The Refractive Index

The refractive index is the most-used number in optics, and it wears three faces at once. Operationally it is kinematic: $n=c/v$ , the factor by which a medium slows the phase velocity of a monochromatic wave. Structurally it is electromagnetic: this chapter's two preceding articles establish that Maxwell's equations in a linear dielectric give a wave speed $v=c/\sqrt{\varepsilon_r}$ , so $n=\sqrt{\varepsilon_r}=\sqrt{1+\chi}$ — the index is the square root of one plus the susceptibility. And underneath both it is microscopic: $\chi(\omega)$ is set by how the bound electrons of the material respond to a driving field, which is why $n$ is never truly a constant but a function of frequency. This article is the synthesis of the O4 head — it takes $n=\sqrt{\varepsilon_r}$ from aq-optics-maxwell-equations-dielectric-media and $n=\sqrt{1+\chi}$ from aq-optics-dielectric-susceptibility-permittivity and asks what the resulting number actually means and does. Three consequences organize everything that follows. First, because $\chi$ depends on frequency, so does $n$ : the index rises gently from red to blue across the visible in ordinary glass, and that dispersion is what fans a prism and blurs a simple lens. Second, because the polarization response lags, the index is fundamentally complex, $\tilde n=n+i\kappa$ : the real part sets the phase velocity, the imaginary part $\kappa$ sets the absorption, and causality ties them together. Third, the "speed" $n$ measures is the speed of a wave crest, not of a signal — a phase index below one (so $v>c$ ) is routine and violates nothing, because information rides the group velocity. The explicit resonant shape of $n(\omega)$ , and the machinery of normal versus anomalous dispersion, are developed in the articles that follow this one and are recalled here by name, not re-derived.

Key takeaways

The refractive index is $n=c/v$ , the factor by which a medium slows a wave's phase velocity; the same number bends rays at an interface through Snell's law.
Electromagnetically $n=\sqrt{\varepsilon_r}=\sqrt{1+\chi}$ for a non-magnetic medium (recalled from the two preceding O4 articles): the index is fixed by the dielectric susceptibility.
Because $\chi$ depends on frequency, $n=n(\omega)$ — this is dispersion, and it makes prisms split colours and lenses show chromatic aberration.
The index is complex, $\tilde n=n+i\kappa$ : the real part sets phase velocity and refraction, the extinction coefficient $\kappa$ sets absorption via $\alpha=4\pi\kappa/\lambda_0$ .
Real values are tabulated by the Sellmeier equation: BK7 glass has $n=1.51680$ at the yellow d-line and an Abbe number $V_d=64.17$ summarizing its dispersion.
A phase index below 1 ( $v>c$ ) is physical and common above resonances and in the X-ray region; signals still travel at the group velocity $\le c$ .
Key idea: the refractive index is one number seen three ways — kinematic ( $n=c/v$ ), electromagnetic ( $n=\sqrt{1+\chi}$ ), and microscopic (the frequency-dependent polarizability of bound charge) — and its frequency dependence, dispersion, is the subject of the whole rest of the chapter.

The core insight

The refractive index is simultaneously kinematic ( $n=c/v$ ), electromagnetic ( $n=\sqrt{\varepsilon_r}=\sqrt{1+\chi}$ ), and microscopic (fixed by the frequency-dependent response of bound charges). Because that response depends on frequency, $n=n(\omega)$ — the dispersion that fans a prism.

Where this lives today

The index and its dispersion are the working currency of optical design. Lens catalogues specify each glass by its d-line index and Abbe number, and designers combine crown and flint glasses to build achromatic and apochromatic lenses that cancel chromatic aberration — impossible without knowing $n(\lambda)$ to five decimals. In fibre-optic telecommunications it is the group index, not $n$ , that sets a pulse's travel time and hence a link's timing budget, and the derivative $\mathrm{d}n_g/\mathrm{d}\lambda$ (chromatic dispersion) broadens pulses and caps the bit rate. Refractometers exploit the dependence of $n$ on composition to measure sugar in juice, salinity in seawater, and protein in solution to better than one part in $10^4$ ; surface-plasmon sensors read tiny index changes for label-free biosensing. And at the short-wavelength extreme, $n<1$ in the X-ray band is what makes grazing-incidence mirrors and X-ray telescopes possible — the same physics that would seem to "break relativity" put to work.

