Refraction at a Spherical Surface

Press your face to the side of a fish tank and the fish hangs a little closer than it really is; look down into a swimming pool and the bottom floats up toward you. Both illusions are made by a single curved sheet of one transparent material meeting another — and that one interface, a spherical refracting surface, is the atom from which every lens is built. Grind a blank of glass and the natural shape your tools produce is a sphere, so nearly all conventional optics reduces to stacking these surfaces and letting each bend the rays further. At a curved boundary each ray obeys Snell's law, and the geometry does something remarkable: a fan of rays leaving one object point can be turned back toward a single image point. That reconvergence is imaging, and it works cleanly only when we agree to look at rays that hug the axis. In the paraxial (Gaussian) regime — small angles, strike points near the vertex — $\sin\theta$ and $\tan\theta$ both collapse to $\theta$ , Snell's law becomes linear, and every object point maps to exactly one image point. The whole surface is then captured by one algebraic relation among the object distance, the image distance, the two refractive indices, and the radius. That relation, $n_1/s_o + n_2/s_i = (n_2-n_1)/R$ , is the master equation of geometrical imaging. Push the object to infinity and it hands you a focal length; flatten the surface and it gives the pool-looks-shallow shift; chain two of them and it becomes the lensmaker's equation. Its accuracy is set by how small the angles stay — the moment they grow, the single image point smears into aberration.

Key takeaways

A single spherical interface of radius $R$ between media $n_1$ (object side) and $n_2$ (image side) obeys the paraxial relation $\dfrac{n_1}{s_o}+\dfrac{n_2}{s_i}=\dfrac{n_2-n_1}{R}$ .
It follows from Snell's law in the small-angle limit applied where a ray of height $h$ crosses the surface; because $h$ cancels, all paraxial rays meet at one point — the image is stigmatic.
Sign convention (Hecht): light runs left→right; $s_o>0$ for a real object to the left; $s_i>0$ for a real image to the right; $R>0$ when the centre of curvature $C$ lies to the right of the vertex.
The surface power is $P=(n_2-n_1)/R$ , and its two focal lengths are unequal, $f_i/f_o=n_2/n_1$ — the fingerprint of a refracting (not reflecting) surface.
Even a flat interface ( $R\to\infty$ ) images: $s_i=-(n_2/n_1)s_o$ , the apparent-depth shift that makes the fish look closer.
Key idea: apply Snell's law at the vertex of a sphere in the small-angle limit and the ray, the normal through $C$ , and the axis combine into one relation $\dfrac{n_1}{s_o}+\dfrac{n_2}{s_i}=\dfrac{n_2-n_1}{R}$ valid for all paraxial rays — the single building block of every lens.

The core insight

A spherical surface images because its normal, which always points to the centre of curvature $C$ , tilts more the farther a ray strikes from the vertex — exactly enough to bend a diverging fan back toward one point. Linearise Snell's law for small angles and that geometry becomes the master equation $\dfrac{n_1}{s_o}+\dfrac{n_2}{s_i}=\dfrac{n_2-n_1}{R}$ . Chain two such surfaces and you have a lens.

Where this lives today

The most important single refracting surface most of us own is the front of the eye. Two-thirds of the eye's roughly $60\ \mathrm{D}$ of focusing power comes not from the crystalline lens but from the single curved air–cornea interface, where the index steps from $1.00$ to about $1.376$ across a radius near $7.8\ \mathrm{mm}$ : its power is $P=(n_2-n_1)/R\approx 0.376/0.0078\ \mathrm{m}\approx 48\ \mathrm{D}$ . That is why refractive surgery reshapes the cornea, not the lens — a laser retunes $R$ by microns, and the master equation predicts the new focus. The same equation runs quietly inside every optical-design program. Ray-tracing software applies single-surface refraction, surface after surface, to lay out cameras, microscopes, and telescopes before any glass is ground — a paraxial pass fixes focal lengths and magnifications, then an aberration pass restores the $\theta^3$ terms this article dropped. Ball lenses coupling lasers into optical fibres are designed by placing the back focus $f_i=n_2R/(n_2-n_1)$ exactly on the fibre facet, and immersion objectives match the oil's index to the coverslip to shrink the first surface's index step. One spherical interface, understood paraxially, is still the first line of every imaging calculation made today.

