Tool · Fiber and telecom

Fiber Numerical Aperture and V Number Calculator

What cone of light does a fiber accept, is it single mode at your wavelength, and how large is its guided mode? From the NA or the core and cladding indices, the core diameter and the wavelength, the calculator gives the acceptance angle, the V number and guided modes, the cutoff wavelength and the mode field diameter. Background: numerical aperture, acceptance angle, V number, cutoff wavelength, mode field diameter, and types of optical fiber.

Fiber

NA from

Numerical aperture NA

Core index n_core

Cladding index n_clad

Core diameter 2a (µm)

Index profile

Wavelength λ (nm)

Index in front of the fiber n₀

With the NA given directly, the core index only sets Δ, the cladding index and the critical angle. For single-mode fiber use the NA of the index step, not the far-field NA on the datasheet. n₀ is 1 in air and 1.333 in water.

Presets

The single-mode presets use an 8.2 µm core and the NA of a 0.36% index step, 0.123, as for ITU-T G.652 fiber. The multimode presets use the nominal NA of each fiber type.

Readouts

Acceptance cone

V number and mode cutoffs against wavelength

The blue curve is V = 2πa·NA/λ; the dashed lines are the values of V at which each higher LP mode starts to be guided in a step-index core. The dot is the current wavelength.

Mode field diameter against wavelength

Learn with it

Three short experiments. Each one sets the inputs, says where to look, and asks for a prediction before it shows the result.

Checked against

These checks run in your browser on every load. The NA, acceptance angle, V number, cutoff wavelength and mode counts are compared with values worked out by hand, the mode cutoffs with tabulated zeros of the Bessel functions, and the mode field diameters with Marcuse’s formula and the figures in the site’s fiber entries.

Check	Expected	Computed	Tolerance

The expected values follow Gloge, “Weakly guiding fibers” (1971), and Marcuse, “Loss analysis of single-mode fiber splices” (1977), evaluated by hand for the stated cases; the Bessel zeros are those tabulated in Abramowitz and Stegun. The tolerance is the largest difference from Expected that still passes, relative to Expected or to 1, whichever is larger.

The model

A step-index fiber with core index $n_\text{core}$ and cladding index $n_\text{clad}$ guides rays that meet the core boundary beyond the critical angle. Snell’s law at the end face then limits the rays it accepts from a medium of index $n_0$ to

\text{NA} = n_0 \sin\theta_\text{max} = \sqrt{n_\text{core}^2 - n_\text{clad}^2}, \qquad V = \frac{2\pi a}{\lambda}\,\text{NA}

where $a$ is the core radius. In the weakly guiding approximation the guided modes are the LP modes, each cut off at a zero of a Bessel function: the fiber guides only LP₀₁ while $V < 2.405$ , which gives the cutoff wavelength $\lambda_c = 2\pi a\,\text{NA}/2.405$ , and a large core guides about $V^2/2$ modes, or $V^2/4$ for a parabolic graded-index core. The mode field radius $w$ of the fundamental mode follows Marcuse’s fit,

\frac{w}{a} \approx 0.65 + \frac{1.619}{V^{3/2}} + \frac{2.879}{V^6},

accurate to about 1% for $1.2 < V < 2.4$ , and the mode field diameter is $2w$ . The model assumes an ideal step-index core; real fibers have graded or shaped profiles, so the measured mode field diameter of standard single-mode fiber is a few percent larger than the formula gives. For coupling a free-space beam into the mode see the fiber coupling efficiency calculator.

Worked example

Standard single-mode fiber has a core 8.2 µm across and a 0.36% index step, an NA of 0.123. At 1550 nm, $V = 2\pi \times 4.1 \times 0.123 / 1.55 = 2.044$ , so the fiber is single mode, and Marcuse’s formula gives a mode field diameter of 10.2 µm. The theoretical cutoff is 1318 nm; at 1310 nm $V = 2.419$ and the LP₁₁ mode is only weakly guided, which bends remove within meters of fiber. The acceptance half-angle in air is 7.07°.

References: D. Gloge, “Weakly guiding fibers,” Applied Optics 10, 2252 (1971). D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell System Technical Journal 56, 703 (1977). A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983). M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (1964), table 9.5.