The physical picture

An incident wave sweeps across a sheet of atoms; each atom's electrons are driven into oscillation and re-radiate a small wavelet slightly out of step with the driver. Sum every wavelet onto the original wave and the transmitted crests are shifted back in time — the wave has slowed. Over many atomic layers the accumulated slowing is exactly the factor $n$ , and the same slowing, forced to keep wavefronts continuous at a boundary, is what bends rays toward the normal in a denser medium. The first figure makes "slower crests" and "compressed wavelength" one picture.

Figure

A wave crossing into glass (n = 1.50): the frequency ν is unchanged, but the phase velocity drops to v = c/n = 2.0×10⁸ m/s and the wavelength compresses to λ = λ₀/n, so 500 nm in vacuum becomes 333 nm in the glass

The index changes with colour because the electronic oscillators respond differently to different drive frequencies: blue light is driven nearer the material's ultraviolet resonances than red light, so it is polarized more strongly and slowed more. That is dispersion, and it is the reason a prism spreads white light and a cheap lens rings its images with colour. The imaginary part of the index is the unavoidable companion of dispersion — an oscillator that responds out of phase (bending light) must also respond in quadrature (absorbing it) somewhere in the spectrum, the content of the Kramers–Kronig relations. The second figure is the quantity a lens designer actually reaches for: the measured dispersion curve of a real optical glass.

Figure

Measured dispersion of Schott N-BK7 glass across the visible from the Sellmeier equation: n falls from blue to red — n_F = 1.52238 (486.1 nm), n_d = 1.51680 (587.6 nm), n_C = 1.51432 (656.3 nm) — giving Abbe number V_d = (n_d−1)/(n_F−n_C) = 64.17

The core equations

The operational definition is kinematic: the index is how much slower a wave crest moves in the medium than in vacuum. Everything else in this article is a consequence of, or a computation of, this ratio.

n=\frac{c}{v} \qquad (1)

where: $n$ — (phase) refractive index, dimensionless; $c$ — vacuum speed of light, $299\,792\,458\ \mathrm{m/s}$ ; $v$ — phase velocity of the monochromatic wave in the medium. A larger $n$ means a slower crest and, at an interface, a ray bent more strongly toward the normal. The electromagnetic identity is the synthesis this chapter has been building toward. It is recalled by name — $n=\sqrt{\varepsilon_r}$ from aq-optics-maxwell-equations-dielectric-media, and $\varepsilon_r=1+\chi$ from aq-optics-dielectric-susceptibility-permittivity — and combined below.

n=\sqrt{\varepsilon_r\,\mu_r}\;\xrightarrow{\ \mu_r=1\ }\;\sqrt{\varepsilon_r}=\sqrt{1+\chi} \qquad (2)

where: $\varepsilon_r$ — relative permittivity at the optical frequency; $\mu_r\approx1$ optically; $\chi$ — electric susceptibility. Because $\chi=\chi(\omega)$ , equation (2) already contains dispersion: $n$ inherits the frequency dependence of the susceptibility. Inside the medium the frequency is fixed but the wavenumber and wavelength scale with $n$ , which is the bookkeeping behind "the wavelength shrinks in glass".

k=n\,\frac{\omega}{c}=\frac{2\pi n}{\lambda_0},\qquad \lambda=\frac{\lambda_0}{n} \qquad (3)

where: $k$ — wavenumber in the medium; $\omega$ — angular frequency (continuous across an interface, so unchanged); $\lambda_0$ — vacuum wavelength; $\lambda$ — wavelength in the medium. The frequency, set at the source, does not change; only the spatial scale does. For a lossy medium the index must be complex, so that a single quantity carries both the phase advance and the amplitude decay of a propagating wave.

\tilde n=n+i\kappa,\qquad \alpha=\frac{2\kappa\omega}{c}=\frac{4\pi\kappa}{\lambda_0} \qquad (4)

where: $n$ — real part, sets phase velocity and refraction; $\kappa$ — extinction coefficient, sets attenuation; $\alpha$ — intensity absorption coefficient (Beer–Lambert). Derivation B shows how $\kappa$ becomes $\alpha$ . For real materials the frequency dependence is captured empirically by the Sellmeier equation — a multi-resonance fit to $n(\lambda)$ across the transparency window, and the formula used to draw the BK7 curve above.

