Chapter 4 Formation of Centrally Focused Beam via Intracavity Second
4.2 Experimental Observations
4.2.2 Generation of Centrally Focused Beams
Figure 4.2.2 and Fig. 4.2.3 display respectively the far-field patterns of a variety of fundamental standing-wave LG0,l modes and their frequency-doubled counterparts.
The experimental observations show good agreement with the theoretical results as illustrated in Fig. 4.1.1 and Fig. 4.1.2. Moreover, the formation of the centrally focused beam corresponding to Fig. 4.1.3 has been confirmed via the measurement of
the transverse intensity profiles at different z planes as shown in Fig. 4.2.4. The experimental tomographic transverse patterns are found to be in accordance with the theoretical simulations in Fig. 4.1.3. It is worth to mention that the focusing phenomenon is a linear interference of the wave itself as we have validated in previous section and not the result of the nonlinear self-focusing effect. Since focused beams have always been a subject of practical interest, the second-harmonic waves presented here might be utilized for further applications.
Cavity mirror
Gain medium
Screen
CCD camera Focusing lens
Fiber coupled laser diode
KTP
Output coupler Filter
Fig. 4.2.1 Experimental setup of the diode-pumped solid-state laser with intracavity SHG.
0 , 11
0 , 12
0 , 17
0 , 20
0 , 1
0 , 2
0 , 3
0 , 4
p l,
p l,
(a)
(c) (b)
(d)
(e)
(g) (f)
(h)
Fig. 4.2.2 Observed far-field patterns of the standing-wave LG0,l modes at the fundamental wavelength.
0 , 1
0 , 2
0 , 3
0 , 4
0 , 11
0 , 12
0 , 17
0 , 20
(a)
(c) (b)
(d)
(e)
(g) (f)
(h)
p l,
p l,
Fig. 4.2.3 Frequency-doubled counterparts of the fundamental standing-wave LG0,l
modes in Fig. 4.2.2.
0
Rz z
0.2 z
R0.4 z
R0.6 z
R0.8
Rz z
2 z
R4 z
R10 z
R(a)
(c) (b)
(d)
(e)
(g) (f)
(h)
Fig. 4.2.4 Observed transverse intensity profiles along the longitudinal axis.
Reference
[1] L. Allen, M.W. Beijersbergen, R.J.C. Spreeuw, J.P. Woerdman, Phys. Rev. A 45 8185 (1992).
[2] K. T. Gahagan and G. A. Swartzlander, Jr. Opt. Lett. 21, 827 (1996).
[3] H. Adachi, S. Akahoshi, and K. Miyakawa, Phys. Rev. A 75, 063409 (2007).
[4] A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[5] P. Senthilkumaran, Appl. Opt. 42, 6314 (2003).
[6] G. Grynberg, A. Maitre, and A. Petrossian, Phys, Rev. Lett. 72 2378 (1994).
[7] Y. C. Lin, T. H. Lu, K. F. Huang, and Y. F. Chen, Opt. Express 19 10293 (2011).
[8] C. Green, G. B. Mindlin, E. J. D’Angelo, H. G. Solari, and J. R. Tredicce, Phys.
Rev. Lett. 65, 3124 (1990).
[9] S. F. Pereira, M. B. Willemsen, M. P. van Exter, and J.P. Woerdman, Appl. Phys.
Lett.73, 2239 (1998).
[10] Q. Deng, H. Deng, and D. G. Deppe, Opt. Lett. 22, 463 (1997).
[11] Y. F. Chen, and Y. P. Lan, Phys. Rev. A 63, 063807 (2001).
[12] M. P. Thirugnanasambandam, Yu. Senatsky, and K. Ueda, Laser Phys. Lett. 7, 637 (2010).
[13] A. Ito, Y. Kozawa, and S. Sato, J. Opt. Soc. Am. A 27, 2072 (2010).
[14] K. Dholakia, N. B. Simpson, and M. J. Padgett, Phys. Rev. A 54, R3742 (1996).
[15] J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, Phys. Rev. A 56, 4193 (1997).
