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An achromatic lens combines optical lens elements with different optical dispersion so that two specified wavelengths have the same focal length in the paraxial approximation. This brings their parallel incoming light rays to a common optical focus, helping an optical camera form a sharper image under mixed-colour illumination. A common implementation is an achromatic doublet: a positive crown glass element paired with a negative flint glass element. ^[ref].

The achromatic doublet does not require either material to have zero optical dispersion. Instead, the positive element gains positive optical power toward blue, while the negative element becomes more negative. Suitable Abbe numbers and signed optical powers let these changes cancel even though their reference optical powers do not cancel. For example, the achromatic doublet power balance calculation below combines $+10\mathrm{D}$ with $-5\mathrm{D}$ to obtain $+5\mathrm{D}$, or focal length $0.2\mathrm{m}$, with zero primary F–C optical power difference. The smaller Abbe number of the negative element makes its weaker reference optical power sufficient to oppose the positive element's colour change.

An achromatic lens is corrected at two chosen wavelengths, not necessarily at every wavelength between them. The remaining focal length variation is the secondary spectrum. Correcting axial chromatic aberration also does not automatically remove lateral chromatic aberration throughout an imaging system, spherical aberration, or diffraction. Actual designs therefore need ray tracing with the intended wavelengths, element thicknesses, separations and apertures. ^[ref]; ^[ref].

The Airy disk is the central bright region, up to the first dark ring, in the scalar diffraction pattern of a uniformly illuminated, unobscured circular aperture. The whole pattern includes the weaker surrounding rings; its radius is distinct from its full width at half maximum.

For a circular optical pupil of diameter $D$ under normal, uniform monochromatic plane wave illumination, Fraunhofer diffraction givesHere $I_0$ is the on-axis light intensity, $\theta$ is the angle from the optical axis, $\lambda$ is the wavelength in the propagation medium, and $J_1$ is the order-one Bessel function of the first kind. The paraxial approximation replaces $\sin \theta$ by $\theta$, measured in radians. The first positive zero $j_{1,1}=3.831705970\dots$ givesThis is the angular radius of the Airy disk; the angular diameter is twice as large. The Airy disk derivation explains the origin of the Bessel function of the first kind and of the numerical coefficient.^[ref]

The image and both radial curves show the same calculated light intensity, divided by its peak. The image's logarithmic scale makes the weak rings visible; its colour is a display choice. The linear curve shows how small the ring peaks are compared with the centre. A dark ring marks a zero of wave amplitude, produced by cancellation across the optical pupil; it is not the edge of all transmitted light. The first Airy disk contains about $83.78\%$ of the total radiant flux, as calculated in Airy disk normalization and encircled energy.

For an original design example, take an ideal optical lens in air with focal length $f=100\mathrm{mm}$, circular optical pupil diameter $D=5.00\mathrm{mm}$ and wavelength $\lambda =550\mathrm{nm}$. In the focal plane, the radial coordinate $r\simeq f\theta$ givesThe first-zero diameter is $26.8327\mu \mathrm{m}$. Solving $P(u_{1/2})=1/2$ gives $u_{1/2}=1.616339948\dots$, so the diameter measured by full width at half maximum is onlyThus a quoted “spot diameter” is incomplete without its definition. Doubling $D$ halves both widths at fixed wavelength and focal length; increasing magnification alone changes the image scale without improving object-space discrimination.

The Airy disk supplies a calculable reference for a point spread function, allowing diffraction to be distinguished from defocus and optical aberrations. Its beauty is that one circular boundary determines an entire sequence of bright and dark rings through a Fourier transform. The ideal formula assumes an unobscured pupil, uniform illumination, ideal optical focus, a single wavelength, and a scalar description with small output angles. A blocked centre, nonuniform pupil transmission or a significant pupil wave phase error changes the response. High numerical aperture may require polarization of an electromagnetic wave and a vector field description. The circular-aperture diffraction laboratory experiment tests the far-field pattern directly; the Rayleigh criterion uses its first zero to define a two-source convention.

An aperture is an opening that admits light into or through an optical instrument. Its shape and size select the transmitted light rays; an aperture alone does not bend them as an optical lens does. The aperture stop is the opening that most restricts the bundle from an on-axis object point. The entrance pupil is the image of that aperture stop formed by the preceding optical lens elements, viewed from the object side. Consequently the effective collecting diameter $D$ is the entrance pupil diameter, which can differ from the physical hole's diameter. For example, preceding optical lens elements that enlarge a $10\mathrm{mm}$ aperture stop by a factor of two give $D=20\mathrm{mm}$; measuring the physical hole alone would misidentify the focal ratio. ^[ref].

