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Quasi-geometrical Optics Approximation in Gravitational Lensing
Astrophysics · Submitted 7 Feb 2004
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim By analyzing this correction term, we find that (1) the lensing magnification $\mu$ is modified to $\mu ~(1+\delta)$, where $\delta$ is of the order of $(\lambda/M)^2$, and (2) if the lens has cuspy (or singular) density profile at the center $\rho(r) \propto r^{-\alpha}$ ($0 < \alpha \leq 2$), the diffracted image is formed at the lens center with the magnification $\mu \sim (\lambda/M)^{3-\alpha}$.
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The abstract, sentence by sentence
The gravitational lensing of gravitational waves should be treated in the wave optics instead of the geometrical optics when the wave length $\lambda$ of the gravitational waves is larger than the Schwarzschild radius of the lens mass $M$. The wave optics is based on the diffraction integral which represents the amplification of the wave amplitude by lensing. We study the asymptotic expansion of the diffraction integral in the powers of the wave length $\lambda$. The first term, arising from the short wavelength limit $\lambda \to 0$, corresponds to the geometrical optics limit. The second term, being of the order of $\lambda/M$, is the leading correction term arising from the diffraction effect. By analyzing this correction term, we find that (1) the lensing magnification $\mu$ is modified to $\mu ~(1+\delta)$, where $\delta$ is of the order of $(\lambda/M)^2$, and (2) if the lens has cuspy (or singular) density profile at the center $\rho(r) \propto r^{-\alpha}$ ($0 < \alpha \leq 2$), the diffracted image is formed at the lens center with the magnification $\mu \sim (\lambda/M)^{3-\alpha}$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The gravitational lensing of gravitational waves should be treated in the wave optics instead of the geometrical optics when the wave length $\lambda$ of the gravitational waves is larger than the Schwarzschild radius of the lens mass $M$. | background 1.00 | 0.12 | 0.27 | asserted 1.00 |
| s1 | The wave optics is based on the diffraction integral which represents the amplification of the wave amplitude by lensing. | background 0.90 | 0.06 | 0.03 | asserted 1.00 |
| s2 | We study the asymptotic expansion of the diffraction integral in the powers of the wave length $\lambda$. | contribution 0.57 | 0.93 | 0.04 | asserted 0.95 |
| s3 | The first term, arising from the short wavelength limit $\lambda \to 0$, corresponds to the geometrical optics limit. | result 0.44 | 0.37 | 0.06 | asserted 1.00 |
| s4 | The second term, being of the order of $\lambda/M$, is the leading correction term arising from the diffraction effect. | result 0.60 | 0.69 | 0.11 | asserted 1.00 |
| s5 | By analyzing this correction term, we find that (1) the lensing magnification $\mu$ is modified to $\mu ~(1+\delta)$, where $\delta$ is of the order of $(\lambda/M)^2$, and (2) if the lens has cuspy (or singular) density profile at the center $\rho(r) \propto r^{-\alpha}$ ($0 < \alpha \leq 2$), the diffracted image is formed at the lens center with the magnification $\mu \sim (\lambda/M)^{3-\alpha}$. | result 1.00 | 0.95 | 0.81 | asserted 1.00 |
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The paper
Original abstract
The gravitational lensing of gravitational waves should be treated in the wave optics instead of the geometrical optics when the wave length $\lambda$ of the gravitational waves is larger than the Schwarzschild radius of the lens mass $M$. The wave optics is based on the diffraction integral which represents the amplification of the wave amplitude by lensing. We study the asymptotic expansion of the diffraction integral in the powers of the wave length $\lambda$. The first term, arising from the short wavelength limit $\lambda \to 0$, corresponds to the geometrical optics limit. The second term, being of the order of $\lambda/M$, is the leading correction term arising from the diffraction effect. By analyzing this correction term, we find that (1) the lensing magnification $\mu$ is modified to $\mu ~(1+\delta)$, where $\delta$ is of the order of $(\lambda/M)^2$, and (2) if the lens has cuspy (or singular) density profile at the center $\rho(r) \propto r^{-\alpha}$ ($0 < \alpha \leq 2$), the diffracted image is formed at the lens center with the magnification $\mu \sim (\lambda/M)^{3-\alpha}$.
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References
How the literature uses it
21 arXiv papers cite this in 48 sentences
Papers citing it · 30 sentences without a confident label
Uses its method or tool 3 papers
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Expanding the Taylor expansion of the Fermat potential in Eq. ([ref]) to fourth order in $x⃗̃⃗$ yields [ref] [equation] where [equation] Since $T_ab$ is a real, symmetric $2 ×2$ matrix, one can construct the matrix $A_ab$ whose columns are its eigenvectors; this matrix satisfies [equation] where $λ_(c)$ are the eigenvalues of $A_ab$.
Identifying multiple images of gravitational-wave sources lensed by elliptical lensing potentials · Quasi-geometrical optics approximation · 0.84
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Therefore, $dF_c$ can be expressed as [ref] [equation] where $Γ(x)$ is the gamma function.
Gravitational Lensing of Gravitational Wave for Generalized Navarro-Frenk-White Profile and Einasto Profile · (Quasi-)Geometrical Optics Approximation · 0.83
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In calculating the geometrical optics effects, beyond the leading order effect, we will consider the sub-leading order effect (i.e. the post-geometric correction) following the method in [ref] .
Detecting Strong Gravitational Lensing of Gravitational Waves with TianQin · Introduction · 0.99
Compares with it 1 paper
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For the SIS, Eq. [ref] matches with Eq. (21) of [ref] for small $y$.
Lensing of gravitational waves: efficient wave-optics methods and validation with symmetric lenses · Contribution from the cusp · 1.00
Cites it as background 8 papers
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The current body of literature about wave optics effects is based on the so-called diffraction integral, derived in the context of gravitational wave's lensing in [ref] and further developed in subsequent works [ref] .
Proper time path integrals for gravitational waves: an improved wave optics framework · Introduction · 0.91
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The standard framework is provided by the diffraction integral, a highly oscillatory integral over the lens plane, which coherently sums the contributions of different propagation paths [ref] .
Wave-optics gravitational wave lensing in modified gravity · Introduction · 0.82
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Wave-optics effects in gravitational lensing of GWs have been extensively studied in the literature [ref] .
Observability of lensing of gravitational waves from massive black hole binaries with LISA · Introduction · 1.00
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