[16] T. Hasegawa, and T. Shimizu, Opt. Commun. 160, 103 (1999).
[17] A. Ashkin and J. M. Dziedzic, Science 235, 1517 (1987).
[18] P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N.
Christodoulides, and Z. Chen, Opt. Lett. 36 2883 (2011).
[19] D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, Opt.
Lett. 36 1842 (2011).
[20] A. E. Siegman, Lasers (University Science, 1986).
[21] N. N. Lebedev and R. A. Silverman, Special functions and their applications, (Prentice-Hall, 1965).
Chapter 5
Weak Localization in Disordered Systems with Conical Second
Harmonic Generation
5.0 Introduction
Wave localization, which originally results from the peculiar interference of waves scattered by disorders, is an intriguing phenomenon beyond diffusion theory and transfer treatment [1-3]. Since the fundamental processes of scattering and interference are identical for classical and quantum waves, the phenomena of wave localization have been extensively investigated in different physical systems [4-7].
Recent developments have led to much interest in various disordered media specified by weak (WL) [8-14] or strong localization (SL) [5,15-17]. It could be found [1-17]
that the localization phenomenon is still an important issue and deserves further investigations.
Theoretical analyses and experimental observations for the disordered wave functions are the straightforward procedures to determine the extent of wave localization. Numerous theoretical models [18-22] have been constructed to explore the extent of wave localization. Recently, the nonlinear sigma models based on the supersymmetry theory have been employed to investigate the statistical properties of disordered wave functions [22]. The zero-dimensional (0D) nonlinear sigma model has been shown to be equivalent to the random matrix method [22] in the diffusive limit of disordered systems. In the weakly disordered systems, the wave functions are
widely spread over space, corresponding to the so-called extended states. With the one-dimensional (1D) nonlinear sigma model, the density distributions of the extended states can be expressed as an analytical formula related to the well-known Porter-Thomas (P-T) distribution [23]. On the other hand, the wave functions of the strongly disordered systems display log-normal asymptotic forms and long-tail characteristics in the density distributions [24,25], corresponding to the so-called pre-localized states. Fal’ko and Efetov [20] developed the reduced version of the nonlinear sigma model (RV-NLS model) to analyze the long-tail density distributions of the pre-localized states. Although the RV-NLS model seems to be applicable to quantify the varying extent of WL, detailed comparisons with experimental observations have not been performed as yet.
In experiments, the disordered wave functions were measured in a microwave cavity to show the influence of chaos and localization in disordered quantum billiards [13]. In 2006, Chen et al. [26] demonstrated the spatial structure of two-dimensional (2D) disordered wave functions from exploring the near-field patterns of conical second harmonic generation (SHG) in a GdCa4O(BO) (GdCOB) nonlinear crystal with moderate defect domains. So far, experimental results for the disordered wave functions only covered a partial WL regime and did not provide a comprehensive analysis of the transition from extended to pre-localized states.
In this chapter we experimentally generate the 2D disordered wave functions by systematically scanning a GdCOB nonlinear crystal in the conical SHG process to explore the characteristics of WL. We numerically confirm that the RV-NLS model model can provide statistical analyses to agree very well with the experimental wave functions with various localizations. Furthermore, we find that the density distributions of the disordered wave functions can be analytically expressed as the chi-square distributions with fractional parameters. Since the parameters in the formal
expression of chi-square distributions are only integers for the integral degrees of freedom [27], we use the terminology of fractional chi-square distribution to distinguish the difference. Finally, we construct the relationship between the RV-NLS model and the fractional chi-square distributions to reveal the characteristics of the fractional degrees of freedom in the disordered wave functions. Although the present results focus on the regime of WL, the fractional chi-square distribution might be useful for the full crossover of localization. We also believe that the present model can be employed to study the degree of localization in various disordered systems [8-14]
such as scattering powder, cold atoms, randomized laser materials, liquid crystal, scattered systems, microcavities, and graphene.
