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The expression is a scalar diffraction theory integral for a monochromatic wave. The complex quantity ${\psi }_P$ represents a field complex amplitude, with physical field obtained by taking the real part after multiplication by $e^{-i\omega t}$. The light intensity is proportional to $|{\psi }_P{|}^2$. The coordinates $x,y$ label transverse positions in the aperture plane, $\Sigma$ is its transmitting region, and $h(x,y)$ is the incident field multiplied by the aperture's complex transmission. The wavenumber is $k=2\pi /\lambda$ in the propagation medium, and $R$ is the axial propagation distance if $h$ already includes the incident field.

To derive the Fresnel diffraction, start from Huygens-Fresnel principle contributions with propagation factor $e^{iks}/s$. With the coordinate origin at the projection of $P$ into the aperture plane,The paraxial approximation requires $\rho /R\ll 1$, where ${\rho }^2=x^2+y^2$. Replace the slowly varying $1/s$ and the obliquity factor by their axial values, but retain the quadratic variation of wave phase. Neglect of the next wave phase term requires $k{\rho }^4/(8R^3)\ll 1$ over the relevant aperture. Thus, apart from a common constant,The quadratic wave phase may be of order one or much larger; setting it to zero would require the stronger conditions for Fraunhofer diffraction. The scalar treatment also neglects the detailed vector field at aperture edges and assumes dimensions large enough compared with wavelength for this approximation to be useful.

For a finite-distance axial point source at distance $p$ and an observer at distance $q$, the incident field contributes its own quadratic wave phase. If $h$ instead denotes only the aperture transmission, the combined curvature is described by an effective distanceThese are two equivalent conventions for distributing the incident wave phase between $h$ and the exponential. Both distances must be included in the later calculation.

Let the source remain at $(0,0,-p)$ and let the observation point be $(X,Y,q)$. The paraxial approximation to the source-to-aperture-to-observer optical path length isCompleting the squares givesThe point $(x_0,y_0)$ is where the straight source-observer line intersects the aperture. Define the four dimensionless edge positionsOn the Cornu spiral $F(u)=C(u)+iS(u)$, draw the directed chord from $F(u_{x,-})$ to $F(u_{x,+})$, and a second directed chord from $F(u_{y,-})$ to $F(u_{y,+})$. Denote their complex differences by $D_x$ and $D_y$. The complex amplitude is proportional to their product: multiply their lengths and add their arguments, including the common propagation phase factor.

More explicitly, normalize by the unobscured field ${\psi }_0(X,Y)$ at this same observation point. The constant wave phase from completing the square is then already included in ${\psi }_0$, andEach directed chord is a one-dimensional Fresnel diffraction factor. One must multiply the two factors, rather than add the chords or use just one chord length for the square aperture. Setting $X=Y=0$ recovers $D_x=D_y=2F(w)$ and the on-axis answer.

Use Fresnel diffraction, since the detector moves along the axis through successive near-field interference extrema. Write $z$ for its distance from the opening and $\rho$ for the integration coordinate within the opening. For a uniform incident plane wave, the on-axis complex amplitude, in the paraxial approximation, is proportional toThus the light intensity relative to the incident value isThe Fresnel number $F=r^2/(\lambda z)$ is an odd integer at a maximum and an even integer at a minimum. It decreases as the detector moves away. Adjacent extrema therefore differ by one:ConsequentlyAs a check, $F=5$ at $z=1\mathrm{m}$ and $F=4$ at $z=1.25\mathrm{m}$, so these are indeed a maximum followed by the next minimum. Also $r/z\simeq 1.5\times 10^{-3}$, consistent with the paraxial approximation.

For the normally incident plane wave, the incident complex amplitude and phase are uniform across the aperture. Treat the field using scalar diffraction, with the aperture represented by its prescribed amplitude transmittance $h(x,y)$. Let $R$ be the distance from the aperture centre to the observation point and let ${\ell }_x,{\ell }_y$ be the transverse direction cosines of that point.

