Beam Frequency Diagram Overview

• Beam Frequency Diagram is a graphical tool representing frequency-dependent beam responses including resonance, dispersion, and spatial effects.
• It is used to analyze applications from dynamic metasurface antennas and periodic beam lattices to structural and astrophysical systems.
• Methodologies include both numerical simulations and analytic models to visualize beam squint, bandwidth variation, and modal shifts.

A Beam Frequency Diagram is a graphical or computational representation that characterizes the relationship between the frequency-dependent response of beams, beam arrays, or systems with beam-like components in a variety of physical, engineering, and applied-mathematical contexts. Such diagrams encode the frequency selectivity, spatial dispersion, bandstructure, or resonance characteristics of the underlying beam system. This concept is foundational in fields including metasurface antennas, structural engineering, waveguides, molecular beam control, and even radio astronomy, providing a direct route to understanding, optimizing, and diagnosing frequency-dependent beam phenomena.

1. Frequency-Selective Beamforming: Models and Diagrams

In dynamic metasurface antennas (DMAs) and frequency-selective array systems, the Beam Frequency Diagram quantifies how beamforming gain $P(\theta,f)$ varies across operating frequency and steering angle. For a DMA with $N$ reconfigurable elements, each with design resonant frequency $f_{r,n}$ and waveguide phase shift $\varphi_{n}^{\rm wg}(f)$, the per-element response is given by a Lorentzian

$$\alpha_{M,n}(f) = \frac{F\,(2\pi\,f)^2}{2\pi\,f_{r,n}^2-2\pi\,f^2+\Gamma\,f}$$

with complex phase $\psi_n(f)$ and frequency-dependent steering vector $$a_n(\theta,f) = \exp\left(-j \frac{2\pi f}{c} (n-1)d\sin\theta\right)$$. The total beamforming gain is

$G(\theta, f) = \bigl|a(\theta,f)^T w(f)\bigr|^2$

where $w(f)$ is the frequency-dependent weight vector with entries coupled in both magnitude and phase via the Lorentzian model (Deshpande et al., 2024).

A Beam Frequency Diagram is constructed by plotting $P(\theta_0, f)$ at fixed direction $\theta_0$ across a frequency interval, or, more generally, as a density plot $(\theta, f) \mapsto |a(\theta,f)^T w(f)|^2$. This reveals critical features including beam "squint" (frequency-dependent steering), beamwidth variation, and frequency selectivity bandwidth.

Algorithmic steps for diagram computation:

• Discretize frequency $f$ (e.g., over $K$ OFDM subcarriers) and angle $\theta$.
• Compute $P(\theta_m, f_k)$ for all $m,k$.
• Visualize via \texttt{imagesc} in MATLAB or \texttt{pcolormesh} in Python.

Beam frequency diagrams directly inform single-shot beam training protocols, such as dividing a DMA into sub-arrays, assigning each sub-array a resonant frequency $f_\ell^*$ corresponding to specific angular sectors and using a single OFDM symbol to rapidly estimate angle of departure (AoD) and optimal steering frequency (Deshpande et al., 2024).

2. Bandstructure and Dispersion in Periodic Beam Lattices

In periodic elastic structures (metamaterials or phononic crystals) built from beam networks (Timoshenko or Rayleigh beams), the beam frequency diagram typically refers to the bandstructure: frequency $\omega$ as a function of wavevector $\mathbf{k}$, revealing both allowed and forbidden frequency bands (bandgaps).

The dispersion relation for a square-periodic Timoshenko-beam lattice, for example, is determined by the generalized eigenproblem $[K(\mathbf{k}) - \omega^2 M(\mathbf{k})]U = 0$, with quasi-periodicity via Bloch-Floquet conditions. Numerical solution yields $\omega_n(\mathbf{k})$ along high-symmetry paths ($\Gamma$–$X$–$M$–$\Gamma$), forming the beam frequency diagram for the system (Kamotski et al., 2018).