The physical picture

Everything hinges on the surface normal. At the point a ray strikes the sphere, the normal is the line running out to the center of curvature $C$ ; as you slide that strike point up from the vertex, the normal swings away from the axis. A ray far from the axis therefore meets a more steeply tilted normal than one near it, refracts through a larger angle, and is bent harder toward the axis. That position-dependent tilt is precisely what a lens needs: it takes a diverging bundle from one object point and folds every ray back onto a common image point.

Figure

A paraxial ray from object O images to point I through one spherical surface — the normal runs through the center of curvature C

A convex surface (as the light sees it) fronting a denser medium, $n_2>n_1$ , is converging: the wavefront is slowed most where the glass reaches farthest toward the axis, so the rays tip inward. Reverse either the curvature or the index step, and the surface diverges. The single quantity deciding the behavior is the optical power $(n_2-n_1)/R$ — curvature times index step. It helps to read the master equation as a ledger of vergences: $n_1/s_o$ arriving, $n_2/s_i$ leaving, and $(n_2-n_1)/R$ the power the surface adds — imaging is just balancing that ledger across the interface. But the entire construction stands on one approximation — that the angles are small enough for $\sin\theta$ and $\tan\theta$ to be replaced by $\theta$ . That is not a hand-wave; it is a quantitative promissory note, and we can read off exactly how much it costs. The relative error of $\sin\theta\approx\theta$ grows as $\theta^2$ , and plotting it turns the whole paraxial edifice into a single honest picture of where it is trustworthy and where it starts to lie.

Figure

The paraxial approximation, quantified — the relative error of sinθ ≈ θ stays under 1% out to ~10°, then the discarded θ³ terms grow into aberration

The blue curve is the promissory note being honored. Below about $10^\circ$ the error of $\sin\theta\approx\theta$ is under half a percent — this is why designers can treat an instrument's whole first-order layout as exact arithmetic. Push past $20^\circ$ , and it crosses $2\%$ (the amber $\tan\theta\approx\theta$ curve, which controls where a ray lands, misbehaves twice as fast). What the approximation throws away is the $\theta^3$ term in $\sin\theta = \theta - \theta^3/6 + \cdots$ , and that cubic term is not noise: when fed back into the imaging, it becomes the five Seidel aberrations — spherical aberration, coma, astigmatism, field curvature, and distortion. The paraxial equation is the linear world; aberration theory is its cubic correction.

The core equations

The behaviour of the whole surface is contained in one relation between the two conjugate distances, the two indices, and the radius. With the sign convention fixed — light travelling left to right, $s_o>0$ for a real object to the left, $s_i>0$ for a real image to the right, and $R>0$ when the centre of curvature lies to the right of the vertex — the master equation reads:

\frac{n_1}{s_o}+\frac{n_2}{s_i}=\frac{n_2-n_1}{R} \qquad (1)

where: $n_1$ — index of the object-side (incident) medium; $n_2$ — index of the image-side (transmitted) medium; $s_o$ — object distance from the vertex; $s_i$ — image distance from the vertex; $R$ — radius of curvature (positive when $C$ is to the right). Read physically, the left side is the incoming plus outgoing reduced vergence and the right side is the power the interface supplies; a positive result for $s_i$ means a real image formed in the second medium, a negative one a virtual image on the incoming side. The right-hand side alone — curvature times index step — is the intrinsic strength of the surface, its refracting power, measured in dioptres when $R$ is in metres:

P=\frac{n_2-n_1}{R} \qquad (2)

where: $P$ — optical power ( $>0$ converging, $<0$ diverging). Because the object and image spaces sit in different media, the surface has two focal lengths — the object-side (front) and image-side (back) values found by sending the conjugate to infinity:

f_o=\frac{n_1 R}{n_2-n_1}=\frac{n_1}{P}, \qquad f_i=\frac{n_2 R}{n_2-n_1}=\frac{n_2}{P} \qquad (3)

where: $f_o$ — front focal length (object at $f_o$ emerges collimated); $f_i$ — back focal length (collimated light focuses at $f_i$ in the second medium). Their ratio is $f_i/f_o=n_2/n_1\neq 1$ — a single refracting surface is inherently asymmetric, unlike a mirror or a thin lens sitting in one medium. Finally, the transverse magnification carries an index factor the thin-lens formula does not:

m=\frac{y_i}{y_o}=-\frac{n_1 s_i}{n_2 s_o} \qquad (4)

where: $m$ — transverse magnification ( $m<0$ inverted, $|m|<1$ minified); $y_o,y_i$ — object and image heights. The extra $n_1/n_2$ relative to the mirror/lens form $-s_i/s_o$ is a direct expression of the optical (Smith–Helmholtz) invariant $n_1 y_o\theta_o = n_2 y_i\theta_i$ : refraction into a denser medium compresses ray angles, so a given transverse size subtends different angular content on the two sides.

The two derivations

Derivation AThe single-surface equation from Snell's law

We want to show that the master equation is not a definition but a consequence: that Snell's law, applied at one point of a sphere and linearised for small angles, forces every paraxial ray from an axial object point through a single image point. The plan is to trace one ray from the axial object $O$ to a strike point $Q$ at height $h$ on the surface, express the incidence and refraction angles through the geometry of the normal (which passes through the centre of curvature $C$ ), impose paraxial Snell's law, and watch the ray height $h$ cancel. That cancellation is the whole physical payload — it is what makes the image stigmatic.

Step 1. Fix the sign convention (light left→right, $s_o,s_i>0$ for real object/image, $R>0$ when $C$ is right of the vertex) and let a ray from $O$ meet the surface at $Q$ , height $h$ above the axis. Let $\alpha$ , $\gamma$ , and $\varphi$ be the small angles that the object ray, the image ray, and the surface normal make with the axis. Why: near the vertex every relevant length is the shared height $h$ , so all three angles are set by the same geometry and can be written with one variable.

Step 2. By the exterior-angle theorem applied to the triangles $O\,Q\,C$ and $C\,Q\,I$ , the angle of incidence and the angle of refraction (both measured from the normal) are $\theta_1=\alpha+\varphi$ and $\theta_2=\varphi-\gamma$ . Why: the normal is the line $QC$ , so the incidence angle is the object ray's tilt plus the normal's tilt, while on the far side the refracted ray's tilt subtracts from the normal's.

Step 3. Impose paraxial Snell's law $n_1\theta_1=n_2\theta_2$ , the small-angle form of $n_1\sin\theta_1=n_2\sin\theta_2$ , giving $n_1(\alpha+\varphi)=n_2(\varphi-\gamma)$ . Why: replacing $\sin\theta$ by $\theta$ is the single linearising step of the whole subject — everything downstream is exact algebra once this is granted.

Step 4. Substitute the small-angle geometry $\alpha\approx h/s_o$ , $\gamma\approx h/s_i$ , $\varphi\approx h/R$ : $\;n_1\!\left(\dfrac{h}{s_o}+\dfrac{h}{R}\right)=n_2\!\left(\dfrac{h}{R}-\dfrac{h}{s_i}\right)$ . Why: each angle is (transverse height)/(axial distance) for a thin triangle, and the normal's tilt $h/R$ enters because $C$ lies a distance $R$ from the vertex.

Step 5. Divide through by $h$ and collect terms: $\;\dfrac{n_1}{s_o}+\dfrac{n_2}{s_i}=\dfrac{n_2-n_1}{R}$ . The height $h$ has vanished, so every paraxial ray — whatever its launch height — arrives at the same $s_i$ . Why: $h$ -independence is the definition of a stigmatic (point-to-point) image; it is exactly what will fail once the neglected $\theta^3$ terms are restored, producing spherical aberration.

