DOI: 10.5061/dryad.fqz612k5z
title: Coherent control of optical spin–orbit interactions

description: Optical spin-orbit interactions (SOI) are an intriguing physical
 phenomenon where the spin (circular polarization) of light influences and
 governs its spatial degrees of freedom. Yet, achieving real-time and
 flexible control over SOI remains a formidable challenge. Here, we propose
 a coherent control method for the flexible manipulation of optical SOI
 using a thin isotropic crystal. The output horizontally and vertically
 polarized waves from the crystal exhibit strong and distinct dependencies
 on the phase delay between the input signal and control waves, leading to
 substantial spin-orbit beam shifts. By tuning the relative phase between
 the signal beam and the control beam, we observe transitions from Gaussian
 to first-order orbital angular momentum (OAM) vortex modes. The achievable
 transition speed reaches 11.5 GHz, currently limited by the phase
 modulator in the control arm. Notably, the dual-port architecture offers
 coherent and reversible selection of spin–OAM states, with the output port
 uniquely determined by the phase delay between two counter-propagating
 input beams. This ultrafast coherent control method opens avenues for
 advanced manipulation of SOI.
description: # Coherent control of optical spin–orbit interactions Dataset DOI:
 [10.5061/dryad.fqz612k5z](https://doi.org/10.5061/dryad.fqz612k5z) ##
 Description of the data and file structure \&gt; Dataset Description \&gt;
 This dataset supports the findings in the manuscript titled "Coherent
 Control of Optical Spin–Orbit Interactions". The data files provided
 correspond to the experimental and theoretical results presented in
 Figures 2, 3, 4, 5, and 6 of the main manuscript. The dataset covers the
 characterization of the coherent spin-orbit interaction (SOI) effects,
 including reflection/transmission spectra, phase calculations, spin-orbit
 beam shifts, system bandwidth, and high-speed optical switching
 performance. ### Files and variables #### File: dataset.zip ##### File
 Formats and Access * **File Extension**: `.csv` (Comma Separated Values) *
 **How to Open**: These files are standard text-based data files. They can
 be opened and viewed using any text editor (e.g., Notepad, TextEdit),
 spreadsheet software (e.g., Microsoft Excel, Google Sheets), or analyzed
 using programming languages such as Python (pandas), MATLAB, or R.Dataset
 Contents & Usage * **List of Files and Variable Definitions** ###
 Figure 2 Files (Basic Characterization) #### 1.
 `figure2cd_single_incident_processed.csv` **Description**: Corresponds to
 **Figure 2C and 2D**. It contains the theoretical and experimental
 reflection and transmission spectra for a single plane wave incident on
 the silicon wafer. * `Incident_Angle_deg`: The angle of incidence of the
 beam (theta_1) in degrees. * `Theory_Reflectance_p`: Calculated
 reflectance (*R_p*) for horizontally polarized (p-polarized) light. *
 `Theory_Transmittance_p`: Calculated transmittance (*T_p*) for
 horizontally polarized (p-polarized) light. * `Exp_Transmitted_Power_p_W`:
 Measured optical power transmitted through the wafer for p-polarization,
 in Watts. * `Exp_Transmittance_p`: Experimental transmittance for
 p-polarization (normalized). * `Exp_Reflected_Power_p_W`: Measured optical
 power reflected from the wafer for p-polarization, in Watts. *
 `Exp_Reflectance_p`: Experimental reflectance for p-polarization
 (normalized). * `Theory_Reflectance_s`: Calculated reflectance (*R_s*) for
 vertically polarized (s-polarized) light. * `Theory_Transmittance_s`:
 Calculated transmittance (*T_s*) for vertically polarized (s-polarized)
 light. * `Exp_Transmitted_Power_s_W`: Measured optical power transmitted
 through the wafer for s-polarization, in Watts. * `Exp_Transmittance_s`:
 Experimental transmittance for s-polarization (normalized). *
 `Exp_Reflected_Power_s_W`: Measured optical power reflected from the wafer
 for s-polarization, in Watts. * `Exp_Reflectance_s`: Experimental
 reflectance for s-polarization (normalized). #### 2.
