1. Introduction

More than 60 years ago, Pancharatnam [1] pointed out that polarized light can accumulate a phase, in addition to that associated with propagation, when the polarization state is changed over a non-closed trajectory between two points on the Poincare sphere. This geometric phase was shown to be calculated via the so called Pancharatnam connection, demonstrating an intrinsic dependence on the geometric properties of the path taken in polarization space. A special case is when the polarization state undergoes a closed trajectory over several points and returns to its original value. The resulting geometric phase, γ, is then associated with the solid angle, Ω, enclosed by the geodesic polygon on the Poincare sphere, namely $\gamma =-\frac{1}{2}\mathrm{\Omega }$.

A few decades later, Berry [2, 3] demonstrated that an eigenstate of a quantum-mechanical (QM) Hamiltonian, following an adiabatic closed trajectory in the parameter space, accumulates a geometric phase equal to $\gamma =-\frac{1}{2}\mathrm{\Omega }$ for spin-1/2 particles in an adiabatically rotating magnetic field. He later realized [4] that Pancharatnam’s phase for polarized light was equivalent to the adiabatic phase for spin-1/2 particles, due to the similar algebra describing the two phenomena. This polarization-induced geometric phase was thereafter called the Pancharatnam-Berry (PB) phase (where the latter is commonly referred to in the context of non-closed trajectories of the polarization state [5]). Indeed, realizations of the PB phase by means of engineered birefringence of optical devices has given rise to numerous applications, such as light guiding [6] and wavefront shaping [7–9].

Chiao et al. [10, 11] have also demonstrated a scheme for the realization of an adiabatic geometric phase for light. They utilized the photon’s helicity (h = ±1) following an adiabatic change in the propagation direction, $\widehat{k}$, along a helical fiber with N windings. The resulting geometric phase was then given by γ = ∓NΩ, where Ω is now the solid angle enclosed by the trajectory of the k-vector in reciprocal space. The two aforementioned schemes for obtaining geometric phases for light are demonstrated in Figs. (1a)–Figs. (1a)(1b), for the case γ = −π.

 

Fig. 1 Different schemes for obtaining a geometric phase of γ = −π for light. (a) The closed-trajectory Pancharatnam-Berry phase: a circularly-polarized light beam undergoes subsequent birefringence via 2 half-wavelength plates, enclosing a solid angle Ω = 2π on the Poincare sphere, and where $\gamma =-\frac{1}{2}\mathrm{\Omega }$. (b) The adiabatic geometric phase for the photon helicity. A guided mode experiences an adiabatic change in direction when it propagates along a helical fiber with N = 2 windings. This path encloses a solid angle Ω = π/2 on the k-vector sphere, where now γ = −NΩ. (c) The adiabatic nonlinear optical geometric phase. A mutual beam eigenstate in the idler frequency, |ω₊〉 = |ω_i〉, follows a closed trajectory of the magnetic field equivalent on the SFG sphere, enclosing a solid angle Ω = 2π, where $\gamma =-\frac{1}{2}\mathrm{\Omega }$. The closed trajectory is enabled through a specific modulation of the QPM parameters: the phase mismatch and poling duty cycle.

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In this paper, we introduce a different manifestation of the adiabatic geometric phase for light, that will neither include changes in the polarization nor in the direction of the photons. Rather, we shall harness the equivalence between the dynamics of the nonlinear sum-frequency-generation (SFG) process, and the one describing spin-1/2 particles in a magnetic field. In the presence of an adiabatic rotation of the magnetic field equivalent in parameter space, the two SFG eigenstates, or mutual beams, will each acquire a corresponding adiabatic geometric phase. A special case is then considered where the initial eigenstate is the incident wave itself in the idler frequency. The resulting geometric phase is then present in the form of a pure phase factor that can be experimentally measured.

Analogies between SFG and 2-level system dynamics have already given rise to robust, broadband and efficient adiabatic frequency conversion [12–17], where the Hamiltonian was changed adiabatically such that it induces the equivalent of population inversion. Geometric phase analogues have also been studied in nonlinear optics, in the context of four-wave-mixing dynamics [18], as well as with artificially engineered plasmonic metasurfaces [19, 20]. Recently, it was pointed out [21] that a non-adiabatic PB phase exists in the SFG dynamics, thus enabling through unidirectional pumping to realize non-reciprocal propagation, which can be useful for optical isolators. In this paper we focus instead on adiabatic geometric phase analysis. In addition to being robust, the use of adiabatic closed trajectories in parameter space enables the implementation of methods for controlling the geometric phase in nonlinear optics.

