Thick Lenses and Principal Planes

A telephoto lens is a paradox in a barrel: it delivers the long focal length of a distant-object lens while being physically far too short to contain it. A diver's eye, a microscope objective, a camera zoom — none of them is thin, yet we insist on describing all of them with the single tidy relation $1/s_o+1/s_i=1/f$ that was derived for a lens of zero thickness. The reconciliation is the central achievement of Gaussian optics: any centred system, however thick or compound, images exactly like a thin lens of the right focal length — provided you measure object and image distances not from the glass, but from two specially chosen reference planes, the principal planes. A real lens refracts light twice, once at each surface, with a physical path of glass in between. Trace a ray parallel to the axis: it travels straight until it seems to kink at the back principal plane $H'$ and heads for the back focal point $F'$ . Trace a ray from the front focal point $F$ : it emerges parallel, appearing to bend at the front principal plane $H$ . Those two "effective refraction" planes, together with the two focal points and two nodal points, are the six cardinal points that completely specify first-order imaging. The principal planes are unit-magnification conjugates — a ray crosses from $H$ to $H'$ at the same height — which is precisely why the thin-lens equation survives with distances referred to them. Two tools locate the cardinal points, and they agree. Direct algebra composes the two single-surface refractions and the internal glass path into a thick-lens focal length with a thickness correction that vanishes as the lens thins. The ray-transfer (matrix) method multiplies three simple matrices and reads the same focal length and plane positions straight off the product. Get either right and telephoto barrels, retrofocus wide-angles, and the schematic eye all fall into the same one-line imaging law.

Key takeaways

A thick lens refracts at two separated surfaces; its imaging is still governed by $1/s_o+1/s_i=1/f$ provided distances are measured from the principal planes $H$ (objects) and $H'$ (images).
The six cardinal points — two focal, two principal, two nodal — fully specify first-order (Gaussian) imaging of any centred system.
The principal planes are unit-magnification conjugates; a ray keeps its height crossing from $H$ to $H'$ , so they act as the effective refracting planes.
The thick lensmaker's equation $1/f=(n-1)\!\left[1/R_1-1/R_2+(n-1)d/(nR_1R_2)\right]$ adds a thickness term that vanishes as $d\to0$ , recovering the thin lens.
Principal-plane offsets $x_H=-f(n-1)d/(nR_2)$ , $x_{H'}=-f(n-1)d/(nR_1)$ ; for a biconvex lens both planes sit inside the glass.
Two separated thin lenses combine by Gullstrand's equation $1/f=1/f_1+1/f_2-d/(f_1f_2)$ — the design freedom behind telephoto and retrofocus lenses.
Key idea: measure from the principal planes and every thick or compound system images like a thin lens of the same effective focal length; the cardinal points relocate the reference planes and set that focal length, but the imaging law itself is unchanged.

The core insight

Every centred optical system, however thick or compound, images exactly like a thin lens of the same effective focal length once object and image distances are measured from its two principal planes $H$ and $H'$ . Those planes — unit-magnification conjugates — together with the focal and nodal points are the six cardinal points that fully specify first-order (Gaussian) imaging. The thickness merely relocates the reference planes and modifies the focal length.

Where this lives today

The principal-plane idea is what makes modern lens design possible. A telephoto lens combines a converging front group with a diverging rear group so Gullstrand's equation pushes $H'$ ahead of the barrel — a $400\,\mathrm{mm}$ focal length in a $200\,\mathrm{mm}$ tube — while a retrofocus (inverted-telephoto) wide-angle does the reverse, placing $H'$ behind the last element to leave clearance for a camera's mirror box. Both are impossible to describe without cardinal points, because their focal length and physical length are deliberately decoupled. The human eye is a thick system with air in front ( $n_o=1$ ) and vitreous humour behind ( $n_i\approx1.336$ ); its front and back focal lengths differ by that ratio, and retinal image size is computed from the nodal points (about $7\,\mathrm{mm}$ behind the cornea), since a ray through a nodal point is undeviated. Opticians still measure an unknown lens on a nodal slide, rotating it until the image stops moving — the signature that the rotation axis passes through the rear nodal point — to read off the effective focal length directly. Every zoom lens, every microscope tube length, every eyepiece-to-objective spacing is a cardinal-point calculation.