Across an aperture of characteristic size $a$, the Fraunhofer diffraction assumptions are that $a/R\ll 1$, so $1/r\simeq 1/R$ and the obliquity factor is nearly constant, and that the quadratic contribution to the propagation phase is negligible:The remaining path-length dependence is linear:A coherent monochromatic wave and an aperture large enough relative to wavelength for the scalar aperture model complete the usual physical assumptions. The complex amplitude becomeswhere $C$ includes the common propagation phase, distance factor and nearly constant obliquity factor. For the central part of a planar screen in the paraxial approximation, ${\ell }_x\simeq X/R$ and ${\ell }_y\simeq Y/R$. Small observation angles are useful for that screen-coordinate formula; a general far-field Fourier transform can instead use the exact direction cosines. A lens can also produce this pattern in its focal plane without requiring a physically very distant screen.

The far-field complex amplitude is the two-dimensional Fourier transform of the aperture amplitude transmittance, evaluated at transverse spatial frequencies $q_x=k{\ell }_x$, $q_y=k{\ell }_y$. The light intensity is proportional to the squared magnitude of this transform.

Use linearity to add the transforms of simpler aperture components, preserving their relative phases. A translated component acquires a complex exponential factor. Repeated identical openings can be expressed as the convolution of one opening with a sum of Dirac delta functions; the convolution theorem then factors the field into a single-opening transform and an array factor. Multiplying aperture functions instead convolves their transforms. These operations simplify complex apertures without repeatedly evaluating the full diffraction integral. It is the complex amplitudes, rather than their wave intensities, that are added for coherent illumination.

A slit of width $w$ has the single-slit diffraction complex amplitude factor $\sin \alpha /\alpha$, where $\alpha =\pi w\sin \theta /\lambda$. Thus for $w=25\mu \mathrm{m}=d/2$,The interference factor keeps the same zeros and the same nominal principal-order positions as in part (i), while the single-slit diffraction wave envelope suppresses off-axis wave intensity. Its zeros areAt nominal principal diffraction orders $\sin {\theta }_m=m\lambda /d$, the wave envelope factor is $[\sin (m\pi /2)/(m\pi /2){]}^2$. ConsequentlyAll nonzero even principal orders are missing. The central single-slit diffraction maximum extends between the envelope zeros near $X=\pm 20\mathrm{mm}$ and includes the central and the $m=\pm 1$ principal orders. Because there are only three slits, the maxima of the product can shift slightly from the nominal nonzero-order positions; the quoted values are the exact product at those positions. The lower panel of the figure shows the full product.

Both panels normalize to their own central wave intensity. An absolute comparison would require the unspecified widths and transmitted power of the initially “narrow” slits; the normalized pattern change is fully determined.

Take the quoted separation to mean adjacent slit centre spacing $d=50\mu \mathrm{m}$. For three equal very narrow slits at $-d,0,d$, the Fraunhofer diffraction complex amplitude in the transverse direction isWith $I_1$ denoting one narrow slit's contribution, the light intensity isThe quotient is interpreted by its limit where both numerator and denominator vanish. At small angles on a screen at $R=1\mathrm{m}$, $\sin \theta \simeq X/R$ and $\beta =\pi X/(10\mathrm{mm})$. Principal interference fringes occur atBetween successive principal maxima there are two zeros, at $X=(m+1/3)(10\mathrm{mm})$ and $X=(m+2/3)(10\mathrm{mm})$. Halfway between them is a smaller maximum $I=I_1$, one ninth of a principal peak. The pattern is symmetric about $X=0$. The positions quoted in millimetres use the paraxial approximation; exactly $d\sin {\theta }_m=m\lambda$ and $X_m=R\tan {\theta }_m$.

For $N$ equally spaced illuminated slits, with $N\simeq W/D$, summing a geometric series of complex amplitudes givesHere $I_1(\theta )$ is the one-slit single-slit diffraction wave envelope. The grating equation for principal maxima iswith height $N^2{I}_1({\theta }_m)$ and much weaker subsidiary maxima between orders. The first zeros around order $m$ are $\beta =m\pi \pm \pi /N$, so the angular distance from a maximum to its first minimum is, for large $N$,Changing wavelength by $\delta \lambda$ moves the same diffraction order byThe Rayleigh criterion places the maximum for one wavelength at the first zero for its neighbour. Equating these separations givesThis is the ideal resolving power of a diffraction grating, for an illuminated and transmitted order with sufficiently narrow source and instrumental broadening. The zeroth order has no wavelength separation and therefore no useful spectral resolving power. The distinction between $W=ND$ and the span $(N-1)D$ of the outer slit centres is a small edge convention; the exact state count in the resolving-power formula is $N$.

