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J. Eur. Opt. Society-Rapid Publ. 21, 25( 2025)
a consequence of the single-pass configuration BWOPOs does not suffer from back-conversion and can hence provide improved energy conversion efficiency. These features make BWOPOs a promising avenue for generating high-quality, tunable light in specialized applications, particularly where narrow linewidths or precise control over spatial beam characteristics are necessary.
This paper is structured as follows: First, we provide background on nonlinear optics with a focus on QPM, followed by a discussion of the fundamental principles of BWOPOs. We then present specific examples of BWOPO configurations, including cascaded BWOPOs, phase-locked degenerate BWOPOs, and BWOPOs implemented in waveguides. Applications of BWOPOs are subsequently explored, with particular emphasis on gas sensing and CO 2 monitoring. The paper concludes with a summary and an outlook on future research directions.
2 Background on nonlinear optics using QPM towards the backward-wave optical parametric oscillator
Nonlinear optics involves the interaction of light with matter to produce new frequencies or amplify existing signals. Three-wave mixing processes, governed by energy conservation( x 1 + x 2 = x 3), are fundamental to devices like OPOs. QPM allows engineers to optimize these interactions by tailoring the periodicity of nonlinear materials, enabling efficient energy transfer while maintaining phase coherence [ 4 ]. The phase mismatch between interacting waves is in QPM compensated by periodically reversing the sign of the nonlinear coefficient of the material. This periodic modulation allows the waves to remain in phase over longer distances, enhancing the energy transfer.
Efficient energy transfer requires phase-matching and for QPM the condition can be expressed as,
k 1 ¼ k 2 þ k 3 þ K g;
where k i are the are the wavevectors of the interacting waves and K g is the grating vector, defined as,
K g ¼ 2pm K; ð2Þ
ð1Þ
describing the periodic modulation of the nonlinear coefficient. K is the grating vector imposed by periodic poling of the nonlinear medium and m is the order parameter. When one of the generated waves is to be traveling in the opposite direction to the driving pump wave, as in the case of the BWOPO, the phase-matching condition becomes. k p ¼ k f � k b þ K g:
where the subscripts p, f, and b denote pump wave, forward wave, and backward wave, respectively. The grating vector K g ¼ 2pm, then becomes very large as can be seen in
K
Figure 1a. The grating period becomes correspondingly small, in most cases in the sub-lm range, and in the order of half of the wavelength of the counterpropagating wave.
ð3Þ
The field of QPM nonlinear optics took off with the demonstration of domain engineered lithium niobate and KTiOPO 4( KTP) waveguides in 1989 [ 11 – 14 ]. It led forward to the development of electric field poling allowing QPM to be realized in bulk samples [ 15, 16 ]. Today this technology is well-established and periodically poled lithium niobate( PPLN) and periodically poled KTP( PPKTP) are commercially available from several vendors and used in many products.
PPKTP and periodically poled Rubidium doped KTP( PPRKTP) are two materials that have been widely used in nonlinear optics. They are thermally stable and show high damage thresholds and can hence operate under high-intensity conditions, which is crucial for BWOPOs. PPKTP and PPRKTP are also less susceptible to photorefractive damage, blue and green induced IR absorption than for example PPLN [ 17, 18 ]. Additional important properties of crystals from the KTP family are their wide transparency range( roughly 350 – 4500 nm) and their high nonlinear optical coefficients [ 19, 20 ].
So far, the only materials in which BWOPOs have been demonstrated are PPKTP and PPRKTP. This falls back on the material’ s quasi-one-dimensional crystal structure which enables fabrication of the dense domain gratings required for backward processes. The technology for making dense domain gratings was developed by Zukauskas et al. [ 21, 22 ] and is based on fabrication of a coercive field grating by ion-exchange followed by electric field poling. This has enabled fabrication of gratings with periods as short as 317 nm in 1 mm thick PPRKTP [ 23 ]. It corresponds to a domain aspect ratio of 6000:1( height / width) and 60,000 domains in a 10 mm long sample.
3 Principles of the backward-wave optical parametric oscillator
The BWOPO differs from the conventional copropagating OPO by generating photons counterpropagating to the pump. This geometry relies on self-established distributed feedback where no external cavity is needed to build up the parametric waves. As no cavity mirrors are needed the device has also been referred to as a mirrorless optical parametric oscillator or MOPO [ 8 ]. Harris was the first to devise a BWOPO in 1966 [ 9 ], and the first demonstration was done by Canalias and Pasiskevicius in 2007 [ 8 ]. A BWOPO setup can be very simple and compact with just a pump laser and basic optics, as can be seen in Figure 2.
The frequency dependence of the forward and backward waves is obtained by differentiating equation( 1) with respect to the pump frequency, v p:
1 v p
¼ @ k f @ x p
� @ k b @ x p
; ð4Þ
which can be transformed to the following two expressions,
@ x f
¼ v �
gf v gb þ v gp
� 1 þ e; ð5Þ @ x p v gp v gb þ v gf