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The angular spectrum representation expresses a monochromatic wave as a superposition of plane waves. In a homogeneous, lossless, source-free half-space with real wavenumber $k=2\pi /\lambda >0$, transverse Fourier transform components propagate independently, providing a useful solution of the Helmholtz equation.^[ref]

Use angular frequency $\omega$ and time dependence $e^{-i\omega t}$, and define $\tilde{U}(k_x,k_y;0)=\int \int U(x,y,0)e^{-i(k_{x}x+k_{y}y)}dxdy$. Then
The outgoing branch of the angular spectrum has nonnegative real and imaginary parts of $k_z$ for propagation toward $z>0$. A propagating angular-spectrum component has real $k_z$; an evanescent angular-spectrum component has $k_z=i\kappa$ and decays as $e^{-\kappa z}$. The source-plane amplitudes and their complex arguments matter, not just their squared magnitudes.^[ref]

For an original example, a transverse period $\lambda /2$ gives $k_x=2k$, $k_y=0$ and $\kappa =\sqrt{3}k$. After distance $z=\lambda /4$, its amplitude is multiplied by $e^{-\pi \sqrt{3}/2}\simeq 0.0658$. Such fine spatial detail does not propagate with unchanged amplitude to the far field.

The paraxial angular-spectrum propagator uses $k_z\simeq k-(k_x^2+k_y^2)/(2k)$. For the source-plane Gaussian beam $U_0=e^{-(x^2+y^2)/w_0^2}$, its transverse spectrum is proportional to $e^{-w_0^2(k_x^2+k_y^2)/4}$. The on-axis Gaussian integral therefore gives
The denominator produces the axial Gouy phase $-\arctan (z/z_R)$. This connects the phase shift with interference of transverse spectral components, rather than an abrupt phase jump at the focus.^[ref]

The scalar propagation formula does not authorize arbitrary independent vector components. A physical electromagnetic field must satisfy the transverse polarization constraint in an angular spectrum; in a uniform source-free medium,$\mathbf{k}\cdot \tilde{\mathbf{E}}=0$. Boundary conditions and source consistency accompany a vector reconstruction.

An optical physicist who coauthored an interactive teaching text on wave optics. Its polarization and coherence chapters relate field components and correlations to measurable intensity, helping explain why combining beams does not always produce stable interference fringes.^[ref]

An evanescent angular-spectrum component has transverse spatial frequency greater than the medium wavenumber. In a homogeneous lossless medium, with $q=\sqrt{k_x^2+k_y^2}>k$, its forward longitudinal wavenumber is imaginary:It oscillates transversely while decaying away from the source plane. The bounded sign is fixed by the outgoing branch of the angular spectrum.^[ref]

For an original transverse period $d=\lambda /2$, take $k_x=2k$, $k_y=0$. Then $\kappa =\sqrt{3}k$ and the $1/e$ amplitude-decay distance is $1/\kappa =\lambda /(2\pi \sqrt{3})\simeq 0.09189\lambda$. At $z=\lambda /4$, the amplitude is about 0.0658 of its source value, while its squared magnitude is about 0.00433. Amplitude and squared-amplitude attenuation must not be interchanged.

This explains a basic limitation of far-field reconstruction: fine transverse structure can be present at a boundary yet be strongly suppressed before reaching a distant detector. Inverting the factor $e^{-\kappa z}$ also amplifies errors. A tiny measured high-frequency component may therefore be an unstable basis for reconstructing fine source detail.

The electromagnetic field is still constrained by Maxwell equations. In particular, $\mathbf{k}\cdot \tilde{\mathbf{E}}=0$ remains valid for complex $\mathbf{k}$. A single ideal evanescent wave in a lossless medium does not carry time-averaged energy flux normally away from its interface in the same way as a propagating wave. Near-field coupling or a superposition involving another boundary can nevertheless transfer energy, so decay is not a universal prohibition on energy exchange.

An illustrative experiment compares the same patterned source with detectors at increasing distances and accounts for the detector's own spatial response. The decay predicted here concerns a resolved spectral component under the stated homogeneous-medium model, not a guarantee that every observed loss of contrast is evanescence. Absorption, finite aperture and instrument resolution can also suppress structure.

