How is focused spot size calculated?

For a Gaussian beam, the focused spot diameter (1/e²) is d = 4M²λf/(πD), where f is focal length, D is input beam diameter, λ is wavelength, and M² is beam quality factor. Smaller f, larger D, shorter λ, and lower M² all give smaller spots.

What is peak intensity?

Peak intensity I₀ = 8P/(πd²) for a Gaussian beam, where P is total power and d is the focused spot diameter (1/e²). This is the intensity at the center of the spot. For pulsed lasers, use peak (not average) power for the peak intensity calculation.

What is depth of focus?

The longitudinal extent over which the beam stays small. DOF = 2 × Rayleigh range = (π/(8M²)) × d²/λ. Tight focus = shallow depth; loose focus = deep depth. For a 1 µm focused spot at 1064 nm with M²=1: DOF ≈ 2.7 µm.

Can I focus a beam to a point?

No. Diffraction sets a minimum spot size for any beam, regardless of focusing optic. The fundamental limit is d ≈ λ × M² / NA (where NA = sin of half-angle). For visible light and NA = 1, the minimum spot is roughly 1 wavelength across.

How does wavelength affect focused spot size?

Linearly — half the wavelength = half the spot size at the same M² and NA. This is why UV lasers (193 nm) are used in lithography to make tiny features that visible lasers can't match. Going from 1064 nm to 532 nm (frequency doubling) halves the spot at no cost to the focusing optics.

Why does M² matter for focused spot?

M² is the multiplier between ideal Gaussian beam minimum spot and real beam minimum spot. M² = 2 means the focused spot is 2× larger than diffraction-limited. For applications needing tight focus (lithography, laser surgery), single-mode (M² ≈ 1) lasers are essential. Multimode lasers (M² > 2) are cheap and powerful but can't focus as tightly.

How do I make a smaller focused spot?

Multiple ways: (1) Use shorter wavelength (UV instead of IR). (2) Use higher-quality beam (smaller M²) — single-mode laser or spatial filter. (3) Use lower f-number focusing (larger input beam or shorter focal length, but watch for aberrations). (4) Use immersion fluid (water or oil) to push NA higher than 1.0.

Laser Spot Size Calculator

Determine the diffraction-limited spot size when a laser beam is focused by a lens. Calculates spot diameter, depth of focus (Rayleigh range), peak intensity, and f-number of the focusing system.

When a laser beam is focused by a lens, it converges to a minimum-radius waist near the focal point — but never to a true point. The smallest achievable spot is set by the wave nature of light: diffraction. The focused spot diameter for a Gaussian beam is d = 4 × M² × λ × f / (π × D), where M² is beam quality, λ is wavelength, f is focal length, and D is the input beam diameter at the lens. Tight focus requires short wavelength, high-quality beam (M² near 1), and a large input beam (or short focal length).

This calculator returns the focused spot diameter, the depth of focus (twice the Rayleigh range, or the longitudinal extent over which the beam stays small), the peak intensity at the spot center, and the f-number of the focusing system. Use it for laser machining (where spot size sets resolution), laser surgery (depth of focus sets treatable layer thickness), nonlinear optics (where peak intensity drives the desired effect), and microscopy (focused beam for two-photon imaging or STED).

The fundamental trade-off: tight focus = shallow depth. A 1 µm spot at 1064 nm has only a few µm depth of focus, demanding precise z-positioning. A 100 µm spot is much more forgiving but doesn't concentrate intensity as much. Peak intensity scales as 1/spot_area, so halving the spot diameter quadruples the intensity at the same total power.

