J. Eur. Opt. Society-Rapid Publ. 21, 25( 2025) 245
@ x b
¼ v �
gb v gp � v gf
� �e; ð6Þ @ x p v gp v gb þ v gf
with the group velocity v g @ x. It can be seen in equation
@ k
( 6) that the dimensionless parameter e is proportional to the group velocity difference between the forward and the backward wave. When the BWOPO operates far from resonance and in the low dispersion regime it implies that | e | 1. Consequently, the wavelength of the forward wave tunes with the pump frequency and the backward wavelength becomes almost insensitive to the pump frequency, as illustrated in Figure 1b. In the case of broadband multilongitudinal pump, the forward wave spectrum becomes broadband, while the backward wave stays narrow.
Similar to a conventional OPO, the BWOPO exhibits an oscillation threshold. As it is rather high the experimentally demonstrated BWOPOs have been based on pulsed pump lasers [ 24 ], while CW operation is still to be demonstrated [ 25 ]. As the backward wave interacts with the pump inside the crystal in a single pass it is beneficial if the pump pulse duration Dt p is longer than the time it takes the pump to travel to the end of the crystal, and the backward wave back to the entrance,
t p L 1 þ 1: ð7Þ v gp v gb
It is possible to use shorter pulses, but the efficiency is then drastically reduced. Godard et al. [ 26 ] have developed an analytical model for the threshold under pulsed and CW operation. The latter can be written,
I CW th
¼ e 0cn b n f n p k b k f 32d 2 eff L2: ð8Þ
where n b, n f, and n p are the refractive indices for the backward, forward and pump wave, respectively. k b and k f are the backward and forward wavelengths and d eff and L are the effective nonlinearity and sample length, respectively.
An example of how the energy and spectrum is transferred from the pump to the signal and idler for a BWOPO is shown in Figure 3. A PPKTP sample with 800 nm grating period was pumped with 480 ps stretched pulses from a regeneratively amplified Ti: sapphire laser. The pump was centered at 814.5 nm and had a frequency bandwidth of 1.21 THz. The forward wave( signal) had a central wavelength of 1123 nm and a FWMH spectral width of 410 GHz, while the backward wave is centered at 2952 nm with a narrow linewidth of only 13 GHz. One can also see how the spectrum is depleted at the longer wavelength of the pump pulse, which is then mirrored for the signal [ 27 ].
With Q-switched nanosecond pulses it is possible to obtain high pulse energies. Liljestrand et al. demonstrated a BWOPO in PPRKTP pumped at 1064 nm [ 28 ]. It had a forward propagating signal at 1740 nm and a counterpropagating idler at 2741 nm, with mJ outputs for both beams and a total signal-and-idler conversion efficiency of 47 %, see Figure 4a. Both generated waves had narrow spectral bandwidths, thanks to the unique properties of the counter-propagating nonlinear interaction and the single frequency pump laser. Liljestrand compared the spectrum of the backward wave with a regular singly resonant OPO( SRO) designed for the same wavelength and measured an almost 100-fold reduction in linewidth( see Fig. 4b).
As can be seen in Figure 4 there is no role-off in the conversion efficiency curve for the BWOPO. This means that there is no back-conversion, which is a consequence of the single pass configuration and promise that the conversion efficiency can be scaled further. Mølster et al. recently demonstrated a BWOPO with more than 70 % conversion efficiency at high repetition rate, using a diode laser seeded Yb-amplifier system [ 29 ]. This system was built as a prototype for atmospheric greenhouse gas monitoring, and it could be tuned with the seed laser on-off the absorption lines of carbon dioxide.
For certain applications, like long distance LIDAR sensing, further energy scaling might still be necessary. To evaluate this, a BWOPO was combined with a three-stage optical parametric amplifier using a single pump laser, see Figure 5. The pump was a single-longitudinal mode, injection seeded, 100 Hz, 10.5 ns, Q-switched diode-pumped Nd: YAG laser power amplifier system operating at 1064 nm with a maximum output energy of 110 mJ. The pump beam was split in two, where the weaker part pumped the BWOPO and the remaining pump was sent to the three-stage amplifier section to boost the output energy. Energies exceeding 20 mJ per pulse was obtained, while maintaining the narrow linewidth and high stability of the BWOPO [ 30 ]. As the OPA crystals had broad bandwidth the output could in this figuration be easily tuned just by changing the temperature of the BWOPO crystal.
3.1 Cascaded BWOPOs
When a BWOPO is pumped hard the generated signal can become strong enough to act as the pump for consecutive BWOPOs. This cascade effect enables the generation of additional signals and idlers in a stepwise manner. The energy condition for the first BWOPO is given as:
x p1 ¼ x s1 þ x i1; ð9Þ
where the x p1, x s1, and x i1 are the pump, first signal( forward) and first idler( backward) frequencies. x s1 then becomes the pump generating a second signal and idler as;
x s1 ¼ x p2 ¼ x s2 þ x i2: The corresponding phase-matching conditions are:
and k p ¼ k s1 � k i1 þ K g
k s1 ¼ k s2 � k i2 þ K g: ð10Þ
ð11Þ
ð12Þ
With strong enough pumping the second signal can then even drive a third BWOPO.
An example of such a cascaded BWOPO used a PPRKTP sample with a period 755 nm which was pumped with 240 ps pulses at 800 nm [ 31 ]. It generated a first signal at 1125 nm and an idler at 2763 nm. With further pumping