Compute the V-number (normalized frequency) from core radius, NA, and wavelength, and watch single-mode vs multimode operation, the V=2.405 cutoff, cutoff wavelength, and mode count come alive as fiber cross-section mode patterns.
Fiber Presets
Waveguide Parameters
Core radius a
µm
SMF: ~4.1 µm / MMF: ~25 µm
Numerical aperture NA
NA = √(n_core² − n_clad²). SMF: ~0.12 / MMF: ~0.20
Wavelength λ
µm
Common bands: 0.85 / 1.31 / 1.55 µm
Design tip
When V is below 2.405 only the fundamental mode LP01 is guided (single-mode); above 2.405 higher-order modes switch on (multimode). Watch the cross-section on the right to see how the mode count grows as you tune a, NA, and λ.
While paused, move the sliders to update the result instantly.
What is Fiber Waveguide Design (V-Number & Modes)?
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What is this "V-number" that fiber designers keep mentioning? I hear it decides single-mode versus multimode — but it sounds intimidating.
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Think of the V-number (normalized frequency) as a single number that tells you how many shapes (modes) the fiber can carry light in. It's $V=(2\pi a/\lambda)\cdot\mathrm{NA}$ — set entirely by the core radius $a$, the wavelength $\lambda$, and the numerical aperture NA. The magic threshold is V = 2.405: below it only the fundamental mode LP01 propagates (single-mode); above it higher-order modes switch on (multimode). Drag the core radius slider up and watch a second mode appear in the cross-section the instant V crosses 2.405.
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So crossing the boundary adds modes. What is the "cutoff wavelength" then?
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Good question. The cutoff wavelength $\lambda_c$ is the wavelength at which V equals exactly 2.405. Solving the formula gives $\lambda_c = 2\pi a\cdot\mathrm{NA}/2.405$. If your operating wavelength is longer than $\lambda_c$, then V < 2.405 and you're single-mode; if it's shorter, higher-order modes appear. A standard SMF is designed with $\lambda_c \approx 1.26\,\mu m$, so the 1.31 µm and 1.55 µm bands are nicely single-mode. Slide the wavelength toward shorter values and the same fiber flips to multimode.
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And in multimode, roughly how many modes are there?
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For a step-index fiber with a large V, the guided-mode count is approximately N ≈ V²/2. So V = 10 gives ~50, V = 20 gives ~200. A 50 µm-core, NA 0.2 MMF at 850 nm has V ≈ 37 and carries several hundred modes. That huge collecting area makes MMF easy to couple into, but the modes arrive at different times (modal dispersion), which limits reach and bandwidth. Hit the MMF preset and watch the cross-section fill up with patterns.
Physical Model & Key Equations
The number of modes a fiber guides is set by the normalized frequency (V-number). From the core radius $a$, the wavelength $\lambda$, and the numerical aperture $\mathrm{NA}=\sqrt{n_{co}^2-n_{cl}^2}$:
The first higher-order mode (LP11) cuts on at $V=2.405$ (the first zero of the Bessel function $J_0$), which is exactly the single-mode/multimode boundary.
The cutoff wavelength $\lambda_c$ (the wavelength where V = 2.405) and the large-V mode count are:
For $\lambda>\lambda_c$ only the fundamental mode (two polarizations) propagates. Check: $a=4.5\,\mu m,\ \mathrm{NA}=0.12,\ \lambda=1.31\,\mu m$ gives $V\approx2.59$, just above the single-mode boundary (weakly multimode).
Frequently Asked Questions
V = 2.405 is the cutoff of the LP11 mode. Once V slightly exceeds it, the second mode (LP11) starts to propagate and can cause double imaging and modal dispersion. For robust single-mode operation, target V ≈ 2.0–2.3 at the operating wavelength so you have margin. Use the live V-number readout while fine-tuning core radius, NA, and wavelength to keep a safe distance from the boundary.
The V-number is inversely proportional to wavelength (V ∝ 1/λ). Shortening the wavelength raises V, and once it drops below the cutoff wavelength λc the fiber becomes multimode. For example, an SMF with λc ≈ 1.26 µm is single-mode at 1.31 µm and 1.55 µm but multimode at 850 nm, where V exceeds 2.405. Move the wavelength slider and watch the marker cross the boundary on the V-number scale.
Since V = (2πa/λ)·NA, lowering NA (which corresponds to a smaller core/clad index contrast) directly reduces V, making the single-mode condition V < 2.405 easier to satisfy. However, too low an NA broadens the mode field and increases bend and coupling loss. Practical SMFs use NA ≈ 0.12 and core radius ≈ 4 µm, sized just below cutoff at the operating wavelength.
