Snell's Law of Refraction

Dip a straight straw into a glass of water and it snaps — the submerged half hangs at a visibly wrong angle, as if the straw were broken at the surface. Stand at the edge of a clear pool and the bottom floats up toward you; the water is always deeper than it looks. Try to spear a fish and, unless you aim low, you miss. All three are the same fact: when light crosses from one transparent medium into another it changes direction at the boundary, and our brains — which insist that light travels in straight lines — misplace whatever the bent rays came from. Snell's law puts a number on the bend. Measuring both angles from the normal, the line perpendicular to the surface, the sine of the angle of incidence and the sine of the angle of refraction stand in a fixed ratio set only by the two media: $n_1\sin\theta_1 = n_2\sin\theta_2$ . The refractive index $n = c/v$ measures how much a medium slows light; a ray entering a slower, denser medium bends toward the normal, and one entering a faster, rarer medium bends away. The deep reason is that a wavefront crossing the boundary obliquely reaches it one edge first, and the side that enters the slow medium falls behind — so the whole front pivots, exactly like a marching column wheeling as it steps from firm ground onto mud. That single ray-level rule is at the base of geometrical optics. Refraction at a curved surface, the thin lens, the prism's spectrum, the sparkle of a diamond, and every ray traced through a camera design are Snell's law applied surface by surface. Master the bend and the rest is bookkeeping.

Key takeaways

Snell's law, $n_1\sin\theta_1 = n_2\sin\theta_2$ , relates the angle of incidence and the angle of refraction of a ray crossing a boundary between media of index $n_1$ and $n_2$ — both angles measured from the surface normal.
The refractive index $n = c/v = \lambda_0/\lambda$ measures how much the medium slows light; the frequency is fixed by the source, so the wavelength shrinks in a denser medium and the wave bends.
A ray bends toward the normal entering a denser (higher- $n$ ) medium and away from it entering a rarer medium; going denser → rarer, beyond the critical angle $\theta_c = \arcsin(n_2/n_1)$ , there is no refracted ray at all.
Plot $\sin\theta_2$ against $\sin\theta_1$ for a fixed pair of media and you get an exact straight line of slope $n_1/n_2$ — the law is linear in the sines.
It follows geometrically from the Huygens wavefront construction, and equivalently from Fermat's principle of least time and the electromagnetic boundary conditions — three routes to one result.
Key idea: at a flat interface the quantity $n\sin\theta$ is conserved. Because $n$ is a pure speed ratio and the frequency cannot change at the boundary, only $n\sin\theta$ carries across — and that single conserved number is Snell's law.

The core insight

At any interface the quantity $n\sin\theta$ is the same on both sides. It is conserved because the frequency is pinned by the source while the wave speed changes as $v = c/n$ , so the wavefront can stay continuous across the boundary only if $n\sin\theta$ matches. A ray therefore tips toward the normal when it slows (enters higher $n$ ) and away when it speeds up — and everything from lenses to rainbows is this one rule applied surface by surface.

Where this lives today

Every lens designed today is designed by ray-tracing: software launches thousands of rays through each surface and bends every one of them by the exact Snell's law, predicting focal spots, distortions, and colour errors even before a single piece of glass is manufactured. Optical fibres carry the internet on the flip side of the same equation — light beyond the critical angle at the core–cladding boundary is totally internally reflected and races down the fibre without escaping. And measurement instruments live directly on the punchline: an Abbe refractometer reads a single critical-angle boundary to pin a liquid's refractive index to four decimal places, which is how winemakers gauge sugar content, gemmologists tell diamond from paste, and clinical labs check the protein in a drop of serum. The reach is wider than light. The same $n\sin\theta$ bookkeeping — a wave slowing and bending as it changes medium — governs sound in the ocean's layered water, seismic waves refracting through the Earth's mantle, and radio in the ionosphere. Snell's law is the template for what any wave does at a boundary, and geometrical optics is just its most familiar home.

The physical picture

Think of wavefronts, not rays. A wavefront is a surface of constant phase — the crest line of the wave. Where a tilted wavefront meets the interface, one end crosses into the slow medium first and immediately dawdles, while the far end is still racing along in the fast medium. The front pivots about the boundary, and the ray, which always points perpendicular to the front, swings with it. This is the marching-column picture made precise: a rank of soldiers stepping obliquely from pavement onto sand wheels toward the sand, because the men on the sand take shorter steps while their comrades on the pavement stride on. The conserved quantity behind the pivot is the trace speed of the wavefront along the surface — the speed at which the point where the front meets the interface slides sideways. On side 1 that speed is $v_1/\sin\theta_1$ ; on side 2 it is $v_2/\sin\theta_2$ . They must be equal, or the wave would tear apart at the boundary, and setting them equal gives $\sin\theta_1/\sin\theta_2 = v_1/v_2 = n_2/n_1$ at a stroke. The Huygens construction below turns that sentence into a picture you can measure.