The displacement is along $x$, transverse to propagation in the membrane's $yz$ plane. Write $A=|A|e^{i\varphi }$; then $d=|A|\cos (\omega t-k_{y}y-k_{z}z+\varphi )$. Here $|A|$ is the displacement amplitude, $\varphi$ the initial phase, $\omega$ the angular frequency and $\mathbf{k}$ the wavevector normal to constant-phase lines. Its magnitude is $2\pi /\lambda$, and its direction specifies phase propagation for the chosen sign convention.

For equal-phase two-wave membrane modes, two equal complex amplitudes with opposite $k_y$ sum toThe edge condition gives $\cos (k_{y}a/2)=0$, hence $k_y=(2j+1)\pi /a$, $j=0,1,\dots$, for this even-in-$y$ family. Opposite relative phase gives the sine family; together they yield all Dirichlet transverse modes $k_y=n\pi /a$, $n\ge 1$. For the equal-phase family,The lowest mode has $k_y=\pi /a$. ThereforeIt approaches $c$ for $\lambda \to 0$ and diverges as $\lambda \to 2a^{-}$; there is no real travelling-mode $k_z$ beyond the cutoff. This is an axial phase velocity, not the group velocity: $v_g=c\sqrt{1-(\lambda /2a{)}^2}<c$. Here $c$ denotes the membrane wave speed, not the speed of light.

For the evanescent membrane response below cutoff, below $f_c=c/(2a)$, put $k_z=-i\kappa$ for decay along positive $z$, with $\kappa =\sqrt{(\pi /a{)}^2-(\omega /c{)}^2}$. A localized central drive creates an evanescent, sinusoidally oscillating deformation decaying away in both axial directions. Far enough from the drive the leading profile is proportional to $\cos (\pi y/a)e^{-\kappa |z-z_0|}\cos (\omega t+{\varphi }_0)$. Higher evanescent modes contribute near the point source. The pattern has no steadily advancing axial phase front or far-field propagating wave; decreasing frequency increases its spatial localization.

First intensity minimum. In Fraunhofer diffraction, the path difference between adjacent slits is $d\sin \theta$, so the relative wave phase is $\delta =2\pi d\sin \theta /{\lambda }_1$. Summing the $N$ equal complex amplitudes as a geometric series gives the diffraction grating interference factorThe central maximum has $\delta =0$. Its nearest zero occurs at $N\delta /2=\pi$, where the denominator is nonzero. ThusThere is a symmetric first zero on each side of the incident direction. The narrow slits are treated as having a sufficiently broad single-slit diffraction envelope.

Wavelength resolution and the missing order specification. Adopt the Rayleigh criterion: the principal maximum of one wavelength lies at the adjacent minimum of the other in the same nonzero diffraction order $m$. For ${\lambda }_2>{\lambda }_1$ and $m>0$, those directions obeyConsequently the ideal resolving power of a diffraction grating is ${\lambda }_1/\Delta \lambda =mN$, givingThe usual first-order reading therefore gives $64\mathrm{nm}$. Zero diffraction order cannot separate the two wavelengths, because both central maxima coincide.

The paper does not specify which diffraction order is observed, so $64\mathrm{nm}$ is not an order-independent minimum. In the ideal infinitely narrow-slit model, the largest usable order for this positive wavelength separation must satisfy $(m+1/N){\lambda }_1\le d$. Since $d/{\lambda }_1=125$, this permits $m\le 124$, giving the formal all-order resultThat pair meets the criterion at $\theta \simeq 83.1^{\circ }$. Order $125$ of ${\lambda }_1$ is at grazing incidence and the corresponding longer-wavelength peak cannot propagate. In a real device, finite slit width, weak high-order intensity and screen acceptance may prevent such an extreme order from being used; those data are absent. Thus the specified-order expression, and $64\mathrm{nm}$ when first order is intended, are the defensible resolution answers under the stated criterion.

