Observation of quantum-classical correspondence from high-order transverse patterns

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Observation of quantum-classical correspondence from high-order transverse patterns

Y. F. Chen,*Y. P. Lan, and K. F. Huang

Department of Electrophysics, National Chiao Tung University, Hsinchu, Taiwan, Republic of China

共Received 23 October 2002; published 6 October 2003兲

We experimentally observe the formation of high-order transverse patterns in a laser resonator with a high degree of frequency degeneracy. It is found that the transverse patterns are well localized on the Lissajous orbits. The connection between the wave functions and the classical periodic orbits is analytically constructed by using the representation of SU共2兲 coherent states. With this connection, the observed transverse patterns are reconstructed very well. The nice reconstruction suggests that the laser resonators can be deliberately designed to attain a more thorough understanding of the quantum-classical connection.

DOI: 10.1103/PhysRevA.68.043803 PACS number共s兲: 42.60.Jf, 03.65.⫺w, 05.45.Mt

It is well known that the paraxial wave equation for the spherical laser resonators has the identical form with the Schro¨dinger equation for the two-dimensional 共2D兲 har-monic oscillator关1,2兴. The eigenfunction of the 2D quantum harmonic oscillator can be analytically expressed as a Hermite-Gaussian function with Cartesian symmetry共x,y兲 or a Laguerre-Gaussian function with cylindrical symmetry (r,␾) 关1–3兴. Since the functional forms of the 2D quantum oscillator and the spherical resonators are similar, the higher transverse modes of the spherical resonators can be in terms of Hermite-Gaussian 共HG兲 modes or Laguerre-Gaussian

共LG兲 modes.

The wave functions of HG mode native to a spherical resonator are given by

⌽m,n共HG兲_共x,y;␼_o兲⫽ 1

冑

2m⫹n⫺1_␲_m!n! 1 ␼o Hm

冉

&x ␼o

冊

Hn

冉

&y ␼o

冊

⫻exp

冋

⫺共x 2_⫹y2_兲 ␼o2

册

, 共1兲

with the resonance frequencies

␯l,m,n⫽l共⌬␯L兲⫹共m⫹n⫹1兲共⌬␯t兲, 共2兲

where Hn(•) is a Hermite polynomial of order n, ␼o is the

laser beam waist, l is the longitudinal mode index, m and n are the transverse mode indices, ⌬␯L is the longitudinal

mode spacing, and⌬␯Tis the transverse mode spacing. For a

plano-concave resonator, as shown in Fig. 1, the transverse mode spacing is given by

⌬␯T⫽⌬␯L

冋

1

␲cos⫺1

冉

冑

1⫺

d

R

冊册

, 共3兲

where d is the cavity length and R is the radius of curvature of the output coupler. Recently, we use a doughnut pump profile to generate the LG0⫾Nmodes in an a-cut Nd:YVO4 laser 关4兴 and to generate the elliptical modes in a c-cut Nd:YVO4laser关5兴. The emission from a-cut Nd:YVO4 crys-tals is linearly polarized, whereas the emission is usually a random polarization for c-cut crystals. The polarization

prop-erty is the main difference between a-cut and c-cut crystals. Therefore, when a linear polarized emission is desired, a-cut crystals are used. The LG₀_⫾N modes are formed by the superposition of the degenerate HG eigenmodes

⌽K,N(HG)_⫺K_{(x,y ;}_␼ o), where K⫽0,1,2, . . . ,N 关3兴 ⌽0,共LG兲⫾N共x,y;␼兲⫽2⫺N/2

兺

K_⫽0 N

冉

N K

冊

1/2 共⫾i兲K_⌽K,N ⫺k 共HG兲 _共x,y;␼_o兲. 共4兲

To generate LG0⫾N modes we setup the resonator length to be as short as possible for reaching single-longitudinal mode operation and ⌬␯LⰇ⌬␯T.