For a step-index fiber at large V, the total number of guided modes (including polarization and degeneracy) is approximately N ≈ V²/2. A parabolic graded-index fiber (α = 2) has roughly half as many, N ≈ V²/4. The simulator's mode count is the step-index estimate; combined with the number of cross-section patterns, it gives an intuitive feel for how multimode the design is.
Real-World Applications
Long-haul SMF design: Choose core radius and NA so the fiber is firmly single-mode (V < 2.405) at the operating band (1.31/1.55 µm). Single-mode operation eliminates modal dispersion, which is essential for long-distance, high-bit-rate links. The cutoff wavelength is usually set just below the operating wavelength.
Data-center MMF design: A 50 µm-core, NA 0.2 MMF has V ≈ 37 at 850 nm and guides several hundred modes. The large core makes coupling easy and low-cost, but modal dispersion limits bandwidth, so graded-index profiles (OM3/OM4/OM5) are used to minimize differential mode delay.
Specialty fibers for sensing and fiber lasers: Few-mode (FMF) and large-mode-area (LMA) fibers deliberately place V just above 2.405 to guide only two to a few modes. Understanding the mapping between V-number and mode pattern is the starting point for space-division multiplexing (SDM) and beam-quality design.
Education and experimentation: The clean single threshold at V = 2.405 is the heart of waveguide theory. Whether you change core radius, NA, or wavelength, they all act on the same boundary through the V-number — and the cross-section mode patterns make that visible at a glance.
Common Misconceptions and Points to Note
First, are you assuming "a bigger core is always better"? That is a major misconception. Increasing the core radius $a$ raises V and quickly pushes you into multimode (V > 2.405). For long-distance, high-speed links, modal dispersion is fatal, so the goal is usually the opposite — keep the core small (a ≈ 4 µm) to stay single-mode. Raise the core-radius slider and you will see the second mode appear the moment V crosses 2.405. A big core only helps light collection and coupling; it is a separate issue from transmission quality.
Second, many beginners get the direction of the cutoff wavelength backwards. Because V ∝ 1/λ, a longer wavelength makes V smaller and is therefore more likely to be single-mode. So λ > λc is single-mode and λ < λc is multimode. If you remember it the wrong way ("shorter wavelengths are more single-mode"), you will pick the wrong band. Use the slider and confirm which way the marker moves on the V-number scale.
Finally, do not assume the formula N ≈ V²/2 works near the boundary. This approximation is valid only in the large-V multimode regime, not in the few-mode region just above V = 2.405, where only LP01 and LP11 (a handful of groups) actually propagate. This tool counts modes from the real LP cutoffs (2.405, 3.832, …) at small V and switches to the V²/2 estimate only at large V. Keep this range of validity in mind when estimating by hand.
Pick a fiber preset (SMF 1310/1550 nm, Near SM boundary, MMF 850 nm), or enter core radius a, numerical aperture NA, and wavelength λ directly
The V-number, single/multimode verdict, guided-mode count, and cutoff wavelength update instantly; V = 2.405 is the single/multimode boundary
Watch the cross-section animation for the number and shape of mode patterns, and move a, NA, or λ to see the marker cross the boundary on the V-number scale
Worked Example
Single-mode boundary design: enter core radius a = 4.5 µm, numerical aperture NA = 0.12, wavelength λ = 1.31 µm. Then V = (2π × 4.5 / 1.31) × 0.12 ≈ 2.59, just above the single-mode boundary V = 2.405 (weakly multimode). The cutoff wavelength is λc = 2π × 4.5 × 0.12 / 2.405 ≈ 1.41 µm, so at the operating wavelength 1.31 µm (< λc) the LP11 mode also propagates. To guarantee single-mode operation, reduce the core radius to a ≈ 4.1 µm, giving V ≈ 2.36 (λc ≈ 1.29 µm).
Practical Notes
Designing just below cutoff (V ≈ 2.0–2.3) at the operating wavelength balances mode-field diameter against bend loss; sitting too close to 2.405 risks LP11 switching on with temperature or manufacturing spread
The estimate N ≈ V²/2 applies to step-index fibers at large V; graded-index fibers give N ≈ V²/4, and near the boundary you should count from the real LP cutoffs (2.405, 3.832, …)
NA corresponds to the index contrast. Raising NA eases coupling and collection but increases V and pushes toward multimode; SMFs are designed around NA ≈ 0.12