominant longitudinal wave vector kz of VCSEL are important to directly observe the near-field pattern for analogously studying a high-order wave pattern in quantum billiard. In this work, we use large-aperture square and equal triangular VCSELs to study the wave pattern of each shape of billiard system. Addition to the fundamental eigenmodes such as the Chessboard-like (bouncing ball) mode in square billiard and honeycomb mode in triangular billiard, when the temperature of VCSEL is decreased, the pumping current distribution of the VCSEL causes superscar modes which has interference pattern localized on classical periodic orbits such as diamond-like orbits in square billiard and (1.1) mode in triangular billiard. These measurements are theoretically reconstructed very well by the eigenfunctions of an infinite potential well and the superposition of coherent states. Furthermore, we study the characteristics of the directional emission in far field and the morphology of the evolution from near field to far field. The measurements of large-aperture equilateral-triangular VCSEL show the directional emission of far field from a honeycomb eigenmode is very similar to the superscar (1,1) mode, although the near fields are completely visually different to each other. As a result, the far-field directional emission from a microcavity is just a necessary not sufficient condition for the emergence of a superscar mode. This result is confirmed in experimental finding and numerical simulation, and has been published in Optics Letters, Volume 34, Number 12, 2009 [1]. Contrast to the classical limit, the free-space propagations of the superscars show an addition direction on the rebounding region parallel to the
103
boundary edge. Furthermore, the interference structures on the diffraction pattern result to the fascinating Star of David on the far field from large-aperture equilateral-triangular VCSEL.
In chapter 3 and 4, we study the water splash generated by laser-induced breakdown beneath a free surface. The water jet can be divided into a thin jet and a thick jet with crown-like structure on its top. The mechanisms and features of each part of the water jet with different bubble depths are studied in detail and have been published in Optics Express, Volume 21, Number 1, 2013 [2]. The crown-like structure becomes a pair of arm when the laser is horizontally focused beneath the free surface. Based on this morphology changing, we can clearly observe the formation of this crown-like structure and two-arm splash. Addition to the two-arm splash, the water jet shows a rotating thin jet and sheet thin jet when the laser is horizontally focused beneath the free surface.
A well control water jet induced by laser is necessary for several laser applications such as laser printing [3] and drug delivery [4]. Based on the knowledge of the mechanism of the water jet by this work, we can further explore the methods for controlling the water jet, for example, bubble shaping or implemented a transverse boundary beneath the free surface to change the structure of the water jet, as shown below in which the boundary is a equilateral triangular. As we can see, the crown-like structure forms three arms which are perpendicular to the three edges of the triangular boundary, respectively.
104
Fig. 5.1 A laser-induced water jet is generated by vertically focusing the laser beam beneath a free surface with a lateral equilateral triangular boundary inserted below the free surface. There are three arms forms on the crown, which are perpendicular to the three edges of the triangle, respectively.
105
Reference
[1] Ross C. C. Chen, Y. T. Yu, Y. J. Huang, C. C. Chen, Y. F. Chen, and K. F. Huang,
“Exploring the origin of the directional emission from a microcavity with a large-aperture surface-emitting laser,” Opt. Lett. 34(12) (2009).
[2] Ross C. C. Chen, Y. T. Yu, K. W. Su, J. F. Chen, and Y. F. Chen, “Exploration of water jet generated by Q-switched laser induced water breakdown with different depths beneath a flat free surface,” Opt. Express, 21, 445-453 (2013).
[3] M. Duocastella, A. Patrascioiu, J. M. Fernández-Pradas, J. L. Morenza, and P.
Serra, “Film-free laser forward printing of transparent and weakly absorbing liquids,” Opt. Express 18(21), 21815–21825 (2010).
[4] T. Hirano, M. Komatsu, H. Uenohara, A. Takahashi, K. Takayama, and T.
Yoshimoto, “A novel method of drug delivery for fibrinolysis with Ho:YAG laser-induced liquid jet,” Lasers Med. Sci. 17, 165–172 (2002).
106
Appendix A
Free space propagation of a paraxial ray
In free space propagation, specially, the paraxial optic such as laser optic, the approximation from wavelength on longitudinal smaller than the one on transverse leads to the Fresnel diffraction. By the analogy between paraxial optics and non-relativistic quantum mechanics, it was showed that the transient wave function has remarkable temporary interference pattern analogous to the spatial diffraction pattern of light diffracted by a sharp edge. This property of diffraction in time space, as result, called diffraction in time which is first proposed by Moshinsky [1]. The experiment of diffraction in time can be extended to several systems for example atom cooling, neutrons [2], electrons [3], Bose-Einstein condensates [4]. The significant in diffraction in time is not only the scientific studies but also the application related to the transient response of abrupt changes of potential in quantum device [5,6].
For an electromagnetic wave in free space, assume the vector field is time harmonic with an amplitude
y
( , , )x y z , the amplitudey
( , , )x y z term would obey Helmholtz equation:2y k2y 0
∇ + = (A.1)
Apply the paraxial wave which the propagation vector k is inclined by a small angle with respect to the z axis, that kz can be approximated as below,
107
2 2
Because conversion between the coordinate and momentum space is the Fourier transform, the U k k0( , )x y can be displayed,
The integral can be carried out by completion of the square in the exponent.