As indicated in Eq.共3兲, adjusting the cavity length d may result in the ratio⌬␯L/⌬␯Tto be an integer S. These cavity

configurations constitute a high degree of frequency degen-eracy. From Eq.共2兲 it can be seen that lowering 共raising兲 the longitudinal mode index l by K, while simultaneously raising

共lowering兲 the sum of the transverse mode indices n⫹m by

S⫻K, will leave the frequency unaltered. It has been shown that configurations with a high degree of frequency degen-eracy allow closed geometric trajectories 关6兴. So far, the transverse modes suited in a frequency degenerate cavity are focused on one dimension关7兴. Here we use the cavity shown

*Email address: [email protected]

FIG. 1. Schematic of a fiber-coupled diode-end-pumped micro-chip laser; a typical pump profile of a fiber-coupled laser diode away from the focal plane; the cavity length d is set at ⌬␯L/

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in Fig. 1 to investigate 2D transverse modes in a frequency degenerate cavity with an a-cut Nd:YVO4 microchip laser. The radius of curvature of the output coupler is 10 mm. The pump radius on the crystal is controlled to be around ␼p

⫽0.16– 0.22 mm. Experimental measurement reveals that

the thickness of the pump ring can be approximately ex-pressed as⌬␼p⫽0.025⫹0.16␼p (mm). In other words, the thickness of the pump profile relatively increases with in-creasing the pump radius. The fundamental mode size in the present cavity is ␼o⬇0.04 mm. From the formula Fr

⫽␻p

2

/(␲␻₀2), the Fresnel number can be estimated to be 5–10. Since the emission from an a-cut crystal is naturally linearly polarized, the experimental pattern is a pure scalar field. Slightly adjusting the cavity length in the vicinity of

⌬␯L/⌬␯T⫽3 and controlling the pump spot size, several

typical sharp patterns on the concave mirror are obtained and shown in Fig. 2. The sharp patterns outside the resonator are found to be preserved in free-space propagation. The preser-vation of the experimental patterns consists of the property that HG modes remain HG field patterns as they propagate. The incident angle of the pump beam is controlled to be within⫾5° with respect to the longitudinal axis. The critical points to obtain a locked pattern consist in the adjustment of the cavity length and the use of the doughnut pump profile. The fine modification of the pump angle can further enhance the sharp patterns, however, it is very difficult to precisely define the optimal incident angle for each pattern. Although each different sharp pattern is obtained at a different cavity length, the change of the cavity lengths is rather short. The difference between the cavity lengths of two sharp patterns is approximately 30 ␮m. The difference of the cavity lengths for different sharp patterns mainly arises from the fact that the effective cavity length depends on the order of the trans-verse mode, even though the dependence is very weak. Spe-cifically there is unavoidable astigmatism in the present cav-ity because of the thermal lensing effect and anisotropic properties of the gain medium. Astigmatism-induced split-ting of the two degenerate mode frequencies can lead to a significant influence on laser dynamics关4兴. Here we believe that the inevitable astigmatism plays an important role not only for the dependence of the effective cavity length on the transverse order but also for the origin of the symmetry

2ប/(my␻y). The

ei-genvalues associated with the eigenfunctions

⌽m,n(x,y ;␼x,␼y) are given by

Em,n⫽

冉

m⫹1

2

冊

ប␻x⫹

冉

n⫹ 1

2

冊

ប␻y. 共7兲 As is well known, the classical trajectory for 2D anisotropic harmonic oscillator with commensurate frequencies is a pe-riodic orbit, called a Lissajous figure关9兴. However, the con-ventional eigenstates⌽m,n(x,y ;␼x,␼y) do not manifest the

characteristics of classical periodic orbits even in the corre-spondence limit of large quantum numbers.

FIG. 2. Experimental results for the typical transverse patterns observed in a cavity length near⌬␯L/⌬␯T⫽3.

