Novel non-planar ring cavity for enhanced beam quality in high-pulse-energy optical parametric oscillators

Stefano Bigotta; Georg Stöppler; Jörg Schöner; Martin Schellhorn; Marc Eichhorn

doi:10.1364/OME.4.000411

1. Introduction

As soon as the first lasers have been realized, the beam quality of the laser radiation, i.e. its ability to focus to a small spot at a given divergence angle or to propagate over long distances with low divergence, has gained in importance. It has since become a challenging task for laser source development to obtain good beam quality, e.g. by properly designing the laser resonator and taking into account dynamic effects like thermal lensing. It is well known that mode matching, i.e. designing a resonator in such way that its fundamental mode diameter 2w₀ becomes close to the diameter of the pumped volume, is important to avoid the appearance of higher-order modes and thus a decrease in beam quality. Although this can often be achieved in solid-state lasers, it becomes a problem in nano-second optical parametric oscillators (OPOs) with a need of simultaneous generation of high pulse energies and high beam quality. OPOs with such properties are important for various applications where non-linear conversion is necessary as a result of a lack of directly-emitting or unsuitable laser transitions. Especially the non-linear conversion in the 2 μm and 3–5 μm range is of rapidly increasing interest for applications in long-range detection, medicine and countermeasures, with a strong trend towards high pulse energies and average powers. Substantial progress in mid-IR non-linear generation for these applications has been achieved in the last years [1–5]. The difficulty in obtaining high beam quality at high pulse energies arises from the usually low optical damage thresholds of nonlinear media on the order of several J/cm². A high pulse energy therefore is related to a large beam size in the non-linear crystal, resulting in high Fresnel numbers of the associated OPO cavity. This causes a low cavity transverse-mode discrimination and thus a low beam quality. Especially when efficiency and compactness of an OPO are key parameters, an increase in the cavity length cannot be used to enhance mode discrimination. To circumvent this problem, a 90° image rotation is usually employed and finally resulted in the invention of the monolythic RISTRA cavity [6]. Therein, the beam quality enhancement in the critical plane of a critically phase-matched non-linear crystal is used in both transverse directions of the cavity beam. Owing to the 90° image rotation in a RISTRA, the cavity mode effectively needs four round-trips to be self-consistent in terms of its field distribution in the transverse plane. Moreover, the combination of the image rotation with walk off between signal and idler further improves the beam quality. However, at larger pump diameters, the output beam may show a 4-fold (square-like) symmetry due to the image-rotation effect [2] which, even if it disappears during propagation, is sign of beam degradation.

In this paper we present a novel non-planar ring cavity causing a fractional image rotation, i.e. a rotation whose angle cannot be expressed as 2π/n, where n is a small integer, which is chosen in such a way, that much more (up to an infinite number) of round-trips would be theoretically necessary to re-establish image self-consistency, which further enhances beam quality.

The enhancement of beam quality by such a fractional image rotation is, however, linked to a much more fundamental effect than the critical plane acceptance angle restriction the RISTRA is based on. Therefore, it will also provide good beam quality for non-critically phase-matched non-linear materials.

The beam-quality enhancement effect is tied to the nature of the cavity itself. It is known that the photons are delocalized in the cavity and the quantum states in which they can distribute are given by the geometry of the cavity itself. Analogally, we can state that for a given resonator a defined set of mathematical eigenmodes exist. With respect to the theory discussed in this paper, the transverse modes are of importance as they define the beam quality. The fractional image rotation then suppresses many higher-order modes as a result of the need of self consistency after a round-trip propagation. Owing to its circular symmetry, the fundamental Gaussian mode will always be self-consistent with itself after one round-trip for any given image rotation angle. Therefore, the fundamental Gaussian beam is also an eigenmode of the new type of resonator. Higher order modes which are self-consistent after one round-trip, however, need to provide for the rotational symmetry of the cavity image rotation in its field distribution. Thus, theoretically, a true fractional image rotation that reproduces a beam only after an infinite number of round-trips excludes many higher-order mode, leaving only those with true circular symmetry (Laguerre-Gaussian TEM_lp modes with l=0). Therefore, for a given diffraction loss caused by the stronger divergence of the higher-order modes, the number of possible modes is greatly reduced, resulting in a good beam quality.

