Numerical Aperture and Acceptance Angle

Total internal reflection tells you when a ray already inside a fiber stays trapped: strike the core–cladding wall past the critical angle $\phi_c$ and none of the light escapes. But a fiber is useless until you can get light into it, and that is a different question — which incoming rays, launched through the flat polished end face from the air, will meet that wall steeply enough to be guided? Numerical aperture is the number that answers it. Take the critical-angle condition that total internal reflection hands you, run it backwards through Snell's law at the entrance face, and the internal "stay trapped" requirement turns into an external cone: every ray entering within a half-angle $\theta_a$ of the axis is captured and guided, and every ray outside that cone leaks into the cladding within a few bounces. The sine of that half-angle is the numerical aperture, $\mathrm{NA}=n_0\sin\theta_a$ . What makes NA more than a restatement of the critical angle is where the geometry lands. Carry the algebra through and the core radius cancels, the wavelength cancels, and the acceptance cone collapses to a startlingly compact statement about the two glasses alone: $\mathrm{NA}=\sqrt{n_1^2-n_2^2}$ . That one number then propagates through the whole of fiber optics — it fixes how much power a diode or LED couples in, multiplies into the V-parameter that counts the guided modes, and through the spread of ray path lengths caps the bandwidth. NA is where the ray picture of guiding hands off to the wave picture of modes.

Key takeaways

Numerical aperture is the sine of the acceptance half-angle: $\mathrm{NA}=n_0\sin\theta_a$ , the boundary between rays a fiber captures and rays it leaks.
Tracing total internal reflection back through the end face gives the compact result $\mathrm{NA}=\sqrt{n_1^2-n_2^2}$ — set by the core and cladding indices alone, independent of core radius and wavelength.
For small index steps it is usually written $\mathrm{NA}\approx n_1\sqrt{2\Delta}$ with the fractional contrast $\Delta=(n_1-n_2)/n_1$ .
NA multiplies straight into the V-parameter $V=(2\pi a/\lambda_0)\,\mathrm{NA}$ , which counts guided modes ( $M\approx V^2/2$ ) and sets the single-mode cutoff $V<2.405$ .
High NA is a trade, not a goal: it gathers more light and eases alignment, but admits more modes and worsens modal dispersion, so long links use low NA.
Key idea: numerical aperture repackages the internal total-internal-reflection guiding condition as an external light-gathering cone, $\mathrm{NA}=n_0\sin\theta_a=\sqrt{n_1^2-n_2^2}$ , fixed by the core–cladding index contrast alone.

The core insight

Numerical aperture converts the internal total-internal-reflection guiding condition into an external light-gathering cone: $\mathrm{NA}=n_0\sin\theta_a=\sqrt{n_1^2-n_2^2}$ , fixed by the core–cladding index contrast alone and independent of core radius or wavelength.

Where this lives today

Numerical aperture is the first number on any fiber datasheet because it silently sets three budgets at once. In a data-center link on OM3/OM4 multimode fiber, the specified $\mathrm{NA}\approx0.2$ lets a cheap VCSEL or LED couple enough power through a loosely-toleranced connector — and also caps the reach, because the several hundred modes it admits disperse the pulse. Telecom long-haul fiber goes the other way: a deliberately tiny $\mathrm{NA}\approx0.14$ and a few-micron core drop $V$ below $2.405$ , so the fiber is single-mode and modal dispersion vanishes, at the price of a coupling job that demands a laser and precise alignment. At the extreme, double-clad and air-clad fibers for high-power lasers push NA toward and past unity so their inner cladding can scoop up the messy multimode pump light of diode bars. And the concept reaches beyond glass: the NA of a microscope objective still fixes its resolution by Abbe's $\lambda/(2\,\mathrm{NA})$ , and in laser spectroscopy matching the collection NA to the fluorescence solid angle decides whether a faint signal is seen at all. Every one is $n\sin\theta$ doing the same job — quantifying how greedily an optical system reaches for light.