108
(
x2/2)
( 2/2 ) 2/2 2 ( 2/2 )The above equation (A.8) introduced in (A.5) gives the well know Fresnel diffraction integral in the paraxial approximation
Next, Fraunhofer diffraction is the limit of the Fresnel diffraction when the distance of z from the plane with u x y0( , )0 0 approaches to infinity. In this limit, the term
(
x x− 0) (
2+ y y− 0)
2 in the exponential term in equation (A.9) is approximated:(
x x− 0) (
2+ y y− 0)
2 (
x2+y2)
−2xx0−2yy0 (A.10)Because the far-field pattern ideally expands to infinity which means
0 0
( ) ( )
x y x y . Actually, the Fraunhofer diffraction is valid enough when the amplitude distribution in the near-field pattern extends over a transverse dimension
d z
k .
The approximation in (A.10) gives the amplitude distribution in far field:
109
( /2 )
(
2 2)
( )(/ 0 0)0 0 0 0 0
( , , ) j j k z x y ( , ) j k z xx yy
u x y z e dx dy u x y e
λ
z∞ ∞
− + +
−∞ −∞
=
∫ ∫
(A.11)Apparently, the integral overx0and y0 is the same as the Fourier transform of
0( , )0 0
u x y . That is why the correlation between near field and far field is related to the coordinate and momentum space.
110
Appendix B
Boundary Integral Method
Figure B.1 shows the schematic representation of the bubble inside a liquid domain and beneath an infinity expansion free surface. The problem is an axisymmetric cylindrical coordinate system in which the r and z denote the radial and vertical axis of a cylindrical polar coordinate. It should be noted that the normal vectors ˆn are outward from liquid across the free space to air domain or the vapor cavity of the bubble.
For simplifying the simulation, we assume the fluid domain is incompressible which is valid when the shock wave generation is poor or its speed in liquid are much smaller than the speed of sound in liquid. For incompressible fluid, the potential of velocity must be satisfied the Laplace’s equation
2φ 0
∇ = . (B.1)
The other assumptions are non-viscous and no surface tension. However, the process of smoothing on the boundary for convergent results (not discussed in detail) has the same effect with surface tension. Next, we introduce the Navier-Stokes equation for time evolution applied to the dynamic of bubble in an axisymmetric cylindrical coordinate.
The solution of Laplace’s equation of a smooth boundary surface S in a domain Ω can be derived by the Green’s integral formula from the Green’s second identity, as shown below:
111
( ) ( , )
Due to the cylindrical symmetry, the integral by surface area can be simplified by firstly integrating the parameter θ . As a result,
2 2
of the first and second kind, respectively.
For numerically calculating the integration, we take N collocation points with index j on the surface, and the surface integral is then divided into a set of segments in which the end points on each segment are the N collocation points.
( , )
The curve in each segment of line integration is approximated by cubic spline. The potential
φ
and then
φ
∂
∂ is approximated by linear interpolation from the end points of each segment. Finally, a matrix form H G
n
φ
= ∂φ
∂ is derived and can be used to calculate the normal derivate of velocity potential which is just the normal velocity on the boundary. Combined with the tangential velocity which is derived from the approximation of cubic spline, the velocity on the boundary can be derived for equation of motion in the time evolution.
113
The boundary dynamics of bubble and free surface are derived from the central point of the bubble in our simulation.
2
The pressure in the liquid domain on the boundary of bubble can be a constant pressure vapor bubble (Rayleigh, 1917 [7]), a vapor bubble (Theofanous, 1969 [8]) and ideal gas bubble with different polytropic indices (Soh and Shervani-Tabar, 1992a, 1992b and 1994 [9-11]). The unsteady Bernoulli equation of velocity potential is rewrite,
The polytropic indices κ is 1.2~1.3 for gaseous explosion products resulting from material explosion like TNT explosion and about 1.4 for diatomic ideal gas.
114
Use the velocity calculated by the method which is described above chapter with the unsteady Bernoulli equation, the finite element method in time:
1 0 0
The procedure of calculation is as following:
(1) Assuming the initial state of location and potential on the boundary;
(2) Calculate the velocity on the boundary by the Eq. B.9;
(3) The next position and potential in a small time step t is derived from the unsteady Bernoulli equation,
Repeat from step (2) for the next time step.
115
Fig. B.1 The geometry of scheme chosen for constructing the model of pulsation bubble beneath a free surface. The normal vector points out from the fluid domain.
The azimuth angle is θ .
116
Reference for Appendix
[1] M. Moshinsky, “Diffraction in Time,” Phys. Rev. 88, 625-631 (1952).
[2] T. Hils, J. Felberg, R. Gähler, W. Gläser, R. Golub, K. Habicht, and P. Wille,
“Matter-wave optics in the time domain: Results of a cold-neutron experiment,”
Phys. Rev. A 58, 4784-4790 (1998).