For an unobscured circular entrance pupil under a uniform beam normal to its plane, the collecting area and admitted radiant flux arewhere $E_{\mathrm{pupil}}$ is incident irradiance. Doubling $D$ quadruples admitted radiant flux. Obstructions, nonuniform illumination and losses after the entrance pupil require the actual illuminated area and transmission to be included. The same $D$ also sets an angular diffraction-limited resolution scale $\lambda /D$ at wavelength $\lambda$. Collection and diffraction therefore answer different questions from field of view: an aperture selects the bundle from each object point, while the receiving plane's extent helps select which object directions appear in the image.

NASA/JPL's published pinhole demonstration uses two white cards, foil taped over a window in one card, and a small hole pierced in the foil. Sunlight passes through the hole onto the second card; the projected image is viewed on that card. The activity describes a larger image when the card separation increases, improved visibility when the receiving card is shaded, and a separate image from each of several holes. These are qualitative observations of image formation, rather than calibrated measurements of radiant flux or diffraction. ^[ref].

A proposed quantitative extension replaces sunlight by a steadily illuminated asymmetric target and puts the foil and a translucent receiving screen in a dark box. Record target distance $u$, hole-to-screen distance $b$, hole diameter $d$, image height, orientation and edge blur. Measure screen brightness with a fixed photodetector, or record the screen with an optical camera whose integration time and gain stay fixed. Exchange centred foils of measured $d$ at fixed $u,b$; then vary $b$ with the same foil. Geometrical optics predicts an inverted image with transverse magnification $m=-b/u$ and a point's geometric blur diameter $d(1+b/u)$. Thus a larger hole admits more radiant flux without changing the central image scale, whereas increasing $b$ enlarges the image. These are predictions to test, not reported data. Keep target illumination and detector position fixed, subtract a hole-covered dark reading, repeat the original foil to check drift, and avoid clipping the detector response. Foil thickness and ragged edges can block oblique light rays; at sufficiently small $d$, diffraction invalidates the claim that ever smaller holes always sharpen the image. A lensless optical camera has no optical lens focal length: $b$ is a propagation distance, not that focal length.

For uniform coherent plane-wave illumination, an ideal binary amplitude zone plate transmits alternate Fresnel half-period zones. If $N_o$ complete odd zones are open, every one contributes approximately twice the unobstructed axial complex amplitude, in the same phase, and $I_f/I_0\simeq 4N_o^2$. This is a local light intensity gain relative to removing the plate and outer stop. It differs from comparison with a finite clear aperture: an aperture containing an odd integer number of half-period zones has axial intensity $4I_0$, giving plate-to-aperture ratio $N_o^2$. Neither gain is a total power efficiency.

Basic radiance is radiance divided by the square of the refractive index. For radiance $L$ at a fixed wavelength,Lossless propagation conserves reduced radiance along each light ray; $L$ itself is conserved only when the endpoint refractive indices agree. A larger irradiance at a smaller image can result from admitting more directions without increasing reduced radiance. Uniform wave absorption can lower radiant flux while leaving the occupied geometric etendue unchanged; an opaque aperture removes part of that occupied set. ^[ref]

For small measurement errors, write $f=L/4-d^2/(4L)$. Its partial derivatives and first-order propagation of uncertainty areHere the $\sigma$ values are standard deviations used to describe uncertainty, and $\mathrm{Cov}(L,d)$ is the covariance. Independent carrier-reading errors give ${\sigma }_d^2={\sigma }_{r_{-}}^2+{\sigma }_{r_{+}}^2$; correlated readings require the term $-2\mathrm{Cov}(r_{-},r_{+})$. Averaging repeated independent pairs reduces the random component of the uncertainty of the mean, but does not remove calibration or model errors.

As a stipulated uncertainty calculation, let $L=45\mathrm{cm}$, $d=15\mathrm{cm}$, ${\sigma }_L=0.10\mathrm{cm}$ and ${\sigma }_{r_{-}}={\sigma }_{r_{+}}=0.05\mathrm{cm}$, with all three input errors independent. Then ${\sigma }_d=0.0707\mathrm{cm}$, $A=5/18$, $B=-1/6$ and ${\sigma }_f\simeq 0.0302\mathrm{cm}$. The result would be $f=10.000\mathrm{cm}$ with this standard uncertainty in the stipulated thin lens model; these numbers are not a measured uncertainty budget. A small neglected principal-plane separation gives an additional first-order bias $f_{\mathrm{thin}}-f\simeq Ah$, distinct from that random budget.