The physical picture

Point a diverging beam at the polished end of a fiber and watch what different rays do. A ray entering nearly along the axis refracts only slightly and strikes the core–cladding boundary at a glancing, nearly- $90°$ angle — well past the critical angle, so it is totally internally reflected and zig-zags down the fiber. Tilt the incoming ray off-axis and it refracts to a steeper internal angle, meeting the wall more head-on. At some input angle $\theta_a$ it hits the wall at exactly the critical angle $\phi_c$ : that is the last ray still guided. Any steeper, and the wall angle drops below $\phi_c$ , so the ray refracts into the cladding and is gone within a bounce or two. The captured rays fill a cone of half-angle $\theta_a$ , and $\sin\theta_a$ is the numerical aperture.

Figure

Acceptance cone of a step-index fiber with n₁ = 1.48 and n₂ = 1.46. A ray entering inside the half-angle θ_a ≈ 14° refracts, meets the wall past the critical angle, and is guided by total internal reflection green. A ray entering too steeply hits the wall below φ_c and leaks into the cladding red. Here NA = √(n₁² − n₂²) = 0.24

The width of that cone is governed entirely by the jump in index at the wall. A larger contrast pulls the critical angle further from grazing, so a ray can be bent more steeply inside the core and still clear $\phi_c$ — a bigger index step buys a wider cone and a larger NA. This is why NA is best read as the index contrast re-expressed as an angle, and why the result is so clean: only the two indices enter the critical-angle condition, so only the two indices survive into NA. That single number then sets the character of the whole fiber. Think of NA as the size of a light bucket: a high-NA fiber is wide, easy to fill from a diffuse LED and forgiving of a misaligned connector, but every ray it catches travels a different zig-zag path length, so a sharp pulse arrives smeared. A low-NA fiber is narrow — fussy to couple into, but its surviving rays travel nearly the same distance, keeping pulses crisp over long hauls. The plot below is what an engineer reads off: acceptance half-angle versus NA, with the working fibers marked.

Figure

Acceptance half-angle θ_a = arcsin NA versus numerical aperture. Real fibers sit low on the curve: telecom single-mode NA = 0.14 gives θ_a ≈ 8°, standard multimode NA = 0.20 gives ≈ 11.5°, high-NA fiber NA = 0.30 gives ≈ 17.5°. Only an air-clad fiber at NA = 1 opens to the full hemisphere θ_a = 90°

The core equations

The operational definition is the acceptance cone itself: the numerical aperture is the sine of the largest angle, measured from the axis in the launch medium, at which a ray is still captured and guided. Carrying the critical-angle condition back through Snell's law at the end face — the derivation below — collapses it to the two glasses.

\mathrm{NA}=n_0\sin\theta_a=\sqrt{n_1^2-n_2^2} \qquad (1)

where: $\mathrm{NA}$ — numerical aperture, dimensionless; $n_0$ — index of the external launch medium (air, $\approx1$ ); $\theta_a$ — maximum acceptance half-angle in that medium; $n_1$ — core index; $n_2$ — cladding index, with $n_2<n_1$ . The middle expression is what you measure (the cone in air); the right expression is what the fiber is (a property of the two glasses). They are equal because Snell's law at the face and the critical angle at the wall are the same physics seen twice. To recover the physical cone you invert the sine, remembering that $\theta_a$ is a half-angle — the full captured cone spans $2\theta_a$ . In air ( $n_0\approx1$ ) the acceptance angle is just $\arcsin\mathrm{NA}$ .