The outline of this paper is given as follows: in section 2, we present the theoretical model unveiling the analogy between the spin-1/2 adiabatic geometric phase and the equivalent case in SFG; in section 3, we present different schemes for obtaining and controlling the geometric phase in nonlinear optics; in section 4, we discuss future applications for wavefront shaping, and then finally conclude this paper in section 5.

2. Model

We shall now present the equivalence between spin-1/2 dynamics in a magnetic field and the classical nonlinear optical system. The time-independent coupled wave equations (CWEs) for SFG under the approximations of diffraction-free propagation, slowly varying envelope and undepleted pump field are given by [22]

Substituting Eq. (3) into Eqs. (1) and (2), we keep in the right hand side only one of the terms m = ±1 which are assumed to oscillate closest to phase matching, similar to the rotating wave approximation. Next, we make a transformation to a rotating frame $\left({\tilde{A}}_i,{\tilde{A}}_s\right)$ given by (4)

 

Fig. 2 The control over the equivalent magnetic field is possible via the variation of the QPM parameters. The $\widehat{z}$ component of B is proportional to the phase mismatch Δk, the transverse radial component is given by the product of the nonlinear coupling κ and duty cycle parameter $\overline{d}$, and the relative phase between the pump and the poling pattern, ϕ, determines the azimuthal angle.

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The vector ${\left({\tilde{A}}_i{\tilde{A}}_s\right)}^T$ can be described as a state vector using spin-1/2 nomenclature. Adopting the braket notation, we write this state as $|\psi 〉=\left({\tilde{A}}_i|{\omega }_i〉+{\tilde{A}}_s|{\omega }_s〉\right)/\sqrt{{\left|{\tilde{A}}_i\right|}^2+{\left|{\tilde{A}}_s\right|}^2}$, where ${\left|{\tilde{A}}_i\right|}^2+{\left|{\tilde{A}}_s\right|}^2=const.$, due to the Manley-Rowe relations [22]. Different frequency kets |ω_j〉 are orthogonal in the sense that they cannot produce an observable interference intensity pattern, and they are also distinguishable by means of spectral filtering. Explicitly, if different carrier frequencies are assumed to be located far apart (with respect to their bandwidth) in the spectral domain, we can write $〈{\omega }_j|{\omega }_k〉={\displaystyle \int d{\omega }^{\prime }{S}_j^{*}\left({\omega }^{\prime }\right)S_k\left({\omega }^{\prime }\right)}={\delta }_{jk}$. In the latter relation, S_j(ω′) ≡ 〈ω′|ω_j〉 is the normalized (∫ dω′|S(ω′)|² = 1) spectral amplitude of the |ω_j〉 wave, at frequency ω′.

We call the state |ψ〉 a mutual beam, since it comprises of two co-propagating frequency components. The ket states |ω_i〉 and |ω_s〉 can be identified as the equivalents of the spin-up |↑z〉 and spin-down |↓_z〉 in the z-basis of the spin, respectively. All the possible states of the system can now be mapped onto a Bloch sphere, with the spin-up (idler wave) and spin-down (signal wave) states located at the north and south poles of the sphere, respectively. In that sense, we can interpret this representation of the state of light as a new type of polarization, namely spectral polarization.

Elementary QM dictates that two mutual beam eigenstates of the Hamiltonian H = −σ · B correspond to a magnetic field equivalent in the direction $\widehat{\mathbf{B}}$. These eigenstates are given by

Now, if the direction of the magnetic field equivalent was to be changed adiabatically round a closed trajectory in parameter space, i.e. $\widehat{\mathbf{B}}=\widehat{\mathbf{B}}\left(\tau \right)$, each eigenstate would have accumulated a geometric phase, γ_±, in addition to the dynamic phase B₀τ, such that

