Wave-Optics Diffraction

Geometric optics replaces a lens by discrete stationary rays. Wave optics keeps the phase across the entire lens plane, so it remains finite at caustics and describes interference when the gravitational wavelength is not negligible compared with the lens scale. The governing observable is a complex amplification factor $F$: its magnitude changes strain amplitude and its argument changes phase. This page fixes the delay zero and dimensionless frequency explicitly, derives the low- and high-frequency limits, and separates diffraction from curvature tails and extra gravitational degrees of freedom. Detector likelihoods are outside scope.

Required background. Waves, Green Functions, and Geometric Optics supplies Fresnel integrals and stationary phase; Curvature supplies the weak lens geometry and null-ray limit.

Helpful background. Einstein Equations supplies the linearized wave equation on a lens background; Relativistic Cosmology supplies angular-diameter distances and redshifted lens scales.

The normalized amplification integral

Work in the thin-lens, small-angle, weak-potential approximation. Use detector-frame frequency $f>0$ with

$$h(t)=\int_{-\infty}^{\infty}\tilde h(f)e^{-2\pi ift}\,df, \qquad \tilde h_L(f)=F(f,\boldsymbol{y})\tilde h(f).$$

Let the dimensionless Fermat potential be

$$T(\boldsymbol{x},\boldsymbol{y}) =\frac12\lvert\boldsymbol{x}-\boldsymbol{y}\rvert^2 -\psi(\boldsymbol{x})-\phi_m(\boldsymbol{y}).$$

The function $\phi_m(\boldsymbol{y})$ is the minimum of the unshifted geometric-plus-potential terms, so $\min_{\boldsymbol{x}}T=0$. The explicit minus sign is part of this page’s convention. For a point lens of rest mass $M_L$ at redshift $z_L$, define

$$w:=8\pi G M_L(1+z_L)f, \qquad t_d=4GM_L(1+z_L)T.$$

Thus the phase $2\pi f t_d=wT$. The mass is local to the lens frame, while $f$ is measured in the cosmological detector frame; the factor $1+z_L$ translates between them. The dimensionless amplification integral is

$$F(w,\boldsymbol{y}) =\frac{w}{2\pi i} \int_{\mathbb R^2}d^2\boldsymbol{x}\, e^{iwT(\boldsymbol{x},\boldsymbol{y})}.$$

The prefactor is chosen so that $F=1$ when the lens potential is removed, with the corresponding free Fresnel phase convention. Takahashi and Nakamura derive the dimensional thin-lens integral, its cosmological redshift factor, and the declared minimum-delay zero in Takahashi and Nakamura 2003, §2, pp. 3–6, eqs. (8)–(14), PDF.

The integral is oscillatory rather than absolutely convergent. One may define it by adding a positive Gaussian damping factor and taking that regulator to zero after integration. With $\psi=0$ and $\phi_m=0$, translation by $\boldsymbol y$ gives $\int d^2x\,e^{iw|x-y|^2/2}=2\pi i/w$ in this prescription, verifying the free-wave normalization. The formulas below use $w\gt0$; the physical transfer function at negative frequency is $F(-f)=F(f)^*$ for each real polarization.

Changing $T\mapsto T+C(\boldsymbol{y})$ would multiply $F$ by $e^{iwC}$. It would not change $\lvert F\rvert$, but it would change a quoted waveform phase and obscure comparisons. Delay differences are physical; subtracting $\phi_m$ to set $T_{\min}=0$ is the bookkeeping convention that fixes the otherwise arbitrary common phase.

Long wavelengths: the diffraction limit

For the point lens, $\psi=\ln\lvert\boldsymbol{x}\rvert$, and the integral can be evaluated exactly:

$$\begin{aligned} F(w,y)={}& \exp\!\left[ \frac{\pi w}{4} +\frac{iw}{2} \left(\ln\frac{w}{2}-2\phi_m(y)\right) \right]\\ &\times \Gamma\!\left(1-\frac{iw}{2}\right) {}_1F_1\!\left( \frac{iw}{2},1;\frac{iwy^2}{2} \right). \end{aligned}$$

Here

$$x_m=\frac12\left(y+\sqrt{y^2+4}\right), \qquad \phi_m(y)=\frac12(x_m-y)^2-\ln x_m,$$

so the minimum delay is zero. As $w\to0$, every displayed factor approaches one: $w\ln w\to0$, $\Gamma(1-iw/2)\to1$, and the hypergeometric function tends to one. Hence

$$F(w,y)\longrightarrow1.$$

A wavelength much longer than the lens’s gravitational time scale cannot resolve separate paths; diffraction washes out the geometric magnification. The crossover is $w\sim1$, equivalently a wavelength comparable to the redshifted Schwarzschild scale up to the declared $2\pi$ convention. This is a scale comparison, not a claim that the lens curvature disappears.