5.1 Experimental Observations
5.1.1 Experimental Setup and Results
Figure 5.1.1 shows the experimental setup that is a diode-pumped actively Q-switched Nd: YAG laser with intracavity SHG in the GdCOB crystal. The gain medium is a 0.8-at. % Nd3+:YAG crystal with a length of 10 mm. The GdCOB crystal was cut for type Ⅰ frequency doubling in the XY planes (θ = 90°, φ = 46°) with a length of 2 mm and a cross section of 3 mm3 mm. All crystals were coated for antireflection (R<2%) at 1064 nm on their both sides. The radius of curvature of the concave-front mirror is 50 cm with coating of antireflection (R<0.2%) at 808 nm, high-reflection (R>99.8%) at 1064 nm, and 532 nm on the entrance side and high-transmission (T>90%) at 808 nm on the other side. The output coupler is a plant mirror with coating of high-reflection (R>99.8%) at 1064 nm and high-transmission at 532 nm (T>85%). The pump source is a 10 W 808 nm fiber-coupled laser diode with a core diameter of 800 µm. A focusing lens with a focal length of 2.5 cm and 90% coupling
efficiency was employed to reimage the pump beam into the laser gain medium. The acoustic-optic Q switch with a length of 30 mm has coating with antireflection at 1064 nm on both sides and is driven at a 27.12-MHz center frequency with 15.0 W of rf power. An object lens was used to reimage the near-field patterns on the screen.
It has been shown that GdCOB crystals possess various random defect domains which can be used to generate the intensities
r 2of 2D disordered wave functions in the SHG process [26]. Here we find that the extent of random defect domains significantly depend on the transverse position of the GdCOB crystal. With this feature, we can scan all transverse positions of the GdCOB crystal to generate a variety of disordered wave functions from extended to pre-localized states as shown in Figs. 5.1.2(a)-5.1.2(f).5.2 Statistical Analyses
5.2.1 Porter-Thomas Distribution and 1D Nonlinear Sigma Model
To determine the extent of localization, the density probability distribution
2
P r is illustrated to specify the localization of wave functions. For extended states in quantum chaotic systems, random-matrix method and equivalent 0D nonlinear σ model have been verified to give good explanations of universal statistic behaviors with the P-T distribution [22]. For weakly disordered systems, density probability of the normalized disordered wave functions can be expressed with 1D nonlinear σ model as [19,22,26]
Fig. 5.1.1 Experimental setup for the generation of disordered wave functions with the diode-pumped Q-switched Nd:YAG laser of intracavity SHG in the GdCOB crystal.
Screen
Front mirror
Nd:YAG Laser diode
Focusing lens A.O. Q-switch
Output Coupler
GdCOB Object lens
Screen
Front mirror
Nd:YAG Laser diode
Focusing lens A.O. Q-switch
Output Coupler
GdCOB Object lens
Fig. 5.1.2 (a)-(f) Experimental observation of near-field wave patterns measured at different transverse positions of the GdCOB crystal.
a
c d
b
e f
440 m
a
c d
b
e f
440 m
where PPT I expI 2 2I is the expression of the P-T distribution, and IPR
I d r2 2 is the inverse participation ratio associated with the extent of localization. For P-T distribution, the IPR can be directly achieved to be
2
IPR
0I PPT I dI3.0 indicating the chaotic systems. The larger the IPR value, the stronger the extent of localization is. As a result, the IPR values for disordered systems are greater than 3.0 in general. The IPR values for the experimental data in Fig. 5.2.1(a) and 5.2.1(b) are 3.3 and 5.72, respectively. Evidently, the fitting curve of 1D nonlinear σ model is validated to be consistent with the experimental data which displays small deviation to the P-T distribution in Fig. 5.2.1(a). However, as depicted in Fig. 5.2.1(b), the use of the perturbative result according to the 1D nonlinear σ model is violated obviously in the region where the deviation from the P-T distribution is considerable. Negative quantities of the density distribution P I
canbe obtained for IPR values greater than 7.0. We numerically confirm that the 1D nonlinear σ model is only appropriate for the disordered wave function with IPR<5.5.
For stronger disorder, higher densities of the distribution functions decay more slowly in the region where 1D nonlinear σ models break down. Therefore, a more appropriate model should be given to clarify the varying extent of localization.