Non-planar ring resonators, NPROs [7], are also known for solid-state lasers. The most recent ones (see e.g. [8]), consisting of all-planar mirrors stabilized by thermal lensing, are similar to the cavity investigated in this work. They allow for extremely narrow linewidth in fundamental mode operation with high beam quality. It is worth of note that NPROs have been proposed because their non-planar cavity in presence of a magnetic field forces the unidirectional operation in the ring and consequently achieves high-quality single-frequency laser beams. However, the effect of the fractional image rotation has not been studied.

In this paper a theory is presented that explains the high beam quality obtained in (non-planar) image-rotating resonators and a novel non-planar ring cavity causing a fractional, i.e. non-90°, image rotation, which is chosen in such a way, that much more (up to an infinite number) of round-trips would be theoretically necessary to re-establish image self-consistency, is presented, which further enhances beam quality.

We show that the proposed new resonator further limits the number of modes that can oscillate in the cavity itself, and we relate the decrease of the oscillating modes to the enhancement of the output beam quality factor. We then compare the results obtained with a ZnGeP₂ (ZGP) crystal for mid-IR generation pumped by a Ho:LiLuF₄ (LLF) MOPA system at 2.05 μm in this new cavity to those realized with the same crystal and pump arrangement using a standard RISTRA cavity.

2. Theory of fractional image rotating resonators

As already pointed out degradation in beam quality arises from excitation of higher order modes. This may be caused by phase distortion as a result of thermal effects or by a gain volume area much larger than the fundamental mode diameter. Due to the complexity of the system, for the theory presented here we limit the analysis to the case of a resonator with all planar mirror and with a small thermal lens without aberration that stabilizes the cavity. The general case of the general astigmatic resonator described by Arnaud in [9] will be the subject of further investigations. Under these conditions it is convenient to use cylindrical coordinates (r, φ) and Laguerre-Gaussian modes, whose transverse electric field in the focal plane is described by [10]

{LG}_{l, p} (ρ, φ) \propto e^{i l φ} {(2 ρ)}^{| l |} L_{p}^{| l |} (4 ρ^{2}) e^{- ρ^{2}} e^{- i (\frac{π R}{λ} - \frac{π}{4} (2 p + l + 1))} \propto f (ρ) e^{i l φ}

It separates into a radial factor f (ρ) with ρ = r/w, where w is the waist, and an azimuthal dependence in φ. It is important that these modes form a complete set, such that any field distribution can be decomposed into a linear combination of these modes.

As the Laguerre-Gaussian modes are eigensolutions of the round-trip wave equation of the resonator, the final eigenmodes of a resonator with a total round-trip image rotation θ are therefore given by those Laguerre-Gaussian modes that are simultaneously also eigenmodes of the rotation operator

U (θ) = e^{- θ \frac{\partial}{\partial φ}} = \sum_{k = 0}^{\infty} \frac{1}{k!} {(- θ)}^{k} \frac{\partial^{k}}{\partial φ^{k}}

with a unity eigenvalue. Applying U(θ) on Eq. (1) results in

\begin{array}{l} U (θ) {LG}_{l, p} (ρ, φ) & \propto f (ρ) \sum_{k = 0}^{\infty} \frac{1}{k!} {(- θ)}^{k} \frac{\partial^{k}}{\partial φ^{k}} e^{i l φ} \\ \propto f (ρ) e^{i l φ} \sum_{k = 0}^{\infty} \frac{1}{k!} {(- θ)}^{k} {(i l)}^{k} \\ \propto f (ρ) e^{i l φ} e^{- i l θ}, \end{array}

showing that the Laguerre-Gaussian modes are eigenmodes of the rotation operator with eigenvalue e^−ilθ. Self-consistency after one round-trip therefore imposes

e^{- i l θ} = 1 \Rightarrow l = \frac{2 π n}{θ} with n \in {\dots, - 2, - 1, 0, 1, 2, \dots} .