[3] F. Lindner, M. G. Schätzel, H. Walther, A. Baltuška, E. Gouliemakis, F. Krausz, D. B. Milošević, D. Bauer, W. Becker, and G. G. Paaulus, “Attosecond Double-Slit Experiment,” Phys. Rev. Lett. 95, 040401 (2005).
[4] Y. Colombe, B. Mercier, H. Perrin, and V. Lorent, “Diffraction of a Bose-Einstein condensate in the time domain,” Phys. Rev. A 72, 061601R (2005).
[5] F. Delgado, H. Cruz, and J. G. Muga, “The transient response of a quantum wave to an instantaneous potential step switching,” J. Phys, A: Math Gen. 35(48), 10377-10389 (2002).
[6] F. Delgado, J. G. Muga, D. G. Austing, and G. García-Calderón, “Resonant tunneling transients and decay for a one-dimensional double barrier potential,” J.
Appl. Phys. 97, 013705 (2005).
[7] L. Rayleigh, “On the pressure developed in a liquid during the collapse of a spherical cavity,” Phil. Mag. 34, 94-98. (1917).
[8] T. Theofanous, L. Biasi, and H. S. Isbin, “A theoretical study on bubble growth in constant and time-dependent pressure fields,” Chem. Eng. Sci. 24 885-897 (1969).
[9] W. K. Soh and M. T. Shervani-Tabar, “Computer study of unsteady flow around a cavity bubble,” Computational Methods in Engineering; Advances and Applications, Singapore, 529-534 (1992a).
117
[10] W. K. Soh and M. T. Shervani-Tabar, “Computer study on the rebound of a vapour cavity bubble,” 11th Australasian Fluid Mechanics Conference, University of Tsmania, Hobart, Australia, 199-202 (1992b).
[11] W. K. Soh and M. T. Shervani-Tabar, “Computer model for a pulsating vapour bubble near a rigid surface,” Computational Fluid Dynamics Journal, 3(1), 223-236 (1994).
118
Curriculum Vitae
2002-2005 National HsinChu Senior High School, HsinChu, Taiwan.
2005-2009 B.S. in Department of Electrophysics, National Chiao Tung University, HsinChu, Taiwan.
2009-2013 Ph.D. in Department of Electrophysics, National Chiao Tung University, HsinChu, Taiwan.
Publication List
1. H. C. Liang, Ross C. C. Chen, Y. J. Huang, K. W. Su, and Y. F. Chen, “Compact efficient multi-GHz Kerr-lens modelocked diode-pumped Nd:YVO4 laser, ” Opt.
Express, 16(25), 21149-21154 (2008).
2. Ross C. C. Chen, Y. T. Yu, Y. J. Huang, C. C. Chen, Y. F. Chen, and K. F. Huang
“Exploring the origin of the directional emission from a microcavity with a large-aperture surface-emitting laser, ” Opt. Lett. 34(12), 1810-1812 (2009).
3. C. C. Chen, Y. T. Yu, Ross C. C. Chen, Y. J. Huang, K.W. Su, Y. F. Chen, and K.
F. Huang, “Transient Dynamics of Coherent Waves Released from Quantum Billiards and Analogous Observation from Free-Space Propagation of Laser Modes, ” Phys. Rev. Lett. 102, 044101 (2009).
4. J. F. Chen, Ross C. C. Chen, C. H. Chiang, Y. F. Chen, and Y. H. Wu, “Bimodel onset strain relaxation in InAs quantum dots with an InGaAs capping layer, ” Appl. Phys. Lett. 97, 092110, (2010).
5. J. F. Chen, Ross C. C. Chen, C. H. Chiang, M. C. Hsieh, and Y. C. Chang,
“Compensation effect and differential capacitance analysis of electronic energy band structure in relaxed InAs quantum dots,” J. Appl. Phys. 108, 063705 (2010).
6. C.H. Chiang, Y.H. Wu, M.C. Hsieh, C.H. Yang, J.F. Wang, Ross C.C. Chen, L.
Chang, and J. F. Chen, “Improving the photoluminescence properties of self-assembled InAs surface quantum dots by incorporation of antimony” Appl.
Surf. Sci. 257, 8784–8787 (2011).
7. Meng-Chien Hsieh, Jia-Feng Wang, Yu-Shou Wang, Cheng-Hong Yang, and
Ross C. C. Chen, “Role of the N-related localized states in the electron emission
properties of a GaAsN quantum well, ” J. Appl. Phys. 110, 103709 (2011).120
8. J. F. Chen, Y. C. Lin, C. H. Chiang, Ross C. C. Chen, and Y. F. Chen, “How do InAs quantum dots relax when the InAs growth thickness exceeds the dislocation-induced critical thickness?, ” J. Appl. Phys. 111, 013709 (2012).
9. Ross C. C. Chen, Y. T. Yu, K. W. Su, J. F. Chen, and Y. F. Chen, “Exploration of water jet generated by Q-switched laser induced water breakdown with different depths beneath a flat free surface,” Opt. Express, 21, 445-453 (2013).
10. Ross C. C. Chen, Y. T. Yu, K. W. Su, and Y. F. Chen, Phys. Rev. E (be prepared).
121