High-contrast components (e.g., "soft" beams with much lower stiffness in each cell) induce low-frequency bandgaps, visible in the diagram as frequency intervals where no real Bloch wave exists, with their location and width analytically predicted from homogenization theory. The formation of these gaps is governed by the frequency-dependent "effective mass" matrix $\beta(\lambda)$, whose sign determines the existence of propagating solutions (Kamotski et al., 2018).

Representative tabulation:

k-point	ω₁ℓ	ω₂ℓ	ω₃ℓ	...
Γ	0	0	3.12	...
X	1.12	2.34	3.45	...
M	1.65	2.78	4.01	...
Γ	0	0	3.12	...

Bandgap regions on the diagram are intervals in $k$ where neither $\omega_1$ nor $\omega_2$ provide real frequencies (i.e., $\beta_1(\lambda) < 0$ and $\beta_2(\lambda)<0$) (Kamotski et al., 2018).

Similar approaches build frequency diagrams for grid-based Rayleigh-beam structures, including analytic identification of flat bands and Dirac cone degeneracies in the dispersion, as well as conditions for isotropization and vibration localization (Bordiga et al., 2018, Piccolroaz et al., 2016).

3. Resonance, Transfer Functions, and Mechanical Frequency Response

For a single beam (cantilever, simply supported, or with attachments), the Beam Frequency Diagram often refers to the magnitude and phase of the system’s frequency-domain transfer function, as in a Bode plot. For an Euler-Bernoulli beam in ambient fluid, the frequency-dependent response

$$H(\omega) = \frac{\cosh(kL)\,\sin(kL) - \sinh(kL)\,\cos(kL)}{\left[\sinh(kL)+\sin(kL)\right]\,kL}$$

with $k(\omega)$ reflecting both fluid added mass and viscous drag, captures the resonance phenomena, modal damping, and fluid-induced peak broadening (Metzger et al., 2016).

In the presence of local mass, damper, and spring attachments, as for a beam with lumped control actuators, the transfer function can be written analytically in Laplace/frequency space (see Propositions 3.1 & 4.1 in (Zuyev et al., 29 Jul 2025)), producing diagrams that visualize system resonances, bandwidth, and the effects of damping or model choice (Timoshenko vs. Euler–Bernoulli) on resonance shifts and mode structure.

Algorithmic plot construction involves:

• Computing $G(i\omega)$ for a logarithmic frequency sweep.
• Plotting $20\log_{10}|G(i\omega)|$ (magnitude) and $\arg G(i\omega)$ (phase).
• Identifying peak frequencies (modes), breakpoints, and slope changes.

4. Parameter Sweeps: Beam Geometry and Modal Frequency Diagrams

In structural applications, beam frequency diagrams are used to explore how modal frequencies depend on geometric parameters (length, width, thickness). For a micro-scale piezoelectric cantilever, finite element analysis (COMSOL) is used to compute the eigenfrequencies of the first several modes as one varies length $L$, width $W$, and height $H$ (Fanse, 2021).

Key findings include:

• $L, H$ sweeps: All modes display continuous frequency variation (eigenfrequencies decrease monotonically with $L$; increase with $H$).
• $W$ sweep: Some modes (notably 2,3,4,6) show discontinuous jumps with $W$, revealing complex cross-sectional mode coupling.

This type of diagram—plots of $f_n$ vs. $L$, $W$, $H$—directly guides device design and modal engineering, validating analytical scalings (e.g., $f_n \sim L^{-2}$ for classical cantilevers) and revealing geometry-dependent splitting and crossing of resonance branches (Fanse, 2021).

5. Application to Photonic and Quantum Beams: Sum-Frequency and Laser-Chirped Beam Diagrams

In nonlinear optics, the beam frequency diagram can express the sum-frequency process $$\omega_{\rm up} = \omega_{\rm ir} + \omega_{\rm pump}$$, annotated as:

ω_ir (1550 nm) ───┐ │ [SUM]──►ω_up (525 nm) │ ω_pump (795 nm)───┘

This frequency-domain representation clarifies the allowed frequency conversion pathways in a χ⁽²⁾ crystal and explains the effect of spatial beam (pump) profile on up-converted image fidelity. Flat-top pump profiles yield more faithful images due to uniform multiplicative transfer in Eq. (3): $$I_{\rm up}(x,y) = K\,I_{\rm ir}(Mx,My) I_{\rm pump}(x,y)$$ (Yang et al., 2019).