This result explains already that, independent of the existence of a critical plane in an OPO crystal, the θ = 90° image rotation in a RISTRA excludes many normal resonator modes to be eigenmodes of the RISTRA cavity. Furthermore, we immediately see that by choosing θ such that 2π/θ becomes a non-integer fraction where 2πn/θ is non-integer for all n > 0, only l = 0 Laguerre-Gaussian modes will be eigenmodes of the resonator. In case this is true for at least all small 0 < n < n_max, such that the divergence of the mode with l = 2πn_max/θ causes enough diffraction losses to keep it below threshold, an identical beam-quality enhancement results as in the case where l is restricted to l = 0 only. We call this the fractional-image-rotation-enhancement (FIRE) resonator.

For Laguerre-Gaussian beams an effective mode radius can be defined by

w_{eff} = w_{0} \sqrt{2 p + | l | + 1},

where w₀ is the 1/e² intensity mode radius of the fundamental Gaussian beam. If we assume that all modes will be excited that fill the pump area of the amplifying medium (laser or OPO crystal) up to a beam radius w_max, we can count the number of possible modes (l, p) for a given amplification radius w_max

\sqrt{2 p + | l | + 1} \leq \frac{w_{\max}}{w_{0}} = x .

Figure 1 shows the strong reduction in possible eigenmodes as a function of the maximum beam radius with respect to the fundamental mode size. It is clear that both the RISTRA and the FIRE resonator allows the oscillation of the fundamental mode only for values of w_max larger than the case without image rotation. For even larger values of w_max, it can be shown that the asymptotic number of modes decreases in the RISTRA case with respect to the case without image rotation, but the number of modes are still proportional to ∼ x⁴, while in the FIRE case only to x²/2. This strong reduction in the number of modes is already a hint that the beam quality could be positively affected, but in order to confirm that, the knowledge of the mode distribution has to be determined.

Fig. 1: Calculated number of modes for the different types of resonators as a function of the beam size relative to the fundamental mode size.

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Taking into account the relation between the real spot size of a laser beam and its corresponding fundamental spot size we can deduce the M² beam propagation factor for each mode as

M^{2} = {(\frac{w_{eff}}{w_{0}})}^{2} = 2 p + | l | + 1 .

In case the real beam is comprised of a superposition of different modes

E (r, φ) = \sum_{n = 0}^{- \infty} \sum_{p = 0}^{\infty} r_{n, p} ψ_{l, p} (r, φ)

E (r, φ) = \sum_{n = 0}^{- \infty} \sum_{p = 0}^{\infty} r_{n, p} ψ_{4 n, p} (r, φ)

E (r, φ) = \sum_{p = 0}^{\infty} f_{p} ψ_{0, p} (r, φ)

for a standard resonator (Eq. (8)), the RISTRA (Eq. (9)) and the FIRE (Eq. (10)) case, with ψ_l,p(r, φ) being power-normalized Laguerre-Gaussian field amplitudes, the M² value of this superposition can be calculated from the complex amplitudes c_l,p, r_n,p and f_p as

\begin{array}{l} M^{2} = \sum_{l = 0}^{- \infty} \sum_{p = 0}^{\infty} {| c_{l, p} |}^{2} (2 p + | l | + 1) \\ M_{R}^{2} = \sum_{n = 0}^{- \infty} \sum_{p = 0}^{\infty} {| r_{n, p} |}^{2} (2 p + 4 n + 1) \\ M_{F}^{2} = \sum_{p = 0}^{\infty} {| f_{p} |}^{2} (2 p + 1) \end{array}

The distribution of the complex amplitudes c_l,p, r_n,p and f_p will in general depend not only on the cavity parameters, but also (and mainly) on other several laser parameter (gain...). For the purpose of this work, we suppose that all modes will be excited that fill the pump area of the amplifying medium (laser or OPO crystal) and, for simplicity, we assume that the pump distribution has a flattened Gaussian profile. While this assumption is not strictly verified, since the output beam will not fully mimic the pump profile, this analysis is useful to evaluate how the modes redistribute within the different cavities. A different approach could be using a numerical model to investigate more general cases. However, current numerical models simply propagate the electric field in the unfolded cavity, thus transforming the cavity in a periodic focusing system. While this approximation can be used in most cases without problems, they do not usually take into account the boundary conditions imposed by the cavity, meaning that the models do not check whether the final solution is an eigenmode or a superposition of eigen-modes of the resonator. A complete numerical model that takes the boundary condition of the cavity into account is currently under investigation and it will be the subject of a further work. Since the information about the phase profile of the real laser beam inside the cavity is not available, we consider the problem of finding the real and positive coefficients λ_m,n, equivalent to the |c_l,p|², |r_n,p|² and |f_p|² terms appearing in Eqs. 11 representing the power carried by each mode defined as