The physical picture

We keep the Hecht sign convention: light travels left to right, $s_o>0$ for a real object, $s_i>0$ for a real image, $f>0$ for a converging system — but now $s_o$ is measured from $H$ and $s_i$ from $H'$ , not from the glass. Collapse the messy internal path of a thick lens onto two imaginary planes. A ray parallel to the axis runs straight in, and refracts, in effect, only at the back principal plane $H'$ , from where it dives to the focus $F'$ one focal length away. A ray from the front focus $F$ emerges parallel, having "bent" at the front plane $H$ . Those two bend locations are the system's effective refracting surfaces.

Figure

A thick lens replaced by its cardinal points: a parallel ray appears to refract at the back principal plane H′ and crosses the axis at F′ a distance f = 20.87 cm away; the effective focal length is measured from H′, not from the vertex

The principal planes are unit-magnification conjugates: a tiny object placed at $H$ would be imaged erect and same-size at $H'$ , which is why a ray keeps its height crossing between them. For a symmetric biconvex lens both planes sit inside the glass, roughly a third of the way in from each face. For a telephoto design, $H'$ is pushed out in front of the whole assembly — so the effective focal length exceeds the barrel length. That decoupling of focal length from physical length is the entire punchline of the thick-lens picture.

Figure

Punchline — effective focal length of an equiconvex lens (n = 1.5, R = ±20 cm) grows with thickness d: the thin-lens limit f = 20.00 cm at d = 0 rises to f = 20.87 cm at d = 5 cm (not 20.4 — a common arithmetic slip) and 24.0 cm at d = 20 cm

The core equations

The imaging law is unchanged in form; only the origin of measurement moves. Two or three sentences of context: refer object and image distances to $H$ and $H'$ , and the whole thick system behaves as one equivalent refractor of power $1/f$ , because the principal planes carry unit magnification and absorb the internal path into the definitions of $H,H',f$ .

\frac{1}{s_o}+\frac{1}{s_i}=\frac{1}{f} \qquad (1)

where: $s_o$ — object distance from the front principal plane $H$ ; $s_i$ — image distance from the back principal plane $H'$ ; $f$ — effective focal length, from a principal plane to its focal point. Same equation as a thin lens, new reference planes. The effective focal length itself carries a thickness correction. The context: composing the two surface powers is not simply additive because the second surface acts on a ray already bent and displaced by the first, and that coupling is the $d$ -term:

\frac{1}{f}=(n-1)\!\left[\frac{1}{R_1}-\frac{1}{R_2}+\frac{(n-1)\,d}{n\,R_1 R_2}\right] \qquad (2)

where: $n$ — lens index (in air); $R_1,R_2$ — signed front/back radii; $d$ — vertex-to-vertex thickness. Setting $d=0$ deletes the last term and recovers the thin-lens lensmaker's equation. The whole bracket is multiplied by a single $(n-1)$ ; the thickness term already contains a second $(n-1)$ inside — a place where arithmetic goes wrong (see the warning). The principal planes are located relative to the two glass vertices by:

x_H=-\frac{f(n-1)\,d}{n\,R_2},\qquad x_{H'}=-\frac{f(n-1)\,d}{n\,R_1} \qquad (3)

where: $x_H$ — offset of $H$ from the front vertex $V_1$ (positive toward the image); $x_{H'}$ — offset of $H'$ from the back vertex $V_2$ . For a biconvex lens ( $R_1>0$ , $R_2<0$ ) both offsets place the planes inside the glass. Two separated elements combine by Gullstrand's equation — the same relation whether the gap is glass (one thick lens) or air (two lenses):

P=\frac{1}{f}=P_1+P_2-\frac{d}{n}\,P_1P_2 \qquad (4)

where: $P=1/f$ — total power; $P_1,P_2$ — component powers; $d$ — separation; $n$ — index between them ( $n=1$ for two air-spaced lenses). For two thin lenses, $1/f=1/f_1+1/f_2-d/(f_1f_2)$ . Choosing $P_1>0$ , $P_2<0$ pushes $H'$ far in front of the assembly — the telephoto trick.

Derivation AThe thick lensmaker's equation by composing two spherical surfaces

This is the owned structural result: we build the finite-thickness focal length from scratch by refracting at the front surface, translating the intermediate image across the glass, and refracting again at the back surface — recalling the single-surface relation of refraction at a spherical surface without re-deriving it. The reward is the thickness term $(n-1)d/(nR_1R_2)$ , the one piece of physics that separates a real lens from its idealisation, and a clean check that it vanishes as the glass thins. The bookkeeping is a warning in miniature: the thickness correction already carries its own factor of $(n-1)$ .

