1. Introduction

Transverse modes in a laser with a geometrically stable cavity are usually determined by the resonator configuration (boundary conditions) and exhibit definite field distributions. For example, in a cavity with the round-trip ray matrix $T=\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]$ starting from a certain plane z, the spot size w(z) and the radius of curvature R(z) of the Gaussian beam can be found by the ABCD law $q=\frac{Aq+B}{Cq+D}$ that the complex radius-of-curvature q(z) repeats itself after one round trip, where $\frac{1}{q\left(z\right)}\equiv \frac{1}{R\left(z\right)}-j\frac{\lambda }{\pi w^2\left(z\right)}$. In a confocal cavity, however, its round-trip ray matrix T starting from the focal plane is easily determined to be T=-I and T²=I, the identity matrix. Thus any starting field distribution that has inverse symmetry with respect to the optical axis of the resonator will exactly reproduce after one round trip. Transverse modes in a laser with a confocal resonator are therefore determined mainly by the gain effect instead of by the resonator configuration. This property has been applied for tailoring the beam distribution inside the resonator in order to optimize the extraction efficiency of solid-state lasers [1]. In addition, any arbitrary paraxial off-axis beam will also return to its initial position and direction after two round-trips. This resonator is referred to as a self-imaging, half-degenerate system [2]. It has been employed in the mode-locked laser to produce femtosecond pulses with interpulse period equal to twice the cavity round-trip time [3].

In a plane-concave cavity, the round-trip ray matrix T of the hemispherical resonator with a reference plane located at the planar mirror is $T=\left[\begin{array}{cc}-1& 0\\ -\frac{2}{R}& -1\end{array}\right]$, a self-imaging system (B=0), and $T^2=\left[\begin{array}{cc}1& 0\\ 4⁄R& 1\end{array}\right]$. Thus, any optical rays starting from the center of the sphere located on the intersection between the cavity axis and plane mirror will repeat themselves after two round trips. However, it is also easily found that no value of complex radius-of-curvature q can satisfy the ABCD law in this cavity. For a Gaussian beam with a complex radius-of-curvature q propagating along the cavity after one round trip, the new complex radius-of-curvature q ^′ becomes $\frac{1}{q\text{'}}=\frac{2}{R}+\frac{1}{q}$. Hence, any Gaussian beam propagating through a hemispherical cavity after one round trip will retain its beam spot size but change the radius of curvature. In addition, the hemispherical resonator is also a critical resonator that has the cavity length on the edge of the geometrically stable region. For a near-hemispherical stable resonator, the spot size at the concave mirror end can be made as large as desired, whereas the spot size at the flat mirror end becomes corresponding tiny. It is therefore very interesting to examine the behavior of lasers with a plane-concave cavity near the hemispherical resonator configuration.

Here, the cavity-length-dependent characteristics of axially pumped Nd:YVO₄ lasers with a plane-concave resonator near the hemispherical resonator configuration is experimentally investigated. I find that when the cavity length is slightly changed so as to approach the hemispherical resonator configuration, the beam pattern extends at first along the optical axis of the Nd:YVO₄ crystal and then extends along the perpendicular direction in sequence. By inserting an object into the resonator, I further find that these patterns are not the high-order transverse modes but consist of individual spots. These spots if off-axis always exist in pairs on opposite sides of the cavity axis. Therefore, they are off-axis beams, which reproduce themselves after two round trips in a V-shape path. One can even select and manipulate the desired beam spots to oscillate by placing a mask inside the cavity. I also study the output power as a function of cavity length at varying pump power without or with an object placed on-axis inside the cavity. The results show that higher pump power, or equivalently higher gain, is able to support off-axis beams in a wider range of cavity length near the hemispherical resonator configuration.

 

Fig. 1. Schematic of the experimental setup. LD, laser diode; CL, collimating lens; FL, focusing lens; LC laser crystal; OC, output coupler; S, screen; F, filter; CCD, charge coupled device.

