Abstract

A pulsed self-seeded double-ring optical parametric oscillator realized with periodically poled KTiOPO4 is demonstrated. When pumped from two opposite directions, the cavity supported two automatically aligned, independent, counter-propagating parametric wave pairs whose wavelength could be continuously tuned by varying a single degree of freedom. The tuning range from 1189 nm to 1267 nm has been achieved for the resonant idler waves. The parametric rings could be cross-seeded by using a feedback arrangement. Here, Fourier-domain filtering was utilized to spectrally manipulate the output spectrum of the seeded ring oscillator. Parametric ring oscillator efficiencies of 46 % were achieved.

© 2004 Optical Society of America

1. Introduction

Coherent light sources which can provide wavelength-tuning agility within a given spectral region are required by a number of applications, especially in spectroscopy and remote sensing. Optical parametric oscillators (OPO’s) employing periodically poled crystals such as PPKTP, PPLN, PPLT allows one to choose the spectral range due to the unique possibility to engineer noncritically phase-matched second-order interactions over the transparency region of these materials. Wavelength-agility, on the other hand, implies capability of rapid tuning of the output wavelength of the OPO, preferably, by adjusting a single degree of freedom. Several solutions for OPO tuning employing quasi-phase-matched (QPM) nonlinear materials have been demonstrated by changing crystal temperature [1], rotating a cylindrically polished nonlinear crystal inside the OPO cavity [2], changing the cavity angle or pump angle in noncollinear OPO arrangement [35], or electro-optically induce a phase shift inside the gain medium [6]. Temperature-tuning is relatively slow, while the cylindrical OPO requires a special geometry of the nonlinear crystal and the pump beam, in the case of angle tuning in a linear OPO, consisting of two cavity mirrors, adjusting of more than one degree of freedom is required. When tuning an OPO electro-optically, fields of the order of 2 kV/mm are necessary to induce significant wavelength change. In spectroscopy, coherent light sources with tailored spectrum and usually narrow bandwidth are required. There are numerous ways of reducing spectral bandwidth in parametric devices, to mention a few: OPOs with intracavity grating or birefringent filter [7], externally injection-seeding an OPO by a tunable diode laser [8] or a spectrally filtered seed-OPO [9], or self-seeding an optical parametric generator (OPG) by delaying the generated signal pulse in a 22 km long fibre [10].

The noncollinear parametric interaction offers a number of advantages such as easy separation of the pump and the parametric beams for easy seeding arrangement, drastic reduction of the backconversion, and thus the possibility to generate output spectra without spectral amplitude modulation. Moreover, by properly choosing the pump beam radius the efficiency of the noncollinear interaction can be as high as in the collinear case [11]. In this work we utilized the properties of noncollinear interaction in a single PPKTP crystal to demonstrate a self-aligned double ring parametric oscillator with a broad tunability achieved by changing only one degree of freedom. Moreover, we employed the inherently unidirectional nature of the parametric ring oscillators in order to modify spectrum generated in one direction and used this output for cross-seeding the ring oscillator in the opposite direction. By adding Fourier spectral filtering inside the ring cavity we demonstrate a self-aligned OPO tunable with a single degree of freedom and whose output spectra can be modified at will.

2. Experiment, results and discussion

Parametric interaction in QPM nonlinear structures is characterized by the angular dispersion of the signal and the idler waves, which can be derived from the vector phase-matching equations [3]. In the case of the pump wave-vector kp being fixed along the QPM grating vector Kg=2π/Λ, where Λ is the QPM period, the phase-matched signal and idler angles in the small-angle limit can be expressed by:

θi2=2ks(Kg+kg+kikp)kiks+ki2.
θs=kiksθi

where ki, ks are the moduli for the idler wave and signal wave-vectors, respectively. These dependences have been verified in a simple, 16 mm-long linear OPO cavity, built around the 8 mm-long AR-coated PPKTP crystal with the QPM period of Λ=9.1 µm. The OPO mirrors, had reflectivities of R1≈80 % and R2≈99 % for the idler wave and were AR-coated for the signal and the pump beams. The main reasons for choosing idler-resonant OPO configuration were the smaller angular dispersion and, at the same time, the larger diffraction of the idler wave, allowing for generation of narrower spectrum and higher spatial quality beams [11]. The OPO was pumped with a frequency-doubled Nd:YAG laser producing 5-ns long pulses of 532 nm radiation at 20 Hz, having a beam-quality parameter of M2=9. The pump beam was focused to a 260 µm radius (e-2 intensity) inside the PPKTP crystal. By tilting the cavity mirrors the OPO idler (signal) was tuned from 1189 nm (968 nm) to 1305 nm (898 nm), this corresponds to the phase-matching angles spanning from θi=0 (θs=0) to θi=45.6 mrad (θs=25.9 mrad). Due to the relatively loose focussing conditions the saturated OPO efficiency did not change essentially in the noncollinear configuration. For instance, for the phase-matched angles of θi=0 mrad, 18 mrad and 37 mrad the measured OPO efficiencies at the same pump intensity of 104 MW/cm2, were 67 %, 67 % and 61 %, respectively.

The noncollinear parametric interaction can be utilized to simplify the OPO tuning arrangement. Fig. 1 shows a ring OPO cavity that we realized by adding totally reflecting mirror M3 to the above-described linear cavity that originally consisted of mirrors M1 and M2. The physical length of the resonator ring was 170 mm. The pump laser and crystal period was the same as mentioned above [11]. In contrast to the laser ring oscillators, the ring OPO’s are automatically unidirectional due to the vector nature of the parametric interactions. This greatly simplifies the ring cavity design, makes it more compact, and obviates the need for intracavity Faraday isolators. Moreover, once aligned for one wavelength the ring OPO can be tuned by simply rotating one of the cavity mirrors (e.g., M2 as shown in Fig. 1.), i.e., by varying a single degree of freedom. Consequently this ring OPO cavity lends itself to relatively simple automation of the output wavelength tuning.

 

Fig 1. Singly-pumped OPO setup.

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2.1 Singly-pumped ring OPO

In our setup we could continuously tune the OPO idler (signal) from 1189 nm (968 nm) to 1267 nm (917 nm). The limiting factor on the total continuous tuning range was the aperture of the mirror M3. The pump intensity at threshold was higher in the ring OPO as compared to the linear cavity, because of the increased cavity length. It should be noted that the ring cavity was about 10 times longer than the linear cavity. For example, at an internal angle of θi=18 mrad, 22.3 mrad and 30.7 mrad, the measured thresholds were 44.5 MW/cm2, 67 MW/cm2 and 79.5 MW/cm2 or about four-times higher than in the linear cavity at the same noncolinear angles. The parametric gain provided by the 10 mm-long PPKTP was large enough to reach substantial efficiencies of 49%, 48.5% and 34% corresponding to the noncollinear angles of θi=18 mrad, 22.3 mrad and 30.7 mrad, and measured at the constant pump intensity of 190 MW/cm2.

2.2 Double-pumped ring OPO

The unidirectional nature of the parametric ring resonators can be fully exploited by launching a backward pump beam (kp,2 on Fig. 2.) into the cavity from the opposite direction with respect to the forward pump beam (kp,1 on Fig. 2.). When the two beams are overlapping (kp,1=-kp,2), the cavity then supports two independent, automatically aligned and counter-propagating parametric ring oscillators. Moreover, the signal and idler outputs of both oscillators are easily separated as shown in Fig. 2, which is useful in certain experimental situations.

 