Step 1. Refract a collimated ray at surface 1 (air→glass): $\dfrac{n}{s_{i1}}=\dfrac{n-1}{R_1}$ for $s_o\to\infty$ , so the internal image forms at $s_{i1}=\dfrac{nR_1}{n-1}$ from $V_1$ . Why start collimated: a parallel input sends the emergent ray through $F'$ , so the final image distance directly measures the focal length.

Step 2. Translate to surface 2: the image of surface 1 is the object for surface 2, at object distance $s_{o2}=d-s_{i1}$ (a virtual object beyond $V_2$ when $s_{i1}>d$ ). Why subtract $d$ : the glass path of thickness $d$ separates the two refractions, and this separation is exactly what makes the lens "thick".

Step 3. Refract at surface 2 (glass→air) with power $(1-n)/R_2$ , and eliminate the intermediate image. Assembling with reduced angles, the ray-transfer product $S_2\,T(d/n)\,S_1$ has lower-left element $C=-1/f$ , and simplifying gives $\dfrac1f=(n-1)\!\left[\dfrac1{R_1}-\dfrac1{R_2}+\dfrac{(n-1)d}{nR_1R_2}\right]$ . Why the matrix form is cleanest: the coupling of the two surfaces through the glass translation is exactly the matrix product, and $C$ reads off the total power in one line.

Step 4. Check the thin limit: $d\to0$ removes the last bracket term and leaves $\dfrac1f=(n-1)\!\left(\dfrac1{R_1}-\dfrac1{R_2}\right)$ — the ordinary lensmaker's equation, with $H$ and $H'$ collapsing onto the single lens plane. Why this check matters: it certifies that the thick-lens machinery is a genuine generalization, not a different theory. *The ABCD product $S_2 T(d/n) S_1$ gives $-C=(n-1)[1/R_1-1/R_2+(n-1)d/(nR_1R_2)]$ with $\det M=1$ ; the $d\to0$ limit reproduces the thin-lens form; and the equiconvex example ( $n=1.5$ , $R_1=+20$ , $R_2=-20$ , $d=5$ ) gives $f=20.87\,\mathrm{cm}$ .*

Go deeper · Derivation B · 1 minLocating the principal planes and Gullstrand's two-element rule

Derivation BLocating the principal planes and Gullstrand's two-element rule

Once the effective focal length is known, the cardinal-point offsets follow from the same system matrix, and the two-element combination law drops out as a special case — this is what lets a designer decouple focal length from barrel length. The strategy is to read the principal-plane positions off the matrix elements via $x_{H'}=f(1-A)$ and $x_H=f(D-1)$ , then apply the identical algebra to two air-spaced thin lenses to obtain Gullstrand's equation and the offsets that build a telephoto.

Step 1. For the thick lens, the system matrix referred to the vertices has $A=1-\dfrac{(n-1)d}{nR_1}$ and $D=1-\dfrac{(n-1)d}{nR_2}$ ; then $x_{H'}=f(1-A)=-\dfrac{f(n-1)d}{nR_1}$ and $x_H=f(D-1)=-\dfrac{f(n-1)d}{nR_2}$ . Why $f(1-A)$ : a parallel input ray exits at height $Ay$ and crosses at the back focal distance; subtracting $f$ gives the plane where the incoming and outgoing heights coincide — $H'$ .

Step 2. Evaluate for the equiconvex example ( $f=20.87$ , $n=1.5$ , $d=5$ , $R_1=+20$ , $R_2=-20$ ): $x_H=+1.74\,\mathrm{cm}$ and $x_{H'}=-1.74\,\mathrm{cm}$ , both inside the $5\,\mathrm{cm}$ of glass and symmetric. Why symmetric: an equiconvex lens is its own mirror image, so its two principal planes must be placed symmetrically about the centre.

Step 3. Replace the glass by an air gap and the two surfaces by two thin lenses $f_1,f_2$ : the matrix product $L_2\,T(d)\,L_1$ gives $C=-(1/f_1+1/f_2-d/(f_1f_2))$ , i.e. Gullstrand's $1/f=1/f_1+1/f_2-d/(f_1f_2)$ , with $x_{H'}=-fd/f_1$ and $x_H=+fd/f_2$ . Why the same algebra: a thick lens and a two-lens system differ only in whether the internal index is $n$ or $1$ — the composition rule is identical.