The evanescent field in total internal reflection oscillates and advances in wave phase along the boundary while its wave amplitude decays away from it. With the coordinates above and time dependence $e^{-i\omega t}$, a TE polarization has complex amplitudeThe physical electric field is $\mathrm{Re}[\mathbf{E}(x,z)e^{-i\omega t}]$. The decaying sign excludes a field that grows without bound at $z\to \infty$. Dielectric-interface boundary conditions determine $A$; for nongrazing incidence it is generally nonzero.^[ref]

Zero normal energy flux follows directly from Maxwell equations. The wavevector is $(q,0,i\kappa )$; the relation $\mathbf{H}=(\mathbf{k}\times \mathbf{E})/(\omega {\mu }_0)$ givesConsequently, the cycle-averaged Poynting vector isThe electric field and $H_x$ are a quarter-cycle apart, so their product has zero cycle average. Tangential energy flux survives. This is a local calculation for the single decaying evanescent wave in the infinite second half-space.

To check reflected power independently, put $a=n_1{k}_0\cos \theta >0$. The existing transverse-electric Fresnel reflection coefficient becomes $r=(a-i\kappa )/(a+i\kappa )$. Its numerator and denominator are complex conjugates, so $|r{|}^2=1$, while $1+r=2a/(a+i\kappa )\ne 0$ supplies the surface electric field relative to the incident wave amplitude. Thus a unit optical reflectance does not require a zero transmitted field. The same normal power balance holds for TM polarization; its reflected wave phase generally differs from the TE polarization phase. A mixture can therefore change its polarization of an electromagnetic wave on reflection even though all normal incident power returns.

An optical physicist who coauthored a teaching text connecting electromagnetic wave descriptions with optical measurement. Its accounts of polarization and interference make the coherence assumptions behind an intensity prediction explicit, an essential step when applying ideal wave models to real sources.^[ref]

Chapter of Interactive Optics by Sander Konijnenberg, Aurèle J.L. Adam and H. Paul Urbach. Whole-work author credits apply; no separate chapter author byline is established. No chapter publication or revision date is established; the URL path does not establish a date or edition.

Online text: interactivetextbooks.tudelft.nl/interactive-optics/content/Chap6_Diffraction/DiffractiveOptics_2022_01Clean.html

Cited passages: Scalar Diffraction Optics: Fresnel Approximation, equations (381)–(385), following paraxial equation remark; Fresnel and Fraunhofer Approximations, equations (386)–(392): omitted aperture quadratic phase, full-diameter Fresnel number; About the authors: biographies and final whole-book author credit.

Chapter of Interactive Optics by Sander Konijnenberg, Aurèle J.L. Adam and H. Paul Urbach. The whole-work authors apply; no separate chapter byline or publication date is assigned. Locators: Coherence of Light Sources, equations (299)–(302); More on Spatial coherence.

The outgoing branch of the angular spectrum selects the longitudinal wavenumber that represents forward radiation and bounded evanescent decay in a specified source-free half-space. For real $k>0$, time dependence $e^{-i\omega t}$ and propagation into $z>0$,The first root is positive real and the second positive imaginary. They accompany the factor $e^{i{k}_{z}z}$.^[ref]

The scalar Helmholtz equation permits both signs. For $q<k$, $e^{i{k}_{z}z}$ and $e^{-i{k}_{z}z}$ describe opposite longitudinal phase propagation under the stated time convention. For $q>k$, the corresponding factors are decaying and growing exponentials. A source-free upper half-space with no incident field from infinity uses the forward, bounded choice.

For an original transverse spatial frequency $q=1.25k$, the decaying root is $k_z=0.75ik$. At $z=\lambda$, amplitude changes by $e^{-1.5\pi }\simeq 0.00898$. Choosing the other sign would instead multiply it by approximately 111.3. A sign error can therefore become an enormous near-field propagation error without changing the input data.

This choice is a statement about the physical region and boundary conditions, not an instruction to delete reflected waves from every problem. An interface or cavity can require both propagation directions in a finite region. Likewise, propagation towards negative $z$ must use a convention consistent with that region. At $q=k$, the root is zero and the component is grazing; formulas that divide by $k_z$ need separate attention.

