Optics Formulas

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

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

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

The wavelength nomogram relates wavenumber, photon energy and wavelength.

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.

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.

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

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)

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 depends on the prism angle α, the refractive index, n, and the angle of incidence θ_i. 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

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.

• i - incident medium
• t - transmitted medium

use Snell’s law to find θ_t

r = (n_i-n_t)/(n_i + n_t)

t = 2n_i/(n_i + n_t)

θ_β = arctan (n_t/n_i)

Only s-polarized light reflected.

θ_TIR > arcsin (n_t/n_i)

n_t < n_i is required for TIR

The field reflection and transmission coefficients are given by:

r = E_r/E_i    t = E_t/E_i

r_s = (n_icosθ_i -n_tcosθ_t)/(n_icosθ_i + n_tcosθ_t)

r_p = (n_tcos θ_i -n_icosθ_t)/n_tcosθ_i + n_icosθ_t)

t_s = 2n_icosθ_i/(n_icosθ_i + n_tcosθ_t)

t_p = 2n_icosθ_i/(n_tcosθ_i + n_icosθ_t)

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

R = r²    T = t²(n_tcosθ_t)/(n_icosθ_i)

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:

I_t = T x I_i(cosθ_i/cosθ_t)

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 (k_i, k_r, k_t) lie in this plane.

Power reflection coefficients R_s and R_p are plotted linearly and logarithmically for light traveling from air (n_i = 1) into BK 7 glass (n_t = 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°.

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.

Gaussian:

1/s₁ + 1/s₂ = 1/F

Newtonian:

x₁x₂ = -F₂

Transverse Magnification:

M_T = Y₂/Y₁ = -S₂/S₁

M_T < 0, image inverted

Longitudinal Magnification:

M_L = ΔX₂/ΔX₁ = -M_T²

M_L <0, no front to back inversion

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.