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Recently, the wave functions associated with the classical elliptical trajectories in a 2D isotropic harmonic oscillator have been analytically constructed by using the representa-tion of SU共2兲 coherent states 关10,11兴. Mathematically, the SU共2兲 coherent states is a superposition of degenerate eigen-states. Here we find that the wave functions related to the Lissajous figures can be constructed by coherent states simi-lar to the SU共2兲 representation. Consider a 2D anisotropic harmonic oscillator with frequencies in the ratio ␻_x:␻_y

⫽q:p, where p and q are integers. The eigenvalues can be

rewritten in the form

Em,n⫽

冋冉

m⫹1

2

冊

q⫹

冉

n⫹ 1

2

冊

p

册

ប␻, 共8兲 where ␻ is the common factor of the frequencies ␼x and ␼y. For q:p anisotropic quantum oscillators, it is explicit

that a family of the eigenstates ⌽pK,q(N_⫺K)(x,y ;␼x,␼y)

with K⫽0,1,2, . . . ,N are degenerate for a given index N and the eigenvalue of these eigenstates is given by EN⫽关pqN

␼y 2 2 . 共12兲

Note that

具

x2

典

and

具

y2

典

are, respectively, the expectation values of x2 and y2 for the coherent state

⌿N p,q

(x, y ;␼x,␼y,Aei␾). Figure 3 shows a comparison

be-tween the SU共2兲 coherent states and the classical Lissajous figures for the frequency ratio of 2:1, 3:2, and 4:3 with A

⫽1, ␾⫽␲/2, ␼x⫽␼y, and N⫽30. It can be seen that the

distributions of 兩⌿N p,q

(x,y ;␼x,␼y,Aei␾)兩2 are in good

agreement with the classical periodic orbits. Moreover, the behavior of 兩⌿_Np,q(x, y ;␼x,␼y,Aei␾)兩2 illustrates

geometri-cally Bohr’s correspondence principle: the velocity of the classical particle is at a minimum at the apogees of the mo-tion, and therefore the probability density has a peak at these points.

Although the number of eigenstates used in the coherent state⌿_Np,q(x,y ;␼x,␼y,Aei␾) is N⫹1, the number of

domi-nant eigenstates for wave localization is rather small for high-order states. To manifest the efficiency of wave local-ization, we modify⌿_Np,q(x,y ;␼x,␼y,Aei␾) to define a

par-tially coherent state as

⌿N, M p,q _共x,y;␼x ,␼y,Aei␾兲 ⫽

冋

兺

K⫽J N⫺J

冉

N K

冊

A 2

册

⫺1/2

冋

K

兺

⫽J N⫺J

冉

N K

冊

1/2 AKeiK␾ ⫻⌽pK,q共N⫺K兲共x,y;␼x,␼y兲

册

, 共13兲

where the index M⫽N⫺2J⫹1 represents the number of eigenstates used in the state ⌿_{N, M}p,q (x, y ;␼x,␼y,Aei␾).

Numerical analyses reveal that the transverse pattern shown in Fig. 2 can be nicely explained by the partially coherent state in Eq. 共13兲. As mentioned earlier, the forma-tion of the observed patterns is a cooperative frequency lock-ing among different transverse orders with the help of differ-ent longitudinal orders. For a cavity near⌬␯L/⌬␯T⫽3, the

FIG. 3. A comparison between the coherent states and the clas-sical Lissajous periodic orbits for p:q to be 2:1, 3:2, and 4:3.

共a兲–共c兲 The coherent states calculated with Eq. 共9兲 and ␶ ⫽exp(i␲/2) and N⫽30. 共d兲–共f兲 The coherent states calculated

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family of the transverse modes ⌽pK,q(N(HG) _⫺K)(x, y ;␼o) with a

given index N can be frequency locked by a different longi-tudinal index l⫽L⫿K with a given index L, where K

⫽0,1,2,...,N and p⫺q⫽⫾3. Substituting m⫽pK, n⫽q(N ⫺K), l⫽L⫿K, p⫺q⫽⫾3, and ⌬␯L/⌬␯T⫽3 into Eq. 共2兲,

the laser frequency of the family ⌽pK,q(N(HG) _⫺K)(x,y ;␼o) can be found to be ␯L,N⫽L(⌬␯L)⫹(qN⫹1)(⌬␯T) independent

of K. From the numerical analysis, the transverse patterns shown in Fig. 2 are found to be associated with the partially coherent states in Eq. 共13兲 with A⫽1 and ␾⫽0. In other words, the wave functions related to the observed patterns can be in terms of HG modes as

