Optics Formulas

Light Right-Hand Rule

Light is a transverse electromagnetic wave. The electric E and magnetic M fields are perpendicular to each other and to the propagation vector k, as shown below.

Power density is given by Poynting’s vector, P, the vector product of E and H. You can easily remember the directions if you “curl” E into H with the fingers of the right hand: your thumb points in the direction of propagation.

Illustration of the right-hand rule of electromagnetic waves
Figure 1. Illustration of the right-hand rule of electromagnetic waves.

Intensity Nomogram

The nomogram below relates E, H, and the light intensity I in vacuum. You may also use it for other area units, for example, [V/mm], [A/mm] and [W/mm2]. If you change the electrical units, remember to change the units of I by the product of the units of E and H: for example [V/m], [mA/m], [mW/m2] or [kV/m], [kA/m], [MW/m2].

Illustration of intensity nomogram
Figure 2. Illustration of intensity nomogram.

Light Intensity

The light intensity, I is measured in Watts/m2, E in Volts/m, and H in Amperes/m. The equations relating I to E and H are quite analogous to OHMS LAW. For peak values these equations are:

Light intensity formulas
Figure 3. Light intensity formulas.

The quantity η0 is the wave impedance of vacuum, and η is the wave impedance of a medium with refractive index n.

Wave Quantity Relationship

Wave quantity relationship formulas
Figure 4. Wave quantity relationship formulas.
  • k: wave vector [radians/m]
  • ν: frequency [Hertz]
  • ω: angular frequency [radians/sec]
  • λ: wavelength [m]
  • λ0: wavelength in vacuum [m]
  • n: refractive index

Light Energy Conversions

Light energy conversion formulas
Figure 5. Light energy conversion formulas.

Wavelength Conversions

1 nm = 10 Angstroms (Å) = 10-9 m = 10-7 cm = 10-3 µm

Plane-Polarized Light

For plane-polarized light the E and H fields remain in perpendicular planes parallel to the propagation vector k, as shown below.

Illustration of plane polarized light
Figure 6. Illustration of plane polarized light.

Both E and H oscillate in time and space as: sin (ωt-kx)

Wavelength Nomogram

The wavelength nomogram relates wavenumber, photon energy and wavelength.

Illustration of wavelength nomogram
Figure 7. Illustration of wavelength nomogram.

Snell’s Law

Snell’s Law describes how a light ray behaves when it passes from a medium with index of refraction n1, to a medium with a different index of refraction, n2. In general, the light will enter the interface between the two media at an angle. This angle is called the angle of incidence. It is the angle measured between the normal to the surface (interface) and the incoming light beam (see figure). In the case that n1 is smaller than n2, the light is bent towards the normal. If n1 is greater than n2, the light is bent away from the normal (see figure below). Snell’s Law is expressed as n1sinθ1 = n2sinθ2.

llustration of Snell’s Law
Figure 8. Illustration of Snell’s Law.

Beam Displacement

A flat piece of glass can be used to displace a light ray laterally without changing its direction. The displacement varies with the angle of incidence; it is zero at normal incidence and equals the thickness h of the flat at grazing incidence.

Grazing incidence is light incident at almost or close to 90° to the normal of the surface.

Illustration of beam displacement
Figure 9. Illustration of beam displacement.

The relationship between the tilt angle of the flat and the two different refractive indices is shown in the graph below.

Displacement/Thickness vs Tilt Angle
Figure 10. Displacement/Thickness vs Tilt Angle.

Beam Deviation

Both displacement and deviation occur if the media on the two sides of the tilted flat are different — for example, a tilted window in a fish tank. The displacement is the same, but the angular deviation δ is given by the formula below. Note:δ is independent of the index of the flat; it is the same as if a single boundary existed between media 1 and 3. (see Figure 9)

Displacement/Thickness vs Tilt Angle

Example: The refractive index of air at STP (Standard Temperature and Pressure) is about 1.0003. The deviation of a light ray passing through a glass Brewster’s angle window on a HeNe laser is then:

δ= (n3 - n1) tan θ

At Brewster’s angle, tan θ= n2

δ= (0.0003) x 1.5 = 0.45 mrad

At 10,000 ft. altitude, air pressure is 2/3 that at sea level; the deviation is 0.30 mrad. This change may misalign the laser if its two windows are symmetrical rather than parallel.