Inputs

Wavelength (nm)

Input Beam Diameter (mm)

1/e² diameter of the collimated beam

Lens Focal Length (mm)

Laser Power (W)

CW power or average power for peak intensity calculation

M² (Beam Quality)

Results

Spot Diameter

27.1 μm

Depth of Focus

1.08 mm

Peak Intensity

346.9 kW/cm²

Laser Spot Size Results

Parameter	Value
Spot Diameter (1/e²)	27.09 μm
Spot Radius w₀	13.55 μm
Depth of Focus (2×z_R)	1.0838 mm
Peak Intensity I₀	346.88 kW/cm²
f-Number	f/20.00
Numerical Aperture	0.0250
M² Factor	1
Input Beam Diameter	5 mm
Focal Length	100 mm
Formula	d = 4M²λf/(πD)

Last updated: May 29, 2026

Formula

**Focused spot diameter (Gaussian beam):** d_focus = 4 × M² × λ × f / (π × D_input) Where: - **M²**: beam quality factor (1.0 for ideal single-mode) - **λ**: wavelength - **f**: lens focal length - **D_input**: input beam diameter (1/e² intensity) at the lens The spot radius is w₀ = d_focus / 2 = 2 × M² × λ × f / (π × D_input). **f-number of the focusing system:** f# = f / D_input The smaller the f-number, the tighter the focus. Equivalent form: d_focus = 4 × M² × λ × f# / π ≈ 1.27 × M² × λ × f# **Depth of focus (Rayleigh range, full depth = 2 × z_R):** DOF = 2 × z_R = 2 × π × w₀² / (M² × λ) = (π / (8 × M²)) × d_focus² / λ So DOF scales as d_focus² / λ — small spots have very shallow depth. **Peak intensity (Gaussian beam, on-axis at waist):** I₀ = 2 × P / (π × w₀²) = 8 × P / (π × d_focus²) Where P is total power. Peak intensity scales inversely with spot area. **Worked example: industrial Nd:YAG laser** λ = 1064 nm, beam D = 5 mm, lens f = 100 mm, M² = 1.5 (multimode). d_focus = 4 × 1.5 × 1064 × 10⁻⁹ × 0.1 / (π × 5 × 10⁻³) = 6.39 × 10⁻⁷ × 0.1 / 1.57 × 10⁻² = 40.6 × 10⁻⁶ m = **40.6 µm**. f# = 100 / 5 = 20 (f/20 focusing — slow). DOF = (π / (8 × 1.5)) × (40.6 × 10⁻⁶)² / 1064 × 10⁻⁹ = 0.262 × 1.65 × 10⁻⁹ / 1.064 × 10⁻⁶ = 4.06 × 10⁻⁴ m = **406 µm** depth. Peak intensity at 1 W: I₀ = 2 × 1 / (π × (20.3 × 10⁻⁶)²) = 1.55 × 10⁹ W/m² = 1.55 × 10⁵ W/cm². Modest by laser standards. **For tight micro-machining focus (f# = 2):** Same wavelength, M² = 1.0, f = 5 mm with 2.5 mm beam: d_focus = 4 × 1 × 1064 × 10⁻⁹ × 0.005 / (π × 2.5 × 10⁻³) = 2.71 × 10⁻⁶ m = **2.7 µm**. DOF = (π / 8) × (2.7 × 10⁻⁶)² / 1064 × 10⁻⁹ = 2.69 × 10⁻⁶ m = **2.7 µm** depth. At 10 W: I₀ = 2 × 10 / (π × (1.35 × 10⁻⁶)²) = 3.5 × 10¹³ W/m² = 3.5 × 10⁹ W/cm². High enough for ablation of most materials. **Diffraction-limited spot vs aberration-limited:** The above formulas give the diffraction-limited spot — the smallest possible at given parameters. Real lenses have aberrations (spherical, coma, astigmatism) that increase the spot beyond this. For high-NA focusing lenses (microscope objectives), choose corrected designs. **Spot size at different wavelengths and NA:** | λ | NA | Spot diameter (M²=1) | |---|---|---| | 1064 nm | 0.10 | 13.5 µm | | 1064 nm | 0.25 | 5.4 µm | | 1064 nm | 0.50 | 2.7 µm | | 532 nm | 0.10 | 6.8 µm | | 532 nm | 0.50 | 1.4 µm | | 248 nm | 0.10 | 3.2 µm | | 193 nm | 0.10 | 2.5 µm | | 1064 nm | 0.95 | 1.4 µm | NA = sin(half-angle) = D/(2f) for paraxial systems. The relationship: d_focus = 1.27 × λ / NA (M²=1). **Intensity required for common laser applications:** | Process | Threshold intensity | |---|---| | LASIK eye surgery (193 nm UV) | 10⁸ W/cm² | | Steel cutting (fiber laser) | 10⁵–10⁶ W/cm² | | Aluminum welding | 10⁵ W/cm² | | Two-photon microscopy | 10⁹ W/cm² peak (pulsed) | | Optical tweezers | 10⁶ W/cm² (CW, low power) | | Lithography (193 nm) | 10⁸ W/cm² | | Nonlinear optics (SHG threshold) | 10⁶–10⁹ W/cm² | | Laser damage threshold (typical) | 10⁹–10¹⁰ W/cm² |

