---
title: Thin Lenses
module: Geometrical Optics
moduleNumber: 11
lessonNumber: 2
order: 1102
summary: >
  A lens gathers the light spreading from one point back onto another, and a single
  paraxial relation $1/s+1/s'=1/f$ predicts where that image lands and how large it
  is. We collapse two refractions into one bending plane, read image position and
  orientation off the three principal rays, and trace focal length back to glass and
  curvature through the lensmaker equation. Sign conventions carry the physics here —
  they separate real from virtual images and upright from inverted — so we drill them
  before chaining lenses in sequence and in contact. The lesson ends on how focal
  length is actually measured on a bench, and where finite thickness, aperture, and
  dispersion break the thin-lens picture.
topics: [Geometrical Optics]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 32 — Optical Images; §32-2"
  - book: Tipler & Mosca
    ref: "Ch. 32 — Optical Images; §32-3"
---

## Thin-Lens Geometry

A lens changes ray direction by refraction at two surfaces. The thin-lens model
replaces those two bends by one effective bend at a plane through the lens center.
It applies when lens thickness is small compared with object and image distances,
the rays are paraxial, and the surface curvatures can be represented by one focal
length. The model predicts paraxial image position and lateral magnification.
Principal-plane shift, aperture clipping, wavelength dependence, and off-axis blur
require a thick-lens or nonparaxial treatment.

The optic axis is the reference line through the lens center. Object distance $s$
is measured from the lens plane to the object along the incoming-light side. Image
distance $s'$ is measured from the lens plane to the final image. The signs below
match the refracting-surface convention used for thin lenses.

$$
\begin{aligned}
s&>0 &&\text{real object on the incident-light side},\\
s'&>0 &&\text{real image on the refracted-light side},\\
f&>0 &&\text{converging lens},\\
f&<0 &&\text{diverging lens},\\
m&=\frac{y'}{y} &&\text{lateral magnification}.
\end{aligned}
$$

The object height $y$ is positive above the axis. An upright image has
$y'>0$ and $m>0$; an inverted image has $y'<0$ and $m<0$. The signs encode
geometric information. Retain them until the image type and orientation have been
identified.

$$
% caption: Thin-lens reference geometry. Object distance $s$ is measured from the
% effective lens plane on the incoming side, image distance $s'$ from the same
% plane on the outgoing side, and both focal points lie one focal length from the
% central plane in the paraxial model.
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$$

### Principal rays and focal geometry

The standard ray construction uses three rays from one object point. Any two
independent principal rays determine the paraxial image point; the third checks the
construction. Their use follows from focal-point reversibility and the central
parallel-surface region of a thin lens.

- **Parallel ray:** leaves the object parallel to the optic axis. A converging lens
  directs it through the outgoing focal point; a diverging lens directs it as though
  it came from the incoming focal point.
- **Central ray:** passes through the lens center and continues without an angular
  change in the thin-lens approximation.
- **Focal ray:** reaches a converging lens through the incoming focal point, or
  reaches a diverging lens directed toward its outgoing focal point. It emerges
  parallel to the axis.

The central ray is a paraxial construction. Near the lens center, the two surfaces
are nearly parallel and their angular deviations cancel. A finite-thickness lens
leaves a small lateral displacement, and a wide beam can depart from the
central-ray construction through aberration.

$$
% caption: The three principal rays for a converging thin lens. Parallel, central,
% and focal-path rays from one object point meet at the same real image in the
% paraxial model, providing both an image construction and an internal consistency
% check.
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$$

Parallel rays entering at a small angle to the axis converge in the focal plane,
not on the axis. A lens therefore maps direction to position over a local field of
view. The relation is approximately $x=f\tan\alpha$, where $\alpha$ is the
incoming angle and $x$ the focal-plane displacement. This relation underlies
collimated-beam alignment and the use of a lens to turn an angular scan into a
position measurement. It holds only over the paraxial angular range.

A diverging lens has three principal rays that leave on paths spreading apart. Their
backward extensions meet at the virtual image. The sign of the focal length
and the location of the focal points reverse relative to the converging case, while
the construction still uses the same axis and central plane.

$$
% caption: Principal rays for a diverging thin lens. Solid rays leave the negative
% lens on diverging paths, and their backward extensions meet at the upright virtual
% image between the lens and the incoming focal point.
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$$

The focal-ray rule requires a direction qualifier. A ray through the incoming focal
point of a positive lens emerges parallel. A ray directed toward the outgoing focal
point of a negative lens emerges parallel. Drawing a ray through the wrong focal
point gives a plausible line with the wrong image location. The side labels on the
first diagram should therefore be carried into every subsequent construction.

## Thin-lens equation, magnification, and lens power

The thin-lens equation follows by applying the spherical-refraction relation to each
surface and then eliminating the intermediate image made by the first surface. The
elimination is valid because lens thickness is neglected: the first image becomes the
object for the second surface at essentially the same plane. The surrounding medium
must be the same on both sides for the common single-index form below.