Angular Deviation of a Prism

Illustration of angular devition in a prism
Figure 11. Illustration of angular devition in a prism.

Angular deviation of a prism depends on the prism angle α, the refractive index, n, and the angle of incidence θi. Minimum deviation occurs when the ray within the prism is normal to the bisector of the prism angle. For small prism angles (optical wedges), the deviation is constant over a fairly wide range of angles around normal incidence. For such wedges the deviation is:

δ ≈ (n - 1)α

Angular devition vs. incident angle for a prism
Figure 12. Angular devition vs. incident angle for a prism.

Prism Total Internal Reflection (TIR)

TIR depends on a clean glass-air interface. Reflective surfaces must be free of foreign materials. TIR may also be defeated by decreasing the incidence angle beyond a critical value. For a right angle prism of index n, rays should enter the prism face at an angle θ:

θ < arcsin (((n2-1)1/2-1)/√2)

In the visible range, θ = 5.8° for BK 7 (n = 1.517) and 2.6° for fused silica (n = 1.46). Finally, prisms increase the optical path. Although effects are minimal in laser applications, focus shift and chromatic effects in divergent beams should be considered.

Fresnel Equations:

  • i - incident medium
  • t - transmitted medium

use Snell’s law to find θt

Normal Incidence:

r = (ni-nt)/(ni + nt)

t = 2ni/(ni + nt)

Brewster's Angle:

θβ = arctan (nt/ni)

Only s-polarized light reflected.

Total Internal Reflection (TIR):

θTIR > arcsin (nt/ni)

nt < ni is required for TIR

Field Reflection and Transmission Coefficients:

The field reflection and transmission coefficients are given by:

r = Er/Ei    t = Et/Ei

Non-Normal Incidence:

rs = (nicosθi -ntcosθt)/(nicosθi + ntcosθt)

rp = (ntcos θi -nicosθt)/ntcosθi + nicosθt)

ts = 2nicosθi/(nicosθi + ntcosθt)

tp = 2nicosθi/(ntcosθi + nicosθt)

Power Reflection:

The power reflection and transmission coefficients are denoted by capital letters:

R = r2    T = t2(ntcosθt)/(nicosθi)

The refractive indices account for the different light velocities in the two media; the cosine ratio corrects for the different cross sectional areas of the beams on the two sides of the boundary.

The intensities (watts/area) must also be corrected by this geometric obliquity factor:

It = T x Ii(cosθi/cosθt)

Conservation of Energy:

R + T = 1

This relation holds for p and s components individually and for total power.


To simplify reflection and transmission calculations, the incident electric field is broken into two plane-polarized components. The “wheel” in the pictures below denotes plane of incidence. The normal to the surface and all propagation vectors (ki, kr, kt) lie in this plane.

E parallel to the plane of incidence; p-polarized
Figure 13. E parallel to the plane of incidence; p-polarized.
E normal to the plane of incidence; s-polarized.
Figure 14. E normal to the plane of incidence; s-polarized.

Power Reflection Coefficients

Power reflection coefficients Rs and Rp are plotted linearly and logarithmically for light traveling from air (ni = 1) into BK 7 glass (nt = 1.51673). Brewster’s angle = 56.60°.


The corresponding reflection coefficients are shown below for light traveling from BK 7 glass into air Brewster’s angle = 33.40°. Critical angle (TIR angle) = 41.25°.


Thin Lens Equations

If a lens can be characterized by a single plane then the lens is “thin”. Various relations hold among the quantities shown in the figure 15.

Diagram of a thin lens
Figure 15. Diagram of a thin lens.


1/s1 + 1/s2 = 1/F


x1x2 = -F2

Transverse Magnification:

MT = Y2/Y1 = -S2/S1

MT < 0, image inverted

Longitudinal Magnification:

ML = ΔX2/ΔX1 = -MT2

ML <0, no front to back inversion

Sign Conventions for Images and Lenses

Quantity + -
s1 real virtual
s2 real virtual
F convex lens concave lens

Lens Types for Minimum Aberration

| s2/s1 | Best lens
<0.2 plano-convex/concave
>5 plano-convex/concave
>0.2 or <5 bi-convex/concave

Thick Lenses

A thick lens cannot be characterized by a single focal length measured from a single plane. A single focal length F may be retained if it is measured from two planes, H1, H2, at distances P1, P2 from the vertices of the lens, V1, V2. The two back focal lengths, BFL1 and BFL2, are measured from the vertices. The thin lens equations may be used, provided all quantities are measured from the principal planes.

