Tunable optical parametric generator based on the pump spatial walk-off

Andrea Cavanna; Felix Just; Polina R. Sharapova; Michael Taheri; Gerd Leuchs; Maria V. Chekhova

doi:10.1364/OL.41.000646

During the last decades, laser technology aims to cover all possible wavelength ranges, from ultraviolet to infrared and terahertz. Although available laser active materials and, accordingly, available lasing wavelengths are limited, one can still fill almost all spectral gaps with the help of nonlinear optics. The common tools to generate radiation at a desired frequency are optical parametric oscillators (OPOs) in which the fields oscillate in a cavity to increase the conversion efficiency. This, however, removes the possibility of generating picosecond and shorter pulses. In this case, one can use pulsed pump and continuous-wave seeding, but this is technically more complicated. Here, we analyze a new OPG, which is based on high-gain parametric downconversion (PDC) along the pump Poynting vector and, therefore, requires neither cavity nor seeding. It is wavelength tunable and spatially single mode and broadband; the spectral width is easily adjustable.

Phase matching for PDC is often achieved through birefringence, with the pump extraordinary polarized and, therefore, subject to spatial walk-off. Inside the crystal, the Poynting vector of the pump is tilted by an angle $ρ$ without affecting the direction of the wavevector. To achieve a high conversion efficiency, it is preferable to have collinear phase matching so that the radiation is self-amplified, in combination with tight focusing. Under these conditions, unless two-crystal compensation schemes are used [1], the pump beam walks off from the signal and idler beams and does not amplify them further. Furthermore, the signal and idler beams can be upconverted to the pump frequency with an overall effect of reducing the spatial beam quality. A common method to prevent the upconversion is to use noncollinear phase matching [2–4], but this reduces drastically the interaction length in the crystal. In OPOs, the cavity compensates for the reduced length of the interaction.

In our system, we generate PDC along the pump Poynting vector, therefore using noncollinear phase matching, as is often done in seeded OPOs and for other nonlinear effects [5,6]. Since, in this configuration, the walk-off is not a limiting factor anymore, a long crystal can be used with tighter pump focusing, the only limitation being the Rayleigh length of the pump. Moreover, upconversion is avoided, and neither a cavity nor seeding is needed as the parametric gain is high enough. As we will show, the waist of the pump determines the bandwidth of the amplified signal beam and, therefore, the corresponding idler one. By tilting the nonlinear crystal, we can also modify the phase matching and tune the central wavelength.

There are several approaches to the description of walk-off effects in low-gain PDC; see, for example [7,8], but at high gain, a different model is needed [9]. This model considers the transverse wavevector spectrum of PDC at a fixed wavelength, and it is applied to all wavelengths within our range of interest. The key characteristic is the two-photon amplitude (TPA), the probability amplitude for the signal photon to be emitted at angle $θ_{s}$ and the idler photon at angle $θ_{i}$ . The standard description [10,11] does not include the effect of the walk-off and, therefore, cannot be used. The TPA with the pump walk-off taken into account has the form [12]

F (θ_{s}, θ_{i}) = \exp (- \frac{Δ k_{x}^{2} σ_{x}^{2}}{2}) sinc [(Δ k_{z} + Δ k_{x} \tan ρ) \frac{L}{2}] .

Here, the

z

axis is assumed to be along the pump wavevector;

Δ k_{x} = k_{s} \sin (θ_{s}) + k_{i} \sin (θ_{i})

is the transverse wavevector mismatch;

σ_{x}

is the pump width (standard deviation of the Gaussian field profile);

Δ k_{z} = k_{p} - k_{s} \cos (θ_{s}) - k_{i} \cos (θ_{i})

is the longitudinal wavevector mismatch;

L

is the length of the crystal; and

k_{p, s, i}

are the pump, signal, and idler wavevectors. To obtain the wavelength-angular distribution of the signal beam intensity at low-gain PDC, the squared modulus of the TPA (1) is integrated over all idler angles

θ_{i}

for each signal wavelength

λ_{s}

. The resulting intensity distribution, with the integration over idler and signal wavelengths, separately normalized, is shown in Fig. 1(a). One can see that even at low gain, the anisotropy manifests itself in the angular asymmetry of the spectrum: for instance, the upper branch is broader than the lower one, but its peak value is lower. This asymmetry has been described in a number of theoretical and experimental papers [13–15].

Fig. 1. Wavelength-angular intensity spectrum for low-gain (left) and high-gain (right) PDC.

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In our experiment, the parametric gain is very high and, hence, the intensity distribution is modified. To describe this change of the TPA, we start by employing the Schmidt decomposition of the TPA [9],

F (θ_{s}, θ_{i}) = \sum_{n} \sqrt{λ_{n}} u_{n} (θ_{s}) v_{n} (θ_{i}) .