n^2(\lambda)=1+\sum_j\frac{B_j\,\lambda^2}{\lambda^2-C_j} \qquad (5)

where: $\lambda$ — vacuum wavelength; $B_j,C_j$ — empirical coefficients fitted per glass. Each term is a resonance of the underlying Lorentz oscillators; the derivation of that Lorentzian $n(\omega)$ is owned by aq-optics-lorentz-oscillator-model and aq-em-absorption-dispersion and recalled here by name. Finally, the speed $n$ measures is the crest speed; the speed of a pulse or signal is set by a separate group index, quoted here and developed in its own article.

n_g=n+\omega\,\frac{\mathrm{d}n}{\mathrm{d}\omega}=n-\lambda\,\frac{\mathrm{d}n}{\mathrm{d}\lambda},\qquad v_g=\frac{c}{n_g} \qquad (6)

where: $n_g$ — group index; $v_g$ — group (pulse) velocity; $\mathrm{d}n/\mathrm{d}\lambda<0$ in normal dispersion. The distinction — crest versus signal — is what saves relativity when $n<1$ .

Derivation AThe refractive index as the square root of the dielectric response

The goal is to assemble the central identity of the chapter, $n=\sqrt{1+\chi}$ , from the two results the preceding articles own — the wave equation's speed $v=c/\sqrt{\varepsilon_r}$ and the susceptibility's repackaging $\varepsilon_r=1+\chi$ — and to make explicit that this single line carries all of dispersion once $\chi$ is a function of frequency. We recall each owned piece by name, combine them, and check the limits, so that the three faces of the index (kinematic, electromagnetic, microscopic) become one equation.

Step 1. From aq-optics-maxwell-equations-dielectric-media, the source-free non-magnetic wave equation gives a phase velocity $v=1/\sqrt{\mu_0\varepsilon}=c/\sqrt{\varepsilon_r}$ , so the kinematic definition $n=c/v$ becomes $n=\sqrt{\varepsilon_r}$ . Why: the wave equation is where "how fast a crest moves" is fixed by the material constant $\varepsilon$ , converting a kinematic definition into an electromagnetic one.

Step 2. From aq-optics-dielectric-susceptibility-permittivity, $\varepsilon_r=1+\chi$ ; substitute to get $n=\sqrt{1+\chi}$ . Why: this exposes the microscopic face — the index is set entirely by the bound-charge response $\chi$ , so anything true of $\chi$ (frequency dependence, complexity, tensor character) is inherited by $n$ .

Step 3. Promote to frequency: since $\chi=\chi(\omega)$ , the index is $n(\omega)=\sqrt{1+\chi(\omega)}$ , which is not a constant — it rises with $\omega$ across a transparency window (normal dispersion) and swings sharply near a resonance. Why: the whole phenomenon of dispersion is contained in this one substitution; a "constant index" is only the leading approximation over a narrow band.

Step 4. Check the limits: $\chi\to0$ gives $n\to1$ (vacuum); $\chi\ll1$ gives $n\approx1+\tfrac12\chi$ (gases); a typical glass $\varepsilon_r=2.25$ gives $n=1.50$ . Why: the identity must reproduce the trivial vacuum case and the measured dilute-gas and glass values, which it does exactly. *Verified: $n=\sqrt{1+\chi}$ ; $n\to1$ as $\chi\to0$ ; glass $\varepsilon_r=2.25\Rightarrow n=1.50$ ; air $n\approx1+\tfrac12\chi$ in verify_aq-optics-refractive-index.py.*

Go deeper · Derivation B · 1 minComplex index and the Beer–Lambert absorption law

Derivation BComplex index and the Beer–Lambert absorption law

The aim is to show that once the index is complex, $\tilde n=n+i\kappa$ , a single plane wave automatically splits into a phase-advancing oscillation and an exponentially decaying envelope, and to extract the intensity absorption coefficient $\alpha$ that a spectrophotometer measures. We carry the complex index through the propagation factor, separate real and imaginary parts, and square the amplitude to get intensity — the standard route from $\kappa$ to $\alpha$ .

Step 1. Write a plane wave with complex index, $E\propto e^{i(\tilde n\omega z/c-\omega t)}$ , and substitute $\tilde n=n+i\kappa$ to factor it into $e^{i(n\omega z/c-\omega t)}\,e^{-\kappa\omega z/c}$ . Why: the imaginary part of the index appears in the exponent as a real negative number, turning part of the phase factor into an amplitude decay — absorption is built into the same object as refraction.