5.2.2 Reduced Version of the Nonlinear Sigma Model
In the following, we employ the experimental data to testify the RV-NLS model that is developed to quantitatively specify different regimes of localization. The RV-NLS model indicated by a dimensionless parameter g is given by [20,22]
||2 1D nonlinear model (IPR=5.72) 1D nonlinear model (IPR=3.3)
a
b
Fig. 5.2.1 (a)-(b) The density distribution P(I) according to experimental data in Fig.
5.1.2(a) and 5.1.2(c), respectively.
where A is a normalized constant, z I
could be solved numerically according to the relation z ez I g, and g is the dimensionless conductance [2,3] used to identify the degree of localization. The parameter g is also called the “Thouless number” which first proposed by Thouless in the discussion on the scaling theories of localization [2,3]. The dimensionless conductance g is adopted by the scaling theory as its only parameter and depends on the dimensionality of the system. For 2D case,
~ ln
g k l L l [2] where k is the wave vector, k2
, l signifies the value of mean free path, and L denotes the size of the system. The formal definition of g is
2 2
g G L e (Ref. 2) where G L
is the conductance of a hypercube of size
Ld, d relates to the dimensionality, is Plank’s constant , and e is the electronic charge. In the diffusive limit of g 1, the density distribution reveals a universality of the statistics of localized waves. The value of g is substantially decreased due to WL which is the precursor of Anderson localization (SL) of g1 [28]. In other words, the scaling parameter g can be exploited to specify the extent of localization for the experimental results. Figures 5.2.2(a)- 5.2.2(f) depict the numerical results of the RV-NLS model for the best fits to the wave patterns shown in Figs.5.1.2(a)-5.1.2(f), where the values of g are found to be 33, 11, 5.5, 3.5, 2.3, and 1.1, respectively. It can be seen that the density distributions generated with the RV-NLS model agree very well with the experimental results for all cases. Actually, K. B.
Efetov [22] has once bought up the idea that the RV-NLS model can be applied to explain the statistical behavior for the disordered wave functions in a microwave cavity [13]. Employing the laser system with the conical SHG operation, we have verified here the practicability of the RV-NLS model in another disordered system.
The fact implies possible extension of RV-NLS model on the studies of different extent of localization in various kinds of disordered systems.
5.2.3 Fractional Chi-square Distribution
Besides the verification of the RV-NLS model, we originally find that the chi-square distributions with fractional parameters can satisfactorily describe the experimental results. The analytic expression of the chi-square distributions is given by [27]
2
where v
0
is a parameter referred to the number of degrees of freedom and
v2
1
is the gamma function which serves to normalize the density distributionsP
CS( vI
; ). The P-T distribution PPT(I )
is the chi-square distribution with one degree of freedom, i.e.P
CS(I
;v
1) [23]. In addition, the exponential distribution exp( -I ) can be referred to the chi-square distribution with two degrees of freedom, i.e.P
CS(I
;v
2). Even though there is no conceptual difficulty to extend an integer value of v to a non-integer, it has not been confirmed that whether non-integer degrees of freedom have any applications in nature. As shown in Figs.5.2.2(a)–5.2.2(f), the chi-square distributions with 0.06 , almost identical to
v
1 the features of the RV-NLS model, can excellently illustrate the experimental results.The values of v for experimental wave patterns in Figs. 5.1.2(a)-5.1.2(f) are 0.774, 0.54, 0.32, 0.20, 0.126, and 0.06, respectively. The evidence shows that the tails of the density distribution decay more slowly at small values of v and the degree of localization becomes larger while the values of v decrease rapidly. The investigation yields a clear result that the fractional chi-square distribution could be a powerful procedure for analyzing the statistical properties of the localization phenomena. It is well-known that the non-integer dimensionality is an important property of most fractals. Our exploration reveals that non-integer or fractional parameters are also valid concepts in statistical distributions of disordered wave functions.
