Intermodal Raman Scattering

• Intermodal Raman scattering is a process that transfers vibrational energy between distinct electromagnetic modes via multi-mode interactions involving pump, Stokes, anti-Stokes, and phonon fields.
• It exhibits nonclassical phenomena such as intermodal entanglement, squeezing, and sub-shot noise, which are tunable through coupling strengths and phase controls.
• Engineering these effects in fibers and nanostructures enables high-performance amplification, frequency conversion, and quantum light source development in modern photonics.

Intermodal Raman scattering is a broad term encompassing physical processes in which the vibrational energy transfer between electromagnetic field modes via the Raman effect occurs not merely within a single optical mode, but between distinct spatial, polarization, or system modes. This includes processes in bulk and structured media, guided-wave systems (fibers and nanostructures), and complex multimode quantum optical settings. Intermodal Raman scattering is pivotal both for fundamental insights into mode-coupled nonlinear quantum optics and for applications in quantum communication, Raman amplification, modal conversion, and sensing.

1. Foundational Hamiltonians and Quantum Correlations in the Intermodal Regime

In quantum optics, intermodal Raman scattering is rigorously described by a fully quantum-mechanical multi-mode Hamiltonian. For processes involving pump ($L$), Stokes ($S$), anti-Stokes ($A$), and vibrational/phonon ($V$) modes, the interaction Hamiltonian takes the form

$$H = \sum_{j=L,S,A,V} \hbar\omega_j a_j^\dagger a_j - [\hbar g\, a_L a_S^\dagger a_V^\dagger + \hbar\chi^* a_L a_V a_A^\dagger + \text{h.c.}]$$

Here, the non-commuting operator terms manifest explicit intermodal interactions by coupling creation and annihilation across several modes, distinguishing this from single-mode or semi-classical approaches (Pathak et al., 2012). Short-time Heisenberg evolution reveals joint quantum correlations and enables analytic calculation of joint photon-phonon and photon-photon distributions, squeezing parameters, and entanglement diagnostics in the Hilbert space of the interacting modes.

Nonclassical features—such as intermodal entanglement, single-mode and compound squeezing, and sub-shot noise—are exhibited across stimulated, spontaneous, and partially spontaneous regimes. The precise structure of quantum correlations depends on coupling strengths (e.g., the ratio $p=|\chi/g|$), time evolution, and input state phases, providing a means to tune intermodal entanglement strength and nonclassicality (Pathak et al., 2012, Sen et al., 2013).

2. Nonclassical Phenomena and Entanglement Diagnostics

Multiple distinct nonclassical effects arise from the intermodal nature of quantum Raman scattering. These include:

• Intermodal Entanglement: Detected via Hillery-Zubairy criteria,

$$\langle N_a N_b \rangle - | \langle a b^\dagger \rangle |^2 < 0 \quad \text{(HZ-1)}$$

and

$$\langle N_a \rangle \langle N_b \rangle - | \langle a b \rangle |^2 < 0 \quad \text{(HZ-2)}$$

as well as quadrature-based Duan’s inseparability criteria,

$$\langle (\Delta u)^2 \rangle + \langle (\Delta v)^2 \rangle < 2$$

for suitable quadrature combinations $u$, $v$ (Sen et al., 2013). Entanglement is observed in a variety of mode pairs including pump–anti-Stokes, pump–phonon, Stokes–phonon, and Stokes–anti-Stokes, and its presence may depend sensitively on the phase of the pump eigenvalue.

• Intermodal Squeezing: Compound squeezing is identified via parameters

$$\lambda_{ij} = 1 + B_i + B_j - 2\operatorname{Re}\bar{D}_{ij} - |C_i + C_j + 2D_{ij}| < 1$$

with negative values indicating quadrature noise reduction below the vacuum level (Pathak et al., 2012).

• Sub-shot Noise and Photon-Phonon Number Correlations: Sub-Poissonian statistics, conditional variances, joint photon–phonon and quasi-intensity distributions reveal subpoissonian difference statistics and negativity in integrated correlations, signifying strong intermodal quantum effects (Pathak et al., 2012).
• Independence of Nonclassical Signatures: Entanglement, squeezing, and antibunching are independent in general intermodal settings; their appearance depends on the specific interaction parameters and can be selectively controlled (Sen et al., 2013, Thapliyal et al., 2017).

3. Modal Phase-Matching and Four-Wave Mixing in Guided Wave Systems

In multimode optical fibers and photonic devices, intermodal phase-matching is a central concept. Nonlinear four-wave mixing (FWM) can generate photon pairs with signal and idler occupying distinct fiber modes; the phase-matching condition,

$$2\beta_p(\omega_p) - \beta_s(\omega_s) - \beta_i(\omega_i) - 2\gamma P = 0$$

with $\beta_j$ the propagation constant for each mode and $\gamma$ the third-order nonlinearity, enables large frequency detunings between the generated pairs and pump (Pourbeyram et al., 2016). This geometric separation in the modal and spectral domains suppresses noise from spontaneous Raman scattering and residual pump light—a critical advantage for quantum information applications.

By tuning fiber parameters and pump conditions, multiple independent, highly pure photon-pair states (i.e., with factorable joint spectral amplitudes) can be generated simultaneously in different modal channels. The modal degree of freedom can also be leveraged for high-dimensional quantum key distribution or wavelength-division multiplexed quantum networks (Pourbeyram et al., 2016).