The probability density is proportional to $|a+ix{|}^2=a^2+x^2$, not to the complex amplitude itself. For equal small intervals approached from inside the support,

In the Lorentz oscillator model, each participating bound electron is displaced by an applied electric field against a restoring force. The model replaces complicated interband transitions by a driven damped harmonic oscillator. If $u$ is the electron displacement along the field and the electron charge is $-e$, its equation of motion isHere ${\omega }_0$ describes binding and $\gamma >0$ describes loss of coherent motion. For $E(t)=\mathrm{Re}(E_0{e}^{-i\omega t})$, the complex amplitude isThe induced electric dipole moment is $-eu$, so the electric polarization amplitude for number density $n$ isUsing $D={ϵ}_{0}E+P={ϵ}_0{ϵ}_{r}E$ givesThe paper's $ϵ$ is this dimensionless relative permittivity, rather than the absolute permittivity ${ϵ}_0{ϵ}_r$. Independent classes of bound electrons contribute additive susceptibilities, usually with different resonance frequencies and optical oscillator strengths.

Writing $D_{\omega }=({\omega }_0^2-{\omega }^2{)}^2+{\gamma }^2{\omega }^2$ separates the response intoAt zero frequency the real part is $1+{\omega }_p^2/{\omega }_0^2$ and the imaginary part vanishes. Near resonance the imaginary part has a positive photon absorption peak and the real part changes rapidly, passing through $1$ at ${\omega }_0$. Above resonance the real part can become negative when damping is sufficiently weak; at very high frequency it approaches $1$ from below while the imaginary part tends to zero. Finite damping removes the undamped divergence.

For a nonmagnetic material take the passive square-root branch $N=\eta +i\kappa =\sqrt{{ϵ}_r}$, with $\kappa \ge 0$. A plane wave then has an intensity attenuation coefficient $\alpha =2\omega \kappa /c$. Thus the imaginary permittivity expresses dissipation, whereas the actual photon absorption coefficient is determined by the imaginary refractive index. The normal-incidence power reflectance isA large refractive-index mismatch produces wave reflection even without loss. In the undamped model ${ϵ}_r<0$ between ${\omega }_0$ and $\sqrt{{\omega }_0^2+{\omega }_p^2}$, so the field is evanescent and wave reflection is total in a semi-infinite sample. With damping the response includes both wave reflection and photon absorption; a peak in the imaginary permittivity should not simply be identified with a reflectance peak.

To infer the short-wavelength background from $R=0.35$, assume weak photon absorption and take the physical semiconductor branch $\eta >1$. ThensoThis is an estimate of the valence-electron background in a frequency window above the doped-carrier response and below the effective bound-electron resonance. One reflectance value alone does not determine both parts of a complex dielectric constant. Even for real ${ϵ}_r$, a reciprocal-index branch gives the same $R$; the ordinary background index greater than unity selects the branch used here.

Using the supplied conventional cubic lattice parameter, the participating valence-electron number density isUsing $m=m_e$ for the stipulated free-electron comparison givesAn angular frequency “in eV” means its quantum energy $\hslash \omega$, not $h\omega$. No doped-carrier effective mass should be inserted into this valence-electron count.

The low-frequency limit of the bound-electron contribution obeysConsequentlyThis effective resonance is much larger than the quoted $0.2\mathrm{eV}$ band gap. There is no contradiction: the single oscillator has been assigned the strength of all sixteen participating valence electrons per conventional cell, whereas the band gap is only the lowest interband excitation threshold. A real spectrum distributes optical oscillator strength over many transitions. Equating the oscillator resonance to the lowest threshold would misrepresent the assumed background response. Here “low frequency” means low relative to ${\omega }_0$; the static response of the doped conducting sample also contains its free carriers.

