For random nonstationary electromagnetic beams, Laatikainen, Friberg, and Setälä (Opt. Lett. 48, 5823 (2023)) proved that cross-spectral purity of a Stokes parameter is consistent with a reduction formula for the corresponding intensity-normalized, time-integrated coherence Stokes parameter, whose product form contains a third factor depending exclusively on polarization, and that purity implies a specific frequency structure for the intensity-normalized spectral coherence Stokes parameters. These results hold under a narrowband assumption in which the frequency dependence of the diffraction factors K1K_1 and K2K_2 in Young's two-pinhole superposition is neglected; the authors state that similar results can be established for broad spectral bandwidth with wavefront-shearing interferometers but do not develop this. The open problem is to complete the intensity-normalized theory for beams of arbitrary spectral bandwidth: derive the reduction formula for the intensity-normalized, time-integrated coherence Stokes parameters, establishing whether the product form with the polarization-dependent normalization factor survives and what additional frequency-dependent factors enter, within an operational shearing/folding-interferometer scheme that eliminates the geometric diffraction factors; and prove the corresponding frequency-structure law for the intensity-normalized spectral coherence Stokes parameters. Any valid answer must reduce to the published narrowband formulas and to the known broadband scalar reduction formula. Later literature (2024-2026) derives broadband nonstationary vector reduction formulas only with per-Stokes-parameter or matrix-element normalizations, which lack the polarization-dependent factor and obey a different frequency-structure law, leaving this core open.

Background

Cross-spectral purity is a classical concept of statistical optics introduced by Mandel for stationary scalar light: two fields with identical normalized spectra are called cross-spectrally pure if there exists a point in their interference pattern where the superposition has the same spectral shape as the constituent beams (L. Mandel, J. Opt. Soc. Am. 51, 1342 (1961)). Mandel showed that purity is tied to a reduction formula: the normalized two-point correlation (complex degree of coherence) factorizes into a spatial-coherence factor and a temporal-coherence factor.

For electromagnetic (vector) light, second-order statistics are described by 2x2 correlation matrices: the mutual coherence matrix Γ(r1,r2,t1,t2)=⟨E∗(r1,t1)ET(r2,t2)⟩\Gamma(\mathbf{r}_1,\mathbf{r}_2,t_1,t_2)=\langle\mathbf{E}^*(\mathbf{r}_1,t_1)\mathbf{E}^T(\mathbf{r}_2,t_2)\rangle in the space-time domain and the cross-spectral density matrix W(r1,r2,ω1,ω2)W(\mathbf{r}_1,\mathbf{r}_2,\omega_1,\omega_2) in the space-frequency domain, where E=(Ex,Ey)T\mathbf{E}=(E_x,E_y)^T is the complex analytic signal of the transverse field and the angle brackets denote ensemble averaging. The coherence Stokes parameters are Pauli-matrix traces of these matrices, Sn=tr[σnΓ]S_n=\mathrm{tr}[\sigma_n\Gamma] and Sn=tr[σnW]S_n=\mathrm{tr}[\sigma_n W], with σ0\sigma_0 the identity and σ1,σ2,σ3\sigma_1,\sigma_2,\sigma_3 the Pauli matrices; S0S_0 is the spectral density (or intensity) and S1,S2,S3S_1,S_2,S_3 represent the polarization state. For nonstationary (e.g. pulsed) fields, whose statistics are not invariant under shifts of the origin of time, the natural two-point quantities are time-integrated: averaging Γ\Gamma over the mean time tˉ=(t1+t2)/2\bar{t}=(t_1+t_2)/2 gives Γˉ(r1,r2,Δt)\bar{\Gamma}(\mathbf{r}_1,\mathbf{r}_2,\Delta t), whose coherence Stokes parameters are Sˉn(r1,r2,Δt)=∫0∞Sn(r1,r2,ω,ω)e−iωΔtdω\bar{S}_n(\mathbf{r}_1,\mathbf{r}_2,\Delta t)=\int_0^\infty S_n(\mathbf{r}_1,\mathbf{r}_2,\omega,\omega)e^{-i\omega\Delta t}d\omega. Laatikainen, Friberg, and Setälä normalized these by the total intensity,