A lens of refractive index $n_{\ell}$ in a medium of index $n_m$, with signed
surface radii $R_1$ and $R_2$, has lensmaker relation

$$

\frac{1}{f}
=\left(\frac{n_{\ell}}{n_m}-1\right)
\left(\frac{1}{R_1}-\frac{1}{R_2}\right).

$$

The radius sign follows the center of curvature. With incoming light from the left,
$R_1>0$ when the first center lies to the right of the first surface, and $R_2<0$
when the second center lies to the left of the second surface. A symmetric
double-convex glass lens in air therefore has $R_1>0$, $R_2<0$, and $f>0$. A
double-concave lens has the reverse radius signs and a negative focal length.

$$
% caption: Signed surface radii for a double-convex thin lens. The first center lies
% on the outgoing side of the first surface and gives $R_1>0$; the second center
% lies on the incoming side of the second surface and gives $R_2<0$, producing a
% positive focal length for a lens with index above its surroundings.
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Lensmaker design separates material and shape. Raising $n_{\ell}/n_m$ increases
power for fixed radii. Reducing either radius magnitude also increases the
curvature contribution. The two surfaces can trade curvature while preserving the
same focal length, which permits mechanical and aberration constraints to influence
the chosen form. Lensmaker calculations require radii in a consistent length unit;
the reciprocal sum then gives focal length in that unit.

Substituting the lensmaker result into the two-surface imaging relation gives the
thin-lens equation.

$$

\frac{1}{s}+\frac{1}{s'}=\frac{1}{f}.

$$

A real object in front of a positive lens has $s>0$. If $s>f$, the solution has
$s'>0$, so physical rays meet on the outgoing side and form a real image. If
$0<s<f$, the solution has $s'<0$, so outgoing rays diverge and their backward
extensions locate a virtual image. A negative lens with a real object has $s'<0$
for every positive object distance.

$$
% caption: Object and image triangles for a thin converging lens. Similar paraxial
% triangles relate object height $y$, image height $y'$, and signed distances
% $s$ and $s'$; the inverted real image has a negative lateral magnification.
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$$

Similar triangles give the lateral magnification.

$$

m=\frac{y'}{y}=-\frac{s'}{s}.

$$

The minus sign distinguishes orientation. A positive $m$ describes an upright
image and a negative $m$ an inverted image. Magnification magnitude determines
relative size: $\lvert m\rvert>1$ enlarges the image, $\lvert m\rvert<1$ reduces
it, and $\lvert m\rvert=1$ preserves height. A large magnification near the focal
distance also amplifies sensitivity to object-position error.

The equation has an asymptote at $s=f$. As an object approaches the focal point
from outside, the real image distance grows without bound. As it approaches from
inside, the virtual image distance grows in magnitude on the incoming side. The
limit corresponds to an outgoing collimated beam; it is a poor configuration for
measuring image distance on a finite bench.

Lens power is reciprocal focal length expressed in meters.

$$

P=\frac{1}{f},
\qquad
[P]=\mathrm{D}=\mathrm{m^{-1}}.

$$

A $0.250\ \mathrm{m}$ positive lens has $P=4.00\ \mathrm{D}$, while a
$0.100\ \mathrm{m}$ positive lens has $P=10.0\ \mathrm{D}$. A negative focal
length produces negative power. Diopters add directly for thin lenses in contact,
which makes power convenient for preliminary optical-system calculations.

## Image regimes for converging and diverging lenses

Solving the thin-lens equation for image distance and magnification gives

$$
s'=\frac{fs}{s-f},
\qquad
m=-\frac{f}{s-f}.
$$

These forms place the focal-distance boundary at $s=f$. A positive lens has separate
real-image and virtual-image regimes on opposite sides of $s=f$. The image
classification follows from both the sign of $s'$ and the sign of $m$. Image
distance alone identifies the side of the lens; magnification sets orientation
and relative size.

For $s>2f$, a converging lens produces a real, inverted, reduced image between
$f$ and $2f$ on the outgoing side. Distant scenes use this regime: a camera
sensor placed near the focal plane receives a small real image. The principal-ray
intersection lies beyond the lens, so a screen can intercept the light there.

At $s=2f$, the image also lies at $s'=2f$. It is real, inverted, and equal in
height to the object. This special geometry works as a calibration check because
the magnification magnitude should be one. Finite lens thickness, uncertain lens
plane position, and nonparaxial rays shift the exact equality in a physical setup.

For $f<s<2f$, the real image lies beyond $2f$, is inverted, and is enlarged.
Projection systems use this regime when the desired screen image must exceed the
object. The image distance becomes increasingly sensitive as the object approaches
the focal point. A small longitudinal movement of the object then produces a large
screen displacement and a large change in magnification.

At $s=f$, the denominator of the image-distance form vanishes. Rays leave
collimated, and the image lies at infinity in the paraxial model. A lens cannot form
a finite sharp screen image from an object placed at its focal plane. This setting
serves to produce or test a collimated beam, but it is unsuitable
for direct object-screen focal-length measurement.

For $0<s<f$, a positive lens produces an upright enlarged virtual image on the
incoming side. The outgoing physical rays diverge, and dashed backward extensions
locate the image. A magnifier uses this regime: the eye receives a larger angular
extent from the virtual image, while a screen cannot receive a sharp image at the
virtual position.

$$
% caption: Upright virtual image for an object inside a positive lens’s focal
% distance. Physical rays diverge after the lens, while their backward extensions
% meet on the object side with $m>0$ and $|m|>1$.
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$$

A negative lens with a real object always gives a virtual upright reduced image.
The image lies between the lens and its incoming focal point. This stable geometry
is used to reduce the convergence of a preceding positive lens or to spread a beam.
The same sign result follows algebraically from $f<0$ and $s>0$, which give
$s'<0$ and $0<m<1$.