Backward reconstruction of a decayed evanescent angular-spectrum component multiplies measurement error by an increasing exponential. That is an inverse problem's conditioning issue, rather than evidence that the forward outgoing branch should grow. Stating the time convention, physical half-space and radiation condition prevents the square-root symbol from hiding these choices.

The paraxial angular-spectrum propagator is the quadratic-phase approximation to forward free propagation of transverse Fourier transform components concentrated near the optical axis. For real medium wavenumber $k>0$, write $q^2=k_x^2+k_y^2$. With time convention $e^{-i\omega t}$, the exact factor is $e^{iz\sqrt{k^2-q^2}}$. Expanding at $q/k\ll 1$ givesafter removing the common carrier $e^{ikz}$.^[ref]

The approximation preserves the magnitude of each retained spectral component and changes its wave phase. Recombining those components produces diffraction; it need not preserve the shape or intensity at an individual spatial point. In coordinate space the envelope satisfies the parabolic wave equation, $2ik{\partial }_{z}U+{\nabla }_{\perp }^{2}U=0$.

Small propagation angle and acceptable accumulated phase error are separate conditions. The first omitted term produces a phase discrepancy of order $z{q}^4/(8k^3)$. For an original component with $q/k=0.1$, the exact difference between approximate and exact phases isIt is about 0.00789 radians at $z=100\lambda$, but about 0.789 radians at $z=10,000\lambda$. The direction is still only about $5.74{}^{\circ }$ off axis. Propagation distance matters when phase accuracy is the observable.

For a source-plane Gaussian beam $U_0=\exp [-r^2/w_0^2]$, the spectral width is of order $1/w_0$. Thus $k{w}_0\gg 1$ concentrates the important components in the paraxial region. That provides a reason for the familiar wide-waist condition rather than treating all Gaussian formulas as exact electromagnetic solutions.

A calculation containing substantial evanescent angular-spectrum components or large-angle components requires the exact branch and, when appropriate, the vector transverse polarization constraint in an angular spectrum. The paraxial transfer factor has unit magnitude even at large $q$, so extending it blindly into the evanescent region replaces physical decay by an artificial phase rotation.

Fresnel diffraction is interference during propagation when curvature of the optical path across an aperture matters. It provides a continuous account of fringes near geometric edges and their evolution into the far-field pattern. A screen that is many aperture widths away is not automatically in the Fraunhofer diffraction limit; wavelength also enters the comparison.

For plane-wave illumination of a transmission $h(\xi ,\eta )$ at $z=0$, the Huygens-Fresnel principle in the scalar paraxial approximation givesIt follows by expanding the propagation distance as $z+[(X-\xi {)}^2+(Y-\eta {)}^2]/(2z)$ while treating the amplitude prefactor as slowly varying. Small propagation angles justify dropping higher path terms. Finite coherent illumination, vector edge fields and very small apertures can require a more complete theory. The Fresnel number $N_F=a^2/(\lambda z)$, with a clearly declared aperture scale $a$, measures whether the aperture's quadratic phase is significant. When it is negligible, the remaining linear phase produces the aperture Fourier transform.^[ref]

Define the optical Fresnel integralsTheir limiting values are $C(\infty )=S(\infty )=1/2$, with odd continuation to negative arguments. The Cornu spiral plots $(C(u),S(u))$: the field contributed by a finite interval is proportional to the chord between its endpoints. Intensities require squared chord lengths; summing zone intensities would lose the cancellation mechanism.

For an opaque straight edge transmitting $\xi >0$ and all $\eta$, let $v=X\sqrt{2/(\lambda z)}$, so $X>0$ is the geometrically illuminated side. Integrating in the unrestricted transverse direction and using the half-line in the other giveshenceFar into the shadow, $I\to 0$; far into the illuminated region, $I\to I_{\mathrm{in}}$. At the geometric shadow boundary, $I(0)=I_{\mathrm{in}}/4$, not half the incident intensity: the field is halved there. On the illuminated side, alternating constructive and destructive contributions give bright and dark fringes whose contrast declines away from the edge. Their positions scale with $\sqrt{\lambda z}$ and are not equally spaced. A finite detector averages fine fringes and tends to recover the geometric-optics step. In a controlled experiment, translate a straight edge through a broad collimated beam, scan a small photodetector across the screen and normalize against an unobstructed reference; vary $z$ to test the square-root scaling. Source size, spectral bandwidth, edge roughness and detector aperture wash out contrast and must be distinguished from a failure of the propagation law.