Figure

The Huygens construction: while the incident front advances v₁τ from B to C, the wavelet from A spreads only v₂τ into the slow medium — the new front CD tilts, giving sinθ₁/sinθ₂ = v₁/v₂ = n₂/n₁

A useful mental limit lives at each end of that ratio. As $n_2 \to \infty$ the refraction angle $\theta_2 \to 0$ : a ray plunges straight in along the normal no matter how it arrived, because the medium is so slow that the sideways progress of the front is negligible. Conversely, entering a rarer medium the ray bends away from the normal and, as the incidence angle climbs, eventually grazes the surface — and past that point Snell's law demands $\sin\theta_2 > 1$ , which is impossible, so the light is thrown back entirely. That is total internal reflection, and it is where fibre optics and a diamond's fire come from. The cleanest way to see that Snell's law is a genuine law and not a fudge is to plot it. Fix the two media and sweep the incidence angle: the sine of the refraction angle rises in exact proportion to the sine of the incidence angle. Not approximately — exactly. The graph is a straight line through the origin whose slope is $n_1/n_2$ , and the slope alone identifies the interface.

Figure

Snell's law is exactly linear in the sines: sinθ₂ against sinθ₁ is a straight line of slope n₁/n₂. For air → glass (n = 1.5) the slope is 0.667, and θ₁ = 30° lands precisely on it at θ₂ = 19.5°

That the punchline is a line — and that its slope is exactly the index ratio — is the whole reason a refractometer works: measure one refraction angle and you have read off $n$ to several decimals. The dashed diagonal is the trivial case $n_1 = n_2$ , an index-matched boundary that bends nothing and is optically invisible; the blue glass line lies below it because glass is slower, so $\theta_2 < \theta_1$ and the ray tips inward.

The core equations

Snell's law is one relation with several equivalent faces; each says the same thing about the geometry of the bend, and which face is handiest depends on whether you are given angles, speeds, or indices. Everything below is that single rule, read four ways. The headline form ties the two angles to the two indices. Both angles are measured from the normal, never from the surface, and the indices are pure numbers ( $n = 1$ in vacuum, $\approx 1.33$ in water, $\approx 1.5$ in ordinary glass). The equation is symmetric: swap the two sides and the ray simply runs backward along the same path.

n_1 \sin\theta_1 = n_2 \sin\theta_2 \qquad (1)

where: $n_1, n_2$ — refractive indices of the incident and transmitting media; $\theta_1$ — angle of incidence, from the normal; $\theta_2$ — angle of refraction, from the normal. Physical reading: the product $n\sin\theta$ takes the same value on both sides — one conserved quantity carried unchanged across the boundary. The index is not a fitting constant but a speed ratio. Light of vacuum wavelength $\lambda_0$ slows to $v = c/n$ inside the medium; its frequency is set by the source and cannot change at a passive boundary, so the wavelength must shrink to $\lambda = \lambda_0/n$ to keep the wave continuous.

n = \frac{c}{v} = \frac{\lambda_0}{\lambda} \qquad (2)

where: $c$ — speed of light in vacuum, $2.998\times10^{8}\ \mathrm{m\,s^{-1}}$ ; $v$ — phase speed in the medium; $\lambda_0, \lambda$ — vacuum and in-medium wavelengths (frequency unchanged). Physical reading: a larger $n$ means slower light and a shorter in-medium wavelength — and it is exactly this slowing that bends the ray. Because only speeds enter the geometry, the law can be written without the indices at all. This is the form the Huygens construction hands you directly, and it makes the direction of the bend obvious.