In molecular beam slowing, the diagram tracks the instantaneous laser frequency $\nu_L(t)$ versus time, visualizing the frequency-chirp used to compress the velocity distribution of a molecular beam. The resonance condition

$$\Delta(t) \simeq -\Delta_D(t) \implies v(t) \simeq -\frac{c}{\nu_0} \Delta(t)$$

maps beam velocity to laser frequency sweep, and the beam frequency diagram plots detuning $\Delta(t)$ (MHz) against $t$ (ms), encoding the trajectory of the Doppler-matched deceleration phase (Truppe et al., 2016).

6. Beam Frequency Diagrams in Accelerator and Radio Astronomy Contexts

In accelerator physics, the so-called "beam frequency diagram" is a frequency map ($\nu_x$, $\nu_y$ tune plane), plotting the betatron tunes extracted from trajectory data as a function of initial amplitudes under nonlinear dynamics. Each resonance condition $m_x \nu_x + m_y \nu_y = n$ appears as a straight line, and the color coding of the diffusion index $D$ reveals stochastic and regular regions, critical resonances, and suppression effects (e.g., through Crab Waist sextupole strengths) (Shatilov et al., 2010).

In pulsar emission studies, a Beam Frequency Diagram refers to the plot of pulse profile width $W(\nu)$ versus frequency $\nu$, used to discriminate between radius-to-frequency mapping ($r(\nu) \sim \nu^{-\xi}$) and multi-altitude "fan beam" emission models. Observed slopes in log-log $(W, \nu)$ space span from negatives (profile narrowing) to positives (broadening), which cannot be explained by pure RFM ($W_{\rm RFM}(\nu) \sim \nu^{-\xi}$). Instead, the diagrams support a scenario where profile width is dominated by frequency-dependent propagation effects within fan-like or patchy emission beams spanning extended magnetospheric altitudes (Jaroenjittichai et al., 15 Sep 2025).

Table: Pulsar Beam Frequency Diagram Interpretations

Observable	RFM Prediction	Observation (Jaroenjittichai et al., 15 Sep 2025)	Fan Beam Model
$W(\nu)$ scaling	$-\xi$ (narrowing)	$\mu$ ranges negative to positive	Flat/weak $\nu$-dependence, can broaden
Component motion	Strong centroid shift	Centroids stable	Can be stationary
Emission height	Drops with $\nu$	No systematic change	Multi-altitude possible

7. Practical Importance and Implications

Beam frequency diagrams provide a unifying framework across disciplines for the direct visualization, quantification, and analysis of frequency-dependent phenomena in beam systems. Key applications include:

• Antenna and Array Optimization: Beam steering and squint assessment, codebook design, beam training, and discrete-state metasurface tuning (Deshpande et al., 2024).
• Metamaterials and Structural Engineering: Bandgap identification, design of vibration isolation, energy localization, and tailoring of photonic/phononic properties (Kamotski et al., 2018, Sharma et al., 2015, Bordiga et al., 2018).
• Quantum and Atomic Physics: Optimizing molecular beam slowing and laser cooling by mapping Doppler-matching protocols directly on the frequency-time diagram for multilevel/multiband systems (Truppe et al., 2016).
• Spectroscopy and Signal Analysis: Characterizing response functions, resonance shifts, and damping mechanisms in fluid-loaded and dissipative beam structures (Metzger et al., 2016, Zuyev et al., 29 Jul 2025).
• Astrophysics: Discriminating emission models in radio pulsars through systematic multi-frequency width analysis, tied directly to theoretical predictions and magnetospheric structure (Jaroenjittichai et al., 15 Sep 2025).

In all cases, the precise construction, interpretation, and application of beam frequency diagrams are indispensable to both understanding and engineering frequency-selective phenomena in modern beam-based and wave-propagating systems.