 `figure2e_phase_calculations.csv` **Description**: Corresponds to **Figure
 2E**. It provides the calculated phase relationships relevant to the
 coherent control mechanism. * `Incident_Angle_deg`: The angle of incidence
 (theta_1) in degrees. * `Cumulative_Phase_delta_deg`: The calculated
 cumulative phase shift (delta) accumulated by the beam during propagation
 through the wafer thickness, in degrees. * `Critical_Phase_delta_p_deg`:
 The critical cumulative phase (Delta_p) required for coherent extinction
 of p-polarized light, in degrees. * `Critical_Phase_delta_s_deg`: The
 critical cumulative phase (Delta_s) required for coherent extinction of
 s-polarized light, in degrees. #### 3.
 `figure2f_dual_port_incident_processed.csv` **Description**: Corresponds
 to **Figure 2F**. It contains data for the coherent dual-beam
 configuration (signal + control beams) with a fixed relative phase delay
 (Delta_varphi = -pi/2). It shows the output intensities at Port 2. *
 `Incident_Angle_deg`: The angle of incidence (theta_1) in degrees. *
 `Theory_Output_Intensity_Port2_p`: Theoretical normalized output intensity
 for p-polarization at Port 2 (|B_2^p|^2). *
 `Theory_Output_Intensity_Port2_s`: Theoretical normalized output intensity
 for s-polarization at Port 2 (|B_2^s|^2). * `Exp_Output_Power_Port2_p_W`:
 Measured output power for p-polarization at Port 2, in Watts. *
 `Exp_Output_Intensity_Port2_p`: Experimental normalized output intensity
 for p-polarization at Port 2. * `Exp_Output_Power_Port2_s_W`: Measured
 output power for s-polarization at Port 2, in Watts. *
 `Exp_Output_Intensity_Port2_s`: Experimental normalized output intensity
 for s-polarization at Port 2. #### 4. `figure2h_experiment_profile.csv`
 **Description**: Corresponds to the experimental data in **Figure 2H**. It
 shows the cross-sectional intensity profiles of the output beam for
 opposite spin states (sigma+ and sigma-) at a fixed incident angle
 (theta_1 = 8.8deg). * `Position_m`: Transverse spatial position across the
 beam profile, in meters. * `Exp_Intensity_SigmaPlus_Normalized`: Measured
 normalized intensity profile for the sigma+ component. *
 `Exp_Intensity_SigmaMinus_Normalized`: Measured normalized intensity
 profile for the sigma- component. #### 5. `figure2h_theory_profile.csv`
 **Description**: Corresponds to the theoretical data in **Figure 2H**. It
 provides the calculated beam profiles matching the experimental conditions
 in the file above. * `Position_m`: Transverse spatial position across the
 beam profile, in meters. * `Theory_Intensity_SigmaPlus_Normalized`:
 Calculated normalized intensity profile for the sigma+ component. *
 `Theory_Intensity_SigmaMinus_Normalized`: Calculated normalized intensity
 profile for the sigma- component. #### 6. `figure2i_experiment_shifts.csv`
 **Description**: Corresponds to the experimental data in **Figure 2I**. It
 contains the measured spin-orbit beam shifts (transverse displacements) as
 a function of incident angle. * `Incident_Angle_deg`: The angle of
 incidence (theta_1) in degrees. * `Y-`: Individual measurements of the
 beam centroid shift for the sigma- component, in micrometers (mu m).
 Multiple columns indicate repeated measurements. *
 `Exp_Shift_SigmaMinus_Mean_um`: The average centroid shift for the sigma-
 component, in micrometers. * `Exp_Shift_SigmaMinus_Std_um`: The standard
 deviation of the shift for the sigma- component, in micrometers. * `Y+`:
 Individual measurements of the beam centroid shift for the sigma+
 component, in micrometers (mu m). * `Exp_Shift_SigmaPlus_Mean_um`: The
 average centroid shift for the sigma+ component, in micrometers. *
 `Exp_Shift_SigmaPlus_Std_um`: The standard deviation of the shift for the
 sigma+ component, in micrometers. #### 7. `figure2i_theory_shifts.csv`
 **Description**: Corresponds to the theoretical prediction in **Figure
 2I**. * `Incident_Angle_deg`: The angle of incidence (theta_1) in degrees.