Fig. 2. Double-pumped OPO setup

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In our experimental setup a dielectric R=0.5 reflectivity mirror was used to split the output of the frequency-doubled Nd:YAG laser into two counter-propagating pump beams: the forward pump (P1) and the backward-pump (P2). Due to additional optical path, the latter was delayed by 1.5-ns with respect to P1. The pump beam P1 (P2) was focused to an e-2 intensity radius of 155 µm (200 µm) inside the PPKTP crystal. A slightly larger beam waist of the backward-pump beam was chosen in order to better mode-match the seed beam and thus increase the seeding efficiency in a cross-seeded ring OPO configuration as will be described below. Also the larger beam size of the backward pump lowers the OPO threshold due to increased effective noncollinear parametric interaction length [3]. For instance, at an internal noncollinear angle of θi=22.3 mrad, the thresholds for the forward-pumped (P1) and the backward-pumped (P2) OPO were 67 MW/cm2 and 44 MW/cm2, respectively. As the counter-propagating ring oscillators share the same physical beam path, the output wavelength generated in both rings can be simultaneously tuned, by varying a single degree of freedom, for instance, rotating the mirror M2 in the plane of the Fig. 2. The independence of the counter-propagating ring OPO’s can also be exploited to generate two signal (idler) waves at distinct and tunable wavelengths in the same cavity by using the forward- and the backward-propagating pump sources at slightly different wavelengths, which are within the phase-matching range of the QPM crystal. For instance, by employing the PPKTP crystals with the periodicity of 9.1 µm and using two frequency-doubled single-frequency Q-switched YAG lasers detuned by 50 GHz as pump sources would produce two counter-propagating signal or idler waves with approximately constant frequency separation of 500 GHz over the tuning range. On the other hand, the double-ring OPO configuration can be used to generate parametric waves with modified spectrum by seeding one of the ring oscillators by the spectrally-filtered output of the other ring, as we demonstrate below.

2.3 Self-seeded double ring OPO

The setup which we used for this purpose is shown in Fig. 3. Here the total reflectivity mirror M3 has been exchanged with a diffraction grating with 900 grooves/mm which diffracted the idler wave generated by the forward-pump (P1) into the minus first order (m=-1) with an efficiency of approximately 35 %.

 

Fig. 3. Self-seeded double-pumped OPO setup.

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At an incident angle of +80°, the diffraction angle was +10.3° with respect to the grating normal. The diffraction grating and the mirror M4 were located in the conjugate planes of the lens with the focal length of f=50 mm. This is a zero-dispersion arrangement, which only reverses the direction of propagation of the optical beam without changing spectral or spatial properties at the point of beam diffraction on the grating [12]. The spectral filtering can be performed in the Fourier plane (plane of the mirror M4) by inserting appropriate spatial masks. Moreover, the additional delay of around 1 ns introduced by this arrangement compensates for the delay of the pump P2 thus increasing the seeding efficiency. Overall, about 10% of the idler generated in the forward-pumped OPO was utilised for seeding the opposite-directional ring OPO.

Figure 4 shows the measured pump depletion in the forward-pumped (P1) OPO (solid circles), unseeded backward-pumped (P2) OPO (solid squares) and seeded backward-pump OPO (open triangles). The pump depletion was measured for the OPO alignment, corresponding to the non-collinear internal angle of θi=22.3 mrad (the conjugate angle for the signal was θs=16.7 mrad). The unseeded backward-pumped ring OPO reached threshold at the of 43 MW/cm2, approximately 1.5 times lower pump intensity than in the forward pumped ring. It is evident from the Fig. 4 that the larger beam waist of the backward pump provides for faster rise in the backward-pumped ring OPO efficiency, the usual feature of the noncollinear parametric interactions [3]. Consequently, the unseeded backward-pumped ring reached efficiency of 46% at a pump intensity of 110 MW/cm2 and was limited by the available pump power. When evaluating the cross-seeded ring configuration we kept the forward pump at a constant intensity of 200 MW/cm2 and varied the power of the backward pump. At this forward pump level the seed energy injected into the opposite-directional ring was 14 µJ. In the seeded configuration the backward pump depletion reached nearly 46 % at 84 MW/cm2 generating 103 µJ of the idler and 136 µJ of the signal energy. The measured spatial profile of the idler beam in the far-field was almost perfectly circular and the intensity distribution was Gaussian, with M2=2.

 

Fig. 4. Measured pump depletion as a function of pump intensity. The forward pumped OPO (solid circles), unseeded backward pumped OPO (solid squares), and the seeded backward pumped (open triangles) OPO. In the latter case the intensity of the forward pump P1 was kept constant at 200 MW/cm2.

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Next we investigated the possibility of manipulating the seed spectrum in the Fourier plane utilising the fact that the spectral frequencies are spatially distributed over mirror M4. First we inserted a vertical 40 µm-wide beam block on to the mirror M4 thus producing a hole in the seed spectrum. The effect on the backward-pumped OPO idler spectrum is shown in Fig. 5(a). The beam block could be translated horizontally thus placing the spectral void at a desired position. The FWHM width of the spectral hole in the OPO output was 295 GHz. Further, we inserted a 20 µm-wide thin-strip mirror instead of the mirror M4 and hence reflected back only a small part of the frequency spectrum, shown in Fig. 5(b).