Go deeper · Derivation B · 2 minThe refraction invariant and the two focal lengths

Derivation BThe refraction invariant and the two focal lengths

The master equation hides a conserved quantity that makes single-surface bookkeeping almost trivial and exposes a notorious sign-convention trap. We want to recast equation (1) as an invariant — a number computed the same on both sides of the surface — then read the front and back focal lengths straight off it. The quantity is the Abbe (refraction) invariant, and its physical meaning is simply that $n\theta/h$ is conserved as a paraxial ray crosses the interface.

Step 1. From Step 3 above, $n_1\theta_1=n_2\theta_2$ with $\theta_1=h(1/s_o+1/R)$ and $\theta_2=h(1/R-1/s_i)$ . Dividing by $h$ , the same number appears on both sides: $\;Q\equiv n_1\!\left(\dfrac{1}{s_o}+\dfrac{1}{R}\right)=n_2\!\left(\dfrac{1}{R}-\dfrac{1}{s_i}\right)$ . Why: $Q=n\theta/h$ is literally paraxial Snell's law at the surface, so it must match across it — its equality is algebraically identical to equation (1).

Step 2. Mind the convention. Many texts write the invariant in the symmetric form $Q=n_1(1/R-1/s)=n_2(1/R-1/s')$ , but that uses the Cartesian sign rule where a real object's distance is negative ( $s=-s_o$ ). In our Hecht convention ( $s_o>0$ ) the object term flips to $+1/s_o$ ; the physics is identical, only the bookkeeping sign of the object distance differs. Mixing the two forms is the single commonest error here. Why: a spherical-surface answer is meaningless without its convention stated — a stray sign turns a converging surface into a diverging one.

Step 3. Set the object at infinity ( $s_o\to\infty$ , collimated input): $n_2/s_i=(n_2-n_1)/R$ , so the image forms at the back focal length $f_i=n_2R/(n_2-n_1)$ . Set the image at infinity instead: the object sits at the front focal length $f_o=n_1R/(n_2-n_1)$ . Why: the focal lengths are just the two conjugates for which one distance is infinite, and each divides the master equation by the power $P$ .

Step 4. Their ratio is $f_i/f_o=n_2/n_1$ — the two focal lengths are unequal whenever the media differ, and both are fixed by the one power $P=(n_2-n_1)/R$ . Why: this index asymmetry is the structural signature of a refracting surface; it disappears only when $n_1=n_2$ (no interface) or when two surfaces in air combine into a thin lens with a single $f$ . *As a consistency check: for the worked glass rod, $Q_\text{obj}=Q_\text{img}=0.625\ \mathrm{cm^{-1}}$ , confirming the invariant is conserved, and the focal lengths come out $f_o=4\ \mathrm{cm}$ , $f_i=6\ \mathrm{cm}$ , with $f_i/f_o=n_2/n_1=1.5$ .*

Limiting cases

Each familiar sub-result is a corner of the master equation:

Case	Result	Meaning
Flat interface, $R\to\infty$	$s_i=-\dfrac{n_2}{n_1}\,s_o$	A plane boundary still images. Looking from air ( $n_2=1$ ) into water ( $n_1=1.33$ ) puts objects at $0.75\,s_o$ — the pool and the fish look shallower. Virtual image, magnification $+1$ .
Object at infinity	$s_i=f_i=\dfrac{n_2 R}{n_2-n_1}$	Collimated light focuses at the back focal length inside the second medium.
Image at infinity	$s_o=f_o=\dfrac{n_1 R}{n_2-n_1}$	An object at the front focus emerges collimated; note $f_i/f_o=n_2/n_1\neq1$ .
Equal indices, $n_2=n_1$	$s_i=-s_o$	No index step means zero power — there is no interface to refract at, so the "image" is just the object.
Two surfaces in series	$\dfrac{1}{f}=(n-1)\!\left(\dfrac{1}{R_1}-\dfrac{1}{R_2}\right)$	Chaining the equation across front and back surfaces gives the Thin Lens Equation and Lensmaker's Equation — this surface is the lens's parent.