 * `Theory_Shift_SigmaPlus_um`: Calculated transverse beam shift for the
 sigma+ component, in micrometers (mu m). * `Theory_Shift_SigmaMinus_um`:
 Calculated transverse beam shift for the sigma- component, in micrometers
 (mu m). ### Figure 3 Files (Coherent Control of SOI) #### 8.
 `figure3b_output_control.csv` **Description**: Corresponds to **Figure
 3B**. It characterizes the output intensity control at Port 2 by varying
 the incident angle. * `Incident_Angle_deg`: The angle of incidence
 (theta_1) in degrees. * `Theory_Output_Intensity_Port2_p_Normalized`:
 Calculated normalized output intensity for p-polarization at Port 2. *
 `Theory_Output_Intensity_Port2_s_Normalized`: Calculated normalized output
 intensity for s-polarization at Port 2. * `Exp_Output_Power_Port2_p_W`:
 Measured output optical power for p-polarization at Port 2, in Watts. *
 `Exp_Output_Intensity_Port2_p_Normalized`: Experimental normalized output
 intensity for p-polarization. * `Exp_Output_Power_Port2_s_W`: Measured
 output optical power for s-polarization at Port 2, in Watts. *
 `Exp_Output_Intensity_Port2_s_Normalized`: Experimental normalized output
 intensity for s-polarization. #### 9.
 `figure3cd_experiment_SigmaMinus_shifts.csv` **Description**: Corresponds
 to the experimental data in **Figure 3C and 3D** for the
 **sigma- **component. * `Incident_Polarization
 Angle_deg`: The polarization angle of the incident beam, in degrees. *
 `X-`, `X-.1`, `X-.2`: Individual measurements of the beam centroid shift
 in the X-direction. * `Y-`, `Y-.1`, `Y-.2`: Individual measurements of the
 beam centroid shift in the Y-direction. *
 `Exp_Shift_SigmaMinus_X_Mean_um`: Average centroid shift in the
 X-direction for sigma-, in micrometers (mu m). *
 `Exp_Shift_SigmaMinus_X_Std_um`: Standard deviation of the X-shift. *
 `Exp_Shift_SigmaMinus_Y_Mean_um`: Average centroid shift in the
 Y-direction for sigma-, in micrometers (mu m). *
 `Exp_Shift_SigmaMinus_Y_Std_um`: Standard deviation of the Y-shift. ####
 10. `figure3cd_experiment_SigmaPlus_shifts.csv` **Description**:
 Corresponds to the experimental data in **Figure 3C and 3D** for the
 **sigma+ **component. * `Incident_Polarization
 Angle_deg`: The polarization angle of the incident beam, in degrees. *
 `X+`, `Y+`: Individual measurements of the beam centroid shift in X and Y
 directions. * `Exp_Shift_SigmaPlus_X_Mean_um`: Average centroid shift in
 the X-direction for sigma+, in micrometers. *
 `Exp_Shift_SigmaPlus_Y_Mean_um`: Average centroid shift in the Y-direction
 for sigma+, in micrometers. #### 11. `figure3cd_theory_shifts.csv`
 **Description**: Corresponds to the theoretical predictions in **Figure 3C
 and 3D**. * `Incident_Polarization Angle_deg`: The polarization angle of
 the incident beam, in degrees. * `Theory_Shift_SigmaMinus_X_um`:
 Calculated X-direction shift for the sigma- component, in micrometers. *
 `Theory_Shift_SigmaMinus_Y_um`: Calculated Y-direction shift for the
 sigma- component, in micrometers. * `Theory_Shift_SigmaPlus_X_um`:
 Calculated X-direction shift for the sigma+ component, in micrometers. *
 `Theory_Shift_SigmaPlus_Y_um`: Calculated Y-direction shift for the sigma+
 component, in micrometers. #### 12. `figure3fg_experiment_profiles.csv`
 **Description**: Corresponds to the experimental beam profiles in **Figure
 3F and 3G**. * `Position_m`: Spatial position along the cross-section (X
 or Y axis), in meters. * `Exp_Normalized_Intensity_SigmaMinus_X`: Measured
 intensity profile of the sigm- component along the X-axis. *
 `Exp_Normalized_Intensity_SigmaPlus_X`: Measured intensity profile of the
 sigma+ component along the X-axis. *
 `Exp_Normalized_Intensity_SigmaMinus_Y`: Measured intensity profile of the
 sigma- component along the Y-axis. *
 `Exp_Normalized_Intensity_SigmaPlus_Y`: Measured intensity profile of the
 sigma+ component along the Y-axis. #### 13.