Step 2. Identify the two roles: the real part $n$ advances the phase (sets $v=c/n$ and Snell bending), while $\kappa$ attenuates the amplitude over a scale $c/(\kappa\omega)$ . Why: keeping the two parts separate is what lets one number, $\tilde n$ , describe a coloured, absorbing medium without extra machinery.

Step 3. Form the intensity $I=|E|^2\propto e^{-2\kappa\omega z/c}\equiv e^{-\alpha z}$ , giving $\alpha=2\kappa\omega/c=4\pi\kappa/\lambda_0$ . Why: intensity is amplitude squared, so the decay exponent doubles; this is exactly the Beer–Lambert law, and it is why measuring transmission versus frequency maps out $\kappa(\omega)$ and hence the imaginary susceptibility. *Verified: $\alpha=4\pi\kappa/\lambda_0=2\kappa\omega/c$ and the factor-of-two intensity-vs-amplitude relation in verify_aq-optics-refractive-index.py.*

Limiting cases

Case	Result	Meaning
Vacuum	$n=1$ exactly	no medium to slow or absorb; $v=c$
Dilute gas	$n\approx1+\tfrac12\chi,\ \chi\ll1$	$n-1\propto$ density; air $n-1\approx2.9\times10^{-4}$ at STP
Transparent (below resonance)	$n(\omega)$ real, $\mathrm{d}n/\mathrm{d}\omega>0$	normal dispersion — the visible regime of ordinary glass; blue slowed more than red
High-frequency / X-ray	$n\to1-\omega_p^2/2\omega^2<1$	electrons act free; index dips below one, $v>c$ , enabling grazing-incidence X-ray mirrors
Near resonance	$\tilde n=n+i\kappa,\ \kappa$ peaks	anomalous dispersion ( $\mathrm{d}n/\mathrm{d}\omega<0$ ) with strong absorption; real and imaginary parts trade off by Kramers–Kronig

Common trap“A refractive index below 1 means light beats c, breaking relativity.”

It does not. An index $n<1$ means the phase velocity $v=c/n$ exceeds $c$ , and this is common and harmless — it happens just above every absorption resonance and throughout the X-ray region (where it makes total external reflection and X-ray telescopes possible). A phase velocity is the speed of an idealized infinite wave crest, which carries no information: you cannot send a message on a feature that was already everywhere. A signal — the leading edge of a pulse — travels at the group velocity $v_g=c/n_g$ , which stays at or below $c$ in exactly the regimes where $n<1$ , because the group index $n_g=n+\omega\,\mathrm{d}n/\mathrm{d}\omega$ is large there. So $n<1$ is a statement about crests, not about causality. (A related trap: "optical density" is an old name for $n$ , but it is a misnomer — a higher index does not mean a physically denser material, as diamond versus water shows.)

Worked examples

1 · The dispersion of an optical glass (confidence builder)

Schott N-BK7 has refractive indices $n_F=1.52238$ (blue F-line, 486.1 nm), $n_d=1.51680$ (yellow d-line, 587.6 nm), and $n_C=1.51432$ (red C-line, 656.3 nm). Find its Abbe number and say what it means for a prism.

Step 1. The Abbe number summarizes dispersion by comparing the index at the yellow reference to the blue-minus-red spread: $V_d=\dfrac{n_d-1}{n_F-n_C}$ . Compute the spread first. The 5-decimal line indices give $n_F-n_C=1.52238-1.51432=0.00806$ , but subtracting two nearly-equal numbers throws away a digit; carrying the full Sellmeier evaluation gives $n_F-n_C=\mathbf{0.008054}$ . Either way the index is higher for blue than red — normal dispersion.

Step 2. Numerator: $n_d-1=1.51680-1=0.51680$ . Ratio: $V_d=0.51680/0.008054=\mathbf{64.17}$ (the Schott catalogue value; the rounded spread $0.00806$ would give $64.1$ , showing how sensitive $V_d$ is to the last digit of a small denominator). A large $V_d$ means low dispersion — BK7 is a crown glass.

Step 3. What it does: because $n$ is larger for blue, a BK7 prism refracts blue light more than red at each face, fanning white light into a spectrum with the blue end deviated most. The angular spread scales with $1/V_d$ , so a flint glass ( $V_d\approx36$ ) of the same shape would spread the colours nearly twice as far — which is exactly why achromatic doublets pair a crown and a flint to cancel dispersion while keeping focusing power. V_d = 64.17, and blue (n_F) is deviated more than red (n_C).