4. Intermodal Gain and Amplification: Fiber and Nanostructure Platforms

Intermodal stimulated Raman scattering (IM-SRS) enables distributed amplification in space-division multiplexed (SDM) systems. By selectively injecting pump and signal into orthogonal spatial modes of a few-mode fiber, the amplification process is described by a set of coupled GNLSEs encompassing group velocity dispersion, random mode coupling, Kerr and Raman nonlinearity, and wavelength-dependent loss:

$$\frac{\partial A_p(z,t)}{\partial z} = i(\beta_0^{(p)} - \beta_0)A_p - (\beta_1^{(p)} - \beta_1) \frac{\partial A_p}{\partial t} + \ldots + i(n_2 \omega_0/c) \sum_{l,m,n} Q_{plmn}[(1-f_R)A_l A_m A_n^* + f_R A_l [h*(A_m A_n^*)]]$$

where the $Q_{plmn}$ encode intermodal coupling efficiencies (Zitelli et al., 19 Mar 2024). IM-SRS is efficient for distributed, low-noise amplification, compensating for modal differential loss and enabling significant gain (e.g., $>3.9$ dB over 10 km at 1 W pump, or $19.3$ dB at 10 W) across multiple spatial signal channels. This strategy is robust to mode-dependent gain and loss, and experimental results validate its utility for unrepeatered long-haul fiber networks.

5. Raman Gain Suppression and the Role of Higher-Order Modes

In gas-filled broadband-guiding hollow-core photonic crystal fibers, a fundamental effect is coherent Raman gain suppression—occurring when the rates of phonon creation (pump-to-Stokes) and phonon annihilation (pump-to-anti-Stokes) are exactly balanced. The phase-matching condition

$\Phi = (\beta_S + \beta_{AS}) - 2\beta_P = 0$

leads to identical phonon coherence waves for both processes; their mutual cancellation quenches the net Raman gain (Hosseini et al., 2016). At high pump intensities, even slight deviations from perfect phase-matching can yield a dramatic reduction in intramodal (e.g., LP$_{01}$–LP$_{01}$) SRS, with the result that intermodal SRS (e.g., LP$_{01}$–LP$_{11}$) can dominate. This effect is especially pronounced in the ultraviolet due to larger intrinsic Raman gain, impacting the performance of Raman shifters, amplifiers, and frequency combs.

6. Engineering and Control of Intermodal Raman Effects in Nanophotonics

Integration of metal nanostructures, such as gratings and nanowires, with active Raman media harnesses both local-field effects and modal engineering. Two resonant regimes are fundamental:

• Plasmonic Resonances: Support highly localized “hot spots” yielding field enhancement factors and Stokes/anti-Stokes conversion efficiencies up to $10^5$–$10^7$ higher than non-structured references. However, extremely narrow-band and spatially localized field profiles challenge efficient overlap of pump and Raman fields and pose phase-matching constraints.
• Fabry–Perot Resonances: Provide broader modal volumes, significant local intensity enhancement (e.g., factors of 600–700), and improved spatial overlap with the Raman-active medium (Scalora et al., 2012).

At very high local intensities, dynamic, intensity-dependent detuning arises: $$[\Gamma + i(\Delta + \delta(I_{\text{total}}))] Q = \text{coupling term}$$ with $Q$ the medium coherence and $I_\text{total}$ the combined local field intensity. The detuning $\delta(I_\text{total})$ increases with intensity, quenching Raman gain and enforcing a gain saturation limit—essential for device reliability and avoiding damage (Scalora et al., 2012).

7. Cascaded and Intermodal Dynamics: Light-Sound Interaction and Modal Control

The interplay between cascaded intramodal Raman (SRLS) and Brillouin-like intermodal (SIMS) scattering in nanostructured waveguides creates complex scattering ladders and frequency combs. In engineered dual-web fibers, phase-matching with flat phonon dispersion synchronizes both processes,

$$\frac{d S_n}{dz} = i K_R (S_{n-1} R_s + S_{n+1} R_s^*) + i K_M (f_{n-1} M_{sf} + f_{n+1} M_{sf}^*)$$

with $S_n$, $f_n$ denoting mode amplitudes and $K_R$, $K_M$ modal optomechanical couplings (Noskov et al., 2017). This enables periodic reversal of energy flow between optical and mechanical domains and supports comb formation across multiple orders and spatial modes, allowing for advanced mode conversion, switching, and on-chip integration.

8. Practical Impact and Applications

Intermodal Raman scattering processes are foundational to a host of modern photonics technologies:

• Quantum Light Sources: Controlled intermodal correlations facilitate generation of entangled photon pairs and pure-state heralded photons for quantum information.
• Distributed Amplification: IM-SRS in SDM fibers promises scalable solutions to the optical network capacity crunch.
• Sensing and Spectroscopy: Nanostructured and waveguide-enhanced Raman scattering platforms offer high sensitivity, broad bandwidth, and efficient modal collection (e.g., near-unity Raman $\beta$-factor) (Fu et al., 2021).
• Frequency Conversion and Comb Generation: Modal engineering and cascaded intermodal processes underpin the formation of broadband frequency combs and tunable lasers, with dynamic control mediated by modal interactions (Kasumie et al., 2019, Noskov et al., 2017).
• Focused Energy Deposition: Wavefront shaping guided by spectrally resolved Raman speckle enables selective delivery and enhancement of energy to specific targets within scattering environments, expanding the reach of Raman diagnostics and treatment (Tian et al., 2022).

Taken together, the diversity of mechanisms and platforms for intermodal Raman scattering provides researchers with powerful strategies for quantum state control, signal amplification and processing, and the development of nonclassical optical devices in both classical and quantum photonics.