How to use this calculator

Enter the laser wavelength (1064 nm IR, 532 nm green, 405 nm blue, 248 nm UV).
Enter the input beam diameter at the lens (1/e² convention).
Enter the focal length of your focusing lens.
Enter the laser power (CW or average for peak intensity calculation).
Enter M² (1.0 for single-mode laser, 1.1–1.5 for typical diode laser, 2–10 for multimode industrial laser).
Read the focused spot diameter, depth of focus, peak intensity, and f-number.

Worked examples

Steel cutting with fiber laser

**Scenario:** A 2 kW fiber laser at 1070 nm with collimator D = 6 mm, M² = 1.1. Focus with 150 mm focal length lens. What spot and intensity for cutting steel? **Calculation:** d_focus = 4 × 1.1 × 1070 × 10⁻⁹ × 0.15 / (π × 6 × 10⁻³) = 37.5 µm. I₀ = 8 × 2000 / (π × (37.5×10⁻⁶)²) = 3.6 × 10⁹ W/m² = 3.6 × 10⁵ W/cm². DOF = ~1.6 mm. **Result:** 37.5 µm spot delivers 3.6 × 10⁵ W/cm² — sufficient to cut 1–2 mm steel sheet at moderate feed rates. The 1.6 mm depth of focus handles typical material thicknesses. Cutting speed depends on assist gas (oxygen for steel, nitrogen for stainless) and beam delivery quality.

Two-photon microscopy

**Scenario:** A 100 fs Ti:Sapphire laser at 800 nm focused by a 1.0 NA water-immersion objective. Average power 50 mW (peak ~50 W). What's the spot and peak intensity? **Calculation:** With NA = 1.0 in water (n=1.33), effective spot ≈ λ/(2×NA) = 800 nm / 2 = 400 nm = 0.4 µm. Peak intensity from peak power: I_peak = 2 × 50 / (π × (0.2 × 10⁻⁶)²) = 7.96 × 10¹⁴ W/m² = 7.96 × 10¹⁰ W/cm². **Result:** 0.4 µm spot at ~10¹¹ W/cm² peak — well into the two-photon excitation regime. Two-photon microscopy uses this confined high-intensity volume to excite fluorophores only at the focal point, giving 3D resolution without optical sectioning. The pulsed laser maintains low average power (no heating) while delivering enough peak intensity for the two-photon process.

Optical tweezers calibration

**Scenario:** Trap a 1 µm polystyrene bead with a 1064 nm laser. Use 100× 1.3 NA oil-immersion objective. What power for stable trap? **Calculation:** Diffraction-limited spot: d ≈ λ × 0.61 / NA = 1064 × 0.61 / 1.3 = 0.499 µm = 0.5 µm. Trap stiffness scales with P/d²; typical needed intensity ~10⁵–10⁶ W/cm² for stable bead trapping. I = 8P/(πd²) → P = I × π × d²/8. For I = 10⁶ W/cm² = 10¹⁰ W/m² and d = 0.5 µm: P = 10¹⁰ × π × (5×10⁻⁷)²/8 = 0.98 mW. **Result:** ~1 mW of laser power at the sample is enough for typical bead trapping. Account for ~50% loss through the objective and additional losses through the microscope; the laser source needs to deliver ~5–10 mW. Higher powers give stiffer traps but risk overheating the sample.