\theta_a=\arcsin\!\left(\frac{\mathrm{NA}}{n_0}\right),\qquad \text{full cone}=2\theta_a \qquad (2)

where: $\theta_a$ — acceptance half-angle; $n_0$ — launch-medium index. Because $\arcsin$ is only defined for arguments up to one, an NA at or above $n_0$ means the fiber accepts light from the entire forward hemisphere — the cone can open no wider than $90°$ . Real telecom fibers have core and cladding indices differing only in the third decimal place, so it pays to rewrite NA in terms of the fractional index contrast $\Delta$ . Factoring the difference of squares and using $n_1\approx n_2$ gives the form catalogues quote — the weakly-guiding approximation, accurate to better than a percent whenever $\Delta\ll1$ .

\Delta=\frac{n_1-n_2}{n_1}\approx\frac{n_1^2-n_2^2}{2n_1^2},\qquad \mathrm{NA}\approx n_1\sqrt{2\Delta} \qquad (3)

where: $\Delta$ — relative index contrast, typically $0.002$ to $0.03$ for silica fibers. Small $\Delta$ is also the regime where guided modes become nearly transverse (the LP-mode picture), so the smallness that makes this formula good also simplifies the wave analysis downstream. NA is the bridge to the wave description through the normalized frequency, or V-parameter, which packages the acceptance cone, core size, and wavelength into one dimensionless number that decides the mode content. NA itself is wavelength-independent, but $V$ is not — the radius and $\lambda_0$ enter here.

V=\frac{2\pi a}{\lambda_0}\,\mathrm{NA}=\frac{2\pi a}{\lambda_0}\sqrt{n_1^2-n_2^2} \qquad (4)

where: $V$ — normalized frequency, dimensionless; $a$ — core radius; $\lambda_0$ — free-space wavelength. A fiber is single-mode when $V<2.405$ (the first zero of the Bessel function $J_0$ ), multimode above. When $V$ is large the fiber is deep in the multimode regime, and a phase-space argument (third derivation) counts the modes: the number of allowed transverse states grows as the square of $V$ .

M\approx\frac{V^2}{2}=\frac{1}{2}\left(\frac{2\pi a}{\lambda_0}\,\mathrm{NA}\right)^2 \qquad (5)

where: $M$ — approximate number of guided modes. Because $M\propto\mathrm{NA}^2$ , doubling the NA roughly quadruples the mode count — the link between a wide acceptance cone and severe modal dispersion. Finally, NA governs how much power you can launch. If the source diverges into a wider cone than the fiber accepts (an overfilled launch, typical of LEDs), the captured fraction scales as the square of the ratio of the two numerical apertures.

\eta\le\left(\frac{\mathrm{NA}_{\text{fiber}}}{\mathrm{NA}_{\text{source}}}\right)^2 \qquad (6)

where: $\eta$ — power-coupling efficiency; $\mathrm{NA}_{\text{source}}$ — sine of the source's divergence half-angle. This bound is why short-reach LED and power-delivery links reach for high-NA multimode fiber.

Derivation ATracing the acceptance cone back from total internal reflection

The goal is to turn the guiding condition that total internal reflection hands us — a ray stays trapped only if it strikes the core–cladding wall at or beyond the critical angle $\phi_c$ — into a statement about which external rays a fiber accepts, and to show the answer depends on nothing but the two indices. We follow one extreme meridional ray (through the axis) from the launch medium, refract it at the end face, walk it to the cylindrical wall, impose the critical-angle limit there, and carry the result back out — and the radius and wavelength both drop out along the way.

Step 1. A ray launched from the external medium at angle $\theta$ to the axis refracts at the flat face according to Snell's law, $n_0\sin\theta=n_1\sin\theta_r$ , where $\theta_r$ is the internal ray angle measured from the axis. Why: the face is the only place the light crosses from $n_0$ into $n_1$ , so this is where the external launch angle is converted into an internal propagation angle.

Step 2. Inside, the ray travels to the core–cladding wall, whose normal is perpendicular to the axis; the angle of incidence there is the complement of the ray angle, $\phi=90°-\theta_r$ . Why: the wall runs parallel to the axis, so a ray tilted by $\theta_r$ from the axis is tilted by $90°-\theta_r$ from the wall normal — the flatter the ray, the more glancing the hit.