U_{N, M}p,q 共x,y兲⫽

冋

兺

K⫽J N_⫺J

冉

N K

冊

册

⫺1/2

冋

K

兺

⫽J N_⫺J

冉

N K

冊

1/2 ⫻⌽共HG兲pK,q共N⫺K兲共x,y;␼o兲

册

共14兲

with p⫺q⫽⫾3. Note that for a spherical cavity, the trans-verse components of the resonance frequencies are not de-generate for each eigenstate in Eq.共14兲 except for p⫽q. For the Kth eigenstate of the coherent state U_{N, M}p,q (x,y ), the transverse components of the resonance frequencies are

关qN⫹1⫹(p⫺q)K兴⌬␯T. If the condition of p⫺q⫽⫾3 is

satisfied, then the frequency of each eigenstate in the coher-ent state U_{N, M}p,q (x, y ) can be possibly locked with the help of different longitudinal component of (L⫿K)⌬␯L because of ⌬␯L/⌬␯T⫽3. To be brief, the formation of the present

trans-verse patterns is a spontaneous frequency locking in the 2

⫹1 dimension, not a pure transverse mode locking. Figure 4

shows the numerically reconstructed patterns for the results shown in Fig. 2, calculated with Eq.共14兲 for several cases of p⫺q⫽⫾3. It is clear that only 3–5 eigenstates are already sufficient to localize the wave patterns on the classical tra-jectories, even for high-order periodic orbits. The present analysis indicates that the wave function obtained as a linear superposition of a few degenerate eigenstates can provide a more physical description of a phenomenon than the true eigenstates in mesoscopic systems关12兴. The good agreement between the experimental and reconstructed patterns con-firms that the interrelation between wave optics and geo-metrical optics is somewhat similar to that between quantum and classical mechanics. Such an analogy enables us to em-ploy quantum theory in analyzing the formation of high-order laser transverse modes.

In conclusion, we have used the representation of the SU共2兲 coherent state to make a connection between the wave functions and the classical trajectories in a 2D anisotropic oscillator with commensurate frequencies. With the analyti-cal wave function, the experimental transverse patterns asso-ciated with the Lissajous trajectories can be explained very well. The nice explanation suggests that the laser resonators with an identical functional form can be used to attain a more thorough understanding of the quantum-classical connection. It is worthwhile to mention that a similar phenomenon is found in a nonintegrable classical system; the wave patterns of the eigenstates are usually concentrated along unstable periodic orbits instead of being randomly distributed 关13– 15兴. In addition, there are some striking phenomena in open quantum ballistic cavities associated with the wave functions in terms of classical periodic orbits 关16–18兴. Therefore, to construct the connection between the wave functions and classical periodic trajectories is not only useful for explain-ing the present transverse patterns but also helpful for under-standing quantum-classical correspondence as well as the quantum transport in mesoscopic systems. Recently, Doya et al.关19,20兴 have introduced the paraxial approximation to establish an analogy between light propagation along a mul-timode fiber and quantum confined systems. We believe that these analogies will continue to be exploited for understand-ing the physics of mesoscopic systems.

The authors thank the National Science Council for finan-cially supporting this research under Contract No. NSC-91-2112-M-009-030.

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FIG. 4. The numerically reconstructed patterns for the results shown in Fig. 2, calculated with Eq. 共14兲. 共a兲兩U_20,54,1(x,y )兩2_; _共b兲

兩U16,5 5,2 (x,y )兩2_;_{共c兲兩U} 10,3 6,3 (x,y )兩2_;_{共d兲兩U} 10,3 7,4

(x,y )兩2_;_{共e兲兩U} 8,3 8,5 (x,y )兩2_; 共f兲兩U8,3 9,6 (x,y )兩2.

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