I (x, y) = \sum_{n, m} λ_{n, m} {| ψ_{n, m} (x, y) |}^{2}

Here we limit ourselves to the standard and the FIRE resonator cases, since they can be easily solved analytically, and the study of the RISTRA case will be the object of further investigations.

For the non rotating image resonator, we follow [11] that solves the problem of Eq. (12) in the case of rectangular symmetry (i.e. when ψ_n,m(x, y) are Hermite-Gaussian (HG) functions), by working in the Fourier space. Indeed, by taking the 2-D Fourier transform of Eq. (12) on both sides and considering the ortho-normality of the base functions, we obtain

λ_{n, m} = 4 π^{4} v_{0}^{4} \int_{0}^{\infty} \int_{0}^{\infty} \tilde{I} (p, q) ℒ_{n} (π^{2} v_{0}^{2} p^{2}) ℒ_{m} (π^{2} v_{0}^{2} q^{2}) p q d p d q

where

ℒ_{n} (t) = L_{n} (t) exp (- t / 2),

Ĩ(p, q) is the Fourier transform of the intensity profile and v₀ is the waist of the fundamental mode of the HG basis functions. We suppose that the pump intensity has a flattened Gaussian (FG) profile:

I (r, w_{0}) = exp [- (N + 1) \frac{r^{2}}{w_{0}^{2}}] \sum_{n = 0}^{N} \frac{1}{n!} {[(N + 1) \frac{r^{2}}{w_{0}^{2}}]}^{n}

where N and w₀ are two positive parameters defining the steepness of the profile and the radius of the top-hat profile towards which the FG tends for N going to infinite, respectively. The Fourier transform of this axially symmetric profile coincide with the Hankel transform and is given by

\tilde{I} (p, q) = \frac{π}{α} L_{N}^{1} [\frac{π^{2}}{α} (p^{2} + q^{2})] exp [- \frac{π^{2}}{α} (p^{2} + q^{2})]

where

α = (N + 1) / w_{0}^{2}

. By using the following property of the Laguerre polynomials [12]

L_{N}^{1} (x + y) = \sum_{k = 0}^{N} L_{k} (x) L_{N - k} (y)

we may rewrite Eq. (16) as

\tilde{I} (p, q) = \frac{π}{α} \sum_{k = 0}^{N} L_{k} (\frac{π^{2} p^{2}}{α}) L_{N - k} (\frac{π^{2} q^{2}}{α}) exp [- \frac{π^{2}}{α} (p^{2} + q^{2})]

Equation (11) (and (12)) leaves the choice of the waist v₀ of the basis functions. In [11], the authors made the additional assumption that

\sqrt{N + 1} v_{0} = \sqrt{2} w_{0}

in order to simplify the calculation and minimize the number of HG(n,m) modes required in Eq. (12), since in this case the λ_n,m coefficients vanish for (n + m) > N. In the present case, however, the v₀ and w₀ are unrelated and depend on the fundamental mode of the cavity and on the dimension of the pump spot, respectively. Introducing Eq. (18) in (13) we obtain

\begin{array}{l} λ_{n, m} & = 4 \frac{π^{5} v_{0}^{4}}{α} \sum_{k = 0}^{N} \int_{0}^{\infty} \int_{0}^{\infty} L_{k} (\frac{π^{2} p^{2}}{α}) L_{N - k} (\frac{π^{2} q^{2}}{α}) \\ \times exp [- \frac{π^{2}}{α} (p^{2} + q^{2})] ℒ_{n} (π^{2} v_{0}^{2} p^{2}) ℒ_{m} (π^{2} v_{0}^{2} q^{2}) p q d p d q \end{array}