Common trapOne spherical surface focuses to a perfect point

It does not, and believing it is the paraxial equation used past its expiry date. Equation (1) says every ray from an axial object point lands at the same $s_i$ , which sounds like a flawless point image. But that clean result survives only because we replaced $\sin\theta$ by $\theta$ and dropped the $\theta^3$ term. Restore it, and rays striking farther from the vertex (larger $h$ , larger angle) focus closer to the surface than paraxial rays do: the single image point smears along the axis into spherical aberration. The punchline figure shows when this bites — the $\sin\theta\approx\theta$ error is under $0.5\%$ at $10^\circ$ but past $2\%$ by $20^\circ$ . A sphere is not the ideal aspheric shape (a Cartesian oval) that would image one conjugate pair perfectly; it is the shape we can grind cheaply, and paraxial optics is the price-of-admission approximation that makes it tractable. Whenever an answer implies a perfect focus at any aperture, the small-angle assumption has been pushed past where it holds.

Worked examples

1 · The glass rod — a real image inside the glass

A long glass rod ( $n_2=1.5$ ) has a convex, polished end of radius $R=+2\ \mathrm{cm}$ . A small object sits on the axis in air ( $n_1=1.0$ ), $s_o=8\ \mathrm{cm}$ in front of that end. Where is the image, is it real or virtual, and how big?

Step 1. Apply equation (1): $\dfrac{1.0}{8}+\dfrac{1.5}{s_i}=\dfrac{1.5-1.0}{2}=0.25\ \mathrm{cm^{-1}}$ .

Step 2. Solve for the image term: $\dfrac{1.5}{s_i}=0.25-0.125=0.125\ \mathrm{cm^{-1}}$ , so $s_i=\dfrac{1.5}{0.125}=\mathbf{+12\ cm}$ . Because $s_i>0$ , the image is real and lies $12\ \mathrm{cm}$ inside the glass, to the right of the vertex.

Step 3. Check it with the refraction invariant. Object side: $Q=n_1(1/s_o+1/R)=1.0(0.125+0.5)=0.625\ \mathrm{cm^{-1}}$ . Image side: $Q=n_2(1/R-1/s_i)=1.5(0.5-0.0833)=\mathbf{0.625\ cm^{-1}}$ . The two match — the invariant is conserved across the surface, confirming the answer.

Step 4. Magnification: $m=-\dfrac{n_1 s_i}{n_2 s_o}=-\dfrac{(1.0)(12)}{(1.5)(8)}=\mathbf{-1.0}$ . The image is inverted and the same size as the object. (Its surface power is $P=(n_2-n_1)/R=0.5/0.02\ \mathrm{m}=\mathbf{25\ D}$ , with back focal length $f_i=n_2R/P'=6\ \mathrm{cm}$ — the object at $8\ \mathrm{cm}$ sits outside $f_o=4\ \mathrm{cm}$ , so a real image is expected.)

2 · The fish that looks closer — a flat surface, a virtual image

A fish hovers $s_o=1.0\ \mathrm{m}$ below the flat surface of a pond ( $n_1=1.33$ ). You look straight down from the air ( $n_2=1.00$ ). At what depth does the fish appear, and is that image real or virtual?

Step 1. A flat surface is the $R\to\infty$ limit, so the power vanishes and equation (1) becomes $\dfrac{n_1}{s_o}+\dfrac{n_2}{s_i}=0$ , i.e. $s_i=-\dfrac{n_2}{n_1}s_o$ .

Step 2. Insert the numbers: $s_i=-\dfrac{1.00}{1.33}(1.0\ \mathrm{m})=\mathbf{-0.75\ m}$ . The negative sign says the image is virtual, sitting $0.75\ \mathrm{m}$ below the surface on the same side as the fish.

Step 3. So the fish appears at only $0.75\ \mathrm{m}$ instead of its true $1.0\ \mathrm{m}$ — raised toward you by the factor $1/n_1=1/1.33$ . This apparent-depth shift is why a submerged stick looks bent and why spearfishers aim short.