The two kinds of sensitivity must be distinguished. At fixed focal length,These derivatives diverge as $L$ approaches $4f$ from above: a small change in object-screen spacing strongly changes the separation of the two settings. Yet the inferred focal length's displacement sensitivity $|B|=d/(2L)$ tends to zero. That does not make coalescing settings easy to measure. For fixed $L$, define the axial mismatch between the predicted real image and the screen as $g(x)=fx/(x-f)-(L-x)$, for $x>f$. At a focused setting,Near $u=v=2f$, moving the optical lens changes this mismatch only weakly, making visual discrimination of the two settings difficult. At the example's settings $g^{\prime }(15\mathrm{cm})=-3$ and $g^{\prime }(30\mathrm{cm})=3/4$, so equal carrier motion produces different axial defocus responses. This is an axial mismatch calculation, not a full prediction of perceived image sharpness.

Choose $L$ sufficiently above $4f$ to separate the two accepted focusing ranges, while retaining both real images on the available screen and rail. There is no universal best $L$: focusing uncertainty itself depends on aperture, transverse magnification, optical aberrations and the target. Close to $d=0$, the nonnegative measured separation and the quadratic term in $d$ also make first-order propagation of uncertainty unreliable if its error is comparable to its value. Repeatability, controls and the principal plane correction determine whether the elegant two-distance formula is an adequate measurement model.

KIT's teaching apparatus uses a glass spoke-wheel or test negative as the object, a halogen lamp with an illuminating optical lens, an imaging optical lens, a screen and a triangular rail with a position scale. The illuminating optical lens concentrates the lamp filament's image on the imaging optical lens's centre; translucent paper checks that illumination. This reduces the contribution of peripheral light rays and their spherical aberration. The instructions require fixed object-screen separation exceeding $4f$, at least ten displacement measurements, and separate red- and blue-filter measurements to test chromatic aberration.^[ref]

The following measurement protocol elaborates that apparatus rather than reporting additional KIT results. Align the target centre, optical lens and screen normal along one optical axis; measure $L$ between the target pattern plane and receiving screen face. Lock the target and screen. Slide the imaging optical lens to obtain the enlarged real image, then the reduced real image, reading the same carrier reference mark at each setting. Record $r_{-},r_{+}$ and $d=r_{+}-r_{-}$ for each pair. Record corresponding target and image intervals to test the reciprocal transverse magnifications independently; merely observing different image sizes is a weaker test.

Use the same central target detail to judge best optical focus. Bracket each setting by moving through both visibly blurred sides, repeat from both travel directions, and record the accepted position range as well as repeat scatter. Rail divisions set a readout limit but do not alone specify focusing uncertainty. Check scale calibration, carrier play and drift; a common additive carrier offset cancels in $d$, whereas scale-factor error remains. Keep illumination, aperture, target, screen and optical lens orientation fixed within a series. Change the aperture and colour filter in separate control series: spherical aberration and chromatic aberration can shift the apparent best optical focus, while a very small aperture introduces stronger diffraction and makes focus discrimination harder. Record these settings alongside the inferred focal length, rather than interpreting all disagreement as random measurement error.

For a rigid thick lens in air, let $h=x_{H^{\prime }}-x_H$ be the signed separation of its rear and front principal planes along the direction of propagation. Then the positive object and image distances measured from the principal planes obey $u+v=L-h$. Translating the whole assembly without reversing it moves both principal planes equally, so the measured carrier displacement still equals the difference of the two solutions for $u$. The same algebra therefore givesHere $f$ is the effective focal length measured from the principal planes. This deduction uses the principal-plane imaging equation, not the uncorrected thin lens formula.^[ref] Mechanical centre thickness is not generally $h$. Record thickness and optical lens orientation, but use a justified bound or model for $h$ if it matters at the claimed precision. The two readings $L,d$ alone cannot determine both $f$ and unknown $h$; cancelling a carrier offset does not cancel this limitation.