For a rectangular aperture of full widths $a,b$, defineThe two transverse integrals separate:andA long slit leaves only one varying Fresnel factor. For a square centred on axis, $a=b$ and $w=(a/2)\sqrt{2/(\lambda z)}$ giveOpposite edge contributions overlap; changing aperture size or propagation distance changes their relative phase. Consequently a geometrically illuminated centre can brighten or darken, rather than remaining a fixed top-hat plateau. The original rectangular-aperture map shows fringes parallel to both edges and their crossing structure. Outside the geometric rectangle the wave also extends into the shadow.

At large Fresnel number an interior region approaches geometric illumination while edge fringes remain on a scale $\sqrt{\lambda z}$. At intermediate values the opposite-edge fringes interact across the whole aperture image. In the far field, the formula tends to the product of sinc-squared factors,The far-field lobe scale $\lambda z/a$ grows linearly with $z$, unlike the local near-edge scale. The transition is controlled by the neglected phase, not an abrupt physical boundary.

To test the main effect, record normalized profiles at several distances, calculate the Fresnel integral with the measured aperture dimensions and compare fringe positions before fitting a free intensity scale. Use the same transverse coordinate origin and check that the incident beam is sufficiently uniform. Babinet's principle relates complementary aperture and obstruction fields through $U_a+U_o=U_{\mathrm{free}}$; their near-field intensities are not generally equal, because the unobstructed field remains present and interferes with the diffracted field. This is why a far-field obstruction argument cannot simply be carried over as equality of Fresnel intensities.

Chapter of Interactive Optics by Sander Konijnenberg, Aurèle J.L. Adam and H. Paul Urbach. Whole-work authors apply. Locators: creating polarization states and wave plates; equation (273) for birefringent relative phase. No separate chapter date is established.

A propagating angular-spectrum component is a transverse Fourier transform component whose longitudinal wavenumber is real in a homogeneous lossless medium. With $q^2=k_x^2+k_y^2<k^2$ and the forward outgoing branch of the angular spectrum,Its propagation factor $e^{i{k}_{z}z}$ has unit magnitude and a distance-dependent wave phase.^[ref]

For an original component with $k_x=0.6k$ and $k_y=0$, $k_z=0.8k$ and $\theta \simeq 36.87{}^{\circ }$. It repeats its phase along $z$ over distance $2\pi /k_z=1.25\lambda$, even though its wavelength measured along its propagation direction remains $\lambda$. The axial phase period is not an energy-transit time.

Preserving each component's magnitude does not preserve the reconstructed spatial intensity. Different values of $k_z$ accumulate different phases; their coherent superposition changes the interference pattern and can spread a beam. This is the angular-spectrum description of diffraction.

Energy crossing a reference plane also depends on direction. For a vacuum plane electromagnetic wave with electric-field magnitude $E_0$, the time-averaged Poynting vector has magnitude $|E_0{|}^2/(2Z_0)$ along the propagation direction. Its normal component through a plane of constant $z$ is that magnitude times $\cos \theta$, where $Z_0$ is the vacuum impedance. Equal electric-field magnitudes at angles zero and $60{}^{\circ }$ therefore give normal fluxes in the ratio $1:1/2$.

The boundary case $q=k$ is grazing, with $k_z=0$ and no forward normal phase advance. For $q>k$, the root becomes imaginary and the component is instead an evanescent angular-spectrum component. These classifications apply to the stated real-wavenumber, lossless model; material absorption requires a consistent complex-wavevector treatment.

A physicist who coauthored Interactive Optics by Sander Konijnenberg, Aurèle J.L. Adam and H. Paul Urbach. Its treatment connects wave polarization, interference and coherence to optical experiments. Writing those topics in a common wave framework helps readers move from a field equation to a prediction for the intensity measured by an instrument.^[ref]

Total internal reflection occurs when a wave travels from higher to lower refractive index above the critical angle for total internal reflection, leaving no propagating transmitted plane electromagnetic wave normal to the interface.