## Lens combinations and intermediate images

Each lens in a sequence acts on the ray bundle arriving at its own plane. Solve the
first lens, locate its image, then treat that image as the object for the second
lens. The intermediate image can be real or virtual. A real intermediate image lies
where rays actually cross before reaching the next lens. A virtual intermediate
image lies where the arriving rays would cross if the next lens were removed.
Keeping this distinction prevents incorrect object-distance signs in multielement
calculations.

For two lenses on one axis separated by distance $d$, measure the first image
distance $s'_1$ from lens 1. The location of that image relative to lens 2 sets
$s_2$. A real image to the left of lens 2 is a positive object for lens 2; a real
image to its right is a virtual object and has negative $s_2$. A drawn axis with
lens positions and signed distances is safer than inferring the sign from a verbal
description.

$$
% caption: Sequential imaging by two thin lenses. Lens 1 forms an intermediate
% image, and that location sets the object distance for lens 2; the second
% calculation uses the intermediate image’s side relative to lens 2, not its label as
% “real” or “virtual” alone.
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$$

The combined lateral magnification is the product of the individual
magnifications.

$$

m_{\rm total}=m_1m_2m_3\cdots .

$$

The product rule tracks orientation. Two negative magnifications produce an upright
final image; an odd number of negative factors produces an inverted final image.
Image location requires the signed image distance from the last lens in addition to
magnification. This separation matters when a diverging lens follows a converging
lens: the second lens can reduce convergence while leaving no real intermediate
screen image.

Two thin lenses in contact have negligible separation. Adding their thin-lens
equations eliminates the intermediate image and gives

$$

\frac{1}{f_{\rm eff}}
=\frac{1}{f_1}+\frac{1}{f_2},
\qquad
P_{\rm eff}=P_1+P_2.

$$

The relation holds when both elements share the same surrounding medium and their
principal planes are effectively coincident. A positive and a negative lens can
therefore form a weaker positive system, a weaker negative system, or zero net power.
Zero net power describes only the first-order paraxial power. Apertures, aberrations,
and principal-plane locations still determine beam clipping, lateral displacement,
and image quality.

For lenses separated by distance $d$ in air, the effective power becomes

$$

\frac{1}{f_{\rm eff}}
=\frac{1}{f_1}+\frac{1}{f_2}
-\frac{d}{f_1f_2}.

$$

The separation term changes system power even when both individual focal lengths
remain fixed. Its size grows when the elements have substantial power or when their
spacing approaches either focal length. Use the contact approximation only when the
separation is negligible on the relevant focal-length scale.

An afocal pair sends collimated input to collimated output. For two positive lenses,
the focal planes coincide when $d=f_1+f_2$. A collimated input then focuses at the
back focal plane of lens 1 and leaves lens 2 as a collimated beam. The angular
magnification of this pair belongs to optical-instrument design, but the shared focal
plane is an alignment condition for lens combinations.

The thin-lens approximation replaces two refractions with an equivalent paraxial
mapping at one plane. In a thick lens, object and image distances are measured from
principal planes inside or near the glass. The simplified model places those planes
at the central plane. A large separation between the real surfaces, a high numerical
aperture, or a short focal length makes the difference measurable.

A converging lens is thicker near its axis than near its edge when its index exceeds
that of the surrounding medium. Parallel rays leave it directed toward a real focal
point on the outgoing side. A diverging lens is thinner near its axis; parallel rays
leave it spreading as though they had originated from a focal point on the incoming
side. Index contrast and surface curvatures together determine the focal-length
sign.

$$
% caption: Parallel-ray response of positive and negative thin lenses. A positive
% lens sends a collimated input toward a real focus on the outgoing side, whereas a
% negative lens produces rays whose backward extensions meet at a virtual focus on
% the incoming side.
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\filldraw[draw=black,fill=black!8] (4.1,1.5) circle (1.4pt);
\draw[->,black,thick] (3.5,1.78)--(4.8,1.78);
\draw[->,black,thick] (3.5,1.5)--(4.8,1.5);
\draw[->,black,thick] (3.5,1.22)--(4.8,1.22);
\draw[black,thick] (4.8,1.78)--(5.9,2.22);
\draw[black,thick] (4.8,1.5)--(5.9,1.5);
\draw[black,thick] (4.8,1.22)--(5.9,.78);
\draw[black,dashed] (4.8,1.78)--(4.1,1.5);
\draw[black,dashed] (4.8,1.22)--(4.1,1.5);
\node[above] at (4.1,1.5) {$F$};
\node[below] at (4.8,.42) {negative};
\end{tikzpicture}
$$

A ray diagram must indicate which arrows represent physical rays and which dashed
lines represent backward extensions. Solid rays carry light away from an object or
toward a detector. Dashed extensions mark the backward geometric continuation that
locates a virtual image or virtual focus. A screen receives an image where physical
rays converge in its plane.