The geometric phase is manifested as a pure phase factor only if the system was initially at an eigenstate of the $\widehat{\mathbf{B}}$ direction [Eqs. (11)–(12)]. In the case where only an idler wave is incident on a non-linear crystal at τ = 0, i.e. |ψ(0)〉 = |ω_i〉, this means that the trajectory of the magnetic field equivalent needs to start (and end) with $\widehat{\mathbf{B}}=\widehat{z}$. As a consequence, the corresponding schemes for the rotation of B should involve a simultaneous adiabatic change in both angles θ and ϕ (and, for now, keeping B₀ constant). Fortunately, these schemes can be specifically tailored since, as we explained previously, the magnetic field analogue is fully controllable. For the discussed case where |ψ(0)〉 = |ω_i〉 and $\widehat{\mathbf{B}}(0)=\widehat{\mathbf{B}}(T)=\widehat{z}$ ($T=\sqrt{k_i{k}_s}L$ is the adiabatic rotation period and where L is the crystal length), we obtain in the laboratory (non rotating) frame

 

Fig. 3 Mach-Zehnder interferometric setup for measuring the geometric phase, consisting of symmetric beam splitters (BS1-2), filters (F1-2) and detectors (D1-2). The pump and idler beams mix in the different arms of the interferometer. Two identical geometric-phase-inducing nonlinear crystals are situated in each arm with oppositely oriented poling patterns with respect to the propagation axes. Due to its geometric properties, the phase difference between the two arms equals twice the geometric phase.

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3. Schemes for adiabatic rotations

In this section we present two different schemes for adiabatic rotations of the magnetic field equivalent, yielding a desired geometric phase. The first involves a circular trajectory about a pre-defined unit vector, while the second is realized by an adiabatic path enclosing a wedge of the unit sphere, which can also be generalized to non-closed trajectories. We then describe the geometric robustness of the second scheme, and discuss the possibilities for optical controllability of the geometric phase, present in the first scheme, via the pump intensity.

3.1. Circular adiabatic rotation

The general parametrization with respect to τ of a circular trajectory of the magnetic field equivalent, rotating in a CCW fashion about some unit vector $\widehat{n}\left({\mathrm{\Theta }}_0\right)$ (Θ₀ is the angle with respect to the z-axis [31]) with frequency ω₀ and starting at $\widehat{z}$, is given(by)

 

Fig. 4 Schemes for adiabatic rotations on the SFG sphere inducing a geometric phase for the idler wave. (a) Circular adiabatic rotation: the magnetic field equivalent precesses about some unit vector $\widehat{n}\left({\mathrm{\Theta }}_0\right)$. (b) Wedged adiabatic rotation: the trajectory of the field encloses a wedge of the unit sphere with opening angle Δϕ. (c)–(d) Normalized QPM modulation parameters corresponding to the circular (with Θ₀ = π/3) and wedged (with Δϕ = π/2) rotation schemes, respectively.

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To illustrate the effect of the adiabatic trajectory on the SFG process, we solve numerically the dynamical equations, Eqs. (1)–(2), in the presence of the QPM modulation given by Eqs. (3) and (19)–(21). The idler, signal and pump wavelengths were chosen as λ_i = 600 nm, λ_s = 432.6 nm and λ_p = 1550 nm, respectively. The idler wave is incident on a 2.5 cm long nonlinear LiNbO₃ crystal with d_eff = 17.2pm/V, pumped with a uniform intensity of I_p = 100 MW/cm². Fig. (5a) shows the normalized photon numbers of the idler and signal waves, given by $N_{i,s}\equiv {\left|A_{i,s}\right|}^2/\sqrt{{\left|A_i\right|}^2+{\left|A_s\right|}^2}$, as well as the accumulation of the geometric phase, along the propagation coordinate z, for Θ₀ = π/3. It is found that the normalized photon number of the idler wave is reduced to 1/4 midway of the crystal, while the signal photon number is simultaneously increased to 3/4. This is an indication that the state traverses the point on the sphere corresponding to 2Θ₀ = 2π/3. Furthermore, the accumulated geometric phase is in good agreement with Eq. (22).

 

Fig. 5 Simulations of the normalized photon numbers of the idler and signal waves (left) and geometric phase accumulation (right) along the propagation coordinate z. (a) Results for the circular adiabatic rotation scheme with Θ₀ = π/3, and thus γ₊ = −π/2. (b) Results for the wedged adiabatic rotation with Δϕ = π/2, giving γ₊ = −π/2.