 

Fig. 5. (a) Output spectrum for the P2 idler spectrum, when the seed was manipulated with a beam block in the Fourier plane.

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Fig. 5. (b) Output spectrum when seeded from a thin strip-mirror in the Fourier plane (high peak, red curve) and unseeded idler spectrum (broader spectrum, black curve).

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The seed’s FWHM bandwidth was 56 GHz and its position could be tuned over the whole frequency range by translating the stripe-mirror in the Fourier plane. For this Fourier filter setup a theoretical calculation, assuming a diffraction-limited optical system, indicated a spectral bandwidth of 38 GHz. For the beams with M2=2 as in the case of the OPO idler, the spectral bandwidth of the filter increases to about 54 GHz which concurs very well with the experimental data. In order to reduce the bandwidth, a grating with larger groove density and a Fourier lens with a longer focal length can be used. The use of diffraction-limited beams would also be preferable. It was not our goal during this investigation to reach narrower bandwidth, but it clearly can be done if applications require. Furthermore, in this configuration the seed energy incident on the PPKTP was only 6 nJ, significantly reduced due to imperfections of the stripe-mirror. The resulting seeded idler spectrum from the backward-pumped ring OPO at full pump power is shown in Fig. 5(b), where it is compared with the unseeded OPO spectrum. The broadband pedestal appearing in the seeded spectrum originates from the onset of the competing unseeded operation of the OPO due to the low seed intensity. So the peak-to pedestal ratio was 35 dB at the backward pump intensity of 33 MW/cm2 and it decreased to 6.4 dB for the pump intensity of 68 MW/cm2. Clearly, the contrast can be increased by using a better quality stripe-mirror and more pump power in the forward-pumped ring OPO. The FWHM of the spectral peak at full pump power was 137 GHz and should be compared with the FWHM of 1.43 THz of the unseeded OPO. In order to further reduce the spectral bandwidth, a diffraction grating with larger groove density and a Fourier lens with a longer focal length can be used. The use of diffraction-limited beams would also be preferable. It was not our goal during this investigation to reach narrower bandwidth, but it clearly can be achieved. In addition to the spectral narrowing, the current ring OPO setup allows one to manipulate and shape the spectrum in other ways as an application requires.

3. Conclusion

In conclusion, we have demonstrated that noncollinear parametric interaction can be conveniently used to design ring OPO cavities with simple tuning arrangement involving adjusting a single degree of freedom. Employing the two independent pump beams gives a unique possibility to realise two independent counter-propagating OPO’s in the same physical cavity with easily separable outputs. In particular we utilized this property of noncollinear ring OPO to cross-seed one of the ring cavities and modify its output spectrum. As a result the ring cavity generated two output signal-idler pairs with distinctly different and a modifiable spectrum. Moreover, the efficiency of the ring OPO approaching 50% has been experimentally achieved.

References and links

1. M. Peltz, U. Bader, A. Borsutzky, R. Wallenstein, J. Hellström, H. Karlsson, V. Pasiskevicius, and F. Laurell, “Optical parametric oscillators for high pulse energy and high average power operation based on large aperture periodically poled KTP and RTA,” Appl. Phys. B 73, 663–670 (2001) [CrossRef]  

2. P. Feve, O. Pacaud, B. Boulanger, B. Menaert, J. Hellström, V. Pasiskevicius, and F. Laurell, “Widely and continuously tunable optical parametric oscillator based on a cylindrical periodically poled KTiOPO4 crystal,” Opt. Let. 26, 1882–1884 (2001). [CrossRef]  

3. V. Smilgevicius, A. Stabinis, A. Piskarskas, V. Pasiskevicius, J. Hellström, S. Wang, and F. Laurell, “Noncollinear optical parametric oscillator with periodically poled KTP,” Opt. Commun. 173, 365–369 (2000). [CrossRef]  

4. M. J. Missey, V. Dominic, and P. E. Powers, “Periodically poled lithium niobate monolithic nanosecond optical parametric oscillators and generators,” Opt. Lett. 24, 1227–1229 (1999). [CrossRef]  