## Focal-Length Measurement

Focal length is usually easier to measure than surface radii. A distant object gives
the simplest estimate: place a screen behind a positive lens and focus an image of a
target far away. The lens-to-screen distance approximates $f$ when the object
distance is much larger than the focal length. The approximation has a known bias:
the exact image distance exceeds $f$ for every finite positive object distance.
Target distance, lens-plane definition, screen positioning, and focus criterion
therefore limit the accuracy of a distant-object measurement.

A finite object and real screen image require measurement of $s$ and $s'$ from
the effective lens plane, followed by calculation of

$$

f=\frac{ss'}{s+s'}.

$$

The object and screen should remain perpendicular to the bench axis, and the lens
should be centered on that axis. Translate the screen across the full sharp region
before choosing the best-focus coordinate. A high-contrast target with fine marks
gives a sharper focus criterion than a broad bright patch. Record the direction and
zero of every translation stage; a lens mount can shift its optical plane from the
scale mark.

Finite-distance measurements gain precision when several object positions are used.
For each pair, calculate a focal-length estimate or fit the linearized relation

$$
\frac{1}{s'}=-\frac{1}{s}+\frac{1}{f}.
$$

The intercept equals inverse focal length, while the expected slope is $-1$. A
slope far from $-1$ flags a distance reference error, a tilted screen, a lens-plane
offset, or an inconsistent focus criterion. Fitting raw distances with an
uncertainty-aware nonlinear model is preferred when uncertainties in both distances
are substantial; the linear plot remains valuable for diagnostic inspection.

Bessel’s method avoids direct measurement from the lens plane. Place object and
screen a fixed distance $L$ apart, with $L>4f$. Two lens positions between them
form sharp screen images: one produces a magnified image and the other a reduced
image. Let $D$ be the distance between those two lens positions. The focal length
is

$$

f=\frac{L^2-D^2}{4L}.

$$

The method uses only the object-screen separation and the displacement between lens
positions. It reduces sensitivity to uncertain principal-plane location, provided
the same lens reference mark is used for both positions. The two sharp settings
coalesce when $L=4f$; a bench set too close to that boundary makes $D$ small and
the two focus positions difficult to resolve.

$$
% caption: Bessel focal-length geometry. A fixed object-screen separation $L$
% admits two sharp lens positions separated by $D$, one with a magnified image and
% one with a reduced image; their separation determines $f=(L^2-D^2)/(4L)$.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0.3,1.5)--(6.1,1.5) node[right] {bench axis};
\draw[black,very thick] (0.7,.6)--(0.7,2.4);
\draw[black,very thick] (5.7,.6)--(5.7,2.4);
\draw[acc,very thick] (2.18,.5) .. controls (1.9,1.02) and (1.9,1.98) .. (2.18,2.5);
\draw[acc,very thick] (2.42,.5) .. controls (2.7,1.02) and (2.7,1.98) .. (2.42,2.5);
\draw[black,very thick] (3.98,.5) .. controls (3.7,1.02) and (3.7,1.98) .. (3.98,2.5);
\draw[black,very thick] (4.22,.5) .. controls (4.5,1.02) and (4.5,1.98) .. (4.22,2.5);
\draw[<->,black] (0.7,.72)--(5.7,.72);
\draw[<->,black] (2.3,2.72)--(4.1,2.72);
\node[below] at (3.2,.72) {$L$};
\node[above] at (3.2,2.72) {$D$};
\node[above] at (0.7,2.4) {object};
\node[above] at (5.7,2.4) {screen};
\node[below] at (2.3,.42) {position A};
\node[below] at (4.1,.42) {position B};
\end{tikzpicture}
$$

### Measurement protocol, uncertainty, and reporting

Set up an optical bench in this order:

1. **Axis:** align the object center, lens center, and screen center to one bench
   line. Record the stage references used for each component.
2. **Aperture:** choose an opening that leaves enough light for a clear focus while
   excluding strongly aberrated outer rays when needed.
3. **Target:** use high-contrast marks with features in both horizontal and vertical
   directions, so tilt and astigmatic blur are visible.
4. **Focus search:** move the screen through the sharp region from both directions,
   record repeated best-focus positions, and account for stage backlash.
5. **Distance reduction:** use the correct effective lens plane or a method such as
   Bessel’s method that reduces principal-plane sensitivity.
6. **Validation:** compare measured image orientation, size, and distance with the
   signed thin-lens prediction before averaging repeated runs.

Eye- or camera-selected screen position estimates a focus coordinate. Image
contrast, target spectrum, detector pixel pitch, and aperture shape the apparent
focus minimum. A reproducible rule can use peak edge contrast, minimum spot width,
or a specified image-metric maximum. Apply the same rule at every object distance.
Changing focus criteria between runs can imitate a focal-length drift.

An uncertainty budget should include more than scale resolution. Lens-plane
reference, object and screen tilt, focus repeatability, wavelength band, target
distance, thermal expansion, and model inadequacy can each contribute. Some terms
are correlated: moving a common bench reference shifts several distances together,
and a lens mount offset enters every finite-distance reading. Model shared offsets
with a calibration parameter or covariance term that preserves their correlation.