Step 3. Total internal reflection guides the ray only if $\phi\ge\phi_c$ , with $\sin\phi_c=n_2/n_1$ . The widest accepted input angle $\theta_a$ is the one that makes the wall angle land exactly on the critical value, $\phi=\phi_c$ , so $\theta_r=90°-\phi_c$ and hence $\sin\theta_r=\cos\phi_c$ . Why: pushing the input angle any wider drives $\phi$ below $\phi_c$ , the light refracts into the cladding, and the ray is lost — so the boundary case defines the edge of the cone.

Step 4. Substitute $\sin\theta_r=\cos\phi_c=\sqrt{1-(n_2/n_1)^2}$ into the face relation from Step 1: $n_0\sin\theta_a=n_1\sqrt{1-(n_2/n_1)^2}=\sqrt{n_1^2-n_2^2}$ , the numerical aperture. Why: the factor of $n_1$ multiplies into the root and cancels the $n_1$ in the denominator, leaving the clean difference of squares — the core index disappears as a standalone parameter and only the contrast survives. *Verified: reproduces $\mathrm{NA}=\sqrt{n_1^2-n_2^2}$ , Saleh and Teich Eq. 9.1-3 and Hecht §5.6; for $n_1=1.48$ , $n_2=1.46$ it gives $\mathrm{NA}=0.2425$ , $\theta_a=14.0°$ .*

Go deeper · Derivation B · 1 minThe weakly-guiding form NA ≈ n₁√(2Δ)

Derivation BThe weakly-guiding form NA ≈ n₁√(2Δ)

The goal is to justify the compact expression every fiber datasheet quotes, $\mathrm{NA}\approx n_1\sqrt{2\Delta}$ , and to fix the condition under which it holds. Telecom silica fibers are built with a whisper of index contrast — a core doped a fraction of a percent above the cladding — because low contrast means low modal dispersion. In that regime, rewriting the exact difference-of-squares formula in terms of the single small parameter $\Delta$ exposes how NA actually scales: like the square root of the contrast, not linearly.

Step 1. Factor the exact NA as a difference of squares, $\mathrm{NA}=\sqrt{(n_1-n_2)(n_1+n_2)}$ , separating the tiny difference from the order-unity sum. Why: the physics lives in the difference $n_1-n_2$ , and factoring isolates it so the two very different scales stop being buried inside one square root.

Step 2. Introduce the fractional contrast $\Delta=(n_1-n_2)/n_1$ , so $n_1-n_2=n_1\Delta$ , and approximate the sum by $n_1+n_2\approx2n_1$ , valid because $n_1\approx n_2$ to a fraction of a percent. Why: when the two indices agree to three decimals, replacing their sum by twice either one costs less than that same fraction of a percent — a controlled approximation, not a guess.

Step 3. Multiply the pieces under the root: $\mathrm{NA}\approx\sqrt{(n_1\Delta)(2n_1)}=n_1\sqrt{2\Delta}$ , the standard weakly-guiding form. Why: halving the contrast does not halve the NA, it drops it by only $\sqrt2$ — which is why squeezing NA down for single-mode operation takes surprisingly large changes in doping. *Verified: for $n_1=1.45$ , $\Delta=0.005$ the formula gives $\mathrm{NA}\approx1.45\sqrt{0.01}=0.145$ , a standard telecom single-mode value and within a percent of the exact difference-of-squares result.*

Go deeper · Derivation C · 1 minCounting guided modes from NA through the V-parameter

Derivation CCounting guided modes from NA through the V-parameter

The goal is to see how a purely geometric quantity — the acceptance cone — ends up controlling the number of guided modes, and to derive the workhorse estimate $M\approx V^2/2$ . The link is that NA caps the transverse component of a guided ray's wavevector: a ray steeper than the acceptance limit is not guided. Counting how many transverse states fit under that cap in the finite core cross-section gives the mode number — the same phase-space bookkeeping that counts states in a two-dimensional box, and why NA and core area appear together in the V-parameter.

