Lenses — Abridged Guide

The essential quick reference for lens types, equations, materials, aberrations, and selection. For the full treatment with worked examples and diagrams, see the Comprehensive Guide.

1.Overview

A lens is a transmissive optical element that uses refraction to converge or diverge light. All lenses work by the same principle: light changes speed (and direction) when passing between materials of different refractive index.

Converging lenses (positive focal length) are thicker at the center and bring parallel rays to a focus. Diverging lenses (negative focal length) are thinner at the center and cause parallel rays to spread.

2.Lens Types & Geometry

Type	Best For	Key Characteristic
Plano-Convex (PCX)	Focusing collimated light, ∞:f conjugate	One flat + one curved surface; orient curved side toward collimated beam
Bi-Convex (BCX)	Finite conjugate imaging (1:1 to 5:1)	Two curved surfaces share the bending, reducing aberration at moderate conjugates
Best-Form	Minimized spherical aberration	Asymmetric curvatures optimized for a specific conjugate ratio
Achromatic Doublet	Broadband or white-light focusing	Crown + flint cemented pair; corrects chromatic and reduces spherical aberration
Aspheric	Diffraction-limited focusing, low f/#	Non-spherical profile eliminates spherical aberration; higher cost
Plano-Concave / Bi-Concave	Beam expansion, virtual images	Negative focal length; diverges light

Conjugate ratio rule of thumb: Use PCX for ∞:f. Use BCX for finite conjugates near 1:1. Use best-form or achromats when aberration performance matters. Use aspheres when you need diffraction-limited performance at f/2 or lower.

Orientation matters. For a plano-convex singlet focusing a collimated beam, always face the curved surface toward the incoming (collimated) beam. This minimizes spherical aberration by distributing refraction across both surfaces.

3.Refraction & Snell's Law

Snell's Law

n_1 \sin(\theta_1) = n_2 \sin(\theta_2)

When light passes from one medium to another, it bends toward the normal (entering a denser medium) or away from the normal (entering a less dense medium). The refractive index n is the ratio of light speed in vacuum to light speed in the material.

Critical Angle (Total Internal Reflection)

\theta_c = \arcsin\!\left(\frac{n_2}{n_1}\right) \quad \text{where } n_1 > n_2

Example: Glass (n=1.52) → Air (n=1.00): θ_c = 41.1°

Typical refractive indices: Air ≈ 1.000, Water ≈ 1.333, N-BK7 glass ≈ 1.517 (@587.6 nm), UV Fused Silica ≈ 1.458, ZnSe ≈ 2.403 (@10.6 µm).

4.Thin Lens Equation

Thin Lens Equation

\frac{1}{f} = \frac{1}{s} + \frac{1}{s'}

f = focal length, s = object distance, s' = image distance

Transverse Magnification

m = -\frac{s'}{s} = \frac{h'}{h}

Negative m → inverted image

Lensmaker's Equation

\frac{1}{f} = (n-1)\left[\frac{1}{R_1} - \frac{1}{R_2}\right]

Thin lens approximation — valid when lens thickness ≪ focal length

Sign convention (real-is-positive): Object distance s is positive when the object is to the left of the lens. Image distance s' is positive when the image forms to the right. Radius R is positive when the center of curvature is to the right of the surface.

5.Thick Lens Considerations

The thin lens model breaks down when lens thickness t is not negligible compared to f. Thick lens theory introduces principal planes (H, H') — the reference planes from which focal length is measured. Object and image distances are measured from H and H', not from the lens surfaces.

Thick Lens EFL

\frac{1}{f} = (n-1)\left[\frac{1}{R_1} - \frac{1}{R_2} + \frac{(n-1)\,t}{n\,R_1 R_2}\right]

When does thick lens matter? When t/f > 0.05 (thickness is more than 5% of focal length), or when precise back focal length is needed for mechanical mounting. Most catalog singlets with f > 50 mm can be treated as thin lenses with minimal error.

6.Optical Materials

Material selection is driven by wavelength. Every optical material has a usable transmission range outside of which it absorbs or scatters light.

Material	Transmission Range	n (typical)	Common Use
N-BK7	350 nm – 2.0 µm	1.517 @588nm	Visible/NIR — general purpose, low cost
UV Fused Silica	185 nm – 2.5 µm	1.458 @588nm	UV through NIR — excimer lasers, broadband
CaF₂	170 nm – 8.0 µm	1.434 @588nm	Deep UV + mid-IR — low dispersion, achromats
ZnSe	0.6 µm – 22 µm	2.403 @10.6µm	CO₂ laser (10.6 µm) — high index, expensive
Germanium	2 µm – 14 µm	4.003 @10.6µm	Thermal IR — very high index, opaque in visible
Silicon	1.2 µm – 7 µm	3.422 @5µm	Mid-IR, SWIR — low cost, good for telecom
Sapphire (Al₂O₃)	150 nm – 5.5 µm	1.768 @588nm	UV to mid-IR — extremely hard, windows & domes
MgF₂	110 nm – 7.5 µm	1.413 @588nm	VUV, birefringent applications — common AR coating material

Abbe Number (Dispersion)

V_d = \frac{n_d - 1}{n_F - n_C}

High V → low dispersion (crown glass). Low V → high dispersion (flint glass).

Wavelength → material decision tree: UV (<350 nm): UV fused silica or CaF₂. Visible/NIR (350 nm–2 µm): N-BK7. Mid-IR (2–8 µm): CaF₂, ZnSe, or Si. Thermal IR (8–14 µm): ZnSe or Ge.