Report a focal length with the method and conditions that define it. A focal-length
result states lens identity, surrounding medium, source wavelength or band, aperture,
temperature when relevant, measurement method, focal-length estimate, uncertainty
coverage, and dominant systematic limitations. For example: “positive glass lens in
air, green source, Bessel method, $f=0.349\ \mathrm{m}$ with stated standard
uncertainty; screen focus repeatability and lens-position reference dominate.” The
statement identifies what another measurement should reproduce.

Compare independent methods when the result will be used for design or calibration.
A distant-target estimate checks the approximate scale. Finite-distance data test
the thin-lens relation over a range of conjugates. Bessel’s method avoids direct
lens-plane measurement. Agreement within a stated uncertainty range supports the
model and alignment. A persistent method-dependent difference points to principal
planes, aberration, wavelength mismatch, or an unmodeled mechanical reference.

The Bessel relation follows from the two object distances. For the two lens
positions, the distances to object and screen interchange.

$$
s_{\rm near}=\frac{L-D}{2},
\qquad
s_{\rm far}=\frac{L+D}{2}.
$$

Their product equals $fL$, giving the stated expression after division by $L$.
The associated magnification magnitudes are reciprocal.
$\lvert m_{\rm near}\rvert=s_{\rm far}/s_{\rm near}$ and
$\lvert m_{\rm far}\rvert=s_{\rm near}/s_{\rm far}$. The pair therefore provides
an internal check on the image measurements as well as on focal length.

> **Worked example (Bessel focal length).** An object and screen sit
> $L=1.70\ \mathrm{m}$ apart, and the two sharp lens positions are separated by
> $D=0.720\ \mathrm{m}$. The focal length follows from the separation alone:
>
> $$
> f=\frac{L^2-D^2}{4L}
> =\frac{(1.70\ \mathrm{m})^2-(0.720\ \mathrm{m})^2}
> {4(1.70\ \mathrm{m})}
> =0.349\ \mathrm{m}.
> $$
>
> The two lens positions lie $s_{\rm near}=(L-D)/2=0.490\ \mathrm{m}$ and
> $s_{\rm far}=(L+D)/2=1.21\ \mathrm{m}$ from the object, so the magnification
> magnitudes are $2.47$ and $0.405$, reciprocal within rounding. That reciprocity
> catches a swapped position label or a distance read from the wrong endpoint.

The Bessel estimate has local uncertainty

$$
u_f^2=
\left[
\frac{L^2+D^2}{4L^2}u_L
\right]^2
+
\left[
\frac{D}{2L}u_D
\right]^2,
$$

when $L$ and $D$ are independent. All terms use the same length unit. The
object-screen separation is often measured more accurately than the small lens
displacement, so the $D$ term can dominate. Repeated focus searches at each lens
position give a practical estimate of $u_D$. A stage backlash measurement should
be included when the two positions are approached from opposite directions.

Lens tilt, decenter, and screen tilt broaden the focus minimum. A tilted lens can
place one side of the target in focus at a different screen coordinate from the
opposite side. Center the object, lens, and screen on one bench axis before recording
distances. Use a target with features across the aperture, observe focus on both
sides, and state the criterion used to select the screen position. Include these
alignment checks in the measurement model.

## Aberration, aperture, and thin-lens model limits

The thin-lens equation is a paraxial result. It assumes rays remain close to the
axis, surface slopes are small, and the lens can be reduced to one plane and one
focal length. A real spherical lens bends marginal rays more strongly than paraxial
rays. The rays then cross the axis over a range of longitudinal positions rather
than at one point. The resulting on-axis blur is spherical aberration.

$$
% caption: Spherical aberration in a positive spherical lens. Marginal rays cross
% the axis closer to the lens than paraxial rays, so no single screen plane makes
% every ray from the on-axis point simultaneously sharp.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0.3,1.5)--(6.1,1.5) node[right] {axis};
\draw[acc,very thick] (2.88,.4) .. controls (2.58,1.0) and (2.58,2.0) .. (2.88,2.6);
\draw[acc,very thick] (3.12,.4) .. controls (3.42,1.0) and (3.42,2.0) .. (3.12,2.6);
\draw[->,black,thick] (0.5,2.3)--(3.0,2.3);
\draw[->,black,thick] (0.5,1.9)--(3.0,1.9);
\draw[->,black,thick] (0.5,1.1)--(3.0,1.1);
\draw[->,black,thick] (0.5,0.7)--(3.0,0.7);
\draw[black,thick] (3.0,2.3)--(4.3,1.5);
\draw[black,thick] (3.0,1.9)--(5.0,1.5);
\draw[black,thick] (3.0,1.1)--(5.0,1.5);
\draw[black,thick] (3.0,0.7)--(4.3,1.5);
\filldraw[draw=black,fill=black!8] (4.3,1.5) circle (1.6pt);
\filldraw[draw=black,fill=black!8] (5.0,1.5) circle (1.6pt);
\node[above] at (1.4,2.42) {edge rays};
\node[above] at (4.02,1.76) {marginal};
\node[above] at (5.34,1.6) {paraxial};
\end{tikzpicture}
$$

An aperture stop limits the ray height at the lens and can reduce spherical blur by
excluding marginal rays. The cost is reduced collected light and increased diffraction
spread at a sufficiently small opening. Aperture selection therefore balances
geometrical blur, signal level, exposure time, and diffraction. The thin-lens
equation contains no aperture diameter, so it can predict the paraxial image plane
while remaining silent about the spot diameter on that plane.