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3.2. Wedged adiabatic rotation

A dependence of the dynamic phase on geometric parameters, such as the one present in Eq. (23), is less desirable when one considers the design of purely-geometric phase-plates. The latter rely on the transverse modulation of the geometric phase’s value, resulting in a shaped phase-front, insensitive of the path length. This is not easily achieved if the dynamical phase is also transversely-variant. Therefore, a scheme where the average z-component of the magnetic field equivalent vanishes (i.e., 〈B_z〉 = 0) is desirable, yielding φ_D = B₀T irrespective of the transverse coordinates. This type of scheme is readily given by

The parametrization of the QPM modulation parameters for this scheme is

3.3. Geometric phase robustness and pump-controlled phase shifts

In the aforementioned schemes, it was assumed that the input pump intensity I_p corresponds to the specific value of κ used in the QPM poling pattern, in either Eq. (19) or Eq. (27). Under this assumption, the vector B remains on the surface of the Bloch sphere [see Eq. (8) for the magnitude of the magnetic field equivalent]. We now investigate how a different input pump intensity $I_p^{{ }^{\prime }}$ affects our derivations and, more interestingly, controls the geometric phase. For convenience, we shall keep the notation κ and I_p as the a-priori selected coupling and intensity values used in the poling pattern of Δk(τ). The input value of the pump intensity, $I_p^{{ }^{\prime }}$, will correspond to a different coupling value denoted as κ′. This change can only affect the x, y-components of the magnetic field equivalent by virtue of Eq. (7)–(10). Explicitly, we now have

 

Fig. 6 Adiabatic rotations on the Bloch spheroid for η > 1. (a) For the circular adiabatic rotation, the trajectory of the field is no longer described by a simple rotation about a constant unit vector. The flux through the surface enclosed by the field’s rotation is different than for the same parametrization on the Bloch sphere (η = 1). (b) From symmetry considerations, in case of wedged adiabatic rotation the field’s trajectory encloses the same wedge as if it were on the unit sphere, retaining the value of the geometric phase.

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The simplest way to calculate the geometric phase now is to consider the Berry connection A(B), given by [3]

Remarkably, it is straightforward to see from symmetry considerations of Gauss’s theorem that the geometric phase of the wedged adiabatic rotation is invariant under changes in the pump intensity [Fig. (4b)]. Alternatively, this can be shown via inserting $\dot{\varphi }=\mathrm{\Delta }\varphi \delta \left(\tau -T/2\right)$ and cos α = cos ω_oτ in Eqs. (32)–(33), and finding γ₊ = −Δϕ, independent of η. As this scheme might be utilized for geometric phase plates, this invariance under the pump intensity is highly desirable.

Moreover, it can be concluded that this scheme may be generalized to a wedged trajectory on an arbitrary surface, provided that the latter has rotational symmetry with respect to the z axis. We remark that the Berry flux, which determines the geometric phase, through a Δϕ-wedge of all rotationally-symmetric surfaces is identical, given that they enclose the origin of parameter space. It is apparent, therefore, that the only condition for the validity of the wedged adiabatic rotation scheme is inherently geometric: all wedge trajectories must lie on a rotationally symmetric surface enclosing the origin. The origin, in turn, lies inside the surface for a relatively wide range of pump wavelengths. This observation ensures that the wedged rotation scheme is broadband, a property also exhibited by adiabatic frequency conversion processes [12].

We now turn our attention to the second rotation scheme, the circular adiabatic rotation. In this case, the change in pump intensity induces a change in the geometric phase given by

3.4. Non-closed trajectories

As it turns out, a manifestation of the PB phase can readily be recognized if one considers the corresponding phenomena in light polarization. A broadly used scheme for PB optical elements relies on the SU(2) rotations imposed by a half wave plate (HWP) with angle α between the optical axis and the x-axis. The eigenvectors of this HWP lie on the equator of the Poincare sphere. A left circularly polarized (LCP) state, positioned on the north pole, is rotated by the HWP about the axis spanned by its eigenvectors, until it reaches the south pole, namely the right circular polarization (RCP). Denoting ϕ = 2α as the corresponding azimuthal angle of the axis of rotation on the Poincare sphere, the resulting state is given by exp(iϕ) |R〉. The phase ϕ is then interpreted as the PB phase acquired by the state. When ϕ is transversely varied, i.e. ϕ = ϕ(x, y), the resulting wavefront is said to be shaped via the geometric phase.