Step 1. A guided ray at internal angle $\theta_r$ carries transverse wavevector $k_T=n_1k_0\sin\theta_r$ , and the guiding limit $\sin\theta_r\le\cos\phi_c=\sqrt{1-(n_2/n_1)^2}$ caps it at $k_T\le k_0\mathrm{NA}$ , with $k_0=2\pi/\lambda_0$ . Why: the same critical-angle condition that bounded the acceptance cone now bounds the transverse momentum — the ray cannot wander off-axis faster than NA permits, or it would refract away.

Step 2. Define the normalized frequency as the maximum transverse phase accumulated across the core radius, $V\equiv a\,k_{T,\max}=(2\pi a/\lambda_0)\mathrm{NA}$ . Why: $V$ measures the core in units of the transverse wavelength — how many half-oscillations of the field fit across the core — which is exactly what decides how many standing-wave patterns (modes) it can hold.

Step 3. Count the transverse states: allowed $\mathbf{k}_T$ values fill a disk of radius $k_{T,\max}$ , and the number in a core of area $\pi a^2$ , times two polarizations, is $M\approx[\pi k_{T,\max}^2\cdot\pi a^2/(2\pi)^2]\times2=\tfrac12(ak_{T,\max})^2$ . Why: the standard density-of-states argument — one state per phase-space cell of size $(2\pi)^2$ — applied to the two transverse dimensions of the guide.

Step 4. Substitute $ak_{T,\max}=V$ to get $M\approx V^2/2$ . Why: it delivers the large- $V$ mode count in one dimensionless number and shows $M\propto\mathrm{NA}^2$ — the price of a wide acceptance cone is a quadratically larger mode population and the dispersion that comes with it. *Verified: matches Saleh and Teich Eq. 9.2-14; the same $V$ gives the single-mode cutoff $V<2.405$ (first zero of $J_0$ ). For $a=25\,\mu\mathrm{m}$ , $\mathrm{NA}=0.20$ , $\lambda_0=850\,\mathrm{nm}$ : $V=37.0$ , $M\approx683$ .*

Limiting cases

Case	Result	Meaning
Weak guidance ( $n_1\approx n_2$ )	$\mathrm{NA}\approx n_1\sqrt{2\Delta},\ \Delta\ll1$	Telecom silica: small index step, small NA, modest $\theta_a$ . Enables the LP-mode approximation used downstream.
High contrast (air cladding)	$\mathrm{NA}=\sqrt{n_1^2-1}$	For $n_1\approx1.45$ , $\mathrm{NA}\approx1.05$ : the cone opens to the full hemisphere. Used in high-power delivery and endoscopy.
Immersion launch ( $n_0>1$ )	$n_0\sin\theta_a=\sqrt{n_1^2-n_2^2}$	The physical cone $\theta_a$ narrows, but the right-hand side is unchanged — NA is a fiber property, not a launch property.
Graded-index core	$\mathrm{NA}(r)=\sqrt{n^2(r)-n_2^2}$	Local NA falls from axis to edge; the on-axis value matches the step-index form with $n_1=n(0)$ .
Single-mode cutoff	$V=(2\pi a/\lambda_0)\mathrm{NA}<2.405$	Small NA and small radius force single-mode operation — no modal dispersion, at the cost of a tiny core.

Common trapNA is a transmission percentage capped at 1

Neither claim is true. NA is the sine of an acceptance angle, not a fraction of transmitted power — a fiber with $\mathrm{NA}=0.2$ does not pass 20 percent of the light. And $\sqrt{n_1^2-n_2^2}$ can exceed unity: an air-clad silica fiber has $\mathrm{NA}\approx1.05$ . There is no contradiction with $\sin\theta_a\le1$ , because when $\mathrm{NA}\ge n_0$ the acceptance angle saturates at $90°$ — the fiber takes in the whole forward hemisphere, and the surplus NA measures confinement, not an impossible angle. A second, subtler trap: NA is independent of core radius and wavelength, but the mode count it produces is not, because the V-parameter carries the radius and $\lambda_0$ that NA itself does not. Widening the core changes how many modes ride the cone; it never changes the cone.