 `figure3fg_theory_profiles.csv` **Description**: Corresponds to the
 theoretical beam profiles in **Figure 3F and 3G**. * `Position_m`: Spatial
 position along the cross-section, in meters. *
 `Theory_Normalized_Intensity_SigmaMinus_X`: Calculated intensity profile
 of the sigma- component along the X-axis. *
 `Theory_Normalized_Intensity_SigmaPlus_X`: Calculated intensity profile of
 the sigma+ component along the X-axis. *
 `Theory_Normalized_Intensity_SigmaMinus_Y`: Calculated intensity profile
 of the sigma- component along the Y-axis. *
 `Theory_Normalized_Intensity_SigmaPlus_Y`: Calculated intensity profile of
 the sigma+ component along the Y-axis. #### 14.
 `figure3h_experiment_SigmaMinus_shifts.csv` **Description**: Corresponds
 to the experimental data in **Figure 3H**. It quantifies the shift for the
 sigma- component as a function of the relative phase delay. *
 `Delta_Phase_pi`: Relative phase delay (Delta_varphi) between the signal
 and control beams, in units of pi radians. * `X-`, `Y-`: Individual
 measurements of the beam centroid shift in X and Y directions. *
 `Exp_Shift_SigmaMinus_X_Mean`: Average centroid shift in the X-direction.
 * `Exp_Shift_SigmaMinus_Y_Mean`: Average centroid shift in the
 Y-direction. #### 15. `figure3i_experiment_SigmaPlus_shifts.csv`
 **Description**: Corresponds to the experimental data in **Figure 3I**. It
 quantifies the shift for the sigma+ component as a function of the
 relative phase delay. * `Delta_Phase_pi`: Relative phase delay
 (Delta_varphi), in units of pi radians. * `X+`, `Y+`: Individual
 measurements of the beam centroid shift in X and Y directions. *
 `Exp_Shift_SigmaPlus_X_Mean`: Average centroid shift in the X-direction. *
 `Exp_Shift_SigmaPlus_Y_Mean`: Average centroid shift in the Y-direction.
 #### 16. `figure3hi_theory_shifts.csv` **Description**: Corresponds to the
 theoretical predictions in **Figure 3H and 3I**. * `Delta_Phase_pi`:
 Relative phase delay (Delta_varphi), in units of pi radians. *
 `Theory_Shift_SigmaMinus_X`: Calculated X-direction shift for sigma-. *
 `Theory_Shift_SigmaPlus_X`: Calculated X-direction shift for sigma+. *
 `Theory_Shift_SigmaMinus_Y`: Calculated Y-direction shift for sigma-. *
 `Theory_Shift_SigmaPinus_Y`: (Note: Header `SigmaPinus` corresponds to
 `SigmaPlus`) Calculated Y-direction shift for sigma+. ### Figure 4 Files
 (Cavity-Length Dependent SOI) #### 17. `figure4b_d_experiment_profile.csv`
 **Description**: Corresponds to the experimental data in **Figure 4B**
 (and 4D). It shows the measured normalized intensity variation mapping to
 wafer thickness. * `Pixel_Index`: The spatial index/position along the
 measurement axis. * `Normalized_Intensity`: Measured normalized output
 intensity. #### 18. `figure4b_d_theory_profile.csv` **Description**:
 Corresponds to the theoretical calculation in **Figure 4B**. * `Wafer
 thickness_um`: The thickness of the silicon wafer (d), in micrometers (mu
 m). * `Theory_Normalized_Intensity`: Calculated normalized output
 intensity. #### 19. `figure4c_experiment_lcp_shift.csv` **Description**:
 Corresponds to the experimental data in **Figure 4C** for the
 **sigma- **component. * `Wafer thickness_d_um`: The
 local thickness of the wafer (d), in micrometers. * `Y-`: Individual
 measurements of the transverse beam shift in the Y-direction. *
 `Exp_Shift_SigmaMinus_Mean_um`: Average centroid shift for sigma-, in