At exact alignment, $y=0$, the hypergeometric factor is one and the gamma-function identity $|\Gamma(1+ix)|^2=\pi x/\sinh(\pi x)$ yields the particularly useful check

$$|F(w,0)|^2=\frac{\pi w}{1-e^{-\pi w}}.$$

It tends to one as $w\to0$ and remains finite at every finite $w$, even though geometric optics has an Einstein-ring caustic there. Its growth as $w\to\infty$ also shows that the high-frequency and exact-caustic limits require care.

Short wavelengths: stationary images

For $w\gg1$, expand $T$ near each nondegenerate stationary point $\boldsymbol{x}_j$:

$$T=T_j+\frac12 \delta\boldsymbol{x}^{\mathsf T} \mathsf H_j\delta\boldsymbol{x}+\cdots, \qquad \boldsymbol{\nabla}T_j=0.$$

The two-dimensional oscillatory Gaussian supplies the determinant of the Hessian and a phase determined by its signature. Because $\mathsf H_j$ is the inverse lens mapping at the image, $\mu_j=1/\det\mathsf H_j$. Stationary phase gives

$$F(w,\boldsymbol{y}) \sim \sum_j \sqrt{\lvert\mu_j\rvert}\, \exp\!\left(iwT_j-i\pi n_j\right),$$

where $n_j=0$, $1/2$, and $1$ for a minimum, saddle, and maximum, respectively. These Morse factors are part of the asymptotic evaluation, not additional source phases. Takahashi and Nakamura show this reduction explicitly in Takahashi and Nakamura 2003, p. 6, eqs. (15)–(18), PDF.

At a caustic, $\det\mathsf H_j=0$, so the Gaussian approximation and its individual image amplitudes diverge. The original diffraction integral remains the correct object; near a fold or cusp it can be reduced to an appropriate uniform canonical integral. The finite wave result is precisely why geometric images cannot be used through their merger point.

Interference is not a propagation tail

For a point lens at high frequency, remove the common minimum delay and write

$$F =\sqrt{\mu_+} -i\sqrt{\lvert\mu_-\rvert} e^{2\pi if\Delta t_d}.$$

The frequency-dependent fringes come from coherent addition of two stationary paths. A curvature tail instead arises from interior-of-the-light-cone support of a retarded Green function. Both can produce delayed waveform structure, but their causal decompositions and approximations differ.

Diffraction also need not introduce a new propagating polarization. Beyond one-ray geometric optics, a field cannot always be represented by one sharply transported screen tensor; finite-wavelength corrections can generate observer-dependent apparent scalar or vector projections. Cusin and Lagos derive those corrections and emphasize their observer dependence in Cusin and Lagos 2020, §IV, pp. 8–9, and Appendix A, pp. 11–12, PDF. Genuine extra polarizations require additional dynamical degrees of freedom in the underlying theory, not merely a smeared wavefront in general relativity.

Exercise

Using the two-image high-frequency expression, compute $\lvert F\rvert^2$. What sets the fringe spacing in $f$, and why can this result not be extrapolated to $w\ll1$?

Solution

Let $a=\sqrt{\mu_+}$, $b=\sqrt{\lvert\mu_-\rvert}$, and $\varphi=2\pi f\Delta t_d$. Then

$$\lvert F\rvert^2 =\lvert a-ib e^{i\varphi}\rvert^2 =a^2+b^2+2ab\sin\varphi.$$

Therefore

$$\lvert F\rvert^2 =\mu_++\lvert\mu_-\rvert +2\sqrt{\mu_+\lvert\mu_-\rvert} \sin(2\pi f\Delta t_d).$$

Adjacent fringes are separated by $\Delta f=1/\Delta t_d$. The calculation assumed two isolated nondegenerate stationary points and $w\gg1$. At $w\ll1$ the lens plane contributes coherently rather than through two localized images; the exact normalized integral instead gives $F\to1$.

References

• Giulia Cusin and Macarena Lagos, “Gravitational Wave Propagation Beyond Geometric Optics,” Physical Review D 101 (2020) 044041, arXiv:1910.13326v2, DOI, Open PDF.
• Ryuichi Takahashi and Takashi Nakamura, “Wave Effects in the Gravitational Lensing of Gravitational Waves from Chirping Binaries,” The Astrophysical Journal 595 (2003) 1039–1051, arXiv:astro-ph/0305055v2, DOI, Open PDF.