An actual teaching setup is described in Laboratory Exercise No. 4: Diffraction, MIT 6.237, Fall 2024, for MIT's 6.237 Modern Optics Project Laboratory, Fall 2024. Preparation includes predicting the aperture-size dependence and the distinction between Fraunhofer diffraction and Fresnel diffraction. A helium–neon laser supplies $632.8\mathrm{nm}$ light; the far-field setup uses a planar incident wavefront, a diffracting object and a distant screen, with beam expansion when needed. Figure 1 also marks an optional CCD camera. Exercise 4.7 asks students to sketch the circular-aperture pattern, measure the Airy disk diameter between correctly identified boundaries, and infer the aperture size. No measured values are supplied. As a regime comparison, exercise 4.9 varies the circular opening's position near a microscope objective optical lens, observes near-field patterns and compares them with the far-field case. This tests whether a change in propagation geometry changes the pattern. The coherent illumination measures one aperture's response, rather than the discrimination of two mutually incoherent optical point sources.^[ref]

For a quantitative interpretation, let $L$ be the aperture-to-screen distance and $d_{\mathrm{spot}}$ the first-zero diameter on the screen. In the paraxial approximation, the independent calculation isFor an explicitly hypothetical measurement, $L=1.80\mathrm{m}$ and $d_{\mathrm{spot}}=5.60\mathrm{mm}$ at $632.8\mathrm{nm}$ would imply $D\simeq 0.496\mathrm{mm}$. The scale $D^2/\lambda \simeq 0.389\mathrm{m}$ is smaller than $L$, but only by a factor of about $4.6$; this alone gives limited confidence in a high-precision Fraunhofer diffraction approximation. It is not a reported MIT measurement.

Additional recommended controls are to keep wavelength fixed, record $L$, centre and uniformly illuminate the aperture, and repeat with another $D$. Compare $d_{\mathrm{spot}}/L$, which should vary inversely with $D$. A camera readout should include blocked-beam background and avoid saturation. A scaled repetition at larger $L$ tests residual Fresnel diffraction; nonuniform illumination, optical aberrations, detector sampling and uncertainty in locating a dark ring limit the inferred $D$. The screen pattern's apparent bright edge depends on display brightness and is not a substitute for the first-zero boundary.

Two planes are optically conjugate when an optical lens system images one onto the other. In confocal microscopy, an illumination pinhole is imaged onto the selected specimen point, and that specimen point is imaged onto the detection pinhole. Thus the two apertures select corresponding positions in different planes; they need not occupy the same physical plane. Modern laser instruments can supply the illumination point with an optical fiber or a suitable focused laser mode instead of a separate metal illumination pinhole. ^[ref]

A fluorescent emitter at the selected optical focus forms a compact point spread function at the detection aperture. An emitter displaced axially produces a broadened spot there through defocus; a smaller fraction of its collected light passes. The emitter still exists and still emits photons. Rejected photons strike the opaque mask or other parts of the instrument instead of contributing to the accepted detector channel. Neither a finite pinhole nor diffraction makes the accepted out-of-focus contribution exactly zero.

Lossless reversible propagation preserves the etendue of the same bundle of light rays. Here the model uses stationary isotropic media, specular reflection and refraction, with every selected light ray reaching the output and with no optical gain or scattering of light. An aperture that blocks some incident light rays changes which bundle is counted; the full incident and transmitted etendue need not be equal. ^[ref]

The underlying phase space has four coordinates. In a planar section with constant refractive index, let $(x,y)$ locate a light ray and setThe Jacobian determinant from spherical coordinates givesThus etendue is a four-dimensional volume, preserved by the optical counterpart of Liouville theorem in Hamiltonian mechanics. Individual position and direction extents may change, and correlations between them matter. The Lagrange optical invariant is a related paraxial approximation invariant for pairs of light rays, rather than the full four-dimensional measure.

A coronagraphic focal plane mask modifies the wave amplitude or wave phase in an intermediate focal plane. Here it is a perfectly opaque circular disk centered on the bright source. A finite aperture produces an extended point spread function, so a finite disk blocks only part of the stellar light. Its sharp edge also changes the transmitted field and hence the later diffraction pattern. The exact model below makes both effects visible.

