Thin Lenses

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 $s^{\prime }$ is measured from the lens plane to the final image. The signs below match the refracting-surface convention used for thin lenses.

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

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.

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.

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

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.

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.

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

Similar triangles give the lateral magnification.

The minus sign distinguishes orientation. A positive $m$ describes an upright image and a negative $m$ an inverted image. Magnification magnitude determines relative size: $|m|>1$ enlarges the image, $|m|<1$ reduces it, and $|m|=1$ 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.

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

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 $s^{\prime }$ 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 $s^{\prime }=2f$. 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.

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 $s^{\prime }<0$ 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 $s_1^{\prime }$ 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.

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

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

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

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.

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 $s^{\prime }$ from the effective lens plane, followed by calculation of

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

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

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.

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.

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

The Bessel estimate has local uncertainty

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.

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.

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 $s^{\prime }$ 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.

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.

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.

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 $s^{\prime }$ near $f$ and a small negative magnification. An object at $2f$ should give $s^{\prime }=2f$ and $m=-1$. An object inside $f$ should give negative $s^{\prime }$ and positive $m$. A negative lens with a real object should give negative $s^{\prime }$, 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 $s^{\prime }$. The magnification magnitude changes from $|s^{\prime }/s|$ 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 $u_{s^{\prime }}$,

The uncertainty coefficients define the preferred object-screen geometry. In a reduced image, $s$ is large and $s^{\prime }$ 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

• 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

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

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

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.