Chromatic aberration follows from dispersion. Lensmaker power changes with
refractive index, and refractive index changes with wavelength. Blue components
usually encounter a larger index than red components in ordinary glass, so they
reach focus closer to a positive lens. A broadband image can therefore show colored
edges or a focus setting that favors one spectral band. The effect survives perfect
mechanical alignment because it arises from the material response.

$$
% caption: Longitudinal chromatic aberration of a positive lens. Wavelength-
% dependent refractive index produces different focal positions for blue and red
% components, so one screen plane cannot place both colors at their paraxial focus.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\definecolor{warm}{HTML}{B23A48}
\draw[->,black] (0.3,1.5)--(6.1,1.5) node[right] {axis};
\draw[acc,very thick] (2.88,.4) .. controls (2.58,1.0) and (2.58,2.0) .. (2.88,2.6);
\draw[acc,very thick] (3.12,.4) .. controls (3.42,1.0) and (3.42,2.0) .. (3.12,2.6);
\draw[->,black,thick] (0.5,1.9)--(3.0,1.9);
\draw[->,black,thick] (0.5,1.1)--(3.0,1.1);
\draw[acc,thick] (3.0,1.9)--(4.2,1.5);
\draw[acc,thick] (3.0,1.1)--(4.2,1.5);
\draw[warm,thick] (3.0,1.9)--(5.1,1.5);
\draw[warm,thick] (3.0,1.1)--(5.1,1.5);
\filldraw[draw=acc,fill=acc!12] (4.2,1.5) circle (1.7pt);
\filldraw[draw=warm,fill=warm!12] (5.1,1.5) circle (1.7pt);
\node[above] at (4.2,1.78) {blue};
\node[above] at (5.15,1.78) {red};
\end{tikzpicture}
$$

Off-axis points introduce additional departures from the simple one-axis image
construction. Coma gives an asymmetric spot, astigmatism gives different sagittal
and tangential focus positions, and distortion changes magnification across an
extended image. These effects grow with aperture and field angle. A central
principal-ray diagram locates the paraxial image; edge sharpness and shape fidelity
require a field-dependent image-quality assessment.

Lens thickness changes the distance reference. A thick lens has front and back
principal planes that can lie inside or outside the glass. Measuring $s$ and
$s'$ from a lens rim or mounting ring then introduces a systematic offset. High
power lenses, immersion in a liquid, and compound systems make principal-plane
location significant. The thin-lens equation remains an adequate first-order
description when distances are measured from effective planes.

Surface quality, internal stress, coating variation, and material inhomogeneity can
add scatter or image defects beyond the ideal model. A measurement report should
separate paraxial focal length from image quality. Two lenses can share the same
measured $f$ while giving different spot size, chromatic focus, transmission, or
off-axis performance. Focal length alone is a necessary but incomplete lens
specification.

## Worked Analysis and Audit

A lensmaker calculation begins with physical orientation.

> **Worked example (Lensmaker focal length).** A glass lens in air has
> $n_{\ell}=1.50$, $R_1=+0.100\ \mathrm{m}$, and $R_2=-0.150\ \mathrm{m}$. The
> positive first radius places the first center of curvature on the outgoing side;
> the negative second radius places the second center on the incoming side.
> Substituting into the lensmaker relation,
>
> $$
> \frac{1}{f}
> =(1.50-1.00)
> \left(
> \frac{1}{0.100\ \mathrm{m}}
> -\frac{1}{-0.150\ \mathrm{m}}
> \right)
> =8.33\ \mathrm{m^{-1}},
> \qquad
> f=0.120\ \mathrm{m}.
> $$
>
> The positive focal length matches the thicker-at-center geometry, and the power is
> $P=8.33\ \mathrm{D}$.

Unit conversion belongs inside the calculation: radii in
centimeters yield focal length in centimeters, but power requires focal length in
meters. A result reported as “8.33 D” after substituting centimeter radii has a
factor-of-one-hundred error.

> **Worked example (Positive-lens image).** Place a real object
> $0.300\ \mathrm{m}$ in front of that same $0.120\ \mathrm{m}$ positive lens. The
> thin-lens equation gives
>
> $$
> \frac{1}{s'}
> =\frac{1}{0.120\ \mathrm{m}}
> -\frac{1}{0.300\ \mathrm{m}},
> \qquad
> s'=0.200\ \mathrm{m},
> \qquad
> m=-\frac{0.200}{0.300}=-0.667.
> $$
>
> The image is real ($s'>0$), inverted ($m<0$), and reduced ($\lvert m\rvert<1$).
> Each conclusion reads off a separate signed quantity.

A
statement such as “the image is small” omits its orientation and side of the lens.
The ray diagram should show an intersection on the outgoing side and a lower image
height below the axis, matching the algebraic signs.

In a separated pair, retain every intermediate sign.