 micrometers (mu ). * `Exp_Shift_SigmaMinus_Std_um`: Standard deviation of
 the shift for sigma-. #### 20. `figure4c_experiment_rcp_shift.csv`
 **Description**: Corresponds to the experimental data in **Figure 4C** for
 the **sigma+ **component. * `Wafer thickness_d_um`: The
 local thickness of the wafer (d), in micrometers. * `Y+`: Individual
 measurements of the transverse beam shift in the Y-direction. *
 `Exp_Shift_SigmaPlus_Mean_um`: Average centroid shift for sigma+, in
 micrometers (mu m). * `Exp_Shift_SigmaPlus_Std_um`: Standard deviation of
 the shift for sigma+. #### 21. `figure4c_theory_shifts.csv`
 **Description**: Corresponds to the theoretical prediction in **Figure
 4C**. * `Wafer thickness_d_um`: The thickness of the silicon wafer (d), in
 micrometers. * `Theory_Shift_SigmaMinus_Y_um`: Calculated transverse shift
 for sigma-. * `Theory_Shift_SigmaPlus_Y_um`: Calculated transverse shift
 for sigma+. #### 22. `figure4e_experiment_profile.csv` **Description**:
 Corresponds to the experimental beam profiles in **Figure 4E**. *
 `Position_m`: Transverse spatial position across the beam profile, in
 meters. * `Exp_Normalized_Intensity_SigmaMinus`: Measured normalized
 intensity profile for sigma-. * `Exp_Normalized_Intensity_SigmaPlus`:
 Measured normalized intensity profile for sigma+. #### 23.
 `figure4e_theory_profile.csv` **Description**: Corresponds to the
 theoretical beam profiles in **Figure 4E**. * `Position_m`: Transverse
 spatial position across the beam profile, in meters. *
 `Theory_Normalized_Intensity_SigmaPlus`: Calculated normalized intensity
 profile for sigma+. * `Theory_Normalized_Intensity_SigmaMinus`: Calculated
 normalized intensity profile for sigma-. ### Figure 5 Files (Dynamic
 Modulation and Bandwidth) #### 24. `figure5c_experiment_tuning_curve.csv`
 **Description**: Corresponds to the experimental data in **Figure 5C**. It
 characterizes the system's ability to modulate output intensity by
 varying the phase delay. * `Delta Phase_pi`: The applied relative phase
 delay (Delta_varphi), normalized by pi. * `Exp_Normalized_Intensity_Mean`:
 The average normalized output intensity. * `Exp_Normalized_Intensity_Std`:
 The standard deviation of the intensity measurements. #### 25.
 `figure5c_theory_tuning_curve.csv` **Description**: Corresponds to the
 theoretical prediction in **Figure 5C**. * `Delta Phase_pi`: The applied
 relative phase delay (Delta_varphi), normalized by pi. *
 `Theory_Normalized_Intensity`: Calculated normalized output intensity.
 #### 26. `figure5d_10mhz_modulation.csv` **Description**: Corresponds to
 **Figure 5D**. Time-domain response under 10 MHz modulation. * `Time_s`:
 Time, in seconds. * `Voltage_Driving_V`: Input driving signal voltage, in
 Volts. * `Voltage_PD Response_V`: Photodetector response voltage (optical
 output), in Volts. #### 27. `figure5f_120mhz_pulse.csv` **Description**:
 Corresponds to **Figure 5F**. Time-domain response under 120 MHz pulse
 modulation. * `Time_s`: Time, in seconds. * `Voltage_Driving_V`: Input
 driving signal voltage, in Volts. * `Voltage_PD Response_V`: Photodetector
 response voltage, in Volts. #### 28. `figure5f_500mhz_modulation.csv`
 **Description**: Corresponds to **Figure 5F**. Time-domain response under
 500 MHz modulation. * `Time_s`: Time, in seconds/indices. *
 `Voltage_Driving_V`: Input driving signal voltage, in Volts. * `Voltage_PD
 Response_V`: Photodetector response voltage, in Volts. #### 29.