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2. Experiment

A diode-pumped Nd:YVO₄ laser with a plano-concave cavity is used in the experiment. The schematic of the experimental setup is shown in Fig. 1. A 1-mm-thick, a-cut Nd:YVO₄ laser crystal is coated at the surface facing the pumping beam for less than 5% reflection at 808 nm and greater than 99.8% reflection at 1064 nm. This surface is also used as the flat end-mirror. The second surface of the crystal is antireflection coated at 1064 nm. The output coupler is a concave mirror with radius of curvature R=100 mm and 90% reflection at 1064 nm. It is mounted on a translatable stage to offer continuous adjustment of the cavity length and equivalently change of the resonator configuration. The pump beam from an 800-mW, fiber (100-µm core) output laser diode (Coherent F808-800-100-SMM) is collimated by a lens with focal length 25.4 mm and focused onto the Nd:YVO₄ crystal by an objective lens with focal length of 8.0-mm. The laser diode is operated at constant current and temperature by a current source and a temperature controller to ensure stable pumping wavelength and power. A screen is placed at a position just outside the output coupler. The output beam profile is then observed by imaging the pattern on the screen onto a CCD camera through a 1064-nm filter.

 

Fig. 2. Beam patterns measured from laser output as a function of cavity length. The cavity length is changed by 30 µm per step from (b) to (o) and by 800 µm from (a) to (b) and (o) to (p).

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Fig. 3. Beam patterns measured for the same conditions as fig. 2 except that the Nd:YVO₄ crystal is rotated by 45°.

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3. Results and discussion

Prior to investigating the cavity-length-dependent characteristics of the Nd:YVO₄ laser near the hemispherical resonator configuration, the laser resonator was aligned such that the moving direction of the translatable stage is parallel to the cavity axis. The laser output pattern was then observed as a function of the cavity length. When the cavity length was adjusted to approach the hemispherical resonator configuration, the spot size of the output beam became larger and larger as theoretically expected. Further increasing the cavity length, I found that the beam pattern at first extended along the optical axis (vertical direction in this experiment) of the vanadate crystal, then broke up into several spots, and eventually returned to a spot and repeated along the direction perpendicular to the optical axis of the vanadate crystal (horizontal direction) as shown in Fig. 2. The cavity length for these patterns is changed from 101.07 mm to 103.06 mm in 30-µm step except for patterns (a) to (b) and (o) to (p) which is changed in a step of 800 µm. Because the Nd:YVO₄ crystal is a birefringent (n_o≅1.957 and n_e≅2.165 at 1064 nm) material, it can introduce significant index anisotropy for the pump beam or the laser beam. I find that the evolution of beam patterns in the experiment is indeed related to the index anisotropy of the Nd:YVO₄ crystal. This is confirmed by observing the patterns after the change of crystal orientation. The beam pattern for rotating the Nd:YVO₄ crystal by 45⁰ is shown in Fig. 3. Now, the beam pattern extends at first along 45° to the vertical direction follow by a 1350° extension.

To understand the properties of these transverse modes, I observe the mode pattern by inserting an object (a wrench about 1-mm in width) into the cavity in front of the concave mirror to block a portion of the beam path. The change of beam patterns as a result of inserting the object at different positions is shown in Fig. 4. It is found that the insertion of the object only results in the disappearance of the spot being blocked and leaves other spots unchanged or even gives rise to new spots in the unblocked region. In addition, the decentered spots always disappear in pairs on opposite sides of the cavity axis when one of them is blocked. These properties show that these patterns are composed of individual spots and contain the off-axis beams in a V shape path. It is worth mentioning that these patterns are distinct from higher-order transverse modes, which are not comprised of individual spots and can’t persist when any portion of the beam is blocked. By inserting a mask into the cavity, one can further select and manipulate the desired off-axis beam. This is demonstrated in Fig. 5. In Figs. 5 (a)-(d), the mask with an opening is used. By inserting the mask to cover a half of the concave mirror, the laser is forced to oscillate through the opening and to form two spots on opposite sides of the cavity axis. The location of the spots can change by moving the position of the opening. In Fig. 5(e), the mask with two openings is used. Now four spots are formed in the output of the laser.