2 · Wavelength and speed of light inside glass

A 500 nm (vacuum) beam enters crown glass of index $n=1.50$ . Find its frequency, wavelength, and phase velocity inside.

Step 1. Frequency is set at the source and is continuous across the interface, so it does not change: $\nu=c/\lambda_0=(2.998\times10^8)/(500\times10^{-9})=\mathbf{6.00\times10^{14}\ Hz}$ .

Step 2. The wavelength contracts by $n$ (equation 3): $\lambda=\lambda_0/n=500/1.50=\mathbf{333\ nm}$ . The crests are packed closer because they move slower while ticking at the same rate.

Step 3. Phase velocity: $v=c/n=(2.998\times10^8)/1.50=\mathbf{2.00\times10^8\ m/s}$ , and consistently $v=\nu\lambda=(6.00\times10^{14})(333\times10^{-9})=2.00\times10^8$ m/s. The three quantities are one statement: same $\nu$ , smaller $\lambda$ , slower $v$ , all by the single factor $n$ . ν = 6.00×10¹⁴ Hz (unchanged), λ = 333 nm, v = 2.00×10⁸ m/s.

Physics of the portrait

The refractive index is where three centuries of optics meet: a number measured by Snel long before anyone knew what it was, explained by Maxwell as $\sqrt{\varepsilon}$ , and finally computed from atomic resonances.

Origins · 1 minrefractive index and how this was discovered

refractive index

the material constant n = c/v = √ε_r · electromagnetic optics of media

The index of refraction is one of the oldest quantitative concepts in optics and one of the last to be explained. Willebrord Snel measured the constant ratio of sines at an interface around 1621, and Descartes published the sine law in 1637, but for two centuries $n$ was a purely empirical number with no physical meaning — the tables listed it, and no one could say why glass had $1.5$ and diamond $2.4$ . Augustin-Louis Cauchy gave the first useful account of its frequency dependence in 1836, fitting $n(\lambda)=A+B/\lambda^2+\cdots$ to explain a prism's spread, though his formula fails near absorption bands. The decisive step was James Clerk Maxwell's electromagnetic theory of light in 1865: if light is an electromagnetic wave, then $n=\sqrt{\varepsilon_r}$ , tying the optical index to the electrical dielectric constant — a prediction Boltzmann confirmed for non-polar gases. Wolfgang Sellmeier in 1871 turned Cauchy's fit into a physical one built from resonances, giving the multi-term formula still used to specify every optical glass, and Lorentz and Lorenz around 1880 grounded it in the polarizability of atoms (the Lorentz–Lorenz relation). The modern picture — $\tilde n(\omega)=\sqrt{1+\chi(\omega)}$ , complex and causal — completes what Snel began: a measured constant is now a computed function of the electrons inside the glass.

History

1621

— Snel measures the constant ratio of sines at a refracting interface

1637

— Descartes publishes the sine law of refraction

1836

— Cauchy fits the empirical dispersion formula n(λ) = A + B/λ² + …

1865

— Maxwell's electromagnetic theory of light gives n = √ε

1871

— Sellmeier casts dispersion as a sum over resonances

1880

— Lorentz and Lorenz relate n² to atomic polarizability

Going further

Texts. Hecht, Optics, 5th ed., §3.5 (index and dispersion) and §4.8 · Saleh & Teich, Fundamentals of Photonics, 3rd ed., Ch. 5 (electromagnetic optics) · Born & Wolf, Principles of Optics, §2.3 (dispersion) · Foot, Atomic Physics, Ch. 7 (light–atom interaction) · Demtröder, Laser Spectroscopy, Vol. 1, §3.1.

Provenance. The identity $n=\sqrt{\varepsilon_r}$ is recalled by name from aq-optics-maxwell-equations-dielectric-media, and $\varepsilon_r=1+\chi$ from aq-optics-dielectric-susceptibility-permittivity; the resonant Lorentzian shape of $n(\omega)$ and the normal-versus-anomalous dispersion it predicts are owned by aq-optics-lorentz-oscillator-model, aq-optics-normal-anomalous-dispersion, and aq-em-absorption-dispersion (forward references, not re-derived here). A verifier should confirm: (1) $n=\sqrt{1+\chi}$ from the wave equation (Hecht §3.5); (2) $\alpha=4\pi\kappa/\lambda_0$ from the complex-index propagation factor; (3) the BK7 Sellmeier values $n_F=1.52238$ , $n_d=1.51680$ , $n_C=1.51432$ and Abbe $V_d=64.17$ from the Schott coefficients; (4) $\lambda=\lambda_0/n$ , $v=c/n$ , $\nu$ unchanged. All indices ( $n_{\text{air}}\approx1.0003$ , water $1.33$ , crown glass $1.5$ , diamond $2.42$ ) are standard tabulated values.