Step 4. Telephoto illustration: take $f_1>0$ (front converging) and $f_2<0$ (rear diverging); with a suitable gap, $x_{H'}$ becomes large and negative, pushing $H'$ well in front of the barrel, so the effective focal length exceeds the physical length. Why it works: the negative rear element bends the marginal ray gently, so the extrapolated principal plane lands ahead of the glass — a long lens in a short tube. *The equiconvex offsets evaluate to $x_H=+1.739$ , $x_{H'}=-1.739\,\mathrm{cm}$ ; and Gullstrand's equation for $f_1=10$ , $f_2=15$ , $d=5$ gives $f=7.5\,\mathrm{cm}$ , $x_{H'}=-3.75$ , $x_H=+2.5\,\mathrm{cm}$ .*

Limiting cases

Case	Result	Meaning
Thin-lens limit, $d\to0$	$1/f=(n-1)(1/R_1-1/R_2)$	thickness term drops; $H,H'$ collapse to one plane — ordinary lensmaker's equation
Same medium both sides	$N\equiv H,\ N'\equiv H',\ f_o=f_i$	nodal points coincide with principal points; a ray through $H$ is undeviated (lens in air)
Unequal media, $n_o\neq n_i$	$f_o/f_i=n_o/n_i$	front and back focal lengths differ; nodal points shift from principal points (the eye)
Thick plane-parallel plate	$1/f\to0$ , shift $\Delta=d(1-1/n)$	zero power, but a lateral image displacement — a pure translation, $M_T=+1$
Telephoto, $P_1>0,\ P_2<0$	$H'$ pushed ahead of the barrel	effective focal length exceeds the physical length

Common trap“The thickness correction gets another factor of (n−1)” — and other origin mistakes

Two errors dominate here. First, where you measure from: the Gaussian equation $1/s_o+1/s_i=1/f$ holds only with distances referred to the principal planes $H$ and $H'$ , not the vertices or the lens centre. For a telephoto these planes may lie entirely outside the glass, and using vertex distances gives a wrong image location and confuses the back focal distance (vertex-to- $F'$ ) with the effective focal length ( $H'$ -to- $F'$ ). Second, the arithmetic of the thickness term: the thick lensmaker's equation is $1/f=(n-1)[\dots+(n-1)d/(nR_1R_2)]$ , and the last term already carries its own $(n-1)$ , so it contributes $(n-1)^2 d/(nR_1R_2)$ to $1/f$ — you must not multiply it by $(n-1)$ a second time. For the equiconvex example ( $n=1.5$ , $R=\pm20$ , $d=5$ ) the correct answer is $f=20.87\,\mathrm{cm}$ ; the slip that double-counts $(n-1)$ produces a spurious $f\approx20.4\,\mathrm{cm}$ . Compute $(n-1)^2 d/(nR_1R_2)$ as a whole and add it to $(n-1)(1/R_1-1/R_2)$ .

Worked examples

1 · The effective focal length of a real thick lens (with the arithmetic done right)

An equiconvex lens has $R_1=+20\ \mathrm{cm}$ , $R_2=-20\ \mathrm{cm}$ , index $n=1.5$ , and thickness $d=5\ \mathrm{cm}$ . Find its effective focal length and its principal-plane positions, and compare with the thin-lens value.

Step 1. Thin part: $(n-1)(1/R_1-1/R_2)=0.5\,(1/20-(-1/20))=0.5\times0.10=0.0500\ \mathrm{cm^{-1}}$ , i.e. $f_\text{thin}=20.00\ \mathrm{cm}$ .

Step 2. Thickness term, as a whole: $\dfrac{(n-1)^2 d}{nR_1R_2}=\dfrac{(0.5)^2(5)}{(1.5)(20)(-20)}=\dfrac{1.25}{-600}=-2.083\times10^{-3}\ \mathrm{cm^{-1}}$ . It is already multiplied by $(n-1)^2$ — do not scale it again.

Step 3. Add: $1/f=0.0500-0.00208=0.04792\ \mathrm{cm^{-1}}$ , so $\boxed{f=20.87\ \mathrm{cm}}$ — slightly longer than the thin-lens value, because the finite thickness weakens a biconvex lens. (The common slip of applying $(n-1)$ twice gives $f\approx20.4\ \mathrm{cm}$ , which is wrong.)