 `figure5g_10ghz_modulation.csv` **Description**: Corresponds to **Figure
 5G**. Time-domain response under 10 GHz modulation. * `Time_s`: Time, in
 seconds. * `Voltage_Driving_V`: Input driving signal voltage, in Volts. *
 `Voltage_PD Response_V`: Photodetector response voltage, in Volts. ####
 30. `figure5g_13.5ghz_modulation.csv` **Description**: Corresponds to
 **Figure 5G**. Time-domain response under 13.5 GHz modulation. * `Time_s`:
 Time, in seconds. * `Voltage_Driving_V`: Input driving signal voltage, in
 Volts. * `Voltage_PD Response_V`: Photodetector response voltage, in
 Volts. #### 31. `figure5h_system_bandwidth.csv` **Description**:
 Corresponds to **Figure 5H**. It presents the frequency response of the
 system. * `Frequency_Hz`: The modulation frequency, in Hertz (Hz). *
 `S12_Magnitude_dB`: The magnitude of the system's frequency response,
 in decibels (dB). ### Figure 6 Files (Optical Switching) #### 32.
 `figure6c_100mhz_switching.csv` **Description**: Corresponds to **Figure
 6C**. Coherent optical switching at 100 MHz. * `Time_s`: Time, in seconds.
 * `Voltage_Driving_V`: Input driving signal voltage, in Volts. *
 `Voltage_Port1_PD Response_V`: Photodetector output voltage at Port 1. *
 `Voltage_Port2_PD Response_V`: Photodetector output voltage at Port 2.
 #### 33. `figure6d_10ghz_switching.csv` **Description**: Corresponds to
 **Figure 6D**. High-speed coherent optical switching at 10 GHz. *
 `Time_s`: Time scale for the photodetector response columns, in seconds. *
 `Voltage_Port1_PD Response_V`: Photodetector output voltage at Port 1. *
 `Voltage_Port2_PD Response_V`: Photodetector output voltage at Port 2. *
 `Time_s.1`: Time scale for the driving voltage column, in seconds. *
 `Voltage_Driving_V`: Input driving signal voltage, in Volts. --- ##
 Contextual Description of Figures * **Figure 2C & 2D**: These figures
 characterize the passive optical response of the silicon wafer under
 single-beam illumination. They compare the measured and calculated
 reflection (*R*) and transmission (*T*) spectra for horizontally (p) and
 vertically (s) polarized light as the incident angle is varied. The
 crossing point of the R and T curves indicates the condition for potential
 coherent perfect absorption or transparency in a dual-beam setup. *
 **Figure 2E**: This figure illustrates the phase matching conditions
 required for coherent control. It plots the calculated cumulative phase
 shift (delta) a beam acquires traversing the wafer and compares it to the
 critical phase (Delta_{p,s}) required to achieve coherent extinction
 (destructive interference) at the output ports. * **Figure 2F**: This
 figure demonstrates the result of coherent interference between two
 counter-propagating beams (signal and control) with a fixed phase delay of
 pi/2. It shows the normalized output intensity at Port 2, revealing a
 distinct dip for p-polarization (destructive interference) and a peak for
 s-polarization (constructive interference) near the critical angle of 7.9
 deg. * **Figure 2H**: This figure presents the spatial intensity profiles
 of the output beam, resolved into its two spin components (sigma+ and
 sigma-). It highlights the "spin-orbit beam shift," where the
 two spin components spatially separate in opposite transverse directions
 due to the coherent Spin-Orbit Interaction (SOI) at a specific incident
 angle (8.8 deg). * **Figure 2I**: This figure quantifies the spin-orbit
 beam shift phenomenon. It plots the transverse centroid displacement
 (shift) of the sigma+ and sigma- beam components as a function of the
 incident angle. The data shows that the separation between the two spin
 states is strongly dependent on the angle of incidence, reaching a maximum
 near the resonance condition. * **Figure 3B**: Illustrates the coherent
 selection of the operating state. By scanning the incident angle, the
 system can be tuned to a condition (typically near the Brewster or
 critical angle) where the coherent interaction allows for maximal control
 over the output intensity of specific polarization states. * **Figure 3C
 & 3D**: These figures demonstrate the **polarization-controlled
 spin-orbit interaction**. They show how rotating the polarization angle of
