Spectral Phase Modulation

📐 Definition

Given phases $\phi(k)$, apply

$$\hat{u}_{\text{mod}}(k) = e^{i\phi(k)} \hat{u}(k)$$

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to modify only the phase of each spectral component while preserving magnitude.

Domain and Codomain

Operates on complex Fourier coefficients $\hat{u}(k)$ over discrete or continuous spectra. Output shares the same domain with identical magnitudes.

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⚙️ Key Properties

Unitary modulation: $|\hat{u}_{\text{mod}}(k)| = |\hat{u}(k)|$. In physical space, corresponds to convolution with a phase kernel given by the inverse transform of $e^{i\phi(k)}$.

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Linear phase $\phi(k) = k \cdot x_0$ produces spatial shifts; quadratic phase creates focusing or defocusing chirps. Zero phase leaves the signal unchanged.

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🎯 Special Cases and Limits

• Linear phase $\phi(k)=k\cdot x_0$ produces spatial shifts.
• Quadratic phase produces chirp-like focusing/defocusing behavior.
• $\phi(k)=0$ leaves the signal unchanged.

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🔗 Related Functions

Dispersion relations apply phase accumulation $\phi(k) = \Omega(k)\Delta t$ during time stepping; spectral filtering alters magnitudes rather than phases.

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Usage in Oakfield

Oakfield applies spectral phase modulation in a few concrete places:

• dispersion operator multiplies Fourier bins by unit-magnitude complex factors $e^{i\phi(k)}$ to model dispersive phase evolution.
• Mixer FM/PM modes apply phase modulation in the “signal processing” sense (e.g. $L\cdot e^{iR}$) for complex fields, which is the pointwise analogue of phase modulation.

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Historical Foundations

📜 Phase as a Spectral Degree of Freedom

Separating magnitude and phase is fundamental in Fourier analysis: magnitude encodes energy distribution and phase encodes alignment, shifts, and propagation.

🌍 Modern Perspective

Phase-only spectral operators are common in wave simulation (dispersion steps), beam steering, and phase-based modulation of signals.

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📚 References

• Stein & Shakarchi, Fourier Analysis
• Oppenheim & Schafer, Discrete-Time Signal Processing