Here,

u_{n} (θ_{s})

and

v_{n} (θ_{i})

are the Schmidt modes of the signal and idler radiation, respectively; and

λ_{n}

are the Schmidt eigenvalues. At high parametric gain, the modes

u_{n} (θ_{s})

and

v_{n} (θ_{i})

are the same as at low gain, while the eigenvalues are redistributed and become [9]

λ_{n}^{'} \propto \sinh^{2} (Γ \sqrt{λ_{n}}) .

The parameter

Γ

can be found experimentally (see Fig. 2) by measuring the output power

P_{PDC}

of the generated PDC as a function of the input pump power

P

. The dependence has the form

P_{PDC} = A \sinh^{2} (B \sqrt{P}),

with

A

and

B

being fitting parameters, from which the single-mode parametric gain

G = B \sqrt{P} = Γ \sqrt{λ_{0}}

is derived. To obtain the high-gain intensity distribution, it is sufficient to calculate the TPA (2) with the new eigenvalues

λ_{n}^{'}

. As a result, as is shown in Fig. 1(b), only two small regions of the distribution are amplified; one is the signal in the IR spectral range centered near the walk-off angle, and the other one is the idler in the visible range [16,17].

Fig. 2. Experimental setup. The lens $L_{1}$ (focal length 50, 75, or 100 cm and the position changed accordingly) focuses the beam on the crystal. Another lens $L_{2}$ , with 10 cm focal length and a $2^{''}$ diameter, collects all the generated radiation. The lens $L_{3}$ has a focal length of 50 cm, and an IR camera is placed in its focal plane. The inset shows the PDC output power versus the input pump power, measured with an 80 μm pump waist. The solid line shows the fit with Eq. (4).

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The experimental setup is shown in Fig. 2. A 10 mm BBO crystal was pumped by the third-harmonic radiation of a Nd:YAG laser at 355 nm wavelength, 1 kHz repetition rate, and 18 ps pulse width. A half-wave plate and a Glan−Thomson prism polarizer were used to ensure the correct polarization of the beam. By using different focusing lenses, it was possible to change the pump beam waist inside the crystal. We used three different lenses with focal lengths of 500, 750, and 1000 mm which corresponded to 80, 130, and 160 μm FWHM waists, respectively. The PDC radiation was then generated under the noncollinear phase matching configuration. The signal in the near infrared (NIR) range was generated at the walk-off angle, while the idler was generated in the conjugated direction. Both beams were collected by a collimating lens with a focal length of 10 cm, while the remaining pump radiation was blocked. The spectra were measured by scanning the beams with fibers coupled to spectrometers with 50 μm spatial resolution. The fiber tips were positioned in the focal plane of the collimating lens to provide far-field intensity distributions.

A flip mirror that reflected the beam toward the beam shaping setup was placed in the infrared arm. Here, the quality of the beam was improved by means of two blazed diffraction gratings, which compensated for the wavelength angle dependence (angular chirp) of PDC and overlapped all wavelength components [18]. Both gratings were reflective with 600 lines per mm. The first one converged all wavelength components, while the second one, placed at the converging point, ensured parallel propagation of all wavelengths. The gratings were followed by a cylindrical lens telescope reducing the beam ellipticity. For instance, in the case of the pump waist 130 μm, the beam had to be magnified three times in the horizontal direction, which was achieved by using cylindrical lenses with focal lengths of 10 and 30 cm.

To evaluate the parametric gain, the output PDC power was measured versus the input pump power and then fitted by Eq. (4). The measurement was performed with the IR beam for all three available pump waists. As expected, for smaller waists, the gain is higher: at 35 mW pumping, $G = 10.5 \pm 0.5$ for an 80 μm waist (Fig. 2), $G = 7.0 \pm 0.8$ for 130 μm, and $G = 5.9 \pm 0.6$ for 160 m. The fit is valid only in the nondepleted pump regime, which was the case only for low pump power. The maximum conversion efficiency observed was 24% and was obtained with a waist of 130 μm and a pump power of 51 mW. Under these conditions, the output pump was strongly depleted.

Next, we analyze the wavelength-angular spectrum of the source. The measurements were performed for all three available pump waists. The wavelength-angular spectra of the signal and idler beams are shown in Figs. 3(b) and 3(d), together with the corresponding numerically calculated spectra [Figs. 3(a) and 3(c)]. In the calculation, the length of the crystal is taken $L = 5 mm$ to take into account the effect of the temporal walk-off due to the different group velocities of the signal and the pump. For a 10 mm crystal, the group delay between the beams is 3.7 ps which is on the order of the coherence time of the laser.

Fig. 3. Wavelength-angular spectrum, calculated (a), (c) and measured (b), (d), of the idler (a), (b) and signal (c), (d) beams generated with the 130 μm pump waist.

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One can notice that the maximum of the emission does not correspond exactly to the walk-off angle (shown by dashed line), but to a slightly larger angle. This is probably due to a narrower angular bandwidth and, accordingly, a higher peak intensity, at larger wavelengths, leading to the shift of the maximum toward larger angles.