The additional mobile carriers in doped indium antimonide contribute a Drude frequency response, obtained by setting their restoring frequency to zero. At infrared frequencies well below the effective valence resonance, the sum is approximatelyAt the shorter wavelengths the carrier term is small and $R$ approaches the background value. As wavelength increases, the negative free-carrier contribution lowers the real permittivity towards unity. For weak damping, refractive-index matching to vacuum gives the Drude-Lorentz reflectivity minimum atThis minimum is near ${ϵ}_r^{\prime }=1$, not its zero. At the slightly lower screened plasma frequency ${\omega }_{pc}/\sqrt{{ϵ}_b}$, the real permittivity crosses zero; at longer wavelengths it becomes negative and the material becomes strongly reflective. Finite damping prevents a perfectly vanishing minimum and rounds the edge. At still lower frequency the large conducting response also drives $R$ towards unity.

For fixed carrier effective mass and background permittivity, ${\omega }_{pc}\propto \sqrt{n_c}$ and ${\lambda }_{\min }\propto n_c^{-1/2}$. The observed shift of the minimum towards shorter wavelength as doping increases is therefore the expected free-carrier trend. The figure is an original model illustration, not a digitization or quantitative fit of the supplied curves; the carrier effective mass, damping and their density dependence have not been specified.

Choose the $e^{-i\omega t}$ convention. The steady displacement in the Lorentz oscillator model has complex amplitudeThe induced electric dipole moment is $p_{\omega }=q{u}_{\omega }$, so the microscopic electric polarizability is $\alpha (\omega )=q^2/[m({\omega }_T^2-{\omega }^2-i\gamma \omega )]$. For $N$ identical oscillators per unit volume, $P_{\omega }=N{p}_{\omega }={ϵ}_0{\chi }_{\omega }{E}_{\omega }$. ConsequentlyHere the dimensionless quantity called polarizability in the paper is the electric susceptibility; the single-electron polarizability $\alpha$ has different units. The sign of the electron charge disappears because both the driving force and dipole moment contain $q$.

Multiplying by the complex conjugate of the denominator givesFor positive damping and positive frequency, ${ϵ}_r^{\prime \prime }\ge 0$, as required for passive wave absorption in this Fourier convention. At zero frequency, ${ϵ}_r^{\prime }=1+{\omega }_p^2/{\omega }_T^2$ and ${ϵ}_r^{\prime \prime }=0$. Near a weakly damped resonance the imaginary part has a narrow positive peak, while the real part rises on the low-frequency side and then falls steeply through ${ϵ}_r^{\prime }=1$ at ${\omega }_T$. It can become negative above resonance for sufficiently strong oscillator strength and weak damping. At large frequency, ${ϵ}_r^{\prime }=1-{\omega }_p^2/{\omega }^2+O({\omega }^{-4})$ and ${ϵ}_r^{\prime \prime }\sim {\omega }_p^2\gamma /{\omega }^3$: both approach the vacuum values. The absorption maximum is close to, but need not be exactly at, ${\omega }_T$ for finite damping.

A phase-conjugate mirror is an optical device that implements phase conjugation at its return surface. It may generate the return using an independently supplied beam, as in digital optical phase conjugation; its name specifies the field transformation, rather than a particular reflecting material. Conjugating the transverse complex amplitude and launching the returning longitudinal carrier produces the complete reversed wavevector. A spatially constant $\gamma$ leaves the shape of the wavefront unchanged; a spatially varying $\gamma$ would add its own wavefront error.

An ordinary flat mirror obeys the law of reflection. At a surface $z=L$, its reflected plane wave has wavevector $(k_x,k_y,-k_z)$, whereas a phase-conjugate mirror returns $(-k_x,-k_y,-k_z)$. Ordinary specular reflection preserves the tangential wavevector. At normal incidence a single plane wave can therefore look identical in the two cases, apart from an overall wave phase and strength; oblique incidence or a curved wavefront reveals the distinction. In boundary-envelope notation an ideal ordinary mirror multiplies the incident complex amplitude by a constant reflection factor $r$, while an ideal phase-conjugate mirror returns its complex conjugate times $\gamma$.