At an infinite planar interface, total internal reflection returns all incident normal power in the reflected plane electromagnetic wave. Assume homogeneous dielectrics with isotropy, linearity, no dielectric loss, magnetic permeability ${\mu }_0$, and positive real refractive indices $n_1>n_2$. Above the critical angle for total internal reflection, both TE polarization and TM polarization have unit optical reflectance despite the external evanescent wave.^[ref]

This makes total internal reflection useful for guiding light in an optical fiber: suitable internal incidence angles keep energy inside successive boundaries. Its more surprising feature is that perfect optical reflectance coexists with an external electric field. The evanescent field in total internal reflection can interact with a nearby object, as the frustrated total internal reflection experiment demonstrates.

For an original idealized example, choose $n_1=1.5$, $n_2=1$ at vacuum wavelength ${\lambda }_0=633\mathrm{nm}$. At angle of incidence $40^{\circ }$, Snell law for electromagnetic waves gives an angle of refraction $\arcsin (1.5\sin 40^{\circ })\approx 74.6186^{\circ }$, so a transmitted light ray still emerges. At ${\theta }_c\approx 41.8103^{\circ }$, its direction becomes tangent to the boundary. At $50^{\circ }$, the proposed sine of a real angle of refraction would be $1.5\sin 50^{\circ }\approx 1.14907>1$: the transmitted wavevector must instead have an imaginary normal component. These are predictions for the stipulated media, not measured properties of a particular block.

A real semicircular-block experiment isolates this threshold. The University of Maryland laboratory instructions place a laser, motor-driven rotation stage and photodetector together; an aperture limits the detector voltage. The semicircular block is centered on the rotation axis, with the photodetector aligned along its flat face. The instructions identify a strong grazing-beam signal and a weaker signal just before the critical angle for total internal reflection. The readout is detector voltage versus stage angle. Retracing the incident laser beam identifies normal incidence for alignment.^[ref]

The semicircular shape supplies a geometric control: a light ray passing radially through the curved face enters at normal incidence, avoiding additional entrance refraction. The weaker signal near the flat face locates a limiting angle of refraction; a vanishing remote-detector signal alone would not establish that the external electric field vanishes. Beam width and alignment limit angular resolution; no measured threshold or decay depth is supplied.^[ref]

The transverse polarization constraint in an angular spectrum requires each source-free electromagnetic plane wave component to have electric field orthogonal to its wavevector in the algebraic sense:For an isotropic homogeneous medium with constant permittivity, this follows by inserting a Fourier component into $\nabla \cdot \mathbf{D}=0$. It restricts the vector fields that can accompany a scalar angular spectrum method calculation.^[ref]

With time dependence $e^{-i\omega t}$ and $\mathbf{k}=(k_x,k_y,k_z)$, the relation becomesWhen $k_z\ne 0$, the longitudinal field is therefore ${\tilde{E}}_z=-(k_x{\tilde{E}}_x+k_y{\tilde{E}}_y)/k_z$. It is not an arbitrary third scalar image that can be propagated independently of the first two.

For an original propagating angular-spectrum component with $\mathbf{k}=k(0.6,0,0.8)$, choosing ${\tilde{E}}_x=1$ and ${\tilde{E}}_y=0$ forces ${\tilde{E}}_z=-0.75$. Polarization along $y$ instead gives a purely transverse-to-axis field for this propagation plane. Thus electromagnetic transversality relative to $\mathbf{k}$ does not require zero field along the laboratory $z$ axis.

For an evanescent angular-spectrum component with $\mathbf{k}=k(1.25,0,0.75i)$, the same choice ${\tilde{E}}_x=1$ gives ${\tilde{E}}_z=(5/3)i$. The dot product is not a Hermitian inner product: there is no complex conjugation of $\mathbf{k}$ in the Fourier-transformed divergence equation. The imaginary longitudinal amplitude represents a phase relationship, not permission to discard the constraint.

At grazing propagation, $k_z=0$, division by $k_z$ fails and the original equation must be used. Boundary conditions, material response and the corresponding magnetic field remain necessary to reconstruct a physical solution. In the strongly paraxial approximation, longitudinal components may be small, but that is an approximation derived from the allowed spectrum rather than a replacement for the vector condition.