Step 4. Magnification: $m=-\dfrac{n_1 s_i}{n_2 s_o}=-\dfrac{(1.33)(-0.75)}{(1.00)(1.0)}=\mathbf{+1.0}$ . The image is erect and the same lateral size — only its apparent distance changes. A flat interface relocates without magnifying, exactly as everyday experience insists.

Physics of the portrait

The hero of this article is not a person but an approximation. Every clean result above — the single image point, the linear master equation, the two tidy focal lengths — exists only because we agreed to keep the angles small. That decision has a name and a lineage. Carl Friedrich Gauss systematised the "first-order" theory in 1841, showing that any centred system behaves paraxially like a black box with a handful of cardinal points; Ernst Abbe pushed it to off-axis points with the 1873 sine condition $n\,y\sin\theta=\text{const}$ ; and the discarded $\theta^3$ corrections, expanded systematically, are the five Seidel aberrations every lens designer spends a career taming.

Origins · 1 minparaxial approximation and how this was discovered

paraxial approximation

the small-angle regime · Gauss 1841, Abbe 1873 · why sinθ ≈ θ

The paraxial approximation is the founding idealisation of geometrical imaging: for rays close to the axis, $\sin\theta\approx\tan\theta\approx\theta$ , and the otherwise transcendental Snell's law collapses into linear algebra. In that regime a spherical surface, a mirror, or a stack of lenses maps every object point to exactly one image point — imaging is stigmatic and computable by hand. Gauss's Dioptrische Untersuchungen (1841) turned this into a complete first-order theory of principal planes and focal points; Abbe's microscope theory (1873) added the sine condition and the diffraction limit bounding it off-axis. The neglected higher-order terms return as the Seidel aberrations, so the paraxial world is the linear backbone of optics and aberration theory its systematic curvature.

History

1611

— Kepler's Dioptrice lays out the refraction of lenses and the theory of the telescope, the first systematic optics of curved surfaces.

1669

— Barrow's Lectiones Opticae analyses where a single refracting surface places an image, an early study of the conjugate points.

1841

— Gauss's Dioptrische Untersuchungen founds first-order (Gaussian) imaging theory with cardinal points and focal lengths.

1873

— Abbe formulates the sine condition and the diffraction theory of the microscope, bounding the paraxial ideal.

Going further

Texts. Hecht, Optics, 5th ed., §5.2.1 (refraction at spherical surfaces) and §6.3 (aberrations) · Saleh & Teich, Fundamentals of Photonics, 3rd ed., §1.2 (simple optical components) · Born & Wolf, Principles of Optics, Ch. 4 (geometrical theory of optical imaging) · Jenkins & White, Fundamentals of Optics, Ch. 3–5 (single surfaces and the Abbe invariant). Sign convention used throughout (Hecht): $s_o>0$ real object left, $s_i>0$ real image right, $R>0$ when $C$ is right of the vertex. Key numerical anchors: the master equation follows from paraxial Snell's law with $h$ cancelling; the refraction invariant is conserved at $0.625\ \mathrm{cm^{-1}}$ for the glass rod (note the Cartesian symmetric form uses a signed object distance); glass-rod results are $s_i=+12\ \mathrm{cm}$ , $m=-1$ , $P=25\ \mathrm{D}$ , $f_o=4\ \mathrm{cm}$ , $f_i=6\ \mathrm{cm}$ ; flat-surface fish apparent depth is $0.75\ \mathrm{m}$ ; the small-angle error table reads $0.13\%$ / $0.51\%$ / $2.06\%$ at $5^\circ$ / $10^\circ$ / $20^\circ$ . Water index $1.33$ and cornea index $1.376$ are standard textbook values.

Prerequisites. Snell's Law of Refraction · Ray Model of Light · Refractive Index and Optical Path Length · Fermat's Principle of Least Time. Leads to. Thin Lens Equation and Lensmaker's Equation · Thick Lenses and Principal Planes · Image Formation and Magnification · Plane and Spherical Mirrors.