 

Fig. 4. Beam patterns measured by inserting a wrench into the cavity downward at different positions to block a portion of beam path.

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Fig. 5. Beam patterns formed by inserting a mask into the laser cavity (a) - (d) with an opening at different positions and (e) with two openings.

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I have calculated the cavity length that corresponds to the hemispherical resonator configuration for different off-axis rays by taking into account the slab of laser crystal as shown in Fig. 6(a). L, R, and l are mirror separation, radius of curvature for the concave mirror, and the thickness of the Nd:YVO₄ crystal, respectively. θ₁ (θ₂) denotes the angle between off-axis rays outside (inside) crystal and the cavity axis. Let y and z represent the optical axis for the Nd:YVO₄ crystal and the cavity axis, respectively. The mirror separation of the hemispherical resonator configuration for an off-axis ray at the y-z plane is $L=R+l\left(1-\frac{\cos {\theta }_1}{\sqrt{{n_e}^2-{\sin }^2{\theta }_1}}\right)$. For the off-axis ray at the x-z plane, the n_e is a function of θ₁ and can be expressed as $n_e\left({\theta }_1\right)=\sqrt{\frac{{{n_e}^2{n}_o}^2}{{n_e}^2{\sin }^2{\theta }_2+{n_o}^2{\cos }^2{\theta }_2}}$, with ${\theta }_2={\tan }^{-1}\frac{n_o\sin {\theta }_1}{\sqrt{{{n_e}^2{n}_o}^2-{n_e}^2{\sin }^2{\theta }_1}}$. The mirror separation which is capable of supporting an off-axis ray at y-z and x-z plane as a function of θ₁ is shown in Fig. 6(b). We can see that the mirror separation for the hemispherical cavity that supports the off-axis rays at θ₁ from 0° to 7° which corresponds to a maximum range of off-axis angle in the experiment has a range less than 5 µm. By referring to the experiments, these calculations show that the off-axis beams not only exist in the hemispherical resonator configuration but also occur in an extended region. To learn more about off-axis beams in this laser, the output power is measured as a function of cavity length with or without an intracavity object on axis at different pump power. The results are shown in Fig. 7. We can see that when an object is placed inside the cavity in front of the concave mirror on axis the laser can only persist within a range of cavity length near the hemispherical resonator configuration. Moreover, the higher the pump power I apply the wider the range of persistence of the laser with an on-axis blocked cavity. By adjusting the profile or spot size of the pump beam, I found that the beam patterns of laser output are changed but off-axis beams still occur around the hemispherical cavity. Similar experiments have been performed in a Nd:YAG laser. It is found that the laser can also maintain oscillation in a region near the hemispherical resonator configuration when an object is inserted inside the cavity on axis although it exhibits different patterns from the Nd: YVO₄ laser. These experiments show that the hemispherical resonator configuration is capable of supporting the off-axis beams in axially pumped solid-state lasers and the gain-related effect is responsible for the formation of various off-axis beam patterns in an extended region.

 

Fig. 6. (a) A plane-concave cavity, which corresponds to a hemispherical resonator for an off-axis ray by taking into account the refraction in the laser crystal. (b) The cavity length corresponds to a hemispherical resonator configuration as a function of the angle θ₁ between the off-axis ray and cavity axis.

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Fig. 7. Laser output power as a function of cavity length. (a) Without and (b) - (d) with an object inserted inside the cavity on axis at different pump powers where (a) and (b) are 360 mW, (c) is 180 mW, and (d) is 50 mW.

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4. Conclusion

In conclusion, I have examined cavity-length-dependent variation of transverse modes from the stable to unstable region in a diode-pumped Nd:YVO₄ laser with a plane-concave resonator. I find that when the cavity length is adjusted around the hemispherical resonator configuration, a variety of off-axis beams can exist inside the cavity in a V shape path. These beams persist even if an object is inserted inside the cavity in front of the concave mirror on axis. The results presented here can be usefully applied to generate manipulable beam arrays in an axially pumped solid-state laser.