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For a symmetric confocal optical resonator, the mirror separation equals either mirror's curvature, $R_2=L_2$, so its symmetric fundamental Gaussian beam hasThe same optical mode matching calculation applies, now with $f_{\min }=\sqrt{z_A{L}_2/2}$. Keeping the example's $L_2=10\mathrm{m}$ gives $z_B=5\mathrm{m}$ and $f_{\min }=0.707107\mathrm{m}$, compared with $0.930605\mathrm{m}$ for the nonconfocal target. With $f=1.5\mathrm{m}$, the two branches becomeAgain only the positive-$v$ branch fits the stated downstream geometry. The confocal target's smaller Rayleigh range means a smaller fundamental beam waist at the same wavelength and a less stringent minimum $f$.

A confocal optical resonator also has transverse normal mode degeneracies: in the ideal paraxial spectrum, increasing transverse order by two shifts the resonance by one longitudinal spacing. Such a cavity can transmit combinations of higher transverse normal modes at a fundamental resonance, so transmitted power alone is a weaker diagnostic of pure fundamental optical mode matching. The degeneracy does not remove the complex beam parameter matching requirement if the objective is specifically the symmetric fundamental Gaussian beam.

Use the time-harmonic wave convention $\mathrm{Re}(\psi e^{-i\omega t})$, with a positive background wavenumber $k$. The scalar Helmholtz equation for a constant-density acoustic model isWriting $\psi =e^{ikx}E$ gives the exact envelope equationThe paraxial approximation neglects $E_{xx}$ relative to $2ik{E}_x$, giving the parabolic wave equationThe choice of carrier and the direction of propagation matter: this is a forward, slowly varying envelope approximation. Sufficient scale conditions are $|E_{xx}|\ll 2k|E_x|$, transverse spectral components $|p|\ll k$, and medium/envelope variation on longitudinal scales large compared with $1/k$. With the carrier fixed at the background $k$, small $|n-1|$ makes the refractive phase vary slowly too. Large-angle propagation, appreciable backscattering, or rapid longitudinal variation violates the approximation. Acoustic models with variable mass density can have additional gradient terms, so the scalar Helmholtz equation itself is a model assumption. For the Gaussian beam in part (b), useful initial conditions are $kD\gg 1$ and $D/|F|\ll 1$, with the second condition absent for an initially uncurved beam. A narrow angular spectrum, rather than merely the label “Gaussian”, justifies the paraxial approximation.

No: the first moment generally depends on the transverse coordinate. At entry to the random medium, the initial data are deterministic, sowhich is a nonconstant Gaussian beam profile. Statistical homogeneity of the medium says that shifting both the medium and the incident data gives correspondingly shifted field statistics. It does not make the response to fixed, localized incident data translation invariant.

In particular, Gaussian coherent-field propagation in the Markov approximation, derived in part (d), givesfor the linear weak-index model. A homogeneous attenuation factor multiplies the varying beam profile. Thus spatially homogeneous medium statistics can coexist with a transversely inhomogeneous coherent field. The first moment is not the mean wave intensity, and disappearance of the coherent component is not itself disappearance of the total wave energy.

In time reversal acoustics, the array records the incoming signal, reverses each recorded time trace, and re-emits it through the same medium. This is phase conjugation in the frequency domain. For the time-harmonic wave convention $e^{-i\omega t}$, reversing a real time trace replaces its positive-frequency wave amplitude by its complex conjugate. The medium must remain unchanged between recording and re-emission.

Let $f(\mathbf{z})={\Psi }_0(0,\mathbf{z})$ and $G_L(\mathbf{\zeta },\mathbf{z})=G(L,\mathbf{\zeta };0,\mathbf{z})$, with the chosen source and array normalizations incorporated into the Green function. Let $a(\mathbf{\zeta })\ge 0$ be the array's aperture weight, equal to the indicator of its receiving region for an ideal uniform array. The recorded field isBy wave reciprocity, back-propagation has the same Green function with the source and receiver exchanged. Therefore the physically re-emitted, back-propagated wave amplitude isIf $Hf=R$ and $P_A$ multiplies by $a$, then ${\Psi }_B=H^T{P}_A\overline{Hf}$, where $T$ denotes the transpose without conjugation. Taking a final complex conjugate instead defines the adjoint reconstruction $\overline{{\Psi }_B}=H^{\ast }{P}_{A}Hf$. This distinction prevents an erroneous conjugation in the time reversal operator.