Why does the single-surface magnification carry an extra factor n₁/n₂ that the thin-lens formula does not?

Because the object and image live in different media. The magnification follows from the chief ray through the vertex, where paraxial Snell's law gives $n_1\,y_o/s_o = n_2\,(-y_i)/s_i$ , so $m=-n_1 s_i/(n_2 s_o)$ . The $n_1/n_2$ is the fingerprint of refraction into a different index: the optical invariant $n\,y\,\theta$ is conserved, so if $n$ jumps the product $y\,\theta$ must adjust. A thin lens in air has the same medium on both sides ( $n_1=n_2$ ), so the factor is $1$ and you recover $-s_i/s_o$ . Reusing the thin-lens form at a single surface is a common mistake.

The Abbe invariant looks different in my textbook — is one of them wrong?

Almost certainly not; you are seeing two sign conventions. The symmetric form $Q=n(1/R-1/s)$ uses the Cartesian rule, where a real object to the left has a negative distance $s=-s_o$ . The Hecht convention this article uses keeps $s_o$ positive, which flips the object term to $+1/s_o$ , giving $Q=n_1(1/s_o+1/R)=n_2(1/R-1/s_i)$ . Plug numbers in and both yield the identical physical quantity — for the worked rod, $0.625\ \mathrm{cm^{-1}}$ on either side. The lesson is iron: never quote a spherical-surface formula without stating which sign convention it lives in, because the same symbols mean opposite things in the two systems.

If the image forms 12 cm inside a solid glass rod, in what sense is it "real"? I can't put a screen there.

It is real in the precise optical sense: the refracted rays physically converge to and cross at that point. Realness is about whether rays actually meet (real) or only appear to diverge from a point (virtual), not about whether you can insert a card. Inside the rod the rays truly cross at $s_i=+12\ \mathrm{cm}$ , and if a second surface intercepts the light before it converges, that crossing point acts as the object for whatever comes next. This is exactly how thick lenses and rod lenses are analysed — the internal real image of the first surface is the object of the second.

Why does even a perfectly flat window or water surface still shift where things appear?

Because refraction depends on the index step, not on curvature. Setting $R\to\infty$ kills the power term but leaves $n_1/s_o+n_2/s_i=0$ , so $s_i=-(n_2/n_1)s_o$ : the image is virtual and displaced. Rays from a submerged point bend away from the normal as they leave the water, and their backward extensions cross nearer the surface, so your eye traces them to a shallower position. A flat interface has zero focusing power but nonzero displacement — it relocates images without magnifying them (magnification exactly $+1$ ).

How does this one surface turn into a lens?

By using it twice. A thin lens is two spherical surfaces back to back with negligible glass between them. Apply equation (1) at the front surface (air→glass) to form an intermediate image, then use that image as the object for the back surface (glass→air). In the thin limit the two vertices coincide, the intermediate-image terms cancel, and the two surface powers simply add: $1/f=(n-1)(1/R_1-1/R_2)$ , the lensmaker's equation. Every property of lenses — focal length, magnification, even chromatic aberration — descends from applying single-surface refraction twice. That derivation is the subject of the Thin Lens Equation and Lensmaker's Equation article.

When exactly does the paraxial equation stop being trustworthy?

When the angles stop being small — and the punchline figure gives the numbers. The relative error of $\sin\theta\approx\theta$ is about $0.13\%$ at $5^\circ$ , $0.51\%$ at $10^\circ$ , and $2.06\%$ at $20^\circ$ , growing as $\theta^2$ . For most instruments the marginal rays stay inside $10^\circ$ , so first-order design is effectively exact and the residual is mopped up as aberration. The practical marker is the numerical aperture: keep it modest and paraxial optics predicts focus and magnification to a fraction of a percent; open it up (a fast camera lens, a high-NA objective) and the discarded $\theta^3$ terms dominate, forcing you to design against the Seidel aberrations.