Fig. 5.2.2 (a)-(f) Experimental and theoretical density distributions P(I) corresponding to experimental data in Fig. 5.1.2(a)-5.1.2(f), respectively.
||2
5.2.4 Relation between Statistical Models
The validity and equivalence between the density distributions
P
CS( vI
; ) 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 g0.85] 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
asg
to indicate no WL effects and the other is vc0.06 with g1 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 ofc 1
v v and 0 v vc, 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.
Reference
[1] P. W. Anderson, Phys. Rev. 109, 1492 (1958).
[2] E. Abrahams, P. W. Anderson, D. C. Licciardelleo, and T. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979).
[3] P. W. Anderson, D. J. Thouless, E. Abrahams, and D. S. Fisher, Phys. Rev. B 22, 3519 (1980).
[4] M. Kaveh, M. Rosenbluh, I. Edrei, and I. Freund, Phys. Rev. Lett. 57, 2049 (1986).
[5] J. Billy, V. Josse, Z. Zho, A. Bernard, B. Hambrecht, P. Lugan, D. Clément, L.
Sanchez-Palencia, P. Bouyer, and A. Aspect, Nature, 453, 891 (2008).
[6] D. Laurent, O. Legrand, P. Sebbah, C. Vanneste, and F. Mortessagne, Phys. Rev.
Lett. 99, 253902 (2007).
[7] H. Hu, A. Strybulevych, J. H. Page, S. E. Skipetrov, and B. A. Van Tiggelen, Nature, 4, 945 (2008).
[8] D. S. Wiersma, M. P. van Albada, B. A. van Triggelen, and A. Lagendijk, Phys.
Rev. Lett. 74, 4193 (1995).
[9] P. E. Wolf, and G. Maret, Phys. Rev. Lett. 55, 2696 (1985).
[10] G. Labeyrie, F. De Tomasi, J. C. Bernard, C. A. Muller, C. Miniatura, and R.
Kaiser, Phys. Rev. Lett. 83, 5266 (1999).
[11] R. Sapienza, S. Mujumdar, C. Cheung, A. G. Yodh, and D. Wiersma, Phys. Rev.
Lett. 92, 033903 (2004).
[12] F. V. Tikhonenko, D. W. Horsell, R. V. Gorbachev, and A. K. Savchenko, Phys.
Rev. Lett. 100, 056802 (2008).
[13] A. Kudrolli, V. Kidambi, and S. Sridhar, Phys. Rev. Lett. 75, 822 (1995).
[14] M. Gurioli, F. Bogani, L. Cavidli, H. Gibbs, G. Khitrova, and D. S. Wiersma,
Phys. Rev. Lett. 94, 183901 (2005).
[15] S. John, Phys. Rev. Lett. 58, 2486 (1987).
[16] T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature, 446, 52 (2007).
[17] G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Nature, 453, 895 (2008).
[18] A. D. Mirlin, and Y. V. Fyodorov, J. Math. A: Math. Gen. 26, L551 (1993).
[19] Y. V. Fyodorov, and A. D. Mirlin, Phys. Rev. B, 51, 13 403 (1995).
[20] V. I. Fal′ko, and K. B. Efetov, Phys. Rev. B, 52, 17 413 (1995).
[21] A. D. Mirlin, Phys. Rev. B, 53, 1186 (1996).
[22] K. B. Efetov, Supersymmetry in Disorder and Chaos (Cambridge University Press, Cambridge, 1997).
[23] C. E. Porter, and R. G. Thomas, Phys. Rev. 104, 483 (1956).
[24] I. I. Kogan, C. Mudry, and A. M. Tsvelik, Phys. Rev. Lett. 77, 707 (1996).
[25] A. Ossipov, T Kottos, and T. Geisel, Phys. Rev. E, 65, 055209 (2002).
[26] Y. F. Chen, K. W. Su, T. H. Lu, and K. F. Huang, Phys. Rev. Lett. 96, 033905 (2006).
[27] G. G. Roussas, A Course in Mathematical Statistics (Academic Press, California, 1997).
[28] A. Lagendijk, B. van Tiggelen, and D. S. Wiersma, Phys. Today, 62, 24 (2009).
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
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