\frac{\sin\theta_1}{\sin\theta_2} = \frac{v_1}{v_2} = \frac{n_2}{n_1} \qquad (3)

where: $v_1, v_2$ — phase speeds in the two media. Physical reading: entering a slower medium ( $v_2 < v_1$ ) forces $\sin\theta_2 < \sin\theta_1$ , so $\theta_2 < \theta_1$ and the ray tips toward the normal; the ratio of sines is nothing but the ratio of speeds. One face of the law is a warning rather than a prediction. Going from a denser to a rarer medium ( $n_1 > n_2$ ), the required $\sin\theta_2 = (n_1/n_2)\sin\theta_1$ grows faster than $\sin\theta_1$ , and at a finite incidence it would demand $\sin\theta_2 = 1$ . Set $\theta_2 = 90^\circ$ and solve for that incidence:

\theta_c = \arcsin\!\left(\frac{n_2}{n_1}\right), \qquad n_1 > n_2 \qquad (4)

where: $\theta_c$ — the critical angle, the incidence beyond which no refracted ray exists. Physical reading: for water → air $\theta_c \approx 48.8^\circ$ , for ordinary glass → air $\approx 41.8^\circ$ , and for diamond $\approx 24.4^\circ$ . Above $\theta_c$ the light is thrown back completely — the physics of Total Internal Reflection and the Critical Angle.

Where the law comes from

Derivation ASnell's law from the Huygens construction

We want to derive the sine law geometrically rather than assert it — to show that the bending of a ray at an interface is forced by nothing more than the different wave speeds on the two sides. The tool is Huygens' principle: treat every point of a wavefront as the source of a secondary spherical wavelet, and the next wavefront as the envelope tangent to all those wavelets. Because the wavelets crawl more slowly in the denser medium, the envelope tilts, and the geometry of that tilt turns out to be exactly $n_1\sin\theta_1 = n_2\sin\theta_2$ .

Step 1. A plane wavefront in medium 1 meets the flat interface obliquely. Take the instant its leading edge just touches the surface at point $A$ ; the trailing edge, at $B$ , still has to travel to the interface and reaches it at $C$ a time $\tau$ later. During $\tau$ the front covers the distance $BC = v_1\tau$ in medium 1. Why: the wavefront is a line of constant phase, so tracking one edge of it from $B$ to $C$ measures precisely how far the wave advances in the fast medium in time $\tau$ .

Step 2. The moment the front reached $A$ , that point began radiating a secondary wavelet into medium 2. After time $\tau$ the wavelet is an arc of radius $AD = v_2\tau$ — smaller than $BC$ because $v_2 < v_1$ . The refracted wavefront is the line $CD$ drawn from $C$ tangent to this wavelet. Why: by Huygens' envelope the new front is the common tangent to every secondary wavelet launched along the interface, and the slower those wavelets spread, the more the tangent tilts.

Step 3. Two right triangles now share the hypotenuse $AC$ on the interface. In triangle $ABC$ (right angle at $B$ ) the angle at $A$ equals the incidence angle $\theta_1$ , so $\sin\theta_1 = BC/AC = v_1\tau/AC$ . In triangle $ADC$ (right angle at $D$ ) the angle at $C$ equals the refraction angle $\theta_2$ , so $\sin\theta_2 = AD/AC = v_2\tau/AC$ . Why: the incidence and refraction angles reappear inside these triangles because each wavefront is perpendicular to its rays, and both triangles are pinned to the same stretch $AC$ of interface.

Step 4. Divide the two sines. The shared hypotenuse $AC$ and the shared time $\tau$ cancel: $\dfrac{\sin\theta_1}{\sin\theta_2} = \dfrac{v_1}{v_2}$ . Using $v = c/n$ this is $v_1/v_2 = n_2/n_1$ , hence $n_1\sin\theta_1 = n_2\sin\theta_2$ . Why: everything geometric cancels, leaving a pure ratio of speeds — and since the speeds are fixed by the indices, Snell's law drops out with no free parameters. *The construction gives $\sin\theta_1/\sin\theta_2 = v_1/v_2 = n_2/n_1$ symbolically, and for the $n = 1.5$ figure the wavelet ratio $BC/AD = 1.5$ as expected.*

Go deeper · Derivation B · 2 minApparent depth in the paraxial limit

Derivation BApparent depth in the paraxial limit

A coin at the bottom of a pool looks closer to the surface than it really is, and a fish looks shallower than where the spear must go. We derive the shift quantitatively: for an object at true depth $d$ beneath a flat interface, viewed from almost straight above, the image sits at an apparent depth $d' = d(n_2/n_1)$ , where $n_1$ is the object's medium and $n_2$ the observer's. The whole effect is Snell's law taken in the small-angle (paraxial) limit, and it is the reason apparent depth is a ratio of indices, not a product.