γˉn(r1,r2,Δt)=Sˉn(r1,r2,Δt)Sˉ0(r1,r1,0)Sˉ0(r2,r2,0),\bar{\gamma}_n(\mathbf{r}_1,\mathbf{r}_2,\Delta t)=\frac{\bar{S}_n(\mathbf{r}_1,\mathbf{r}_2,\Delta t)}{\sqrt{\bar{S}_0(\mathbf{r}_1,\mathbf{r}_1,0)\bar{S}_0(\mathbf{r}_2,\mathbf{r}_2,0)}},

calling them intensity-normalized time-integrated coherence Stokes parameters.

In Young's two-pinhole experiment, the spectral field at the observation point R\mathbf{R} is E(R,ω)=K1E(r1,ω)eiωR1/c+K2E(r2,ω)eiωR2/c\mathbf{E}(\mathbf{R},\omega)=K_1\mathbf{E}(\mathbf{r}_1,\omega)e^{i\omega R_1/c}+K_2\mathbf{E}(\mathbf{r}_2,\omega)e^{i\omega R_2/c} with diffraction factors Kj=−iωA/(2πcRj)K_j=-i\omega A/(2\pi c R_j) that depend on frequency; the Stokes parameters of the superposition then obey the electromagnetic spectral interference law (H. Partanen, B. J. Hoenders, A. T. Friberg, and T. Setälä, J. Opt. Soc. Am. A 35, 1379 (2018)). If the spectrum at the pinholes is sufficiently narrow, K1K_1 and K2K_2 can be treated as frequency-independent, and the narrowband analysis of the source paper proceeds: a Stokes parameter Sn(r,ω)S_n(\mathbf{r},\omega) is cross-spectrally pure when its spectral shape at the two pinholes and at some point on the screen coincide, Sn(r2,ω)=CnSn(r1,ω)=DnSn(Rn,ω)S_n(\mathbf{r}_2,\omega)=C_n S_n(\mathbf{r}_1,\omega)=D_n S_n(\mathbf{R}_n,\omega) with real, positive, frequency-independent constants. Under this condition the paper proves two results. First, a reduction formula

γˉn(r1,r2,Δt)=γˉn(r1,r2,τn)γˉn(r1,r1,0) γˉn(r1,r1,Δt−τn),\bar{\gamma}_n(\mathbf{r}_1,\mathbf{r}_2,\Delta t)=\frac{\bar{\gamma}_n(\mathbf{r}_1,\mathbf{r}_2,\tau_n)}{\bar{\gamma}_n(\mathbf{r}_1,\mathbf{r}_1,0)}\,\bar{\gamma}_n(\mathbf{r}_1,\mathbf{r}_1,\Delta t-\tau_n),

where τn\tau_n is the time difference corresponding to the purity point Rn\mathbf{R}_n, i.e. the time-integrated coherence Stokes parameter factorizes into a two-point part and a single-point temporal part, divided by a third factor γˉn(r1,r1,0)\bar{\gamma}_n(\mathbf{r}_1,\mathbf{r}_1,0), a spectral-density-weighted average of the normalized polarization parameter s^n=Sn/S0\hat{s}_n=S_n/S_0, that is present only for the polarization parameters n=1,2,3n=1,2,3 and is absent from all earlier electromagnetic purity works, which used per-Stokes-parameter normalizations instead. Second, a frequency-structure law: for the intensity-normalized spectral coherence Stokes parameters μn(r1,r2,ω1,ω2)=Sn(r1,r2,ω1,ω2)/S0(r1,ω1)S0(r2,ω2)\mu_n(\mathbf{r}_1,\mathbf{r}_2,\omega_1,\omega_2)=S_n(\mathbf{r}_1,\mathbf{r}_2,\omega_1,\omega_2)/\sqrt{S_0(\mathbf{r}_1,\omega_1)S_0(\mathbf{r}_2,\omega_2)}, the magnitude of μ0\mu_0 is independent of frequency, while for n=1,2,3n=1,2,3 the frequency dependence of ∣μn∣|\mu_n| is exclusively determined by the spectral polarization states s^n(r,ω)\hat{s}_n(\mathbf{r},\omega) at the two points.

