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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.

A submerged object looks shallower because its emerging light rays bend away from the surface normal vector. Their backward extensions give an approximate virtual image nearer the surface. For an object at perpendicular depth $d$ in a medium of refractive index $n_{\mathrm{obj}}$, viewed through a flat boundary from a medium of refractive index $n_{\mathrm{obs}}$, the near-normal paraxial approximation gives $d_{\mathrm{par}}=(n_{\mathrm{obs}}/n_{\mathrm{obj}})d$. Thus the model water depth $12\mathrm{cm}$ appears approximately $9\mathrm{cm}$ from air. This is a bundle-of-light rays statement, not a deflection of the axial light ray.^[ref]

For general transmitted light rays, specify what “depth” means. Put the object at $(0,-d)$ below a horizontal boundary $z=0$. A light ray reaches the boundary at $(x,0)$, with object-side angle ${\theta }_{\mathrm{obj}}$ and observer-side angle ${\theta }_{\mathrm{obs}}$, both measured from the upward normal vector. For $x>0$, its backward extension meets the object's normal line $x=0$ at $z=-d_{\mathrm{axis}}$, whereThis axis-intercept depth is exact for that one transmitted light ray in the stated model. It tends to $d_{\mathrm{par}}$ as the angles tend to zero; at exactly zero the quotient is interpreted as a limit. These elementary tangent, sine and cosine relations show precisely where the small-angle approximation enters.

The original calculations below use $d=12\mathrm{cm}$, $n_{\mathrm{obj}}=4/3$ and $n_{\mathrm{obs}}=1$:

At appreciable obliquity these depths differ, so the full outgoing bundle has no single exact virtual image on the object's normal line. An oblique eye samples a narrow bundle; its local apparent position also depends on that bundle and cannot be inferred by declaring $d_{\mathrm{axis}}$ to be a universal image depth. A plane refracting boundary gives approximate image formation near normal incidence, with departures at larger angles. The construction assumes a flat surface, uniform transparent media with isotropy, and a propagating transmitted light ray; it does not extend to total internal reflection.

Image formation assigns each object point a corresponding image point: outgoing light rays from that object point either converge there, giving a real image, or diverge as though they originated there, giving a virtual image. The requirement applies separately to each point. Bringing unrelated light rays together makes a bright patch, but does not preserve the object's pattern. This point-by-point correspondence explains how an optical camera records a scene. ^[ref].

Use the same convention as thin lens: place the optical lens at $x=0$, with incident light travelling to the right and a real object at $x=-u$, $u>0$. Write the image coordinate as $x=v$, so $v>0$ for a real image and $v<0$ for a virtual image. Heights $h_o,h_i$ are positive above the optical axis and negative below it. In air, within the paraxial approximation, a thin lens obeysHere $f$ is the signed focal length, positive for a converging optical lens and negative for a diverging optical lens; the expression for $v$ assumes $u\ne f$. Object distance $u$ is positive even though the object's coordinate is negative. These coordinate rules describe transmission through the optical lens; a mirror reverses the outgoing direction, so do not identify its image distance with this same rightward coordinate. ^[ref].

The beauty of geometrical optics is that two convenient light rays can locate a point, while a whole admitted bundle forms its image. For a converging thin lens, an incident light ray parallel to the optical axis exits through the rear optical focus; a light ray through the centre continues undeviated. Real optical lens elements do not focus every admitted light ray exactly: optical aberrations and diffraction limit the point correspondence. The diagrams below are original ideal-model constructions, not measured data. ^[ref].

The real image of the specimen supplied to the microscope eyepiece or a recording surface. In the example above, a bare detector placed at $x=60\mathrm{mm}$ records an image five times the specimen's length, inverted. It does not record a fifty-times image: that number compares viewing angles under the stated reference convention. An optical camera receiving the eyepiece's outgoing light needs its own optical lens to form a further real image, with a recording scale determined by that added image formation.

The specimen-facing optical lens assembly of an optical microscope. A microscope objective collects light from the specimen and sets the object-side numerical aperture. In a finite-conjugate microscope it forms the intermediate microscope image directly; in an infinity-corrected microscope a tube lens completes that image formation. Its label alone does not calibrate a specimen length.

The institution-issued AY 105 Lab Experiment 3: Optical aberrations by California Institute of Technology (Spring 2011 offering, Part A, pp. 2–3, 7–8) specifies a concrete optical bench: a lamp, opal diffuser and condensing optical lens illuminate a $25$ or $30\mu \mathrm{m}$ optical pinhole; a photographic telephoto optical lens acts as a collimator, producing an approximately $75\mathrm{mm}$ beam. A test optical lens sits on a rotation stage, followed by an adjustable optical microscope. The procedure establishes the on-axis image, then rotates the test optical lens a few degrees and examines the off-axis image through optical focus. It compares this readout with through-focus ray tracing at the corresponding field angles and offers a Ronchi ruling to calibrate the measuring eyepiece.

Part B (pp. 9–10) substitutes a spherical mirror of $1\mathrm{m}$ radius of curvature, with the optical microscope beside the incoming beam so the image is necessarily off-axis. It calls for measuring the field angle, calibrating the eyepiece and recording the image's length and width at minimum size. These are specified apparatus dimensions and instructions, not supplied measurement results or dimensions of the normalized figures.

A controlled extension would record images at zero and equal positive and negative field angles, repeat zero afterwards, and scan through optical focus at every setting. Keep the illuminated optical pupil and illumination fixed, recenter the optical microscope without clipping the bundle, and use its calibrated scale to separate a mere image displacement from a change in blur shape. The pure Seidel coma pupil mapping predicts reversal and linear growth; a stationary asymmetric central image instead calls for checking alignment. A focus-dependent change between perpendicular elongations suggests an astigmatism contribution. These comparisons test the model rather than assuming every off-axis flare is pure coma.

The optical pinhole has finite angular size, approximately its diameter divided by the collimator's focal length; the collimator and observing optical microscope also contribute their own optical aberrations. A broadband lamp can mix chromatic aberration with the geometric blur, and diffraction limits the smallest visible structure. A narrow spectral band, repeated field settings and a reference image help separate these effects. Consequently a measured length and width are useful checks on off-axis image formation, but they cannot by themselves determine a unique coma coefficient or recover the full point spread function.

An optical aberration is a departure from ideal image formation: light rays from one object point may miss their intended common image point, or image positions may depart from the intended mapping. Spherical aberration, coma, and astigmatism describe different ways of spreading a point's light rays; optical distortion can change image shape without spreading each point. In physical optics, these departures can be described by wavefront error relative to a reference wavefront. Diffraction also gives a finite point spread function in a perfect optical instrument, so an extended image alone does not establish an optical aberration.

An optical bench is a rail or aligned platform carrying an object, optical lens elements and a screen or detector at readable positions. It turns image formation into measurements of position and transverse magnification. Read distances between the optical planes rather than arbitrary holder boundaries; keep heights and optical axis alignment controlled. ^[ref].

A thick lens retains the separation of its refracting surfaces when predicting image formation. In the paraxial approximation with air on both sides, object and image distances measured from the respective principal planes satisfy the familiar imaging equation. Measuring instead from one mechanical centre generally introduces an error.^[ref]