Diagram of a thick lens
Figure 16. Diagram of a thick lens.

Lens Nomogram:

Illustration of a lens nomogram
Figure 17. Illustration of a lens nomogram.

The Lensmaker’s Equation

Convex surfaces facing left have positive radii. Below, R1>0, R2<0. Principal plane offsets, P, are positive to the right. As illustrated, P1>0, P2<0. The thin lens focal length is given when Tc = 0.

The lens maker's equation
Figure 18. The lens maker's equation.

Numerical Aperture

Numeric aperture diagram
Figure 19. Numeric aperture diagram.

φMAX is the full angle of the cone of light rays that can pass through the system.

Numeric aperture equation
Figure 20. Numeric aperture equation.

For small φ:

Numeric aperture equation for small φ
Figure 21. Numeric aperture equation for small φ.

Both f-number and NA refer to the system and not the exit lens.

Constants and Prefixes

Speed of light in vacuum c = 2.998 x 108 m/s
Planck’s const. h = 6.625 x 10-34 Js
Boltzmann’s const. k = 1.308 x 10-23 J/K
Stefan-Boltzmann σ = 5.67 x 10-8 W/m2K4
1 electron volt eV = 1.602 x 10-19 J
exa (E) 1018
peta (P) 1015
tera (T) 1012
giga (G) 109
mega (M) 106
kilo (k) 103
milli (m) 10-3
micro (µ) 10-6
nano (n) 10-9
pico (p) 10-12
femto (f) 10-15
atto (a) 10-18

Wavelengths of Common Lasers

Source (nm)
ArF 193
KrF 248
Nd:YAG(4) 266
XeCl 308
HeCd 325, 441.6
N2 337.1, 427
XeF 351
Nd:YAG(3) 354.7
Ar 488, 514.5, 351.1, 363.8
Cu 510.6, 578.2
Nd:YAG(2) 532
HeNe 632.8, 543.5, 594.1, 611.9, 1153, 1523
Kr 647.1, 676.4
Ruby 694.3
Nd:Glass 1060
Nd:YAG 1064, 1319
Ho:YAG 2100
Er:YAG 2940

Gaussian Intensity Distribution

The Gaussian intensity distribution:

I(r) = I(0) exp(-2r202)

is shown below

Gaussian intensity distribution
Figure 22. Gaussian intensity distribution.

The right hand ordinate gives the fraction of the total power encircled at radius r:

Right hand ordinate of a Gaussian beam equation
Figure 23. Right hand ordinate of a Gaussian beam equation.

The total beam power, P(∞) [watts], and the on-axis intensity I(0) [watts/area] are related by:

Total beam power, and the on-axis intensity of a Gaussian beam equation
Figure 24. Total beam power, and the on-axis intensity of a Gaussian beam equation.


Figure 25 below compares the far-field intensity distributions of a uniformly illuminated slit, a circular hole, and Gaussian distributions with 1/e2 diameters of D and 0.66D (99% of a 0.66D Gaussian will pass through an aperture of diameter D). The point of observation is Y off axis at a distance X>>Y from the source.

Far-field intensity distributions
Figure 25. Far-field intensity distributions.

Focusing a Collimated Gaussian Beam

Focusing a collimated gaussian beam diagram
Figure 26. Focusing a collimated gaussian beam diagram.

In the figure 26 above the 1/e2 radius, ω(x), and the wavefront curvature, R(x), change with x through a beam waist at x = 0. The governing equations are:

Focusing a collimated gaussian beam equations
Figure 27. Focusing a collimated gaussian beam equations.

0 is the waist diameter at the 1/e2 intensity points. The wavefronts are planar at the waist [R(0) = ∞].

At the waist, the distance from the lens will be approximately the focal length: s≈ F.

D = collimated beam diameter or diameter illuminated on lens.

Depth of Focus (DOF)

DOF = (8λ/π)(f/#)2

Only if DOF < F, then

New Waist Diameter

Beam Spread

Optical Density

D = - log (T)


T= 10-D


where R = radius of curvature and Y = radius of the aperture of the surface. SAG is an abbreviation for "sagitta," the Latin word for arrow. Used to specify the distance on the normal from the surface of a concave lens to the center of the curvature. It refers to the height of a curve measured from the chord.