Calculating spatial frequency
September 18, 1998
Recently in class, we introduced the equation for calculating the local spatial frequency
of the interference pattern formed at a point on a two dimensional surface by two rays of
light coming from two mutually coherence sources:
f=
sin θ1 − sinθ
λ
2
This equation is called sometimes called the interference equation. In the equation, λ is
the wavelength of the light from the two sources (commonly measured in nanometers).
The spatial frequency f is measured in cycles per unit length (or, equivalently, line-pairs
per unit length). We will most commonly use cycles per millimeter when talking about
holographic fringe patterns. The two angles are measured as shown below:
plate
source 2
positive angle
θ1
θ2
plate normal
(zero degrees)
negative angle
source 1
What happens when a spatial frequency is negative? Well, a negative spatial frequency is
the essentially the same as a positive spatial frequency; you can in fact consider spatial
frequency to be only positive, and the equation above to include an absolute value sign.
Here are a few simple examples of how the interference equation can be used.
Example 1.
Q: Using the above diagram, say that θ1 is 30 degrees and θ2 is -40 degrees, what is the
spatial frequency of the resulting fringe pattern, at the point on the plate shown, made
with a HeNe laser?
1
A: Plugging directly into the equation, and adjust the units of wavelength to give us our
desired units of cycles/mm:
f=
sin(30) − sin(− 40)
= 1800 cycles / mm
633 × 10 −6 mm
Example 2.
Q: Two HeNe sources are located one meter behind a plate (which is axis-aligned as
shown above). One source is located 1cm above the “z” axis, the other 1cm below. Find
the spatial frequency.
A: In this case, it’s helpful to use the “small angle” approximation, which says that for
small angles measured in radians, θ ≈ sin(θ) ≈ tan(θ). Here, tan(θ) is 0.01, and
sin(–θ) = –sin(θ) so we have:
f≈
2 × 0.01
≈ 30 cycles / mm
633 × 10 −6 mm
Other tips
Remember, angles are always measured with respect to the plate normal. If the plate isn’t
aligned with an axis, you must do a “change of coordinate system” and convert all angles
to a plate-centric coordinate system where the plate normal is 0 degrees.
Using exactly the same mathematics used in the problems shown above, you can
determine angles between sources given an interference pattern of a known spatial
frequency. Problem set #1 asks you to do just that. Since the fringe pattern is coarse (and
easy to measure) when the sources that create it are angularly close together (and
probably harder to measure), this technique can be quite useful as a measurement tool.
2