The Sag (or Sagitta) of a spherical surface is an essential value to calculate when determining the edge and center thickness of a lens. Loosely defined, the Sag is the thickness of material required to accommodate a surface of given radius of curvature with a given aperture (see figure below).

Sag Formula

The Sag of surface (either spherical or cylindrical) may be calculated from:

Sag Formula

Comments on Polarization, Reflection & Refraction
The phenomena of reflection and refraction at dielectric interfaces may be extremely important in designing optical components. In general, different polarization’s of the incident radiation behave
differently under reflection and refraction.

While the formula given below are most useful in calculating the power reflection and transmission co-efficients at uncoated surfaces of optical components such as Brewster Angle windows, prisms, and wedges, the concepts presented should prove useful in understanding qualitatively the effects of polarization state and angle of incidence on coated optical surfaces whose design requires specification of these parameters. These include beamsplitters, non-normal incidence anti-reflection coated optics and non-normal incidence enhanced dielectric mirrors. The polarization state of a light wave refers to the direction of the electric field vector in the wave.

The polarization state is usually characterized as parallel (German-“Parallel”) to the plane of incidence (P-polarized); perpendicular (German-“Senkrecht”) to the plane of incidence (S-polarized); or Random or Natural Polarization (equal amounts of S and P polarization—sometimes this is also called unpolarized light). These are linear polarization states. A light wave linearly polarized in some general direction can be considered a sum of S and P polarized fields having zero phase shift.

If there is a phase shift between the S and P polarized components, the result is elliptical polarization. If the S and P components have equal
amplitudes and the phase shift is exactly 90°, the resulting wave is circularly polarized.

The amount of phase shift between S and P polarized waves is obviously crucial to the design of components such as retardation plates where it is a result of crystal anisotropy. It can also occur in total internal reflection and is crucial to the design of some prism types (e.g. Fresnel rhombs) whose function is to transform a linearly polarized beam into a circularly polarized beam.

If you require assistance in designing “phaseshifting” components, you may call us for more information. On the following pages we present results useful for calculating the power reflection and transmission co-efficients for S and P polarized incident radiation in terms of the indices of refraction and incident angle.

The situation is depicted in the figure to the left. The plane of incidence is the plane of the page (i.e. the plane formed by the incident ray and the surface normal). Usually, we have two cases: the interface is from air (or vacuum) to the substrate (n1=1<n2) or from substrate to air (n1>n2=1).

Angle of Reflection

The basic kinematic properties are:

Angle of reflection = Angle of incidence and n1sinParabola Diagram1
= n2sinParabola Diagram2
(Snell’s law).

The basic dynamic properties are:

S — polarization

s-polarization formula

s-polarization formula

P — polarization

p-polarization formula

p-polarization formula

For normal incidence (Parabola Diagram1 = 0) both the S & P formula yield:

Normal Incidence formula Normal Incidence formula


Brewster’s Angle, Parabola DiagramB, is the angle of incidence for which RP = 0 (i.e., the reflection is entirely s-polarized), or alternatively TP = 1:

Brewster formula

A Brewster Angle Window is depicted below:

Brewster Angle Window Diagram

There are two surfaces which have to be taken into account to calculate the transmission through the window. The Brewster angle for the second surface is:

Brewster Second Surface formula

The incidence angle Parabola Diagrami2 on the second surface is, for parallel surfaces, just the refracted angle Parabola DiagramR of the first surface. It follows then that if:

Incidence Angle Formula


Incidence Angle Formula

Incidence Angle Formula

So one automatically obtains an exact Brewster angle incidence at the second surface.

The reflection and transmission co-efficients may be used to design Brewster stack polarizers and in general to calculate the effects of non-normal incidence on different states of polarization.

Total Internal Reflection occurs at incidence angles greater than or equal to that which yields an angle of refraction greater than 90°. This only occurs if n1 > n2. The angle at which this occurs Parabola DiagramT is found from Snell’s laws:

Incidence Angle Formula

All waves incident at angles greater than Parabola DiagramT will have no transmitted component.

Incidence Angle Formula

Incidence Angle Formula

Computer generated plots showing reflection & transmission (polarized) versus angle of incidence for various refractive index combinations.