For a localized Gaussian beam or acoustic point source in a homogeneous medium, a finite aperture admits a limited range of angles. The focal width is of order $\lambda L/A$ when $A$ denotes the aperture diameter. In a random medium, multiple scattering creates paths with a larger angular spread. Each reversed path retraces its route, and the paths interfere constructively at the source. This can produce a larger effective aperture in time reversal and a narrower focus, even though the unreversed field has a complicated speckle pattern.

This comparison concerns a homogeneous reference medium; a deterministic heterogeneous medium can also provide useful multipath propagation. Suitable scale limits or frequency and spatial averaging can make refocusing self-averaging. Such self-averaging is not automatic for every monochromatic source and every random realization. Wave absorption, changing medium parameters, unresolved paths or poor array coverage can spoil refocusing. With complete capture of the propagating modes and a lossless unitary operator $H$, ideal adjoint reconstruction is already exact in either medium. Random scattering can improve finite-aperture wave focusing through angular diversity. See the regime-dependent analysis in Statistical stability in time reversal.

Because the sphere lies below the optical axis, a ray passing through its upper half samples a brighter part of the Gaussian beam than a comparable ray through its lower half. For a particle with higher refractive index than its surroundings, the stronger upper ray is deflected downward overall and transfers an upward reaction to the particle. The weaker lower ray gives a smaller downward reaction. Their transverse sum is therefore upward, toward the brighter axis.

If the displacement reverses, the force reverses, so the central position is transversely stable. Summing all the rays gives an optical gradient force toward the intensity maximum. There is also a forward optical scattering force. A tightly focused beam can have an axial restoring component that balances it, generally at a small displacement downstream of the ideal focus.^[ref]

Three-dimensional stability requires the restoring axial component to exceed the forward force over part of the trajectory. A Gaussian transverse intensity profile alone does not prove that condition; insufficient focusing can yield transverse confinement while the particle still moves downstream.

The Rayleigh range of a fundamental Gaussian beam in vacuum is the positive axial distance from the beam waist at which its radius grows from $w_0$ to $\sqrt{2}{w}_0$. Thus its contour area doubles and its peak irradiance halves. Within $|z|\le z_R$, the radius stays at most $\sqrt{2}{w}_0$; for $|z|\gg z_R$, beam divergence dominates. At fixed vacuum wavelength, the Rayleigh range scales quadratically with the beam waist radius. The formula here uses the vacuum wavelength, not an unlabelled wavelength in a material.

For a uniform plane-wave beam of cross section $A$, mean intensity is $P/A=c{B}_{\mathrm{rms}}^2/{\mu }_0$. Hence $B_{\mathrm{rms}}=\sqrt{{\mu }_{0}P/(cA)}$. A sinusoidal peak amplitude is $\sqrt{2}$ times the rms amplitude. A Gaussian beam requires a stated waist or diameter convention before inferring its on-axis field from total power.

Two identical concave mirrors of curvature radius R separated by L have half-round-trip ray matrix $A=\left(\begin{array}{cc}1& L\\ -2/R& 1-2L/R\end{array}\right)$. Its determinant is one and its trace is $2(1-L/R)$. The eigenvalues are a conjugate unit-modulus pair for the displayed interval. At the confocal point $L=R$, $A^2=-I$, so it is bounded despite the full-round-trip trace reaching minus two. Matching a Gaussian beam's mirror wavefront gives $z_R^2=(L/2)(R-L/2)$ and $\pi w_0^2=\lambda z_R$. The endpoints are degenerate limiting geometries, not finite confined Gaussian modes.