Two neighboring lines delimit what is known beyond the narrowband regime. For scalar fields the broadband theory is complete: Laatikainen et al. (J. Opt. Soc. Am. A 40, 1260 (2023)) derived the reduction formula for nonstationary scalar fields of arbitrary spectral bandwidth using an ideal field superposition realized by all-reflective wavefront-shearing interferometers, which sidestep the frequency-dependent diffraction factors; Koivurova et al. (Phys. Rev. A 99, 043842 (2019)) had earlier shown that cross-spectral purity is generally not preserved when a finite-bandwidth field propagates in Young-type geometry, which is why shearing-type superpositions are the natural broadband instrument. For vector fields, Joshi and Kanseri (J. Opt. Soc. Am. A 38, 271 (2021); Opt. Commun. 554, 130137 (2024)) analyzed nonstationary cross-spectral purity, the 2024 work deriving a reduction formula valid for any spectral bandwidth in a wavefront-folding interferometer, but for two-time Stokes parameters normalized by their corresponding usual Stokes parameters (per-Stokes normalization), a normalization in which the formula contains no polarization-dependent factor and the normalized spectral quantities are frequency-independent for all nn, in contrast to the intensity-normalized law above. What does not exist in the literature is the broadband version of the intensity-normalized theory: whether, and in what modified form, the polarization-dependent factor and the polarization-state-governed frequency structure survive when the frequency dependence of the geometric factors is no longer negligible.

Problem Statement

For a random nonstationary electromagnetic beam with arbitrary (broad) spectral bandwidth, complete the cross-spectral purity theory of the Stokes parameters in the intensity-normalized formulation of Laatikainen, Friberg, and Setälä. Establish the theory within an operational interferometric scheme realizing the ideal superposition of the fields at two points, such as an all-reflective wavefront-shearing or wavefront-folding interferometer, in which the frequency-dependent diffraction factors are eliminated; any frequency-dependent wavefront-phase factors introduced by the scheme must be carried explicitly. The required results, which jointly constitute the completion of the theory, are: (i) the reduction formula for the intensity-normalized, time-integrated coherence Stokes parameters γˉn(r1,r2,Δt)\bar{\gamma}_n(\mathbf{r}_1,\mathbf{r}_2,\Delta t), n=0,…,3n=0,\ldots,3, for a Stokes parameter that is cross-spectrally pure (same spectral shape at the two input points and at some superposition point), showing whether the product form with the polarization-dependent normalization factor γˉn(r1,r1,0)\bar{\gamma}_n(\mathbf{r}_1,\mathbf{r}_1,0) survives, and identifying any additional frequency-dependent factors that enter in the broadband regime, together with the converse implication as in the narrowband theory; and (ii) the frequency-structure law for the intensity-normalized spectral coherence Stokes parameters μn(r1,r2,ω,ω)\mu_n(\mathbf{r}_1,\mathbf{r}_2,\omega,\omega), stating whether ∣μ0∣|\mu_0| remains independent of frequency and whether the frequency dependence of ∣μn∣|\mu_n|, n=1,2,3n=1,2,3, is still exclusively determined by the spectral polarization states s^n(r,ω)=Sn(r,ω)/S0(r,ω)\hat{s}_n(\mathbf{r},\omega)=S_n(\mathbf{r},\omega)/S_0(\mathbf{r},\omega), or how these statements must be modified. Any valid answer must reproduce, in the narrowband limit, the published narrowband reduction formula and frequency-structure law, and must reduce, for n=0n=0, to the broadband scalar reduction formula obtained with ideal wavefront-shearing superpositions.

References

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