In NIST's 2001 comparison, the Laser-Optimized Cryogenic Radiometer held its cavity near $5.4\mathrm{K}$ and controlled the heat sink separately. Opening the beam reduced heater power at fixed cavity temperature; an AC bridge read the temperature. Tests used stabilized, spatially filtered lasers at $488$, $514$ and $633\mathrm{nm}$, with horizontal input and a movable photodiode transfer detector in the cavity's optical plane. The paper contrasts this with the High Accuracy Cryogenic Radiometer's optical heating followed by two electrical heating cycles and interpolation.^[ref]

The two laboratories compared calibrations of the same transfer detectors, and checked their spatial response. Alignment differences and transported dust limited that comparison; its agreement did not prove universal equivalence. Its correction tables retain window transmission, cavity radiometric absorptance, alignment, electrical readout and optical-electrical heating non-equivalence. The heat sink was controlled, but §3 does not specify its numerical setpoint: the $5.0\mathrm{K}$ reference in the preceding example is stipulated.^[ref]

For a reproducible substitution measurement, record the beam's wavelength and entrance boundary, receiver and reference temperatures, heater terminal voltage and electric current, dark readings before and after illumination, and the settling rule. Check beam collection at the aperture, window transmission, radiometric absorptance, deposited lead heating, and sensitivity to beam position. Repeatability alone misses a stable correction error. Photometric calibration can subsequently use this radiant flux scale, but photometry still needs the appropriate spectral visual weighting; the cold receiver does not itself measure luminous flux.

All four interacting waves have the same frequency. Counterpropagating equal-frequency pumps can give a source proportional to the conjugate signal field, providing phase conjugation. The pump product must have the appropriate uniform spatial phase, and finite apertures and phase mismatch limit reconstruction.

For incoherent imaging with an unobstructed circular aperture of diameter $D$, the intensity-image spatial-frequency cutoff is $D/(\lambda f)$ in the focal plane. The Nyquist–Shannon sampling theorem therefore gives the displayed pitch limit. The Rayleigh criterion instead gives a point-image separation $1.22\lambda f/D$. Consequently two samples per Rayleigh separation and sampling the full optical bandwidth differ by the factor $1.22$. Neither sampling estimate guarantees useful contrast at arbitrarily low signal-to-noise ratio.

Multiple scattering can enlarge the angular diversity reaching a receiver array. Reversing the paths then makes the array act like a larger aperture, improving the focus in appropriate regimes of wave propagation in a random medium.

An electromagnetic waveguide confines an electromagnetic wave across its axis while allowing it to travel along the axis. Conducting walls or changes of refractive index select transverse normal modes: reproducible cross-sectional field patterns with their own longitudinal wavenumbers. A hollow metal pipe and an optical fiber are different realizations of this principle. Their usefulness is to deliver electromagnetic energy along a chosen route instead of letting it spread freely.

For a uniform vacuum-filled conducting guide, write a normal mode's axial and time dependence as $e^{i(\beta z-\omega t)}$. Here $\omega >0$ is angular frequency, $\beta$ is the longitudinal wavenumber, and $k_{\perp }^2$ is the transverse eigenvalue selected by the wall boundary conditions. The electromagnetic wave equation separates intoThe geometry consumes part of the available squared wavenumber: when $\omega /c<k_{\perp }$, this branch is an evanescent wave rather than a propagating normal mode. The waveguide cutoff frequency is therefore a property of a specified branch and geometry. A transverse electromagnetic mode in a suitable two-conductor geometry can have $k_{\perp }=0$ and no positive geometric cutoff. An optical fiber needs its own dielectric boundary conditions, rather than the metal-wall formulas below.

The rectangular electromagnetic waveguide makes these ideas explicit with a single sine profile. The scalar-potential conducting waveguide modes give a broader construction. At an open aperture, a confined field can radiate into the electromagnetic aperture angular spectrum; it need not remain one guided normal mode.

Suppose an extended source fills every position of the entrance aperture with a cone of half-angle ${\theta }_{\mathrm{in}}$, and all these light rays must reach a one-sided planar receiver. If the receiver accepts at most a cone of half-angle ${\theta }_{\mathrm{out}}$, its available etendue bounds the admitted etendue:ConsequentlyThis original derivation combines conservation of etendue with the exact cone integral; the second inequality allows the complete output hemisphere. It bounds capacity and does not prove that a chosen shape attains the bound.