7.Gaussian Beam Focusing

Laser beams are not geometric ray bundles — they are Gaussian beams with a defined waist, divergence, and diffraction-limited behavior. The standard geometric optics equations (thin lens, magnification) must be supplemented with Gaussian beam formulas for accurate results.

Focused Spot Diameter (1/e²)

2w_0' = \frac{4\lambda f}{\pi \cdot 2w_0}

w₀' = focused waist radius, f = focal length, 2w₀ = input beam diameter (1/e²)

Rayleigh Range

z_R = \frac{\pi\, w_0^2}{\lambda}

Distance from waist to where beam area doubles (spot diameter grows by √2)

Beam Divergence (full angle, far field)

\theta = \frac{2\lambda}{\pi\, w_0}

Depth of Focus (Gaussian)

\text{DOF} = 2z_R = \frac{2\pi\, w_0'^{\,2}}{\lambda}

The tighter you focus, the shorter your usable depth of focus

The fundamental trade-off: Smaller spot size requires either shorter focal length or larger input beam diameter. But shorter focal length means shorter working distance, and larger beams require larger (more expensive) optics. There is no free lunch.

f-number rule: f/# = f / D. Faster lenses (lower f/#) produce smaller spots but are more sensitive to aberrations. Below f/2, spherical aberration from singlets becomes significant — consider aspheres or achromats.

8.Aberrations

Real lenses deviate from ideal behavior. The five Seidel (monochromatic) aberrations plus chromatic aberration determine image quality.

Aberration	Cause	Effect	Mitigation
Spherical	Marginal rays focus closer than paraxial rays	Halo around focus, increased spot size	Aspheric lens, stop down (higher f/#), achromat
Coma	Off-axis points imaged asymmetrically	Comet-shaped off-axis spots	Aplanatic design, field stop, symmetric doublets
Astigmatism	Tangential and sagittal foci differ	Line focus instead of point (off-axis)	Field-flattening elements, reduce field angle
Field Curvature	Image plane is curved, not flat	Edges defocused when center is sharp	Petzval correction, field flattener lens
Distortion	Magnification varies with field position	Barrel or pincushion warping	Symmetric lens systems, telecentric design
Chromatic	Refractive index varies with wavelength	Color fringing, wavelength-dependent focus	Achromatic doublet (crown + flint pairing)

Achromatic Doublet Condition

\frac{\phi_1}{V_1} + \frac{\phi_2}{V_2} = 0

φ = lens power (1/f), V = Abbe number — pair a crown (high V) element with a flint (low V) element

When do aberrations matter? Spherical aberration is the dominant concern for on-axis laser focusing with singlets, especially below f/4. Chromatic aberration dominates when using broadband or white-light sources. Off-axis aberrations (coma, astigmatism, field curvature) matter primarily in imaging systems with large field angles.

9.Anti-Reflection Coatings

Every uncoated air-glass interface reflects a fraction of light (Fresnel loss). For a typical N-BK7 lens, each surface reflects about 4% — so a single uncoated lens transmits only ~92%. AR coatings reduce this loss to well below 1% per surface.

Fresnel Reflection (Normal Incidence)

R = \left(\frac{n_1 - n_2}{n_1 + n_2}\right)^2

N-BK7 (n=1.517) in air: R ≈ 4.2% per surface

Coating Type	Typical R (per surface)	Bandwidth	Use Case
Uncoated	~4% (visible glass)	—	Cost-sensitive, disposable, or where loss is acceptable
Single-layer MgF₂	~1.3%	Moderate (centered wavelength)	Budget broadband visible applications
V-Coat	<0.25%	Narrow (±20–50 nm)	Single-laser-line applications
Broadband AR (BBAR)	<0.5%	Wide (e.g., 400–700 nm)	Multi-wavelength or white-light systems
Dual-band AR	<0.5% per band	Two specific bands	Systems using two discrete wavelengths

Quarter-Wave Condition (Single-Layer AR)

n_{\text{coating}} = \sqrt{n_{\text{substrate}}} \qquad t = \frac{\lambda_0}{4\, n_{\text{coating}}}

Ideal single-layer coating: MgF₂ (n≈1.38) on N-BK7 (n≈1.52) is near-optimal

Coating selection rule: Single laser line → V-coat for minimum loss. Broadband source → BBAR. Budget-constrained → MgF₂. High-power laser → verify damage threshold (LIDT) rating with the coating vendor.

10.Lens Selection Workflow

A practical 7-step process for choosing the right lens:

Define your wavelength(s)

This determines material, coating, and available catalog options.

Determine conjugate ratio

∞:f (collimated to focus), finite (imaging), or f:∞ (collimating a point source). This determines lens type.

Calculate required focal length

Use the thin lens equation or Gaussian beam formula depending on your source.

Check f-number and aberration budget

If f/# < 4 with a singlet, spherical aberration is likely significant. Consider achromats or aspheres.

Select material

Match to wavelength transmission range. N-BK7 for visible, fused silica for UV, ZnSe/Ge for IR.

Specify coating

V-coat for single wavelength, BBAR for broadband. Verify LIDT for high-power applications.

Verify mechanical fit

Check lens diameter, edge thickness, and clear aperture against your mount and beam size.

Don't forget the practical constraints: Budget, lead time, minimum order quantity, and whether a catalog lens exists at your target specs. A "perfect" design that requires a custom optic with 8-week lead time is often worse than a slightly-off-spec catalog lens you can have tomorrow.

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The Comprehensive Guide includes 7 worked examples, SVG diagrams, and full mathematical derivations for every formula on this page.