Step 4. Principal planes: $x_H=-f(n-1)d/(nR_2)=+1.74\ \mathrm{cm}$ and $x_{H'}=-f(n-1)d/(nR_1)=-1.74\ \mathrm{cm}$ — both $\boxed{1.74\ \mathrm{cm}\ \text{inside the glass}}$ , symmetric about the centre, separated by $5-2(1.74)=1.52\ \mathrm{cm}$ .

2 · Imaging with a thick lens — a real and a virtual case

Use the same lens ( $f=20.87\ \mathrm{cm}$ , planes as above). (a) An object sits $60\ \mathrm{cm}$ before $H$ . (b) The object is moved to $15\ \mathrm{cm}$ before $H$ . Locate and describe each image (distances from the principal planes).

Step 1. (a) Real case: $\dfrac{1}{s_i}=\dfrac{1}{20.87}-\dfrac{1}{60}=0.04792-0.01667=0.03125\ \mathrm{cm^{-1}}$ , so $\boxed{s_i=+32.0\ \mathrm{cm}}$ measured from $H'$ — a real image.

Step 2. Its magnification: $M_T=-s_i/s_o=-32.0/60=\boxed{-0.53}$ — inverted and reduced, the geometry of a camera photographing a nearby object.

Step 3. (b) Virtual case: $\dfrac{1}{s_i}=\dfrac{1}{20.87}-\dfrac{1}{15}=0.04792-0.06667=-0.01875\ \mathrm{cm^{-1}}$ , so $\boxed{s_i=-53.3\ \mathrm{cm}}$ — negative, a virtual image on the object side (the object is now inside the focal length).

Step 4. Its magnification: $M_T=-(-53.3)/15=\boxed{+3.6}$ — erect and enlarged. The thick lens used as a loupe behaves exactly like a thin one, provided every distance is measured from the principal planes rather than the glass.

Physics of the portrait

The hero of this article is not a person but a number. "Focal length" sounds like a fixed property of a piece of glass, but the thick-lens theory reveals it as something subtler: the invariant that survives when a lens acquires thickness or becomes a whole compound system — provided you agree to measure it from the right planes.

Origins · 1 minfocal length and how this was discovered

focal length

the invariant of a centred imaging system · Gaussian optics

Focal length is the single quantity that lets a camera lens, a microscope objective, and the eye all obey the same imaging law. For a thin lens it is unambiguous — the distance from the lens to where collimated light converges. For any real system it splinters into several distances that a beginner conflates: the back focal distance (last glass surface to $F'$ ), the effective focal length (back principal plane $H'$ to $F'$ ), and, when object and image media differ, two focal lengths $f_o$ and $f_i$ with $f_o/f_i=n_o/n_i$ . Gaussian optics — the first-order theory Gauss set out in 1841 — shows that it is the effective focal length, measured from the principal planes, that plays the role of " $f$ " in $1/s_o+1/s_i=1/f$ . The whole apparatus of cardinal points exists to pin down where this invariant is measured from, so that a telephoto lens can honestly advertise a focal length longer than the barrel that holds it, and the eye's power can be quoted independently of its wet, layered anatomy.

History

1611

— Kepler's Dioptrice treats compound lens systems and the intermediate image

1841

— Gauss, in Dioptrische Untersuchungen, founds first-order optics on the cardinal points and effective focal length

1845

— Listing introduces the nodal points, completing the six cardinal points

1911

— Gullstrand wins the Nobel Prize for the dioptrics of the eye; his equation gives the combined power of two separated systems

Going further

Texts. Hecht, Optics, 5th ed., §6.1 (Thick Lenses and Lens Systems) and §5.7 · Saleh & Teich, Fundamentals of Photonics, 3rd ed., Ch. 1 (ray matrices, Gullstrand) · Born & Wolf, Principles of Optics, §4.3–4.4 (Gaussian optics, cardinal points) · Jenkins & White, Fundamentals of Optics, Ch. 5. A common error applies the factor $(n-1)$ to the thickness term a second time and reports $f\approx20.4\,\mathrm{cm}$ for the equiconvex lens; the thickness term is $(n-1)^2 d/(nR_1R_2)$ and the correct value is $f=20.87\,\mathrm{cm}$ . The ABCD product $S_2T(d/n)S_1$ (with $\det M=1$ ) confirms the thick-lens EFL and the $d\to0$ thin-lens limit; the equiconvex principal-plane offsets are $\pm1.74\,\mathrm{cm}$ ; Gullstrand for two thin lenses with $f_1=10$ , $f_2=15$ , $d=5$ gives $f=7.5\,\mathrm{cm}$ and offsets $-3.75$ , $+2.5\,\mathrm{cm}$ ; and a plane-parallel plate produces an image shift $\Delta=1.667\,\mathrm{cm}$ .