Prerequisites. Maxwell's Equations in Dielectric Media · Dielectric Susceptibility and Permittivity · Plane Waves in Homogeneous Isotropic Media. Leads to. Complex Refractive Index and the Extinction Coefficient · Dispersion: Normal and Anomalous · Group Velocity and Group Index · Sellmeier and Cauchy Dispersion Equations.

Is the refractive index really "how much light slows down"?

Yes, but with one word of care: it is how much the phase velocity slows. A monochromatic wave's crests move at $v=c/n$ , so in glass with $n=1.5$ the crests crawl at two-thirds of $c$ . A pulse or signal, though, moves at the group velocity $v_g=c/n_g$ , which can be faster or slower than $v$ depending on dispersion. For most everyday purposes ("light travels slower in water") the phase picture is what people mean, and it is correct. The subtleties only matter when you care about the timing of pulses, as in fibre communications, where you must use the group index instead.

Why does the frequency stay the same but the wavelength change in a medium?

Because the frequency is imposed by the source and the boundary conditions. At the interface the fields on both sides must oscillate in step at every instant, which is only possible if they share the same frequency — a mismatch would tear the wave apart at the surface. The speed, however, drops to $v=c/n$ , and since $v=\nu\lambda$ with $\nu$ fixed, the wavelength must shrink to $\lambda=\lambda_0/n$ . So the crests pack closer together while ticking past at the same rate. This is why a 500 nm green beam is "green" to your eye whether it is in air or water: your retina responds to frequency, which never changed.

What exactly is the Abbe number, and why do lens makers obsess over it?

The Abbe number $V_d=(n_d-1)/(n_F-n_C)$ is a single figure of merit for how much a glass disperses. The numerator is the refracting power at the yellow d-line; the denominator is the blue-to-red spread of the index. A large $V_d$ (crown glass, $\sim60$ ) means low dispersion — the glass bends all colours by nearly the same amount. A small $V_d$ (flint glass, $\sim35$ ) means strong dispersion. Lens designers pair a positive crown element with a negative flint element so their dispersions cancel while their focusing powers do not, building an achromatic doublet that brings two colours to a common focus. Without the Abbe number there is no systematic way to cancel chromatic aberration.

Where does the imaginary part κ come from if my glass is perfectly clear?

"Perfectly clear" only means $\kappa$ is negligibly small in the visible. No real material has $\kappa=0$ at all frequencies — that would violate Kramers–Kronig, which says any frequency-varying $n(\omega)$ must be accompanied by absorption somewhere. Ordinary glass has strong absorption bands in the ultraviolet (electronic transitions) and the infrared (molecular vibrations); the visible window sits in the transparent gap between them, where $\kappa$ is tiny but not exactly zero. The dispersion you see in the visible is the low-energy tail of those out-of-band resonances, which is exactly why Sellmeier's resonance formula works so well there.

Is "optical density" the same as refractive index?

"Optical density" is an old and misleading name for the refractive index, and you should be wary of it. It suggests that a higher index means a physically denser material, which is false: diamond ( $n=2.42$ ) is far less mass-dense than many high-index glasses, and it is the electronic response near resonances, not the mass density, that sets $n$ . Worse, "optical density" is also used in spectroscopy to mean absorbance ( $\log_{10}$ of the inverse transmission), an entirely different quantity governed by $\kappa$ , not $n$ . Because of this ambiguity, modern usage prefers "refractive index" for $n$ and "absorbance" for the spectroscopic quantity, and reserves "denser medium" only as loose shorthand for "higher index".

How can the same n both slow the wave and bend the ray?

They are two views of one fact: continuity of the wave at a boundary. Picture wavefronts marching toward an interface at an angle. On the far side the phase velocity drops to $c/n$ , so the part of each wavefront that enters first travels a shorter distance before the rest catches up — the wavefront pivots, and the ray perpendicular to it bends toward the normal. Working the geometry out gives $n_1\sin\theta_1=n_2\sin\theta_2$ , Snell's law, with the same $n$ that measures the slowing. "Slower crests" and "bent rays" are the identical statement told from wavefront and ray points of view.