Worked examples

1 · The acceptance cone of a standard fiber (confidence builder)

A step-index fiber has core index $n_1=1.48$ and cladding index $n_2=1.46$ , launched from air. Find its numerical aperture, its acceptance half-angle, and the full angular width of the cone it accepts.

Step 1. Apply the master equation directly, evaluating the squares as a difference to avoid losing precision: $\mathrm{NA}=\sqrt{1.48^2-1.46^2}=\sqrt{(1.48-1.46)(1.48+1.46)}=\sqrt{(0.02)(2.94)}=\sqrt{0.0588}=\mathbf{0.242}$ .

Step 2. The acceptance half-angle in air ( $n_0=1$ ) is $\theta_a=\arcsin(0.242)=\mathbf{14.0°}$ — the largest angle from the axis at which an incoming ray is still captured.

Step 3. The fiber accepts a full cone of $2\theta_a=\mathbf{28°}$ about its axis. A source aimed within $14°$ of the axis couples in; anything steeper leaks into the cladding within a bounce. Note what did not enter the calculation: the core radius and the wavelength are both absent — this cone is fixed by the two glasses alone.

2 · How many modes does a multimode fiber carry?

A 50 µm-core multimode fiber (core radius $a=25\,\mu\mathrm{m}$ ) has $\mathrm{NA}=0.20$ and is used at $\lambda_0=850\,\mathrm{nm}$ . Estimate its V-parameter and its number of guided modes.

Step 1. Compute the normalized frequency, $V=\dfrac{2\pi a}{\lambda_0}\mathrm{NA}=\dfrac{2\pi(25\,\mu\mathrm{m})}{0.85\,\mu\mathrm{m}}(0.20)=184.8\times0.20=\mathbf{37.0}$ . Since $V\gg2.405$ , the fiber is firmly multimode.

Step 2. Use the large- $V$ estimate $M\approx V^2/2=37.0^2/2=\mathbf{683}$ guided modes, each a distinct field pattern with its own axial group velocity.

Step 3. Roughly 680 modes means roughly 680 ray families with different path lengths, so an input pulse fans out in arrival time — modal dispersion, the penalty for the wide, easy-to-fill NA. Halving the NA to $0.10$ would cut $M$ fourfold, to about $170$ modes. V ≈ 37, M ≈ 683 modes.

3 · Coupling an LED into fiber

An LED radiates into a divergence half-angle $30°$ , so $\mathrm{NA}_{\text{source}}=\sin30°=0.50$ . Compare the power coupled into a low-NA single-mode fiber ( $\mathrm{NA}=0.14$ ) versus a high-NA multimode fiber ( $\mathrm{NA}=0.30$ ).

Step 1. The source overfills both fibers, so the captured fraction is bounded by $\eta\le(\mathrm{NA}_{\text{fiber}}/\mathrm{NA}_{\text{source}})^2$ . Single-mode: $\eta\le(0.14/0.50)^2=\mathbf{0.078}$ , under 8 percent. High-NA multimode: $\eta\le(0.30/0.50)^2=\mathbf{0.36}$ , about 36 percent — nearly five times more.

Step 2. No lens can beat this NA-ratio bound (étendue conservation), so a broadly-diverging LED is fundamentally mismatched to a low-NA fiber. This is why short-reach LED and power-delivery links use large NA. About 8 percent into the SMF versus 36 percent into the high-NA fiber.

Physics of the portrait

The very phrase "numerical aperture" was coined not for fibers but for microscope objectives, by the man who first turned the light-gathering angle of a lens into a hard number tied to resolution — and the fiber acceptance cone is that same $n\sin\theta$ in a new guise.