5. S. T. Yang and S. P. Velsko, “Frequency-agile kilohertz repetition-rate optical parametric oscillator based on periodically poled lithium niobate,” Opt. Lett. 24, 133–135 (1999). [CrossRef]  

6. P. Gross, M. E. Klein, H. Ridderbusch, D.-H. Lee, J.-P. Meyn, R. Wallenstein, and K.-J. Boller, “Wide wavelength tuning of an optical parametric oscillator through electro-optic shaping of the gain spectrum,” Opt. Lett. 27, 1433–1435 (2002). [CrossRef]  

7. S. J. Brosnan and R. L. Byer, “Optical parametric oscillator threshold and linewidth studies,” IEEE J. Quantum Elec. 15, 415–431 (1979). [CrossRef]  

8. G. W. Baxter, Y. He, and B. J. Orr, “A pulsed optical parametric oscillator, based on periodically poled lithium niobate (PPLN), for high resolution spectroscopy,” Appl. Phys. B 67, 753–756 (1998). [CrossRef]  

9. A. Borsutzky, “Frequency control of pulsed optical parametric oscillators,” Quantum Semiclass. Opt. 9, 191–207 (1997). [CrossRef]  

10. M. Rahm, G. Anstett, J. Bartschke, T. Bauer, R. Beigang, and R. Wallenstein, “Widely tunable narrow-linewidth nanosecond optical parametric generator with self-injection seeding,” Appl. Phys B 79, 535–538 (2004). [CrossRef]  

11. M. Tiihonen, V. Pasiskevicius, and F. Laurell, “Spectral and spatial limiting in idler-resonant optical parametric oscillator with PPKTP,” in Advanced Solid-State Photonics, none eds., Vol. 94 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 2004), (to be published nov. 2004).

12. J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic Press, 1996), Chap. 7.

References

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  1. M. Peltz, U. Bader, A.Borsutzky, R. Wallenstein, J. Hellström, H. Karlsson, V.Pasiskevicius and F. Laurell, �??Optical parametric oscillators for high pulse energy and high average power operation based on large aperture periodically poled KTP and RTA,�?? Appl. Phys. B 73, 663-670 (2001)
    [CrossRef]
  2. P. Feve, O. Pacaud, B. Boulanger, B. Menaert, J. Hellström, V. Pasiskevicius and F. Laurell, �??Widely and continuously tunable optical parametric oscillator based on a cylindrical periodically poled KTiOPO4 crystal,�?? Opt. Let. 26, 1882-1884 (2001).
    [CrossRef]
  3. V. Smilgevicius, A. Stabinis, A. Piskarskas, V. Pasiskevicius, J. Hellström, S. Wang and F. Laurell, �??Noncollinear optical parametric oscillator with periodically poled KTP,�?? Opt. Commun. 173, 365-369 (2000).
    [CrossRef]
  4. M. J. Missey, V. Dominic and P. E. Powers, �??Periodically poled lithium niobate monolithic nanosecond optical parametric oscillators and generators,�?? Opt. Lett. 24, 1227-1229 (1999).
    [CrossRef]
  5. S. T. Yang and S. P. Velsko, �??Frequency-agile kilohertz repetition-rate optical parametric oscillator based on periodically poled lithium niobate,�?? Opt. Lett. 24, 133-135 (1999).
    [CrossRef]
  6. P. Gross, M. E. Klein, H. Ridderbusch, D.-H. Lee, J.-P. Meyn, R. Wallenstein, K.-J.Boller, �??Wide wavelength tuning of an optical parametric oscillator through electro-optic shaping of the gain spectrum,�?? Opt. Lett. 27, 1433-1435 (2002).
    [CrossRef]
  7. S. J. Brosnan and R. L. Byer, �??Optical parametric oscillator threshold and linewidth studies,�?? IEEE J. Quantum Elec. 15, 415-431 (1979).
    [CrossRef]
  8. G. W. Baxter, Y. He and B. J. Orr, �??A pulsed optical parametric oscillator, based on periodically poled lithium niobate (PPLN), for high resolution spectroscopy,�?? Appl. Phys. B 67, 753-756 (1998).
    [CrossRef]
  9. A. Borsutzky, �??Frequency control of pulsed optical parametric oscillators,�?? Quantum Semiclass. Opt. 9, 191-207 (1997).
    [CrossRef]
  10. M. Rahm, G. Anstett, J. Bartschke, T. Bauer, R. Beigang and R. Wallenstein, �??Widely tunable narrow-linewidth nanosecond optical parametric generator with self-injection seeding,�?? Appl. Phys B 79, 535-538 (2004).
    [CrossRef]
  11. M. Tiihonen, V. Pasiskevicius and F. Laurell, �??Spectral and spatial limiting in idler-resonant optical parametric oscillator with PPKTP,�?? in Advanced Solid-State Photonics, none eds., Vol. 94 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 2004), (to be published nov. 2004).
  12. J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic Press, 1996), Chap. 7.