Phase conjugation creates a returning monochromatic wave whose spatial complex amplitude is proportional to the complex conjugate of the incident field. Fix the time convention throughout:Here $E$ is one scalar component of the electric field, $A\ge 0$ is the wave amplitude, $\omega >0$ is the angular frequency, and $\gamma$ is a spatially constant complex number setting the return strength and reference wave phase. The complex conjugation acts on $U$, including its spatial carrier, while the time factor remains $e^{-i\omega t}$. Conjugating both factors would merely give another representation of the same real field. With $\gamma =1$, the new real field equals $E(\mathbf{r},-t)$ in this ideal monochromatic wave model; that identity does not assert reversal of an arbitrary pulse or of the apparatus producing it.

The spatial wave phase becomes $-\varphi +\arg \gamma$, so the local wavevector changes from $\nabla \varphi$ to $-\nabla \varphi$. For example, $U=A{e}^{i(k_{x}x+k_{z}z)}$ with $k_z>0$ becomes $\gamma A{e}^{-i(k_{x}x+k_{z}z)}$: both components of the wavevector reverse. In a transparent stationary medium with isotropy and positive refractive index, light rays follow the wavevector, giving counterpropagation. The same wavefronts now have opposite propagation normals. This is why phase conjugation can return a distorted beam toward its origin: it reverses the detailed spatial wave phase, rather than requiring that wave phase to be flat.

The attractive application is correction of optical aberrations accumulated on an outward trip. The return uses the distortion itself as the prescription for correction, as the original thin phase-plate cancellation by phase conjugation example makes explicit. An experimentally accessible implementation measures the spatial wave phase and uses a spatial light modulator to generate its negative; digital optical phase conjugation describes an inspected 2010 demonstration of recovery through multiple scattering.

Phase-contrast microscopy converts specimen-dependent phase differences into visible light intensity differences. It can reveal structures in living, unstained biological cells, allowing their changes to be followed without first making an absorbing preparation.^[ref] The key question is how an optical microscope can read a phase when an ordinary detector records light intensity.

Consider a transparent specimen with complex amplitude transmission $t(x,y)=e^{i\varphi (x,y)}$. In ideal in-focus coherent imaging with all relevant spatial frequencies transmitted, its image has $I=|t{|}^2=1$: varying $\varphi$ alone produces no brightness detail. This is a particular model, not a claim that all transparent specimens are invisible. Diffraction, defocus, absorption and a finite optical pupil can already change a real image.

A phase-contrast microscope images an illuminating annulus onto a phase plate ring near the objective's rear focal plane. The aligned ring attenuates and shifts direct light relative to most diffracted light; their interference changes image brightness.^[ref]

The method is beautiful because the bright direct beam becomes its own reference: a small specimen-dependent complex amplitude can interfere with it and change the image at first order. Merely magnifying an invisible phase pattern would leave $|t{|}^2$ unchanged.

Reference wave phases label complex amplitudes without changing measured powers. If $\mathbf{b}=S\mathbf{a}$, choose diagonal matrices $D_{\mathrm{in}},D_{\mathrm{out}}$ whose entries have modulus one, and set ${\mathbf{a}}^{\prime }=D_{\mathrm{in}}\mathbf{a}$, ${\mathbf{b}}^{\prime }=D_{\mathrm{out}}\mathbf{b}$. Then $S^{\prime }=D_{\mathrm{out}}S{D}_{\mathrm{in}}^{-1}$ and each component has unchanged squared modulus. For a beam splitter scattering matrix, transforming $S$ without transforming input complex amplitudes changes the stated physical input.

A phase factor is a unit-modulus complex number $e^{i\varphi }$. Multiplying a complex amplitude by it changes its wave phase by $\varphi$ while preserving its squared modulus. A common phase factor does not change light intensity, while different phase factors of contributions can change their interference.

Phase matching makes wave amplitudes emitted at different depths add coherently. Consider collinear plane waves in a uniform transparent material of length $L$, with fixed polarization of waves and nonzero selected nonlinear optical susceptibility. Assume negligible pump depletion, negligible absorption, an initially absent harmonic, and a wide beam for which diffraction is negligible. The pump's wave amplitude is then constant throughout the interaction.

