---
title: "The exact amplification factor of a point-mass lens"
description: "Derive the diffraction integral for a point mass, its closed form, the two-image limit at high frequency and the smooth limit at low frequency."
canonical_url: "https://sandboxphysics.com/stories/gravitational-wave-lensing/point-lens-amplification-factor/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/gravitational-wave-lensing/point-lens-amplification-factor/"
last_modified: "2026-11-12"
---

[Gravitational-wave lensing and frequency fringes](https://sandboxphysics.com/stories/gravitational-wave-lensing/)

# The exact amplification factor of a point-mass lens

Sandbox Physics · 2026-11-12

A point-mass lens multiplies each frequency of a gravitational wave by one complex number, the amplification factor $F$. At low frequency $F$ tends to 1, because long waves pass the lens almost undisturbed. At high frequency $F$ becomes the sum of two images with a fixed delay, which produces fringes. A single closed formula covers both ends and everything between.

This chapter builds on [why a lens delays a gravitational wave](https://sandboxphysics.com/stories/gravitational-wave-lensing/diffraction-and-the-lens-time-delay/). You need complex exponentials, the Gaussian integral $\int_{-\infty}^{\infty}e^{i a u^2/2}\,\mathrm du=\sqrt{2\pi/|a|}\,e^{i\pi\,\mathrm{sgn}(a)/4}$, and the idea of a Fourier transform.

## Adding up every path

Huygens' principle says each point of the lens plane re-emits the wave. A path through the point $x$ arrives late by $t_d(x)$, so at frequency $f$ it contributes a phase $e^{2\pi i f t_d(x)}$. Summing all points with equal weight and dividing by the same sum without a lens defines the amplification factor. In the dimensionless variables of the last chapter,

$$
F(w,y)=\frac{w}{2\pi i}\int \mathrm d^2x\;\exp\bigl[i\,w\,T(x,y)\bigr],
$$

with $w=8\pi GM_{Lz}f/c^3$ and $T=\tfrac12|x-y|^2-\ln|x|-\phi_m(y)$. This is Eq. (11) of the 2003 paper written with $w$ and $T$. [Takahashi and Nakamura 2003, Sec. 2](https://arxiv.org/abs/astro-ph/0305055v2)

Two conventions sit inside that formula.

- **Normalization.** Without the lens, $T=\tfrac12|x-y|^2$, and the two Gaussian integrals give $2\pi i/w$. The prefactor cancels it, so $F=1$. Every departure from 1 is the lens.
- **Sign of the phase.** We take the Fourier transform of the strain as $\tilde h(f)=\int h(t)\,e^{+2\pi i f t}\,\mathrm dt$, the convention of Cutler and Flanagan. A signal delayed by $\Delta t$ then picks up the factor $e^{+2\pi i f\Delta t}$, which is why the exponent above is $+iwT$. The lensed spectrum is $\tilde h_{\rm L}(f)=F(f)\,\tilde h(f)$. [Cutler and Flanagan 1994, Eq. 20](https://arxiv.org/abs/gr-qc/9402014v1)

## Recovering the two images

When $w$ is large, $e^{iwT}$ oscillates rapidly and almost every part of the integral cancels. Only the neighbourhoods of the points where $T$ is stationary survive. Near such a point $x_j$, expand $T$ to second order along its two principal directions:

$$
T\approx T_j+\tfrac12\lambda_1u_1^2+\tfrac12\lambda_2u_2^2 .
$$

Each direction is a Gaussian integral, $\sqrt{2\pi/(w|\lambda|)}\,e^{i\pi\,\mathrm{sgn}(\lambda)/4}$. Multiplying the two and the prefactor $w/(2\pi i)=(w/2\pi)\,e^{-i\pi/2}$ gives

$$
F_j=\frac{1}{\sqrt{|\lambda_1\lambda_2|}}\;e^{iwT_j}\;e^{\,i\pi(\mathrm{sgn}\lambda_1+\mathrm{sgn}\lambda_2-2)/4}.
$$

The product $\lambda_1\lambda_2$ is the determinant of the second-derivative matrix of $T$, and its inverse is the magnification $\mu_j$. The last factor depends only on the kind of stationary point: 1 at a minimum, $e^{-i\pi/2}=-i$ at a saddle and $-1$ at a maximum. That is Eq. (15) of the 2003 paper,

$$
F_{\rm geo}=\sum_j|\mu_j|^{1/2}\exp\bigl(2\pi i f\,t_{d,j}-i\pi n_j\bigr),\qquad n_j=0,\ \tfrac12,\ 1,
$$

and the same phase appears as Eq. 1 of the LIGO–Virgo–KAGRA lensing search, where $n_j$ is called the Morse index. [LIGO–Virgo–KAGRA 2023, Sec. 3](https://arxiv.org/abs/2304.08393v1)

A point lens has one minimum and one saddle, and no maximum. With the first image at $T=0$,

$$
F_{\rm geo}=|\mu_+|^{1/2}-i\,|\mu_-|^{1/2}\,e^{2\pi i f\Delta t_d},
$$

which is Eq. (18) of the 2003 paper.