Origins · 1 minErnst Abbe and how this was discovered

Ernst Abbe

1840–1905 · resolution, aperture, and the physics of image formation

Ernst Abbe was the theoretical engine behind the Carl Zeiss optical works in Jena, and in the 1870s he explained, quantitatively, what limits a microscope. His answer turned on a single quantity he named the numerical aperture, $\mathrm{NA}=n\sin\theta$ , the product of the index of the medium between object and lens with the sine of the half-angle of rays the lens collects. Abbe's diffraction theory of image formation (from 1873) showed that the smallest resolvable detail is $d\approx\lambda/(2\,\mathrm{NA})$ — resolution is set not by polishing lenses better but by how wide a cone of diffracted light they gather. This is why he pushed oil-immersion objectives: raising $n$ above one lifts NA past unity and sharpens the image, the same immersion effect that appears in this article's limiting cases. The fiber's acceptance cone is Abbe's aperture read backwards — instead of asking what cone a lens collects from a point, we ask what cone a fiber accepts into a guide — but the governing quantity, $n\sin\theta$ , is identical. His NA remains stamped on every objective and, through the acceptance angle, on every fiber.

History

1854

— John Tyndall demonstrates light guided along a curved jet of water by total internal reflection

1873

— Ernst Abbe defines the numerical aperture n sin θ and the diffraction resolution limit for microscope objectives

1954

— van Heel, and Hopkins with Kapany, report clad fiber bundles that transmit images, making low-loss guiding practical

1966

— Kao and Hockham argue that purified glass fiber could carry telecommunications below 20 dB per km

1970

— Corning fabricates the first fiber under that loss threshold, launching optical communications

Going further

Texts. Hecht, Optics, 5th ed., §5.6 (Fiberoptics) · Saleh and Teich, Fundamentals of Photonics, 3rd ed., Ch. 9 (Fiber Optics), Eqs. 9.1-2/9.1-3 and 9.2-14 · Born and Wolf, Principles of Optics, on aperture and resolution · Foot, Atomic Physics, appendix on optical fibers · Demtröder, Laser Spectroscopy, Vol. 1, on light collection.

Provenance. A verifier should recompute $\mathrm{NA}=\sqrt{n_1^2-n_2^2}$ and the acceptance-angle derivation (Snell at the face combined with the critical angle at the wall) against Hecht §5.6 and Saleh and Teich Ch. 9. Check the weakly-guiding form $\mathrm{NA}\approx n_1\sqrt{2\Delta}$ , the V-parameter $V=(2\pi a/\lambda_0)\mathrm{NA}$ , the mode count $M\approx V^2/2$ , and the single-mode cutoff $V=2.405$ (first zero of $J_0$ ). Worked numbers are standard: $\mathrm{NA}=0.242$ and $\theta_a=14.0°$ for $n_1=1.48$ , $n_2=1.46$ ; $\mathrm{NA}=0.145$ for the telecom weakly-guiding case; $V=37$ and $M\approx683$ for the $50\,\mu\mathrm{m}$ multimode example. The NA concept and $\mathrm{NA}=n\sin\theta$ trace to Abbe (1873); the difference-of-squares fiber form is textbook.

Builds on. Total Internal Reflection and Optical Guiding · Step-Index and Graded-Index Optical Fibers. Leads to. Single-Mode and Multimode Operation · Chromatic and Modal Dispersion in Fibers · Fiber Attenuation and Loss Mechanisms · Waveguide Modes and the Wave Equation.

Ask the tutor: Ask me to work through why a fiber's numerical aperture stays fixed while its mode count changes with core size and wavelength, or to size the NA and core for a fiber that is single-mode at your chosen wavelength.

Why does the numerical aperture not depend on the core diameter?