Step 1. Put the object $O$ at depth $d$ below the flat surface. A ray leaves $O$ , strikes the surface a small horizontal distance $x$ from the point straight above $O$ at angle $\theta_1$ (in the water), and refracts to $\theta_2$ (in the air). Straight from the geometry, $\tan\theta_1 = x/d$ . Why: the incidence angle is set purely by how far along the surface the ray emerges relative to its depth — pure trigonometry on the object's side.

Step 2. The eye catches the refracted ray and projects it back in a straight line; that backward extension crosses the vertical through $O$ at the apparent depth $d'$ , so $\tan\theta_2 = x/d'$ . Dividing the two, $d'/d = \tan\theta_1/\tan\theta_2$ . Why: the visual system assumes light travelled straight, so it places the image where the exit ray, run backward, appears to originate.

Step 3. Look nearly straight down, so $\theta_1, \theta_2$ are small and $\tan\theta \approx \sin\theta$ . Then $d'/d \approx \sin\theta_1/\sin\theta_2$ . Why: near-normal viewing is exactly the regime where tangent and sine agree to better than a percent — and it is how you actually peer into a pool.

Step 4. Insert Snell's law, $\sin\theta_1/\sin\theta_2 = n_2/n_1$ , to get $d' = d\,(n_2/n_1)$ . For water ( $n_1 = 1.33$ ) viewed from air ( $n_2 = 1.00$ ) this is $d' = d/1.33$ , about three-quarters of the true depth. Why: because $n_2 < n_1$ the ratio is less than one, so the image is always shallower, never deeper — the entire content of the illusion. *The paraxial result $d' = d(n_2/n_1)$ holds symbolically, and applying it to the coin gives $12\ \mathrm{cm} \to 9.02\ \mathrm{cm}$ .*

Limiting cases

Case	Result	Meaning
Normal incidence ( $\theta_1 = 0$ )	$\theta_2 = 0$	a ray hitting head-on passes straight through undeviated; only its speed and wavelength change
Into a denser medium ( $n_2 > n_1$ )	$\sin\theta_2 = \tfrac{n_1}{n_2}\sin\theta_1,\ \theta_2 < \theta_1$	bends toward the normal; a maximum refraction angle $\arcsin(n_1/n_2)$ is reached at grazing incidence
Into a rarer medium ( $n_2 < n_1$ )	real $\theta_2$ only for $\theta_1 < \theta_c$	bends away from the normal; beyond $\theta_c = \arcsin(n_2/n_1)$ total internal reflection sets in
Equal indices ( $n_1 = n_2$ )	$\theta_2 = \theta_1$	no bend at all — an index-matched boundary is optically invisible
Reverse the ray	$n_2\sin\theta_2 = n_1\sin\theta_1$	the law is symmetric; light retraces its exact path (optical reversibility)

Common trapApparent depth is the real depth divided by n, not multiplied by it

A pool looks shallower than it is, so the correction must make the depth smaller — yet a common reflex is to multiply the real depth by the refractive index, which makes it larger and points the wrong way. The paraxial result is $d' = d\,(n_2/n_1)$ with $n_1$ the object's medium and $n_2$ the viewer's; for a coin under water seen from air that is $d' = d\,(1.00/1.33) = d/1.33$ , about three-quarters of the true depth. Multiply instead and you would predict the coin sinks to $16\ \mathrm{cm}$ under $12\ \mathrm{cm}$ of water — deeper than the beaker, which is nonsense. Hold onto the rule: the image moves toward the observer, so you divide by the index of the object's medium relative to the observer's, never multiply. And, as ever, measure both angles from the normal, not from the surface — the single-most common slip in every refraction problem.

Worked examples

1 · The coin under water (apparent depth)

A coin lies at the bottom of a beaker under $12\ \mathrm{cm}$ of water ( $n = 1.33$ ). Looking almost straight down from the air above, how deep does the coin appear to be?

Step 1. Identify the media in the right order. The object sits in water, $n_1 = 1.33$ ; the observer is in air, $n_2 = 1.00$ . Near-normal viewing means the paraxial apparent-depth formula applies: $d' = d\,(n_2/n_1)$ .

Step 2. Substitute the numbers: $d' = 12\ \mathrm{cm} \times \dfrac{1.00}{1.33} = \dfrac{12}{1.33}\ \mathrm{cm}$ .

Step 3. Evaluate: $d' = \boxed{9.02\ \mathrm{cm}}$ . The coin appears about $3.0\ \mathrm{cm}$ shallower than it truly is.

Step 4. Sense-check the direction. Since $n_2/n_1 < 1$ the image must rise, by a fraction $1 - 1/1.33 \approx 0.25$ — a quarter of the depth. That is why a full pool always looks shallower than a dive reveals, and why a spearfisher, seeing the fish raised, must aim below the image to hit the real fish.