If we define x = p², y = q²,

π^{2} v_{0}^{2} = λ

, π²/α = μ, and remember that 1/2dx = pdp, we obtain

\begin{matrix} λ_{n, m} = \frac{π^{5} v_{0}^{4}}{α} \sum_{k = 0}^{N} [\int_{0}^{\infty} e^{- (μ + λ / 2) x} L_{k} (μ x) L_{n} (λ x) d x \\ \times \int_{0}^{\infty} e^{- (μ + λ / 2) y} L_{N - k} (μ y) L_{m} (λ y) d y] \end{matrix}

We can now use the equality from [12]

\begin{array}{l} \int_{0}^{\infty} e^{- b x} x^{α} L_{n}^{α} (λ x) L_{m}^{α} (μ x) d x = \\ \frac{Γ (m + n + α + 1)}{m! n!} \frac{{(b - λ)}^{n} {(b - μ)}^{m}}{b^{m + n + α + 1}} d x \\ \times F [- m, - n; - m - n - α, \frac{b (b - λ - μ)}{(b - μ) (b - λ)}] \end{array}

where Γ and F(α, β; γ; z) are the incomplete gamma and the Gauss hypergeometric functions, respectively. By taking the special case α = 0 and, from the definition

L_{n}^{0} (x) \equiv L_{n} (x)

[12] we have that

\begin{array}{l} \int_{0}^{\infty} e^{- (μ + λ / 2) b x} L_{n} (λ x) L_{m} (μ x) d x = \\ \frac{Γ (m + n + 1)}{m! n!} \frac{{(μ - λ / 2)}^{n} {(λ / 2)}^{m}}{{(μ + λ / 2)}^{m + n + 1}} \\ \times F [- m, - n; - m - n, \frac{λ / 2 + μ}{λ / 2 - μ}] \end{array}

We can therefore express the λ_n,m coefficients as:

\begin{array}{l} λ_{n, m} & = \frac{π^{5} v_{0}^{4}}{α} \sum_{k = 0}^{N} {c_{k, n, m} F [- k, - n; - k - n, \frac{λ / 2 + μ}{λ / 2 - μ}] \\ \times F [- N - k, - m; - N + k - m, \frac{λ / 2 + μ}{λ / 2 - μ}]} \end{array}

where

c_{k, n, m} = \frac{(k + n)! (N - k + m)!}{k! n! (N - k)! m!} \frac{{(μ - λ / 2)}^{n + m} {(λ / 2)}^{N}}{{(μ + λ / 2)}^{N + m + n + 2}}

For the aim of this paper, it is more convenient to express Eq. (24) in terms of LG modes instead of HG modes. For this reason, we can use the Eqs. (8–11) from [13] to express the HG(m, n) modes in terms of the LG(l, p) modes (Note that all but Eq. (9) in [13] present at least one typo and should be corrected accordingly).

For the FIRE case, we can find the λ_0,_p by noting that all the ψ_0,_p(r, φ) are real (apart from a constant phase factor), implying that also their sum is real. We can therefore calculate the coefficients from their definition

c_{l, p} = \int_{0}^{2 π} \int_{0}^{\infty} E (r, φ) ψ_{l, p}^{*} (r, φ) r d r d φ

by considering the electric field as real

E (r) = \sqrt{I (r)}

, and by noting that the integral over the angular components

c_{l, p} = \int_{0}^{\infty} E (r) ψ_{l, p} (r) r d r \int_{0}^{2 π} e^{i l φ} d φ

for symmetry reasons is zero when l ≠ 0.

We can now demonstrate how the modal weight distribution affects the beam quality by means of a numerical example; in the specific case we suppose to have a pump profile defined by a FG with w₀ = 4v₀ and N = 9. Fig. 2a and 2c show the intensity of the beams obtained by the superposition of the modes (see Eqs. (8) and (10)), respectively. The two profiles are practically indistinguishable, however a look at Fig. 2b and Fig. 2d, which report the calculated values of λ_l,p for the standard non image-rotating resonator and for the FIRE case, respectively, show the striking difference between the two investigated cases. In the standard case the oscillating modes are much more numerous than in the FIRE case. Moreover, in the former case the contribution of the fundamental mode to the final output beam is minimal (the power carried by the TEM₀₀ is almost 0). This is in agreement with the common behaviour of multimode lasers, where the beam with the largest occupancy factor is the dominating one. On the contrary, in the FIRE case the suppression of the modes having l ≠ 0 forces the residual modes to redistribute and the dominant mode is now the fundamental one, with a TEM₀₀ power content of ∼ 12.5%.