For the $30^{\circ }$ input cone with $n_{\mathrm{in}}=n_{\mathrm{out}}=1$, $C\le 4$: a $10\mathrm{mm}$ entrance cannot deliver the entire filled bundle to a circular receiver smaller than $5\mathrm{mm}$ in diameter. If the output half-angle is restricted to $60^{\circ }$, $C\le 3$ and the receiver diameter must be at least $10/\sqrt{3}\approx 5.7735\mathrm{mm}$. For a receiver medium with model refractive index $n_{\mathrm{out}}=1.5$, the full-hemisphere capacity bound instead gives $C\le 9$; that larger value also depends on transmitting the bundle into that medium.

Attempting $C=10$ in the air-to-air example forces rejection of some input. For the uniform radiance already specified, even filling the entire output hemisphere givesThe maximum mean irradiance gain remains $\eta C\le 4$. With nonuniform radiance, rejecting a given fraction of geometric etendue need not reject the same fraction of radiant flux.

These bounds use a filled extended-source bundle in passive reversible geometrical optics, comparing the same wavelength and the same radiation channels. An ideal optical point source does not fill that four-dimensional set. Diffraction, optical coherence, or coupling a Gaussian beam into an optical fiber require a physical optics calculation; the cone bound is not a universal guarantee for a single wave mode. Likewise, optical gain or fluorescence changes the propagation assumptions.

Consider a planar circular aperture of diameter $D$ in constant refractive index $n$. Suppose every position in its area $A=\pi D^2/4$ is filled with directions inside the same rotationally symmetric cone about the unit normal, with half-angle ${\theta }_{\max }$. Using spherical coordinates, the exact etendue iswhere $\mathrm{NA}=n\sin {\theta }_{\max }$ is the appropriate side's numerical aperture. The ordinary cone solid angle is $2\pi (1-\cos {\theta }_{\max })$; its cosine-weighted value is $\pi {\sin }^2{\theta }_{\max }$. A full forward hemisphere therefore gives $G=n^{2}A\pi$, rather than $2\pi n^{2}A$. ^[ref]

For an original model example, choose $n=1$, $D=10\mathrm{mm}$, and ${\theta }_{\max }=30^{\circ }$. ThenIf the radiance is uniform across this filled bundle and equals $L=100\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}$, its radiant flux is $\Phi =LG/n^2\approx 6.1685\mathrm{mW}$. Neither the geometry nor the chosen radiance is an experimental measurement.

Consider two identical, lossless parallel mirrors with intensity reflectivity $0\le R<1$, separation $d$, and a uniform transparent medium of refractive index $n$. Fix one polarization of an electromagnetic wave and an internal angle of incidence $\theta$ measured from the mirror normal vector. Illuminate with a monochromatic wave having planar wavefronts; the emerging contributions must overlap coherently in the detected direction. Define $T=I_{\mathrm{transmitted}}/I_{\mathrm{incident}}$ using the same incident and exit media. Each mirror transmits a fraction $1-R$ of incident power. Its reflected wave amplitude has magnitude $\sqrt{R}$, so two internal specular reflections multiply the circulating complex amplitude by magnitude $R$, not $R^2$.

Let $\lambda$ be the vacuum wavelength and choose the reflection wave phase convention in which the product of the two internal amplitude reflection coefficients is real and positive. The phase difference between successive transmitted contributions, evaluated on a common outgoing wavefront, isWith other mirror coatings or phase conventions, use the total $\delta =4\pi nd\cos \theta /\lambda +{\varphi }_{\mathrm{rt}}$, where ${\varphi }_{\mathrm{rt}}$ is the sum of the two internal reflection wave phases. The stated resonance positions presume ${\varphi }_{\mathrm{rt}}=0$ modulo $2\pi$; a constant nonzero ${\varphi }_{\mathrm{rt}}$ shifts them, while a frequency-dependent one can also change their spacing.

Why does the cosine appear rather than its reciprocal? Inside the gap, the extra two legs have optical path length $2nd/\cos \theta$. But the later emerging contribution starts $2d\tan \theta$ farther along the mirror. Comparing both on one outgoing wavefront subtracts $2nd\tan \theta \sin \theta$ from that path difference, leaving $2nd\cos \theta$. Equivalently, a plane wave's mirror-normal wavevector component is $k_z=(2\pi n/\lambda )\cos \theta$, so the round-trip wave phase is $2k_{z}d$. The diagram uses the same medium outside and inside to show the common-wavefront subtraction directly; at a refracting exit interface, tangential wavevector matching in Snell's law supplies the same result.

