Diffraction of large-number whispering gallery mode by boundary straightening with jump of curvature

Diffraction of a high-frequency large-number whispering gallery mode is studied, which runs along a concave curve turning to a straight line. At the point of straitening, the curvature of the boundary suffers a jump. The parabolic equation method is developed in the problem, and asymptotic formulas are presented for all waves arising in the vicinity of the non-smoothness point of the boundary. The ``ray skeleton’’ of the wavefield is investigated in detail.

💡 Research Summary

The paper investigates the diffraction of a high‑frequency whispering‑gallery mode (WGM) whose mode number is large (n ≫ 1) as it propagates along a concave boundary that abruptly straightens, causing a jump in curvature from 1/a to zero at the junction point O. The authors employ the parabolic‑equation (PE) method, a boundary‑layer asymptotic technique originally developed by V. A. Fock, and extend it to handle two simultaneous large parameters:

1. m = (ka/2)¹ᐟ³, the standard Fock scaling parameter that measures the ratio of the wavelength to the curvature radius a;
2. t, the (large) negative zero of the derivative of the Airy function v(z)=√π Ai(z), i.e., v′(−t)=0, which characterizes the transverse oscillation number of the incident WGM.

The analysis assumes t ≫ 1 and the hierarchy t ≪ m⁴ᐟ⁵, which guarantees that the PE approximation remains valid in a neighbourhood of the non‑smooth point despite the presence of a second large parameter.

Geometrical setting. The boundary S consists of a circular arc S₋ of radius a joined smoothly to a straight line S₊ at O. Local coordinates (s,n) measure arc length along S and normal distance from S, respectively. After nondimensionalisation, the stretched variables σ = m s/a and ν = 2m² n/a are introduced. In these variables the incident WGM takes the form

 u_inc ≈ e^{ik s} U_inc, U_inc(σ,ν)=e^{-i t σ} v(ν−t),

where v(z) is the Airy function. Because v(z) decays exponentially for z>0, the incident field is confined to 0 ≤ ν < t+O(1).

Ray skeleton. A detailed ray analysis shows that, after reaching O, the field splits into two families of rays, ℓ₁ and ℓ₂. Rays of ℓ₁ intersect the straight segment S₊ once more and then travel below the limiting ray l_O (the line y = γ x with γ≈√t/m). Rays of ℓ₂ reflect off the circular part S₋ for the last time, pass through the caustic C (the circle centred at (0,a−b) of radius a−b), and emerge above l_O. The caustic touches l_O at point Q with coordinates (x_Q≈aγ, y_Q≈aγ²). The distance from an observation point M to the caustic is encoded in the combination

 e_ν = t + σ² − ν,

which appears repeatedly in the asymptotic formulas.

Analytical solution. Within the PE framework the authors solve the reduced equation analytically, obtaining explicit expressions for the field in terms of σ, ν, and the Airy function. The eikonal (phase) of the two wave components u₁ (associated with ℓ₁) and u₂ (associated with ℓ₂) is derived both before and after crossing the caustic. For points beyond the caustic the eikonal reads

 k τ₊(M) = k x + (p − ν)²/(4σ) + (2/3)(t − p)^{3/2} + O(…,

where p = 2m² h/a is a stretched vertical coordinate related to the intersection of the ray with the y‑axis. The corresponding expression for τ₋(M) (before the caustic) differs only by the sign of the cubic term. The higher‑order corrections are shown to be negligible under the assumed parameter hierarchy.

Matching and diffraction coefficient. By matching the PE solution to the far‑field cylindrical wave representation

 u_dif = A(φ;k) e^{ik r}/√{k r},

the authors extract the diffraction coefficient A(φ;k). This coefficient depends on the Airy function and its zero t, and its structure is fundamentally different from the small‑mode case previously studied. In particular, the large‑t asymptotics introduce a non‑trivial dependence on (t − p)^{3/2}, which governs the transition from the illuminated region to the shadow region bounded by l_O.

Key findings.

• The presence of a curvature jump together with a large transverse mode number creates a rich ray skeleton consisting of two families and a caustic that separates illuminated and shadow zones.
• The PE method, when supplemented with the second large parameter t, yields uniformly valid asymptotic formulas for the field in all regions near the non‑smooth point.
• The derived diffraction coefficient captures the effect of the curvature discontinuity on high‑order WGMs and reduces to known results in the limit t → 0 (small‑mode case).
• The analytical expressions for the eikonals and amplitudes provide a practical tool for estimating mode conversion, radiation loss, and field distribution in devices where a waveguide or resonator abruptly changes curvature (e.g., bent optical fibers, acoustic waveguides, or microwave resonators).

Overall, the paper extends the classical Fock‑type parabolic‑equation analysis to a previously unexplored regime of large‑order whispering‑gallery modes interacting with a boundary that exhibits a sudden curvature jump, delivering both a detailed geometric picture and explicit asymptotic formulas that can be directly applied in high‑frequency engineering problems.