Fig. 2: Intensity of the beam obtained form Eq. (2b) (non image-rotating case, top left) and Eq. (2d) (FIRE case, bottom left). On the right, the calculated distribution of the LG(l, p) modal weights for the two cases under investigation are shown. Height of the bars are proportional to the corresponding modal weights. Note that since λ₋_l,p = λ_l,p, only the modal weights for modes having l > 0 are reported here.

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Despite the fact that in the two cases the λ_l,p coefficients are so differently distributed, the beams obtained by still do not differ each other, as can be seen from Fig. 2. This result was expected since we supposed that all the modes that fill the amplifying medium are excited. By using Eq. (11) one can then have a measure of the beam quality factor in the two cases: the M² parameter decreases from a value of 17.4 for the standard non image-rotating cavity to 8.9 for the FIRE resonator. Finally we note that, even if we do not have any analytical solution for the RISTRA case, yet we can reasonably argue that the M²of such a cavity should be situated between the values found in the other two cases.

3. Experimental

The experimental set-up used is illustrated in Fig. 3. A Ho³⁺:LLF laser system [14, 15] with an additional amplifier stage was used to pump a ZGP crystal. Since the RISTRA and the FIRE cavities both belong the class of non-planar oscillators, they are insensitive to small angle misalignment [16]. This allows the realization of both cavities out of a monolithic block, and the set-up can easily accommodate and exchange both resonators. The maximum pulse energy of the Ho:LLF MOPA pump system was 82 mJ at a repetition rate of 100 Hz. The spectral profile of the pump laser is reported in Fig. 4 showing multi-longitudinal mode operation. To prevent damage on the ZGP crystal during pumping the OPO, the pump system was limited to 48 mJ at a pulse width of 30 ns. The pump beam quality was measured to $M_{x}^{2} = 1.01$ and $M_{y}^{2} = 1.03$ at a wavelength of 2053 nm (x-axis is parallel to the polarization of the pump beam inside the ZGP crystal). The π-polarized pump beam could be attenuated by a half-wave plate and a polarizer to operate the OPO at constant pump pulse width. A focusing lens with f = 1000 mm was used to create a pump spot diameter of 3.85 × 3.65 mm² in the center of the ZGP crystal resulting in a maximum peak fluence of 0.86 J/cm². The ZGP crystal had a size of 7 × 7 × 16 mm³ and was cut at 56° with respect to the optical axis, which allows for type I phase-matching. It was wrapped with indium foil and fixed in a copper mounting without water cooling. Both cavities were aligned for collinear phase-matching conditions and were placed in a box which can be closed for flushing with dry air. All experiments were done in lab atmosphere at T = 28 °C and a relative humidity (rH) of 40%.

Fig. 3: Schematic of the experimental set-up used fir the current work. The same set-up can either accommodate a RISTRA or a FIRE cavity.

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Fig. 4: Spectral profile of the Ho³⁺:LLF MOPA pump system.

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The RISTRA ring cavity is a monolithic block consisting of four plane mirrors and a half-wave plate for the resonant signal wavelength. The cavity is mechanically as short as possible and measures 130 mm. The output coupler M2 had a reflectivity of 65% for the signal and high transmission for the pump (T > 95%) and idler (T > 98%) wavelength. The other three mirrors were highly transparent for pump (T > 98%) and idler (T > 94%) but highly reflective for the oscillating signal (R > 99%). Residual pump and mid-IR output can be separated in each case by a dichroic mirror.