Absorb the common first-pass wave phase into the incident complex amplitude $E_{\mathrm{in}}$. Successive transmitted contributions are $(1-R)E_{\mathrm{in}}$, $(1-R)E_{\mathrm{in}}R{e}^{i\delta }$, and so on. The superposition principle and the convergent geometric series giveThe last step expands the squared complex modulus as $1+R^2-2R\cos \delta$. This transmission profile is often called the Airy transmission function; it is distinct from the Airy disk, a diffraction image of a circular aperture. The University of Washington's lab handout gives the same transmission profile and distinguishes mirror reflectivity from other sources of peak broadening.^[ref]

For an original example, set $R=3/4$. At $\delta =0$, the amplitudes add to $(1/4)/(1-3/4)=1$, so $T=1$. At $\delta =\pi$, they alternate in sign and add to $(1/4)/(1+3/4)=1/7$, so $T=1/49$. At $\delta =\pi /2$, the amplitude is $4/25+3i/25$ and $T=1/25$. Adding the individual intensities instead would give $(1-R{)}^2{\sum }_j{R}^{2j}=(1-R)/(1+R)=1/7$, independent of wave phase; that discards the wave interference responsible for the peaks.

In general the maximum is $T_{\max }=1$ at $\delta =2\pi m$, with integer $m$, and the minimum is $T_{\min }=[(1-R)/(1+R){]}^2$ halfway between peaks. Perfect resonant transmission requires identical lossless mirrors, coherent illumination, and sufficient time to establish the steady field. Absorption, unequal mirror transmission, finite beams that fail to overlap, and fluctuations of $d$ can reduce it. The exactly reflecting case $R=1$ is excluded: no incident power enters through a perfectly reflecting mirror, and the resonant limit $R\to 1^{-}$ is singular.

The field of view is the region of object directions sampled by an optical instrument in one pointing. State whether it is horizontal, vertical or diagonal, and whether it is a full angle or a half-angle. An object-plane width in millimetres also describes a field of view, but requires a specified object distance. Neither that width nor the angular field of view is the aperture diameter or the diffraction-limited resolution.

For a centred flat rectangular photodetector of width $w$ in the focal plane, with positive effective focal length $f$, infinity focus and ideal rectilinear imaging without optical distortion, the full horizontal field of view isThe model maps an object direction $\theta$ to image height of magnitude $|y|=f|\tan \theta |$; set $|y|=w/2$ and use the inverse tangent. This retains exact right triangle geometry within the stipulated imaging model; a paraxial focal length alone does not guarantee that a real wide-field optical lens obeys it. With detector height $h$, replace $w$ by $h$ for the vertical field of view, and by $\sqrt{w^2+h^2}$ for the diagonal field of view. The small-angle approximation gives ${\Theta }_x\simeq w/f$ in radians. ^[ref].

For an original worked example, a $36\mathrm{mm}$ detector width gives ${\Theta }_x\approx 39.60^{\circ }$ at $f=50\mathrm{mm}$, and $20.41^{\circ }$ at $f=100\mathrm{mm}$. Keeping $D=25\mathrm{mm}$ makes these f/2 and f/4 respectively: the field of view changes because $f$ changes. By contrast, closing $D$ from $25$ to $12.5\mathrm{mm}$ at $f=100\mathrm{mm}$ changes f/4 to f/8 while retaining $20.41^{\circ }$ in this model. The clean separation is useful: entrance pupil area controls collection, focal ratio controls the paraxial cone, and $w/f$ controls angular image coverage. Edge clipping, optical distortion, field curvature and optical aberrations can restrict the practically usable field of view.

A proposed angular readout on an optical bench illuminates a small asymmetric target at the optical focus of a collimator, providing a bundle with a known direction. Rotate the optical camera about the centre of its entrance pupil, reading a graduated rotation stage, and record the target image position on its fixed photodetector. Centre the target to establish zero; turn left and right until its centre reaches each detector edge. The difference between stage readings measures the full horizontal field of view. Repeat with a smaller aperture stop: unchanged edge directions but reduced signal support the distinction between collection and angular coverage. Keep the same active detector width, infinity focus, illumination, integration time and gain; approach each reading from the same direction to check mechanical backlash. A finite target blurs the edge criterion, detector thresholds can imitate lost coverage, and a wide bundle must illuminate the whole entrance pupil at every tested angle. These checks distinguish a geometrical boundary from insufficient signal. The analytical diagram is the prediction for this protocol, not a record of its execution.