> **Worked example (Separated-pair effective focal length).** Take
> $f_1=+0.100\ \mathrm{m}$, $f_2=-0.200\ \mathrm{m}$, and $d=0.050\ \mathrm{m}$.
> The separated-lens power relation gives
>
> $$
> \frac{1}{f_{\rm eff}}
> =10.0\ \mathrm{m^{-1}}
> -5.00\ \mathrm{m^{-1}}
> -\frac{0.050\ \mathrm{m}}{(0.100\ \mathrm{m})(-0.200\ \mathrm{m})}
> =7.50\ \mathrm{m^{-1}},
> \qquad
> f_{\rm eff}=0.133\ \mathrm{m}.
> $$
>
> The spacing term is $+2.50\ \mathrm{D}$, so the pair is stronger than the
> $5.00\ \mathrm{D}$ contact value and remains net converging.

A real object in a multielement system requires sequential lens equations for the
individual image locations needed to check whether the effective-lens approximation
is adequate for the object and screen planes in use.

Use limiting cases before accepting a result. An object far beyond a positive lens
should give $s'$ near $f$ and a small negative magnification. An object at
$2f$ should give $s'=2f$ and $m=-1$. An object inside $f$ should give
negative $s'$ and positive $m$. A negative lens with a real object should give
negative $s'$, positive $m$, and a reduced image. These checks locate sign
errors before a numerical answer is rounded.

Lensmaker limits provide a second audit. If the lens index approaches the surrounding
index, power approaches zero and focal length grows without bound. If both surface
radii grow without bound, the element becomes a plane plate and its thin-lens power
also approaches zero. Reversing illumination preserves focal-length magnitude for a
lens in the same surrounding medium; the signed radii swap their roles consistently.
A calculation violating any of these limits has an index, radius, or unit error.

### Sign, reference-plane, and model audit

Assign a propagation direction and one reference plane before applying the
thin-lens equation. A compact sign audit has five entries:

- **Incoming side:** identify the side from which the ray bundle reaches the lens.
  A physical object on that side has positive object distance.
- **Outgoing side:** locate the physical image or the intersection of backward ray
  extensions. A real image on the outgoing side has positive image distance.
- **Focal sign:** use positive focal length for a converging lens and negative focal
  length for a diverging lens in the same surrounding medium.
- **Image orientation:** classify orientation from the magnification sign after
  solving; compare the result with the ray construction.
- **Distance plane:** state whether all distances are from the thin-lens plane,
  measured principal planes, or a mechanical mount reference.

The same sign audit applies to a virtual object. Suppose rays arrive at a second lens
already converging toward a point beyond that lens. The point lies on the outgoing
side for lens 2, so its object distance is negative. The thin-lens equation then
propagates the converging bundle through the second element. Assigning a positive
object distance changes both the computed image location and the final
magnification sign.

Use physical ray direction as a consistency check. A real image is a location at
which physical outgoing rays meet. A virtual image is a location at which backward
extensions meet. A virtual object is a location toward which incoming rays are
already converging. These three cases have distinct geometry even when a diagram
marks each with a point on the axis. A screen detects a real image only at its
physical plane.

The thin-lens equation is symmetric in object and image distance for a fixed focal
length. Reversing a real object and real image through the same lens interchanges
$s$ and $s'$. The magnification magnitude changes from $\lvert s'/s\rvert$ to
its reciprocal, while the two directions share the same focal length. This
reciprocity gives a direct experimental check for a lens in the same medium on both
sides. A mismatch beyond uncertainty can indicate an incorrect reference plane,
surface stress, an aperture-dependent focus shift, or an unrecorded change of
wavelength.

Finite-distance focal-length estimates have an uncertainty structure that depends on
both measured distances. For independent uncertainties $u_s$ and $u_{s'}$,

$$
u_f^2=
\left[
\frac{s'^2}{(s+s')^2}u_s
\right]^2
+
\left[
\frac{s^2}{(s+s')^2}u_{s'}
\right]^2.
$$

The uncertainty coefficients define the preferred object-screen geometry. In a reduced
image, $s$ is large and $s'$ lies near $f$, so image-distance uncertainty
carries the larger coefficient. In an enlarged image, object-distance uncertainty
dominates. Equal distance uncertainties contribute equally at unit magnification.

### Measurement-domain audit

| item | quantity that must be stated | failure if omitted |
|---|---|---|
| distance origin | thin-lens plane, principal plane, or mount reference | a shared offset biases $s$ and $s'$ |
| surrounding medium | $n_m$ and lens index at the stated wavelength | incorrect focal length or lens sign |
| aperture and conjugates | paraxial range and object-screen geometry | aberration or unstable focus treated as distance noise |
| wavelength and temperature | source band and specimen temperature | chromatic or thermal shift hidden in the focal-length result |
| image acceptance | screen or sensor area and field position | axial focus mistaken for full-field image quality |