Builds on: Refraction at a Spherical Surface · Thin Lens Equation and Lensmaker's Equation · Snell's Law of Refraction · Ray Model of Light. Leads to: Image Formation and Magnification · Stops, Pupils, and Field of View · Plane and Spherical Mirrors.

Why does a thick biconvex lens have a longer focal length than the thin-lens formula predicts?

Because the thickness term subtracts from the power. For a biconvex lens $R_1>0$ and $R_2<0$ , so $R_1R_2<0$ and the term $(n-1)^2 d/(nR_1R_2)$ is negative; adding a negative number to $1/f$ lowers the power and raises $f$ . Physically, the second surface acts on a ray the first surface has already bent inward, so the two refractions are slightly less effective in series than if they happened at one plane. For the worked example the shift is small — $20.00\to20.87\,\mathrm{cm}$ for a hefty $5\,\mathrm{cm}$ of glass — but for meniscus or strongly curved elements it is large and cannot be ignored.

What is the difference between effective focal length and back focal distance?

The effective focal length (EFL) is measured from the back principal plane $H'$ to the back focal point $F'$ ; it is the " $f$ " that appears in $1/s_o+1/s_i=1/f$ and sets magnification. The back focal distance (BFD) is measured from the last glass surface (the vertex $V_2$ ) to $F'$ ; it is what you physically need to position a sensor. They differ by the offset of $H'$ from $V_2$ : $\text{BFD}=\text{EFL}+x_{H'}$ . For an equiconvex lens $x_{H'}$ is a small negative number, so the BFD is a little shorter than the EFL. For a telephoto, $H'$ is far in front of the glass and the BFD is much shorter than the EFL — the whole point of the design.

Why compute retinal image size from the nodal points rather than the lens centre?

Because a ray aimed at the front nodal point $N$ leaves the rear nodal point $N'$ travelling in exactly the same direction — it is undeviated in angle. So the angular size of an object, as set by the chief ray through the nodal point, is preserved into image space, and the retinal image height is simply that angle times the nodal-point-to-retina distance. Using the lens centre or a vertex would misplace the angle. In the eye the media differ ( $n_o=1$ , $n_i\approx1.336$ ), so the nodal points sit about $7\,\mathrm{mm}$ behind the cornea, displaced from the principal points; that displacement is exactly $f_i-f_o$ , and it is why the schematic-eye models quote nodal points explicitly.

When do the nodal points and principal points coincide?

When the medium is the same on both sides of the system — the ordinary case of a lens in air. Then $f_o=f_i$ , the nodal-point offset $f_i-f_o$ vanishes, and $N\equiv H$ , $N'\equiv H'$ ; a ray through the principal point is both unit-magnification-conjugate and undeviated. They separate only when object and image spaces have different indices (a submerged lens, or the eye), by the amount $f_i(1-n_o/n_i)$ . For most bench optics you can treat "nodal" and "principal" as the same planes; for the eye and immersion optics you cannot.

Can I just use the ABCD matrix instead of memorizing the thick-lens formulas?

Yes — and it is often the cleaner route. Build the system matrix as the ordered product $S_2\,T(d/n)\,S_1$ of the two surface-refraction matrices and the internal translation (reduced thickness $d/n$ ). Then the effective focal length is $f=-1/C$ , and the principal-plane offsets are $x_{H'}=f(1-A)$ and $x_H=f(D-1)$ — every cardinal point read straight off the four elements. The thick lensmaker's equation and the offset formulas in this article are just those matrix results written out. The matrix method also scales without extra thought to three, four, or ten elements, which is why lens-design software uses it rather than closed-form lensmaker expressions.

How does Gullstrand's equation relate to the simple “powers add” rule for touching lenses?

Gullstrand's equation $1/f=1/f_1+1/f_2-d/(f_1f_2)$ is the general law for two elements separated by a gap $d$ ; the familiar "powers add," $1/f=1/f_1+1/f_2$ , is just its $d\to0$ limit for lenses in contact. The extra term $-d/(f_1f_2)$ is the coupling: separating the lenses changes the combined power because the second lens intercepts the first lens's already-converging (or diverging) cone at a different height. That single term is the design knob for telephoto and retrofocus lenses — vary $d$ and you move the principal planes and the effective focal length without changing either individual lens.