When defining the FIRE topology, there are many geometric layouts of a non-planar cavity that will provide a given image rotation angle. Additional boundary conditions like a simple setup, an easy alignment and a minimization of needed mirror types and coatings, however, can be met with a design of high symmetry using six mirrors as shown in Fig. 3. For the present FIRE we chose an incident angle on all six mirrors of ∼32.7°, in order to match the beam incident angle of the RISTRA cavity [6]. In this way, the same mirrors and the same coatings used in a RISTRA cavity can be directly employed in the FIRE resonator. The output coupler M2 of the FIRE cavity has a reflectivity of 65% for the signal and high transmission for the pump (T > 95%) and idler (T > 98%) wavelength, exactly as for the RISTRA case. The other five mirrors too have the same coating as their RISTRA analogues, with a high transmission for the pump (T > 98%) and idler (T > 94%) and highly reflective for the oscillating signal (R > 99%). The total cavity round-trip image rotation is achieved by properly tilting the triangles M2-M3-M4 and M5-M6-M1 with respect to the M1-M2-M4-M5 plane. Since all the pump and output beam are on this plane, an easy overall set-up can be achieved. This allows, like for the RISTRA case, the use of a monolithic block that makes the beam alignment means unnecessary. The angle was set to obtain an image rotation of 77.448°. As the result, 14 round-trips are needed before the original transverse image is essentially restored, with an accuracy of ∼1.2%, or 290 round-trips are needed to obtain self-consistency within 0.03%. Also in this case, the length of the cavity as been kept mechanically as short as possible and measures 222 mm.

Figure 5 reports the slope efficiencies of the RISTRA and the FIRE cavity. This figure shows that the slopes are identical in both cases. In the FIRE cavity the threshold is 20 % higher, which is attributed to the longer cavity length of the FIRE cavity (222.4 mm) compared to the RISTRA cavity (129.7 mm). A simulation of the investigated cavities, reported in Fig. 6, has been performed by means of the SNLO software [17] using its ring cavity solver (without taking into account image rotation). Thus the effect of different ring cavity lengths on OPO output, power and threshold can be simulated. Even if it is known [2] that this software is only suited for modelling of single longitudinal-mode pumped OPOs, it can still be used to get some important qualitative information. Figure 6 shows that the relative expected threshold in the FIRE case should be 52% higher than the RISTRA, much higher than the 20% observed experimentally.

Fig. 5: OPO output performance obtained from FIRE and RISTRA cavity under identical conditions (100 Hz, 30 ns pumped at 2.053 μm).

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Fig. 6: OPO output performance simulated using the SNLO software [17].

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The slightly lower outputs obtained in the FIRE case are theregore much likely to be attributed to polarization effects. In the RISTRA case it has been proved that a simple half wave-plate can be used in order to accomplish the required polarization rotation in order to correct the polarization after each round trip. In order to provide self-consistency after one round-trip also in polarization, a waveplate arrangement is inserted into the FIRE cavity in analogy to the RISTRA cavity, however, the lower output could be the result of a non complete compensation. A better compensation system combining half and quarter waveplates is under investigation. Figure 7 compares the M² parameter for the signal and the idler in the RISTRA and FIRE cavities. An enhancement of the beam quality factor can clearly be observed: for the signal $M_{x}^{2} = 1.92 \pm 0.02$ (2.08±0.03), $M_{y}^{2} = 1.99 \pm 0.03$ (2.29±0.03) and for the idler $M_{x}^{2} = 1.88 \pm 0.07$ (2.28±0.1), $M_{y}^{2} = 1.94 \pm 0.07$ (2.16±0.1) were obtained at 20.3 mJ (19.2 mJ) total output energy for FIRE (RISTRA), respectively. While this effect could be partially explained in simple terms of different Fresnel numbers for the two cavities, we believe that this is also due to the mode-selecting properties of the FIRE cavity. Indeed, as already observed theoretically, the larger Fresnel number of the FIRE cavity would also imply a much larger threshold, however experimentally the relative difference of the threshold is much lower than expected. Thus, the same mechanism responsible for the threshold improvement could also be responsible for the beam quality improvement. Further analysis is planned in order to better understand these processes, including the possible role of the thermal effects stabilizing the cavity in improving the beam quality.

Fig. 7: Beam propagation factor measurement for signal and idler in the case of RISTRA (top) and FIRE (bottom) cavities.