2 · The laser through a glass slab (lateral shift)

A laser pointer in air strikes a $t = 10\ \mathrm{mm}$ parallel-faced glass slab ( $n = 1.5$ ) at $45^\circ$ to the normal. It emerges parallel to its original direction but displaced sideways. By how much?

Step 1. Refract at the first face. Snell's law gives $\sin\theta_2 = \dfrac{n_1}{n_2}\sin\theta_1 = \dfrac{1.00}{1.50}\sin45^\circ = 0.4714$ , so $\theta_2 = 28.13^\circ$ . At the second, parallel face the ray refracts back to $45^\circ$ , so the exit beam is parallel to the entry beam — the slab shifts but does not turn it.

Step 2. The perpendicular offset between the undisturbed line and the actual exit ray is $\Delta = t\,\dfrac{\sin(\theta_1 - \theta_2)}{\cos\theta_2}$ , a compact result that follows from the ray's slanted path of length $t/\cos\theta_2$ through the glass.

Step 3. Substitute: $\Delta = 10\ \mathrm{mm} \times \dfrac{\sin(45^\circ - 28.13^\circ)}{\cos 28.13^\circ} = 10 \times \dfrac{\sin 16.87^\circ}{\cos 28.13^\circ} = 10 \times \dfrac{0.2903}{0.8819}$ .

Step 4. Evaluate: $\Delta = \boxed{3.29\ \mathrm{mm}}$ . Thicker glass or a steeper angle shifts the beam more; at normal incidence ( $\theta_1 = 0$ ) the shift vanishes, which is why a window viewed head-on does not offset what you see through it.

Physics of the portrait

The sine law wears one man's name, but its history is a lesson in how discovery, publication, and credit can come apart by centuries.

Origins · 1 minWillebrord Snellius and how this was discovered

Willebrord Snellius

1580–1626 · Leiden

Willebrord Snel van Royen — Latinised to Snellius — was professor of mathematics at Leiden, better known in his own lifetime for geodesy: he measured the size of the Earth by triangulating across the flat Dutch countryside, founding the modern method of survey. Around 1621 he found, in manuscripts he never printed, that refraction obeys a fixed ratio — though he cast it as a constant ratio of cosecants (equivalently, of the lengths cut on the two rays) rather than the modern ratio of sines. The result surfaced only when later scholars, Huygens among them, read through his papers. The law reached the world in 1637 through Descartes' La Dioptrique, stated in the sine form and backed by a corpuscular argument, and a priority quarrel soon followed over whether Descartes had seen Snel's unpublished work. The deeper irony is older still: six centuries earlier the Baghdad mathematician Ibn Sahl, in his 984 manuscript On the Burning Instruments, had already used exactly the sine relation to design lenses and burning mirrors — a discovery historians recovered only in 1990. Snel's name stuck to a law he did not publish, first stated by a man whose work was lost for a thousand years.

History

984

— Ibn Sahl states the sine law of refraction geometrically in On the Burning Instruments, using it to design lenses.

1602

— Thomas Harriot measures refraction angles and finds the sine relation, but leaves it unpublished.

1621

— Willebrord Snellius rediscovers the law at Leiden in a manuscript proof he never prints.

1637

— René Descartes publishes the sine law in La Dioptrique, the first form to reach print.

1662

— Pierre de Fermat derives the same law from the principle of least time.

Going further

Texts. Hecht, Optics, 5th ed., §4.3–4.4 (refraction, Snell's law) · Saleh & Teich, Fundamentals of Photonics, 3rd ed., Ch. 1 (Ray Optics) · Born & Wolf, Principles of Optics, §1.5 (foundations of refraction and the Fresnel relations) · Feynman, Lectures on Physics, Vol. I, Ch. 26 (optics and least time). Key results to keep in mind: the Huygens construction gives $\sin\theta_1/\sin\theta_2 = v_1/v_2 = n_2/n_1$ ; the paraxial apparent-depth formula is $d' = d(n_2/n_1)$ , yielding $12\ \mathrm{cm} \to 9.02\ \mathrm{cm}$ for the coin; the slab lateral shift is $\Delta = t\sin(\theta_1-\theta_2)/\cos\theta_2 = 3.29\ \mathrm{mm}$ at $45^\circ$ through $10\ \mathrm{mm}$ of $n=1.5$ glass. Numerical checks: air → water at $30^\circ$ gives $\theta_2 \approx 22.1^\circ$ ; glass → air critical angle $\arcsin(1/1.5) \approx 41.8^\circ$ ; diamond ( $n = 2.42$ ) critical angle $\arcsin(1/2.42) \approx 24.4^\circ$ . Standard index values: $n_{\text{water}} = 1.33$ , $n_{\text{glass}} \approx 1.5$ , $n_{\text{diamond}} = 2.42$ .

