1. Introduction

In 1970, Ramsay and Degnan showed that a laser cavity configuration with a high degree of frequency degeneracy allows closed geometric trajectories [1]. Such special laser operations have been investigated in semiconfocal cavities and the so-called W mode and the M mode were reported [2–5]. In [4], the authors combined the ray and the wave optics to understand the W mode. They explained the observed fringes in the W mode are the result of interference between two beams overlapping. It is interesting that the relation between the geometric trajectories and the wave distributions can provide analogous insight into the quantum transport in mesoscopic systems [6]. Thus, the W mode was constructed by using SU(2) representation of quantum-mechanics formulism in [5]. Even so, the geometric modes are alternatively viewed as multibounce Gaussian beams traveling in closed off-axis trajectories. Following this thinking, Visser et al. used both an operator and a geometric argument to obtain a wave description by considering the propagation of an off-axial Gaussian beam inside the cavity [7]. They showed that the interference fringes come from only the beams that propagate in the same direction, which differs from the previous research.

When we apply the two wave descriptions of [5] and [7] for the W mode, M mode, and the so-called N mode (see Fig. 3(b) of [7]) in this paper, respectively, nearly the same trajectories and interference fringes can be obtained from the two. For another geometric mode (see Fig. 3(a) of [7]), named VW mode, the interference fringes are absent on the end mirrors of the three-fold degenerate symmetric cavity. However, we found that the fringes should appear there according to SU(2) representation. The difference of these two descriptions needs to be further clarified via observing the VW mode. In this paper, we numerically study the VW mode by using three methods, the aforementioned two theoretical representations and the Fox-Li approach based on Collins diffraction integral together with a rate equation to compare with experimentally generated VW mode. We will show that the VW mode can be generated by three pumping beams in an end-pumped Nd:YVO₄ laser operated near the 1/3-degeneracy with a plano-concave cavity. By observing the transverse patterns of the VW mode, we conclude the operator method in [7] can be extended so that the field of a geometric mode should include simultaneously those of the reverse directional trajectories. On the other hand, the SU(2) representation in [5] is a special one that can not produce the fringes within some transverse patterns. In section 2, we compare the three numerical results for the VW mode. In section 3, the experimental result is compared with our simulations. The conclusions are stated in section 4.

2. The numerical results

According to ray tracing in a plano-concave cavity with three-fold degeneracy or g₁g₂ = 1/4, six-bounce or three round-trip rays will construct a closed trajectory. We plot a specific one that is symmetric with the optical axis in Fig. 1(a), in which z = 0 is the position of the flat end mirror. In Fig. 1(a), the initial ray is indicated starting at (x, z) = (0, 0) with a slope of 2a/L, then it is reflected back to (x, z) = (a, 0) and subsequently retraces itself after two round trips. Here L is the cavity length and x is the transverse coordinate. Using the operator method [7] and choosing the transverse momentum divided by the total momentum q/k = π/180 and the position vector a = 0, we plot the mode trajectories in Fig. 1(b). This is equivalent to considering a fundamental Gaussian beam with its center at (x, z) = (0, 0) and with a slope q/k that retraces itself after three round trips. We can see that the mode trajectories in Fig. 1(b) are the same as the closed rays in Fig. 1(a), which are like overlapped V and W shapes, so that we call it the VW mode in this paper. The VW mode is similar to Fig. 3(a) of [7] except that the distance between the neighboring spots on z = 0 is half of that on z = 6 cm because the plano-concave cavity is used here. In order to compare with the numerical results below, we have implemented the laser wavelength of 1064 nm, the cavity length of 6 cm, the radius of curvature of the curved mirror of 8 cm, and the fundamental Gaussian beam waist is 108 μm .

On the other hand, using the SU(2) representation [5] according to the Schwinger representation [8,9] for a family of the Hermite-Gaussian (HG) modes, the coherent state is given by,

where τ = exp(iϕ), Φ^(HG) _3p,0(x,y,z) is HG mode with the transverse mode index 3p in x direction and p = 0, 1, 2, ..., n. We depict the trajectory of the VW mode with the total wave function being Φ_n(x,y,z;ϕ=π/2)+Φ_n(x,y,z;ϕ=-π/2) and n = 20 in Fig. 1(c). We can see that Fig. 1(c) and 1(b) have the same geometric trajectories. However, in Fig. 1(b) only two regions where two beams propagate in the same direction have the interference fringes but the fringes on the end mirrors preserve for a long distance in Fig. 1(c) which is absent in Fig. 1(b).

 

Fig. 1. (a). Closed ray trajectories symmetric with the optical axis in a three-fold degenerate plano-concave cavity. (b) VW mode trajectories using the multibounce Gaussian beams model in [7]. (c) VW mode trajectories using the SU(2) representation in [5]. (d) VW mode trajectories using the Collins integral together with a rate equation.