The pump electric field has spatial wave phase $k_{\omega }z$; its quadratic source has spatial wave phase $2k_{\omega }z$. The freely propagating harmonic has wave number $k_{2\omega }$. Define the second-harmonic phase mismatch byFor the chosen complex-field convention, the outgoing harmonic wave amplitude, up to an overall constant and propagation phase, is proportional toChoosing the conjugate complex-field convention reverses the exponent's sign without changing the wave intensity. Taking the squared complex modulus givesHere $C$ is fixed for the chosen modes and material; its dimensions accommodate the stated wave intensity convention. The quadratic pump dependence follows because the source wave amplitude is proportional to $E_0^2$, whereas $I_{\omega }$ is proportional to $E_0^2$. The sinc function comes from summing contributions along the whole material, just as a finite aperture produces diffraction through addition of complex amplitudes.

At $\Delta k=0$, the integral is $L$ and the harmonic wave intensity grows as $L^2$. At $\Delta kL=2\pi$, all those contributions cancel. This cancellation needs no absorption: their wave phases wind through a full turn. At high conversion, the constant-pump assumption fails, so the $L^2$ law cannot continue indefinitely. Tight focusing, finite optical pulses, absorption, and transverse separation of modes also require a more complete model.

Optical dispersion generally makes $n_{2\omega }\ne n_{\omega }$. In a suitable crystal, birefringence allows modes with different polarization of waves to experience different refractive indices, making their selected values equal. The appropriate crystal direction and allowed nonlinear optical susceptibility component must both work.^[ref]

For a worked calculation, choose $L_0=1\mathrm{mm}$ and $\Delta k=\pi \mathrm{m}{\mathrm{m}}^{-1}$. At $L=L_0$, the harmonic wave intensity is $4/{\pi }^2\simeq 0.4053$ of its perfectly matched value. At $L=2L_0$ it vanishes in this model. More material can therefore reduce the output. If instead $\Delta k=0$, doubling $L$ multiplies output by four. The two curves retain that common normalization, so their different peak heights are meaningful.

For the overlapping monochromatic wave components, let $d$ be the thickness and ${\lambda }_0$ the vacuum wavelength. Define $\Delta n=n_s-n_f>0$ using the slow and fast refractive indices. With complex amplitude convention $e^{i(kz-\omega t)}$, the accumulated slow-minus-fast phase difference isThis is the optical path difference $\Delta nd$ expressed as a wave phase. It is a signed accumulation convention; with the opposite time convention the sign reverses. The outgoing polarization of an electromagnetic wave depends on $\delta$ modulo $2\pi$. Here the positive slow-minus-fast $\Delta n$ differs from the signed principal-index $n_e-n_o$ used in the parent birefringence entry when $n_e<n_o$.

For an original worked example, stipulate $\Delta n=0.0100$ at ${\lambda }_0=600\mathrm{nm}$ and a uniform thickness $d=15.0\mu \mathrm{m}$. These are design assumptions, not measured material data. The optical path difference is $150\mathrm{nm}$, soIt is a quarter-wave plate. At the same wavelength and refractive indices, doubling $d$ to $30.0\mu \mathrm{m}$ gives $\delta =\pi$, a half-wave plate. A $1.0\mu \mathrm{m}$ thickness increase changes $\delta$ by $\pi /30$, or $6^{\circ }$; retaining the same birefringence at $500\mathrm{nm}$ would instead give $\delta =3\pi /5$ for the $15.0\mu \mathrm{m}$ plate. Real refractive indices also vary with wavelength, so this last calculation explicitly neglects optical dispersion.

For these equal transmitted wave amplitudes, orthogonal projection onto a polarizer crossed with the initial $45^{\circ }$ linear polarization gives a simple independent readout. After removing a common wave phase, the component complex amplitudes are proportional to $(1,e^{i\delta })/\sqrt{2}$; orthogonal projection onto $(1,-1)/\sqrt{2}$ gives $(1-e^{i\delta })/2$. Squaring its modulus yields $I_{\perp }/I_{\mathrm{in}}={\sin }^2(\delta /2)$: zero at $d=0$, one half at $15\mu \mathrm{m}$, and one at $30\mu \mathrm{m}$. This assumes ideal polarizers, negligible specular reflection and wave absorption, overlapping components and stable relative wave phase. It measures a periodic function of phase retardation, so one brightness measurement cannot determine the thickness uniquely.