 the input beam linearly controls the transverse spatial shift (in both X
 and Y directions) of the output sigma+ and sigma- beam components. This
 effectively maps the polarization state (Poincaré sphere) to real-space
 coordinates. * **Figure 3F & 3G**: These figures present the **beam
 profiles** under coherent control. They visualize the spatial distribution
 of the sigma+ and sigma- components along the X and Y axes, confirming
 that the beam retains its quality while undergoing significant spatial
 shifts. * **Figure 3H & 3I**: These figures demonstrate the
 **phase-controlled spin-orbit interaction**. They plot the transverse beam
 shifts as a function of the relative phase delay (Delta_varphi) between
 the two coherent input beams. This confirms that the Spin-Orbit
 Interaction (SOI) and the resulting beam splitting can be continuously
 modulated and reversed purely by adjusting the phase, without moving any
 mechanical parts. * **Figure 4B**: This figure characterizes the
 dependence of the system's optical response on the cavity dimension.
 It plots the output intensity as a function of the silicon wafer thickness
 (d). The observed fringes (Fabry-Pérot resonances) indicate that the
 spin-orbit interaction effects can be tuned by slightly varying the
 thickness or temperature of the crystal. * **Figure 4C**: This figure
 illustrates the **thickness-dependent spin-orbit interaction**. It shows
 that the transverse spatial shift (splitting) between the sigma+ and
 sigma- beam components is not static but oscillates dramatically with the
 wafer thickness. Large beam shifts (giant SOI) are observed near specific
 thickness values corresponding to cavity resonances. * **Figure 4E**: This
 figure provides a snapshot of the beam profiles at a chosen thickness
 where the spin-orbit shift is significant. It confirms that the two spin
 components are spatially separated into distinct lobes while maintaining
 their Gaussian-like profile shape, verifying the high quality of the
 spin-dependent beam splitting. * **Figure 5C**: Establishes the static
 "transfer function" of the device. By sweeping the phase delay
 slowly, this figure maps the relationship between the control parameter
 (phase) and the output intensity, which is essential for determining the
 bias points for linear modulation or switching. * **Figure 5D & 5F**:
 These figures illustrate the **temporal dynamic response** of the system.
 They show time-domain traces of the input electrical signal versus the
 output optical signal at various frequencies (10 MHz, 120 MHz, 500 MHz).
 This proves that the coherent SOI mechanism is not just a static effect
 but can be modulated in real-time. * **Figure 5G**: Demonstrates the
 **high-speed performance** of the system in the microwave frequency range
 (10 GHz and 13.5 GHz). The clear open eye diagrams or sinusoidal waveforms
 at these frequencies indicate the system is suitable for high-speed
 optical communications or signal processing. * **Figure 5H**: Provides a
 quantitative measure of the **system bandwidth**. By plotting the response
 magnitude over a wide frequency range (up to nearly 20 GHz), it
 characterizes the upper frequency limit of the coherent control mechanism,
 showing it is broadband (11.5 GHz) and limited primarily by the phase
 modulators rather than the SOI effect itself. * **Figure 6C**: Illustrates
 the **low-frequency (100 MHz) coherent switching** operation. By
 modulating the relative phase between the input beams, the optical energy
 is dynamically routed between Port 1 and Port 2. The waveforms show that
 when the signal at Port 1 is maximized (constructive interference), the
 signal at Port 2 is minimized (destructive interference), and vice-versa,
 demonstrating efficient 1x2 optical switching. * **Figure 6D**:
 Demonstrates the **high-speed (10 GHz) coherent switching** performance.
 This figure proves that the proposed spin-orbit interaction based
 switching mechanism is robust and fast enough to handle data rates
 compatible with modern optical communication networks (X-band
 frequencies). The clear anti-phase relationship between the two output
 ports is maintained even at this high frequency.