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It is obvious that in Fig. 1(c), the transverse interference fringes at a specified z mainly result from the interference among the high order HG_24,0, HG_30,0, and HG_36,0 modes. However, the fringes along z are due to the Gouy phase differences of these modes. For example, HG_24,0 and HG_36,0 are in phase but HG_24,0 and HG_30,0 are out of phase on the optical axis when $z=\frac{z_R}{\sqrt{3}}=2\mathrm{cm}$, where z_R is the Rayleigh length. When $z=\sqrt{3}{z}_R=6\mathrm{cm}$, the three major modes are again in phase on the optical axis at the curved mirror as at the flat mirror. This leads to that the fringes from z = 0 to 2 cm are similar to those from z = 6 to 2 cm. Such a situation is similar to a bottle beam that comes from the interference of the lower Laguerre-Gaussian modes between LG_0,0 and LG_3,0 [10,11]. On the other hand, the fringes along z in Fig. 1(b) are not related to the Gouy phase but come from the interference of two small-angle tilted co-propagating Gaussian beams. Another difference between these two methods is that the side spot centers on z = 0 can occur at any desired transverse position x for the operator method since the transverse momentum q can be chosen arbitrary within the validity range of the paraxial approximation; however, the two side spots between adjacent order modes of SU(2) are discrete since n is integers.

The numerical model, by using the Collins integral together with a rate equation, can be found in [12]. Here, the Collins integral [13] is written as

where $\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]$ is the round trip transmission matrix, E⁺ _m(x’) and E^- _m+1(x) are the electric fields of the m-th and the (m+1)-th round trips, respectively, at the planes immediately after and before the gain medium; x’ and x are the corresponding transverse coordinates, λ is the wavelength of the laser, 2w is the aperture width on the reference plane that is set on the flat mirror end. To implement the integral by the Romberg method, we divided the 2 mm aperture width into 2048 segments. The used parameters of the gain medium and the cavity configuration are the same as in [12].

Since the spot centers on z = 0 in Fig. 1(c) are at x = -507, 0, +507 μm, we choose the three transverse positions as the pump beam centers. When the three Gaussian pump radii are chosen 108 μm and the pump power ratio are 1: 0.92: 1, the numerical trajectory is shown in Fig. 1(d), in which the scale of x is from -1 mm to +1 mm. We found besides the fringes occurring at the crossings of beams that is similar to Fig. 1(c), they are obvious over all the trajectories in Fig. 1(d). Notice that the fringe spacing of the side spots are 60 μm which are double of the middle spot one of 30μm on z = 0. When the three spots on z = 6 cm are compared with those on z = 0, both the fringe spacing and the spot size are doubled correspondingly. The results are the same as Fig. 1(c) and match with the deduction from the picture of ray and wave optics. As shown in Fig. 1(a), the three angles indicated between two rays intersection at z = 0 are θ,2θ ,θ and those at z = 6 cm are θ/2 ,θ,θ/2. When two beams interfere by a small angle the fringe spacing is inversely proportional to this angle, which explains the data of fringe spacing.

3. Experimental setup and results

The experimental setup is shown in Fig. 2. This laser contains a 1mm-thick a cut 2.0 at% microchip crystal and an output coupler with radius of curvature of 20 cm having 10% transmission at the lasing wavelength of 1064 nm. One face of the crystal facing the pumping beam had a dichroic coating with greater than 99.8% reflection at 1064 nm and greater than 99.5% transmission at the pump wavelength of 808 nm; the other surface was made of antireflection layer at 1064 nm. The pump source was a 1-W fiber-coupled laser diode with a 200μm of core diameter and a numerical aperture of 0.22. A convergent lens with 50 mm focal length and a 20x objective lens were added after the fiber output a distance of 5 cm and 85 cm, respectively to focus the pump beam into the laser crystal. Between the two lenses, the pump beam was split into three beams by two 50/50 beam splitters and two reflective mirrors.

The pump radii were estimated to be ∼150 μm and the neighboring beam centers were separated ∼900 μm apart. After the output coupler we added a transform lens to monitor the beam profile variation with the propagation distance. The laser output was detected with a charge coupled device (CCD).

 

Fig. 2. Schematic diagram of the experimental setup: LD, laser diode; BS, beam splitter; M, reflective mirror; OC, output coupler; LF, line filter; TL, transform lens; other abbreviations defined in text.