Advanced Solid-State Photonics 2004 (1)

M. Tiihonen, V. Pasiskevicius and F. Laurell, �??Spectral and spatial limiting in idler-resonant optical parametric oscillator with PPKTP,�?? in Advanced Solid-State Photonics, none eds., Vol. 94 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 2004), (to be published nov. 2004).

Appl. Phys B (1)

M. Rahm, G. Anstett, J. Bartschke, T. Bauer, R. Beigang and R. Wallenstein, �??Widely tunable narrow-linewidth nanosecond optical parametric generator with self-injection seeding,�?? Appl. Phys B 79, 535-538 (2004).
[CrossRef]

Appl. Phys. B (2)

M. Peltz, U. Bader, A.Borsutzky, R. Wallenstein, J. Hellström, H. Karlsson, V.Pasiskevicius and F. Laurell, �??Optical parametric oscillators for high pulse energy and high average power operation based on large aperture periodically poled KTP and RTA,�?? Appl. Phys. B 73, 663-670 (2001)
[CrossRef]

G. W. Baxter, Y. He and B. J. Orr, �??A pulsed optical parametric oscillator, based on periodically poled lithium niobate (PPLN), for high resolution spectroscopy,�?? Appl. Phys. B 67, 753-756 (1998).
[CrossRef]

IEEE J. Quantum Elec. (1)

S. J. Brosnan and R. L. Byer, �??Optical parametric oscillator threshold and linewidth studies,�?? IEEE J. Quantum Elec. 15, 415-431 (1979).
[CrossRef]

Opt. Commun. (1)

V. Smilgevicius, A. Stabinis, A. Piskarskas, V. Pasiskevicius, J. Hellström, S. Wang and F. Laurell, �??Noncollinear optical parametric oscillator with periodically poled KTP,�?? Opt. Commun. 173, 365-369 (2000).
[CrossRef]

Opt. Let. (1)

P. Feve, O. Pacaud, B. Boulanger, B. Menaert, J. Hellström, V. Pasiskevicius and F. Laurell, �??Widely and continuously tunable optical parametric oscillator based on a cylindrical periodically poled KTiOPO4 crystal,�?? Opt. Let. 26, 1882-1884 (2001).
[CrossRef]

Opt. Lett. (3)

Quantum Semiclass. Opt. (1)

A. Borsutzky, �??Frequency control of pulsed optical parametric oscillators,�?? Quantum Semiclass. Opt. 9, 191-207 (1997).
[CrossRef]

Other (1)

J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic Press, 1996), Chap. 7.

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Figures (6)

Fig 1.
Fig 1.

Singly-pumped OPO setup.

Fig. 2.
Fig. 2.

Double-pumped OPO setup

Fig. 3.
Fig. 3.

Self-seeded double-pumped OPO setup.

Fig. 4.
Fig. 4.

Measured pump depletion as a function of pump intensity. The forward pumped OPO (solid circles), unseeded backward pumped OPO (solid squares), and the seeded backward pumped (open triangles) OPO. In the latter case the intensity of the forward pump P1 was kept constant at 200 MW/cm2.

Fig. 5. (a)
Fig. 5. (a)

Output spectrum for the P2 idler spectrum, when the seed was manipulated with a beam block in the Fourier plane.

Fig. 5. (b)
Fig. 5. (b)

Output spectrum when seeded from a thin strip-mirror in the Fourier plane (high peak, red curve) and unseeded idler spectrum (broader spectrum, black curve).

Equations (2)

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θ i 2 = 2 k s ( K g + k g + k i k p ) k i k s + k i 2 .
θ s = k i k s θ i

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