- **Reference offsets.** A shift $\delta$ in the same lens-plane mark can change
  reported object and image distances together with correlated signs. Repetition does
  not remove that offset. Constrain it with a principal-plane measurement, a Bessel
  comparison, or a calibration lens of known focal length, and list the chosen
  reference with every distance.
- **Medium and spectrum.** Lensmaker calculations use
  $n_{\ell}/n_m-1$, not the air-only factor $n_{\ell}-1$. Immersing glass in a
  liquid with a closer index reduces power and lengthens focal length; an external
  index above the lens index can make an air-converging shape diverge. Dispersion
  changes the result across a visible band, while thermal expansion and
  temperature-dependent index shift curvature and power. Those effects become part
  of the uncertainty model when their size approaches the claimed interval.
- **Approximation domain.** Treat the result as an effective paraxial focal length
  when the lens is thick, the aperture is large, or a conjugate lies close to the
  lens. Principal-plane shifts, spherical aberration, and nonparaxial refraction can
  then exceed the stage resolution. Working distance is a separate mechanical
  clearance measured from a housing or mount; it is not a substitute for the
  principal-plane distance in the thin-lens equation.
- **Image acceptance.** A real image can occur at the predicted axial distance and
  still miss a finite sensor because of object height, field angle, or decenter.
  State whether focus was tested on axis, at one off-axis target, or over an image
  region. Translate the screen on both sides of best focus and retain a focus
  interval when the sharpness metric has a broad plateau. Verify a virtual image
  with ray geometry, a second lens, or a camera receiving the diverging bundle.
- **Ray and equation checks.** Draw the optic axis, lens plane, incoming direction,
  focal points, object, and expected image side. Use principal rays to test signs
  and solve the signed equation for numerical distance and magnification. If they
  disagree, inspect focal-point side, lens type, scale reference, and dashed
  extensions before altering the algebra.
- **Residuals and conjugates.** Store raw positions and calculate focal length for
  every object-screen pair. Plot deviations against object distance, image distance,
  and acquisition order. Angle dependence can indicate target tilt or aperture
  change; time dependence can indicate thermal drift, stage settling, or source
  spectrum. Near the focal distance, small object-position errors produce large
  screen-position changes. At very large object distance, finite-distance correction
  becomes difficult to resolve. A moderate conjugate pair with a clear real image,
  adequate target size, and accurately measured distances is often more stable.
  Bessel’s method formalizes that choice through two sharp positions at fixed
  object-screen separation.
- **Calculation record.** Retain raw distances and repeated focus positions, then
  classify image side, orientation, and relative size from the solved signs and
  magnification. Check distant-object, focal-point, unit-magnification, and
  negative-lens limits. Report medium, wavelength band, aperture, reference plane,
  focus criterion, and finite-thickness limitation before rounding the focal length.

A focal-length statement such as “positive lens, air, stated wavelength band, stated
aperture, paraxial focal length, stated uncertainty” identifies a reproducible
quantity. Diopters supply paraxial power in that stated medium; usable aperture,
chromatic focus, and working distance require separate specifications before a lens
combination or bench image can be predicted.

### Finite-conjugate check

> **Worked example (Finite-conjugate reduction).** A converging lens has paraxial
> focal length $f=0.100\ \mathrm m$, and a real object sits at
> $s=0.300\ \mathrm m$ from the selected lens plane. The signed thin-lens reduction
> gives
>
> $$
> \frac{1}{s'}=\frac{1}{0.100}-\frac{1}{0.300}
> =6.67\ \mathrm{m^{-1}},
> \qquad
> s'=0.150\ \mathrm m,
> \qquad
> m=-\frac{s'}{s}=-0.500.
> $$
>
> The image is real, inverted, and half the object height.

Moving the screen
to $0.150\ \mathrm m$ from the same reference plane tests the axial prediction.
Moving it a known distance on either side tests the focus metric and its plateau.
The calculation is invalid if $0.300\ \mathrm m$ was measured from a housing edge
while $0.150\ \mathrm m$ was measured from a principal plane; both distances must
use the same declared origin.

Repeat the reduction after interchanging the real object and screen locations. The
same lens should produce the reciprocal magnification and the same focal length
within uncertainty. A discrepancy gives a concrete diagnostic sequence: verify the
scale origin, source wavelength, aperture stop, lens orientation, screen normal, and
target alignment before treating it as a change in lens power. A lens whose focal
length is inferred from several conjugates should show residuals without a systematic
trend in either distance.

Bessel's method provides a second finite-conjugate check when object and screen are
fixed a distance $L$ apart and two sharp lens positions are accessible. With their
separation $d$, the paraxial result is

$$
f=\frac{L^2-d^2}{4L},
\qquad L>4f.
$$

The condition $L>4f$ is a geometry requirement, not a numerical preference. Record
both sharp positions, the focus criterion, lens-plane reference, and aperture. Compare
the Bessel result with the direct object-image reduction only after placing both in the
same medium and wavelength band. Agreement tests the distance geometry; disagreement
can expose principal-plane offset, aberration, chromatic shift, or a focus criterion
that varies with image size.

Two thin lenses separated by distance $d$ in one medium provide a related bench
check. Their paraxial powers combine as

$$
P_{\mathrm{eq}}=P_1+P_2-dP_1P_2,
$$

when the stated separation is measured between the thin-lens reference planes. The
last term cannot be omitted when separation is comparable with either focal length.
Use the combined power for a preliminary prediction, then verify the actual image
plane with the signed sequential lens equations. A thick lens group, a changing
medium, or an unrecorded principal-plane shift requires a model beyond this compact
two-thin-lens relation.

Measure the separation from the same reference planes used for the individual focal
lengths. A mechanical gap between housings is not generally the optical separation
between principal planes. Reversing the lens order can leave the paraxial combined
power unchanged in a uniform medium while changing aberration, aperture clipping, or
working distance. Report those practical differences separately from the paraxial
power calculation.