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When the cavity length is tuned near the 1/3-degeneration point of 15 cm and the power of the three pump beams are about 120, 70, and 100 mW, the observed VW mode patterns with the propagation distance are shown in Fig. 3(a). First, we see that the right largest 3-spots pattern has obvious fringe structure and the fringe spacing of the middle spot is half of the other two, which agree with the numerical result in the last paragraph of section 2. Second, the 3-spots patterns indicated with arrows in Fig. 3(a) have asymmetric spot sizes. This is because the two pump sizes are not completely the same, resulting from their different propagation distances before going into the gain medium. The little asymmetric pump sizes do not influence the fringe spacing within the spots. Third, we intentionally adjusted the pumping power a little asymmetric so that we can see the 3-spots patterns have asymmetric intensities. Fourth, we see the 3-spots patterns from left to right exhibit upside down twice.

In order to compare with the experimental results, we simulated the laser patterns by placing a transform lens with a focal length of 5.2 cm a distance of 10.5 cm from the curved mirror (the output coupler in experiment). This is equivalent to propagating the field solution a distance of 16.5 cm and then through the transform lens. Here we neglect the refraction from the finite thickness of the output coupler. The trajectory after the transform lens is shown in Fig. 3(b), in which the pattern variation coincides very well with the experiments. Note that here Z’ = 0 is the position of the transform lens. Due to the design of three pump beam paths, a wider transversal pump region and a longer cavity length were used in experiment. This choice does not influence the coincidence between the experiment and the simulation.

We see that the three spots on z = 0 of Fig. 1(d) were transmitted to Z’ = 7.6 cm. The intra-cavity beam trajectories in Fig. 1(d) were transformed to the respective positions between Z’ = 7.6 and 10.3 cm that follows the Gaussian lens law. Moreover, as compared with the images between Z’ = 7.6 and 10.3 cm, the imaged beam trajectories between Z’ = 6.7 and 7.6 cm shrink and are reversed along the propagation direction. According the Gaussian lens law, the inverse image comes from the objects that are located between 16.5 cm and 22.5 cm before the transform lens, where is just the position of the flat end mirror and the virtual image of the curved end mirror with respect to the position of the flat mirror. Under the situation of asymmetric pump power, our simulation reveals that the two transverse profiles on the two end mirrors in Fig. 1(d) are upside down and then they exhibit upside down twice after the transform lens, which matches with the experiment. This phenomenon can be understood that the VW mode follows the Gaussian lens law not only for their imaging positions but also the reciprocal real images.

 

Fig. 3. The experimental mode patterns (a) and the numerical mode profile variation (b) after the transform lens.

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Furthermore, when the middle pump power is increased to surpass the side pump power, the laser can not generate the VW mode. Contrarily, when we decreased the middle pump, the VW mode patterns exist but blurred with some interference fringes. The best pumping power condition is about 1: 0.5: 1 to 1: 0.95: 1. If one side pump is blocked and the middle pump power is half of the other side pump, one side of the three spots is large and the profile variation is difficult to be identified the VW mode. All the experimental observations match with our simulations.

Finally, we discuss the extension of the operator method and the insufficiency of SU(2) representation for geometric modes. The fringes within 3-spots pattern can be obtained when the six fields of the reversal trajectories are included in the operator method. Since there are six rays leaving the flat mirror end in Fig. 1(b), the total laser field on any transverse plane within the cavity is the accumulated electric fields from the six initial rays, or equivalently both the co-propagating and counter-propagating beams should be considered. This is Eq. (46) of [7] must include twelve fields, six for the rays beginning at (0, 0) with positive slope and six for negative slope. On the other hand, the 6-spots patterns in Fig. 3(a) and the corresponding patterns in Figs. 3(b) and 1(d) have fringes within them. However, in Fig. 1(c) there are no fringes for the two inner trajectories from z = 0.6 to 1.2 cm and from z = 3.2 to 4.2 cm and we know that the absence of fringe is independent of the order n. Hence the inner two spots of the 6-spots pattern implies the insufficiency of the SU(2) representation. The square root of the binomial distribution in Eq. (1) is too special to produce the fringes within all trajectories of the VW mode. Besides, the series of HG function in Eq. (5) of [5] does not satisfy the condition of Eq. (2.13) of [9], in which the total numbers of bosons is constant.

4. Conclusion

We have successfully generated a specific mode, called VW mode, in an end-pumped Nd:YVO₄ laser with a plano-concave cavity near the degeneration point of g₁g₂ = 1/4. The generating condition for the VW mode is somewhat peculiar that is the three pumping beam power must have a proper ratio. Our simulations match with the experiments very well. By observing the 3-spots pattern of the VW mode, we conclude the operator method in [7] can be extended to include simultaneously those of the reverse directional trajectories. Moreover, the SU(2) coherent state representation in [5] is insufficient because it is unable to describe the fringes within the 6-spots pattern of the VW mode.