A similar treatment can be utilized in SFG dynamics. The main difference is, that in the general case the SFG Hamiltonian itself can induce a HWP-like rotation. To see this, we consider the fundamental solution to the SFG dynamics with vanishing phase mismatch (cos θ = 0) and an incident idler wave,

However, by virtue of our result for the wedged adiabatic rotation [Eq. (26) and Fig. (4b)], we find that the geometric phase difference Δϕ between two points on the wavefront is a well defined quantity, as the latter equals the gauge-invariant closed path integral. As in the case of polarization-based PB optical elements, the local phase ϕ(x, y) can be controlled, allowing for robust and broadband wavefront geometric phase shaping in the sum-frequency.

We conclude this section with a reminder that all the above analysis is also applicable to the case where the incident wave is in the signal frequency, i.e. |ψ(0)〉 = |ω_s〉. The accumulated phase of this eigenstate of the SFG Hamiltonian then becomes γ₋ = −γ₊. Thus, every geometric phase effect induced on the idler wave has a conjugate effect on the signal wave. Examples of this feature will be discussed below.

4. Future prospects for wavefront shaping

Future applications of nonlinear geometric phase plates in the scope of current technological capabilities include one dimensional (1D) wavefront shaping and manipulation, in a similar manner to the PB-based linear wavefront shapers that rely on polarization [7, 8]. For example, a Gaussian beam can be approximately transformed into a Hermite-Gaussian (HG) mode of order (n0) [or (0n)] where n is the number of nodes in the wavefront. This can be done via an array of n + 1, γ = π/2 geometric phase nonlinear crystals (GPNLCs) with alternating directionality, generating a wavefront with an alternating phase. In the far field, such a wavefront is expected to approach an HG-(n0) beam.

Another example is a cylindrical geometric phase lens [9], implemented via encoding a continuous, quadratic variation of γ along one transverse direction. If an incident idler beam is expected to accumulate a converging parabolic phase, then such a lens will be converging for the idler wave and diverging for the signal wave. Additionally, flipping the directionality of the lens renders it diverging for the idler wave, and converging for the signal wave. This feature can be attributed to the geometric nature of the lens.

Advancements in the manufacturing technology of 3D nonlinear photonic crystals may open possibilities for 2D wavefront shaping as implemented today by polarization-based geometric phase plates. One application, amongst numerous others, is the nonlinear shaping of beams with orbital angular momentum (OAM) [26, 27], also known as vortex beams. The simplest scheme for such a realization which does not include a complex, continuous variation of the geometric phase is presented in Fig (7). A Gaussian beam (with zero OAM) is incident on four GPNLCs with phases γ = 0, π/2, π and 3π/2 that are placed as a 2 × 2 matrix, where the phase increases in a clockwise fashion. In the far-field, the resulting idler wave is expected to obtain an OAM of value +ħ. A complementary angular momentum of −ħ is expected to be obtained by illuminating the crystal with a signal wave, instead of an idler wave. This nonlinear shaper can actually be realized by diffusion bonding [28] of two GPNLC plates, each containing two different phases.

 

Fig. 7 A vortex beam geometric phase plate. Four GPNLCs with clockwise increasing geometric phase constitute the apparatus. An incident idler beam is expected to accumulate an OAM quantum number of m = 1 in the far-field, whereas an incident signal beam accumulates m = −1.

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The concept of beam shaping can be further extended, beyond the simple examples given above, to realize optically controlled geometric phase holograms, that enable arbitrary shaping of the wavefront by the nonlinear process.

5. Summary and conclusions

In summary, we have proposed and analyzed new ways for obtaining an adiabatic geometric phase for light, via nonlinear interactions. In principle, this scheme is similar to the closed-trajectory version of the PB phase for polarized light. However, whereas light polarization is defined by two complex amplitudes at two orthogonal directions, here we have harnessed a new degree of freedom - the spectral polarization. This type of polarization is represented by the complex amplitudes at two different optical frequencies, that are coupled by the second order nonlinearity of the medium. Consequently, the spectral polarization eigenstates, or mutual beams, can serve as a new basis for describing the state of light [29].

The proposed adiabatic geometric phase in nonlinear optics might as well open doors for new applications. Since the geometric phase can in principle be designed to take different values as a function of the transverse coordinates, it enables new and exciting possibilities for nonlinear beam shaping and nonlinear holography [30]. Furthermore, the geometric phase shift due to the deformation of the Bloch sphere can be further studied either from a fundamental viewpoint or for the realization of an optically-controlled geometric phase. We are hopeful that this new perspective will eventually lead to novel applications for geometric phases in particular, and nonlinear optics in general, both in the classical and quantum domains.