## Fringes in frequency

Square the magnitude. With $a=|\mu_+|^{1/2}$, $b=|\mu_-|^{1/2}$ and $\varphi=2\pi f\Delta t_d$,

$$
|F_{\rm geo}|^2=\mu_++|\mu_-|+2\sqrt{\mu_+|\mu_-|}\,\sin\varphi .
$$

The magnitude swings between $a-b$ and $a+b$, and one full swing takes $\Delta f=1/\Delta t_d$. The $-i$ of the saddle moves the peaks: they sit where $\sin\varphi=1$, a quarter cycle away from where two in-phase copies would add.

**Worked example.** For $y=0.5$, $\mu_+=1.591$ and $|\mu_-|=0.591$, so $a=1.261$ and $b=0.769$. The magnitude runs between 0.49 and 2.03. With $M_{Lz}=1000\,M_\odot$ the delay is 19.9 ms, so the peaks repeat every 50 Hz.

## The exact formula

For an isolated point mass the integral can be done in closed form. The 2003 paper quotes the result, attributing it to Peters (1974):

$$
F(w,y)=\exp\Bigl[\frac{\pi w}{4}+\frac{iw}{2}\Bigl(\ln\frac w2-2\phi_m(y)\Bigr)\Bigr]\;\Gamma\Bigl(1-\frac{iw}{2}\Bigr)\;{}_1F_1\Bigl(\frac{iw}{2},1;\frac{iwy^2}{2}\Bigr),
$$

with $\phi_m(y)=(x_m-y)^2/2-\ln x_m$ and $x_m=(y+\sqrt{y^2+4})/2$. Here $\Gamma$ is the gamma function and ${}_1F_1$ the confluent hypergeometric function. This is Eq. (17) of the 2003 paper. We did not take it on trust: [the recompute guide](https://sandboxphysics.com/stories/gravitational-wave-lensing/recompute-the-point-lens-amplification/) evaluates the integral a second way, without the closed form, and the two agree to $4.2\times10^{-4}$ in real and imaginary parts on 20 test points.

A note on one sign. The paper's Eq. (12) prints $+\phi_m(y)$ in $T$, while its text says $\phi_m$ is chosen so that the earliest arrival is zero. We record this as an observation about notation. Eq. (17) as printed matches the numerical integral with the zero-minimum choice, $-\phi_m$, and that is the convention used throughout this course.

## Long waves: why the factor tends to 1

For small $w$, expand each factor to first order. $\Gamma(1+\epsilon)\approx1-\gamma\epsilon$ with Euler's constant $\gamma=0.5772$, so $\Gamma(1-iw/2)\approx1+i\gamma w/2$. The hypergeometric factor is $1+(iw/2)(iwy^2/2)+\dots$, which differs from 1 only at order $w^2$. Collecting terms,

$$
F\approx1+\frac{\pi w}{4}+\frac{iw}{2}\Bigl[\ln\frac w2+\gamma-2\phi_m(y)\Bigr].
$$

The leading change in magnitude is $|F|\approx1+\pi w/4$, the same for every $y$. A lens much smaller than the wavelength cannot form images. It only raises the amplitude slightly and adds a slow phase drift.

**Worked example.** A $15\,M_\odot$ lens at $f=20$ Hz has $w=1.24\times10^{-4}\times15\times20=0.037$, so $|F|\approx1+0.029=1.029$. The exact formula gives 1.029 at $y=0.5$, while the two-image formula claims 1.50. Ray optics overstates the effect by a factor of about 17 in $|F|-1$.

## Short waves: approaching two images

For large $w$ the exact $F$ approaches $F_{\rm geo}$. How fast depends on $y$. In our reference runs the relative difference at $y=1$ falls from $2.6\times10^{-3}$ at $w=30$ to $2.6\times10^{-5}$ at $w=1000$. At $y=0.3$ it is $3.1\times10^{-2}$ at $w=30$, and on the way down it does not fall steadily: $6.5\times10^{-3}$ at $w=100$ but $8.7\times10^{-3}$ at $w=300$. The leftover difference itself oscillates with frequency, so a single comparison point can mislead. Small $y$ means a nearly aligned source and a saddle image close to the lens, where diffraction lasts longest.

**Worked example.** With $M_{Lz}=1000\,M_\odot$ and $y=0.5$, the exact and two-image magnitudes are 1.825 and 1.828 at 20 Hz, where $w=2.5$, and 1.411 and 1.438 at 100 Hz, where $w=12.4$. Close agreement at one frequency is luck of the fringe phase: at 31.5 Hz, near the first fringe minimum, they are 0.961 and 0.891.