When to use this calculator

**Use focused laser spot calculations for:**

- **Laser micromachining**: cutting, drilling, marking on small features. - **Laser welding**: thermal calculations, heat-affected zone. - **Laser surgery and ophthalmology**: precision tissue ablation, retinal repair. - **Optical tweezers and trapping**: force calibration, single-molecule biology. - **Optical lithography**: critical dimensions of fabricated features. - **Two-photon microscopy and femtosecond biology**: excitation volume calculation. - **Nonlinear optics and harmonic generation**: intensity required for nonlinear effects. - **Laser damage threshold testing**: pulse energy for given spot to reach threshold.

**Trade-offs to balance:**

- **Spot size vs depth**: tighter focus = shallower DOF. Cannot have both. - **Spot size vs working distance**: short focal length = small spot, but less room. - **Spot size vs aberrations**: high-NA focusing demands well-corrected lenses. - **Spot size vs energy density**: at same power, smaller spot = higher fluence/intensity.

**Typical f-numbers for various applications:**

| Application | f# | |---|---| | Industrial laser cutting | f/8 to f/12 | | Laser welding | f/4 to f/8 | | Fine marking | f/8 to f/15 | | Micro-machining | f/2 to f/5 | | Microscopy (high resolution) | f/0.5 to f/1.5 (with corrected objective) | | Laser drilling | f/3 to f/8 |

**Focused spot precision:**

- **At f/1**: spot ≈ 1.3 × λ. (4.5 µm at 1064 nm with M²=1, but high-NA lens design hard.) - **At f/2**: spot ≈ 2.5 × λ. (2.7 µm at 1064 nm.) - **At f/8**: spot ≈ 10 × λ. (11 µm at 1064 nm.) - **At f/20**: spot ≈ 25 × λ. (27 µm at 1064 nm.)

Smaller f-number = tighter focus, larger numerical aperture, lower depth.

**Beam quality matters:**

- **M² = 1**: theoretical minimum. Achievable with single-mode lasers (HeNe, well-built diode). - **M² = 1.1–1.3**: good real-world laser. - **M² = 2–4**: typical multimode laser; spot is 2–4× the theoretical minimum. - **M² = 10+**: bad beam quality; focused spot much larger.

A laser with poor beam quality (high M²) can't be focused as tightly regardless of focusing lens. This is why high-power industrial lasers often invest in "M² conditioning" (spatial filtering, mode cleaners) before delivery.

**Practical safety note:** focused laser intensities easily reach damage thresholds for the eye, even at modest input power. A 5 mW Class 3R pointer focused to 10 µm gives 5 × 10⁴ W/cm² — well above the retinal damage threshold of 100 mW/cm² for prolonged exposure. Always wear proper laser safety eyewear when working with focused beams.

Common mistakes to avoid

Confusing input beam diameter D with input beam radius (off by 2×).
Using M² = 1 for real multimode diode lasers. Beam quality is critical for spot size.
Forgetting that real lens aberrations enlarge the spot beyond diffraction-limited. Use corrected objectives for high NA.
Computing peak intensity from average power for pulsed lasers. Peak ≫ average for short pulses.
Trying to achieve f# below the lens's NA limit. Physical NA ≤ 1.0 for air; ~1.4 for oil immersion.
Ignoring that focal length depends on wavelength (chromatic aberration of single-element lenses).
Not accounting for beam clipping. If beam diameter > lens clear aperture, effective NA decreases.

Frequently Asked Questions

Sources & further reading

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