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4. Conclusion

A novel non-planar ring cavity has been presented allowing for an increase in beam quality compared to a RISTRA-type cavity that cannot be expained in simple terms of Fresnel number effects. The role of the image rotation and the mode selecting properties of this new cavity has been investigated. Owing to its simple geometry with two parallel arms for possible non-linear crystal positions (M1–M2 and M4–M5), an easy overall set-up can be achieved. Especially, all pump and output beams are in one plane (M1-M2-M4-M5). Like for the RISTRA, a monolithic block has been used without the need for mirror alignment means. Further research will be devoted to investigate optimum image rotation angles for maximum beam quality.

References and links

1. A. Dergachev, D. Armstrong, A. Smith, T. Drake, and M. Dubois, “3.4-μm ZGP RISTRA nanosecond optical parametric oscillator pumped by a 2.05-μm Ho:YLF MOPA system,” Opt. Express 15, 14404–14413 (2007). [CrossRef] [PubMed]

2. M. Eichhorn, G. Stoeppler, M. Schellhorn, K. T. Zawilski, and P. G. Schunemann, “Gaussian- versus flat-top-pumping of a mid-IR ZGP RISTRA OPO,” Appl. Phys. B 108, 109–115 (2012). [CrossRef]

3. G. Stöppler, M. Schellhorn, and M. Eichhorn, “Enhanced beam quality for medical applications at 6.45 μm by using a RISTRA ZGP OPO,” Laser Physics 22, 1095–1098 (2012). [CrossRef]

4. G. Marchev, A. Tyazhev, V. Petrov, P. G. Schunemann, K. T. Zawilski, G. Stöppler, and M. Eichhorn, “Optical parametric generation in CdSiP₂at 6.125 μm pumped by 8 ns long pulses at 1064 nm,” Opt. Lett. 37, 740–742 (2012). [CrossRef] [PubMed]

5. C. Kieleck, M. Eichhorn, A. Hirth, D. Faye, and E. Lallier, “High-efficiency 20-50 kHz mid-infrared orientation-patterned GaAs optical parametric oscillator pumped by a 2 μm Holmium laser,” Opt. Lett. 34, 262–264 (2009). [CrossRef] [PubMed]

6. A. V. Smith and D. J. Armstrong, “Nanosecond optical parametric oscillator with 90° image rotation: design and performance,” J. Opt. Soc. Am. B 19, 1801–1814 (2002). [CrossRef]

7. A. C. Nilsson, E. K. Gustafson, and R. L. Byer, “Eigenpolarization theory of monolithic nonplanar ring oscillators,” IEEE J. Quantum Electron. 25, 767–790 (1989). [CrossRef]

8. C. Gao, L. Zhu, R. Wang, M. Gao, Y. Zheng, and L. Wang, “6.1 W single frequency laser output at 1645 nm from a resonantly pumped Er:YAG nonplanar ring oscillator,” Optics Lett. 37, 1859–1861 (2012). [CrossRef]

9. J. A. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Syst. Tech. J. 492311–2348 (1970). [CrossRef]

10. A. E. Siegman, Lasers (University Science Books, 1986).

11. R. Borghi and M. Santarsiero, “Modal structure analysis for a class of axially symmetric flat-topped laser beams,” IEEE J. Quantum Electron , 35, 745–750 (1999). [CrossRef]

12. I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 7th edition (Academic Press, 2007).

13. I. Kimel and L. R. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron , 29, 2562–2567 (1993). [CrossRef]

14. M. Schellhorn, “High-energy, in-band pumped Q-switched Ho³⁺:LuLiF₄ 2 μm laser,” Opt. Lett. 35, 2609–2611 (2010). [CrossRef] [PubMed]

15. M. Schellhorn and Marc Eichhorn, “High-energy Ho:LLF MOPA laser system using a top-hat pump profile for the amplifier stage,” Appl. Phys. B 109, 351–357 (2012). [CrossRef]

16. Y. A. Anan’ev, Laser resonators and the beam divergence problem (Adam Hilger, 1992).

17. SNLO program, http://as-photonics.com/RISTRA-Modeling.html, AS Photonics LLC.

Novel non-planar ring cavity for enhanced beam quality in high-pulse-energy optical parametric oscillators

Abstract

1. Introduction

2. Theory of fractional image rotating resonators

3. Experimental

4. Conclusion

References and links

Cited By

Figures (7)

Equations (26)

Optical Materials Express