## Lab tasks

1. In [the gravitational-wave lens Lab](https://sandboxphysics.com/labs/gravitational-wave-lens-fringes/), choose the preset Waves too long to split. Read the solid and dashed curves of the amplification plot at the left edge, and check $1+\pi w/4$ with $w$ from the lens mass shown.
2. Choose the preset Two paths, clear fringes and count fringes between 50 and 200 Hz. Compare with $150\,\mathrm{Hz}\times\Delta t_d$.
3. Drag the lens toward $y=0.1$ and watch the phase plot. Where do the two curves part, and why does that happen at small $y$ first?

## Exercises

1. Show that the prefactor $w/(2\pi i)$ makes $F=1$ when there is no lens.
2. Derive the phase factor $-i$ for a saddle point from the two one-dimensional Gaussian integrals.
3. For $y=1$, find the range of $|F_{\rm geo}|$ and the fringe spacing when $M_{Lz}=500\,M_\odot$.
4. Use the low-frequency formula at $y=1$, $w=0.1$ to estimate $F$. The exact value is $1.0722-0.0989i$. Where does the remaining difference come from?

Answers

1. $\int e^{iw|u|^2/2}\,\mathrm d^2u$ is the square of $\sqrt{2\pi/w}\,e^{i\pi/4}$, which is $(2\pi/w)\,e^{i\pi/2}=2\pi i/w$. Multiplying by $w/(2\pi i)$ gives 1.
2. A saddle has $\mathrm{sgn}\lambda_1+\mathrm{sgn}\lambda_2=0$. The Gaussian phases cancel, $e^{i\pi/4}e^{-i\pi/4}=1$, and the prefactor contributes $1/i=-i$.
3. $\mu_\pm=\tfrac12\pm3/(2\sqrt5)$ gives $\mu_+=1.171$ and $|\mu_-|=0.171$, so $a=1.082$ and $b=0.413$. $|F_{\rm geo}|$ runs from 0.67 to 1.50. The bracket of the delay formula is $1.118+0.962=2.080$; with the time unit $500\times19.7\,\mu\mathrm s=9.85\,\mathrm{ms}$, $\Delta t_d=20.5$ ms and the fringes are 49 Hz apart.
4. $x_m=1.618$, so $\phi_m=0.191-0.481=-0.290$. The bracket is $\ln0.05+0.577+0.580=-1.839$, giving $F\approx1.0785-0.0919i$. The real part is off by 0.006 and the imaginary part by 0.007. Both are second-order effects. The second-order term of the exponential is $-0.0047-0.0095i$, that of $\Gamma$ is $-0.0025$, the hypergeometric term $-w^2y^2/4$ is $-0.0025$, and products of the first-order terms make up the rest.

Next: [can a detector tell a lensed chirp from an ordinary one?](https://sandboxphysics.com/stories/gravitational-wave-lensing/matched-filtering-a-lensed-chirp/)

## Try it in the Lab

[Gravitational-wave lens fringes](https://sandboxphysics.com/labs/gravitational-wave-lens-fringes/): Compare the solid and dashed amplification curves for a light lens.

## Sources

1. [Takahashi and Nakamura 2003, Sec. 2](https://arxiv.org/abs/astro-ph/0305055v2)
2. [Cutler and Flanagan 1994, Eq. 20](https://arxiv.org/abs/gr-qc/9402014v1)
3. [LIGO–Virgo–KAGRA 2023, Sec. 3](https://arxiv.org/abs/2304.08393v1)

## Keep reading

Gravitational-wave lensing and frequency fringes · Reading path

### From two paths to a lens search

1. [Article **What a detector hears when one chirp takes two paths**](https://sandboxphysics.com/stories/gravitational-wave-lensing/one-chirp-two-paths/)
2. [Article **Why a lens delays a gravitational wave**](https://sandboxphysics.com/stories/gravitational-wave-lensing/diffraction-and-the-lens-time-delay/)
3. [You are here **The exact amplification factor of a point-mass lens**](https://sandboxphysics.com/stories/gravitational-wave-lensing/point-lens-amplification-factor/)
4. [Article **Can a detector tell a lensed chirp from an ordinary one?**](https://sandboxphysics.com/stories/gravitational-wave-lensing/matched-filtering-a-lensed-chirp/)
5. [Article **Recompute point-lens amplification and test a lens search**](https://sandboxphysics.com/stories/gravitational-wave-lensing/recompute-the-point-lens-amplification/)
6. [Article **How to read the O3 search for lensed gravitational waves**](https://sandboxphysics.com/stories/gravitational-wave-lensing/reading-the-o3-microlensing-search/)