Builds on. Ray Model of Light · Refractive Index and Optical Path Length · Law of Reflection. Leads to. Total Internal Reflection and the Critical Angle · Dispersion and Prisms · Refraction at a Spherical Surface · Thin Lens Equation and Lensmaker's Equation.

Why are the angles measured from the normal and not from the surface?

Because the physics lives in the sine of the angle from the normal, and that is the quantity the wavefront geometry conserves. You can measure from the surface — write $\phi = 90^\circ - \theta$ and the law becomes $n_1\cos\phi_1 = n_2\cos\phi_2$ — but it is error-prone and hides the structure. Measuring from the normal also handles normal incidence gracefully ( $\theta = 0$ , no bend) and makes the symmetry with the law of reflection, which uses the same normal, immediate. Every optics text and every ray-tracer uses the normal; adopt it and never convert mid-problem.

Does Snell's law tell me how much light reflects versus refracts?

No — and this is a frequent over-reading. Snell's law fixes only the direction of the refracted ray. How the incident energy splits between the reflected and transmitted beams is a separate question answered by the Fresnel equations, which come from matching the full electromagnetic fields at the boundary and depend on polarization and angle. Some light is always reflected at any interface (about 4% at a plain air–glass surface at normal incidence), even when Snell's law happily provides a refracted ray. Snell tells you where; Fresnel tells you how bright.

Why does the straw look bent exactly at the waterline?

Because the break is where the refraction happens. Light from the submerged part of the straw leaves the water and bends away from the normal as it enters the air, so those rays reach your eye from a slightly different direction than rays from the part in air. Your brain, assuming straight-line travel, projects the underwater portion to a shifted, raised position — and since only the submerged rays are bent, the apparent shift begins abruptly at the surface. The same displacement is the apparent-depth effect seen from the side rather than from above.

Is there always a refracted ray?

No. Going from a rarer to a denser medium there is always a refracted ray, bent toward the normal. But going from denser to rarer ( $n_1 > n_2$ ), Snell's law needs $\sin\theta_2 = (n_1/n_2)\sin\theta_1$ , which exceeds 1 once the incidence passes the critical angle $\theta_c = \arcsin(n_2/n_1)$ . There is then no real refraction angle, and all the light is reflected back — total internal reflection. That is the mechanism behind optical fibres, reflecting prisms in binoculars, and the brilliance of a well-cut diamond, whose small critical angle ( $24.4^\circ$ ) traps light through many internal bounces.

If the wave slows down, why doesn't its frequency change too?

Because frequency is set by the source and enforced across the boundary. The oscillating field on the far side of the interface is driven by the field on the near side, wiggle for wiggle, so the number of crests arriving per second must equal the number leaving — the frequency is continuous. What is free to change is the wavelength: with $v = f\lambda$ and $f$ fixed, a slower speed means a proportionally shorter wavelength, $\lambda = \lambda_0/n$ . It is this wavelength change, not any frequency change, that tilts the wavefronts and bends the ray.

Is Snell's law just Fermat's principle in disguise?

They are two windows onto the same law. Fermat's Principle of Least Time selects the ray path that makes the travel time stationary between two fixed points; carrying out that minimization across an interface yields $n_1\sin\theta_1 = n_2\sin\theta_2$ exactly. The Huygens construction in this article gives the same law from wavefronts and speeds, and the electromagnetic boundary conditions give it a third time from phase continuity. That one relation falls out of a variational principle, a geometric construction, and Maxwell's equations alike is strong evidence it is a genuine law of nature, not an artifact of one picture.

Why does the coin look shallower, not deeper?

Because the rays that reach your eye leave the water bending away from the normal, so they arrive as if from a point higher up. Trace them back and they cross above the real coin, at depth $d' = d(n_2/n_1) = d/1.33$ for water viewed from air — a smaller number than $d$ , hence shallower. The intuition to reject is "denser medium, so multiply by the bigger index": the correction always moves the image toward the viewer, so you divide by the object-medium index. Deeper would require the rays to bend the other way, which only happens looking from water into a denser medium.