Because the acceptance limit is set by the angle a ray makes at the core–cladding wall, and for a meridional ray that angle is independent of core width. A ray tilted by $\theta_r$ from the axis meets the wall at $90°-\theta_r$ whether the core is 4 µm or 400 µm across — a fatter core just means the ray travels further between bounces, not that it strikes the wall differently. Since the critical-angle condition depends only on the two indices and the geometry converting it to an external cone involves no length, NA is a pure function of $n_1$ and $n_2$ . The radius reappears only when you count modes, through the V-parameter, because there the field must fit across the core.

What is the difference between numerical aperture and the acceptance angle?

They are two encodings of the same information. The acceptance angle $\theta_a$ is the physical half-angle of the cone of rays a fiber captures; the numerical aperture is its sine scaled by the launch-medium index, $\mathrm{NA}=n_0\sin\theta_a$ . Working with the sine has two advantages. It makes the coupling formulas clean — power ratios go as $\mathrm{NA}^2$ . And it lets a single number describe launch from any medium: the same fiber has one NA, but its physical acceptance angle shrinks if you launch through oil or index-matching gel instead of air. So NA is the invariant fiber property, and $\theta_a$ is what it looks like in a particular launch medium.

Can the numerical aperture really exceed 1?

Yes. The quantity $\sqrt{n_1^2-n_2^2}$ has no ceiling at one — an air-clad silica fiber, with $n_1\approx1.45$ and $n_2=1$ , has $\mathrm{NA}=\sqrt{1.45^2-1}\approx1.05$ . What cannot exceed one is $\sin\theta_a$ in a given medium, and the two are reconciled by $\mathrm{NA}=n_0\sin\theta_a$ : when NA reaches or passes $n_0$ , the acceptance angle saturates at $90°$ and the fiber gathers light from the entire forward hemisphere. Any NA beyond $n_0$ measures not an impossible angle but how strongly the mode is confined. This is the same reason immersion microscope objectives quote NA above one — the oil raises $n_0$ , so a physical angle under $90°$ still yields $\mathrm{NA}>1$ .

If high NA gathers more light, why not always maximize it?

Because light-gathering is only one of three coupled budgets, and the other two push back. A high NA admits more modes — the count grows as $\mathrm{NA}^2$ — and each mode travels its own path length, so a sharp input pulse arrives spread out in time. That modal dispersion caps the bandwidth-distance product, fatal for long links. So the design splits by application: short reach with cheap incoherent sources (data centers, sensors, power delivery) chooses high NA to win the coupling budget and tolerates the dispersion; long-haul telecom chooses the lowest practical NA to kill dispersion and accepts a coupling job that demands a laser. NA is a dial to be set, never simply cranked up.

How does numerical aperture connect to the number of modes?

Through the V-parameter, $V=(2\pi a/\lambda_0)\,\mathrm{NA}$ . NA caps the transverse component of a guided ray's wavevector at $k_0\mathrm{NA}$ : any ray steeper than the acceptance limit refracts away. The number of distinct transverse field patterns that fit under that cap across a core of radius $a$ is what $V$ measures, and for large $V$ the mode count is $M\approx V^2/2$ — so NA enters quadratically. The single-mode threshold is the clean special case: when $V<2.405$ (the first zero of $J_0$ ), only the fundamental mode survives and there is no modal dispersion. This is why a fiber is made single-mode by shrinking NA, shrinking the core, or lengthening the wavelength — all three lower $V$ .

Does numerical aperture change with wavelength?

To the level of the ray derivation, no — NA comes out of Snell's law and the critical angle, both geometric and colour-blind, so $\mathrm{NA}=\sqrt{n_1^2-n_2^2}$ carries no explicit wavelength. There is a tiny residual dependence because the glass indices themselves disperse, but since NA depends on their difference and core and cladding disperse almost identically, the effect largely cancels and NA is nearly flat across the transparency window. Wavelength matters enormously elsewhere, though: it enters the V-parameter directly, so the same fiber can be multimode at 850 nm and single-mode at 1550 nm even though its NA barely moved.