Using the gratings, it is then possible to eliminate the angular chirp and to combine all wavelengths into a single beam [Fig. 4(a)].

Fig. 4. Wavelength-angular spectrum of the signal after the beam shaping (a) and the spatial intensity distribution of the optimized beam generated with 130 μm pump waist (b).

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The final step is the measurement of the coherence properties of the beam and, in particular, the number of spatial modes. For many applications, a single mode is preferable, for instance, when a single mode fiber has to be used. To estimate the number of modes, we have studied the spatial coherence function $G^{(1)} (x, x^{'})$ [19]. This was done by placing a double slit in the beam and, with the help of a lens, measuring the interference fringes in the far field with an IR CCD camera. The measurement was repeated with many sets of slits, each of 0.4 mm width and with different spacings, ranging between 0.2 to 6 mm. In Young’s double-slit experiment, the visibility of interference for the slits placed at positions $- x$ and $x$ is given by $G^{(1)} (- x, x)$ . As the spacing between the slits is increased, the visibility reduces, the coherence radius corresponding to the spacing for which the fringes visibility drops to 50% [20]. Using different pairs of slits, it is possible to map the anti-diagonal distribution of $G^{(1)} (- x, x)$ . On the other hand, the beam profile corresponds to the diagonal distribution, $G^{(1)} (x, x)$ . Assuming the Gaussian Schell’s model [21], it is possible to fully reconstruct the coherence function:

G^{(1)} (x, x^{'}) = \exp [- \frac{(x + x^{'})}{2 a^{2}}] \exp [- \frac{(x - x^{'})}{2 b^{2}}],

where the standard deviations

a

and

b

are calculated from the beam profile and the visibility distributions, respectively. It is possible to apply Mercer’s decomposition to this function [22]:

G^{(1)} (x, x^{'}) = \sum_{n} s_{n} ϕ_{n} (x) \cdot ϕ_{n}^{*} (x^{'}),

and, from the eigenvalues

s_{n}

, with

\sum_{n} s_{n} = 1

, one can calculate the number of modes

M_{x} = \sum_{n} 1 / s_{n}^{2}

.

The number of modes $M_{x}$ corresponds only to one direction as the subscript indicates. The total number of spatial modes in the beam is given by the product of the mode numbers in two orthogonal directions:

M_{tot} = M_{x} M_{y} .

Figure 5 shows the results obtained with the 130 μm pump waist. Without any frequency filtering, the total number of modes is measured to be

M_{tot} = 2.02

, while including a bandpass filter of 12 nm for the vertical displacement measurement reduces the number of modes to 1.32. At the same time, the number of temporal modes in the source is very large. Indeed, we have measured the normalized second-order correlation function

g^{(2)}

of the beam, which is related to the total number of modes

m

as

g^{(2)} = 1 + 1 / m

provided that a single mode has thermal statistics [23]. Even after a 12 nm interference filter, the measured

m

was found to be 25. Although this was not tested in this Letter, a single mode of signal or idler PDC radiation is known to have thermal statistics [24,25], which makes it very efficient for multi-photon effects such as optical harmonic generation or multiphoton absorption [26]. Therefore, the proposed source will be useful for nonlinear optics provided that the intensity fluctuations are not suppressed by pump depletion.

Fig. 5. Measured horizontal beam profile (green dashed line), the vertical beam profile (blue continuous line), and the visibility for horizontal displacement of the slits without bandpass filter (green triangles) and for vertical displacement of the slits with (red squares) and without (blue circles) bandpass filter of 12 nm for a 130 μm pump waist. The red dotted line represents the Gaussian fit of the square data points. The inset shows the $G^{(1)}$ function for the horizontal direction calculated using the Schell model.

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In conclusion, we have analyzed the spectral and coherence properties of an OPG based on high-gain PDC generated along the pump Poynting vector. The source is very efficient and, in our configuration, provides up to 24% of energy conversion. The radiation is spatially coherent and frequency tunable within the whole transparency range of the crystal. However, in this Letter, the analysis is concentrated around telecom wavelengths that are of great interest for optical fiber technologies. Finally, due to the nature of the PDC process, the signal beam always emerges together with the idler beam, which is its copy in terms of the photon number and is anti-correlated in frequency. Thanks to these correlations, it is possible, for example, to infer the signal beam properties by measuring the idler beam. This is especially important if the former is in a spectral range that is not accessible with measuring devices.

Funding

Seventh Framework Programme (FP7) (308803).

Acknowledgment

The authors are grateful to O. V. Tikhonova for helpful discussions.

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Tunable optical parametric generator based on the pump spatial walk-off

Abstract

Funding

Acknowledgment

REFERENCES

Cited By

Figures (5)

Equations (7)

Optics Letters