STC.UNM v. Intel Corporation

Filing 112

DECLARATION re 110 Brief of Brian Ferrall by Intel Corporation (Attachments: # 1 Exhibit 1, # 2 Exhibit 2, # 3 Exhibit 3, # 4 Exhibit 4, # 5 Exhibit 5, # 6 Exhibit 6, # 7 Exhibit 7, # 8 Exhibit 8, # 9 Exhibit 9)(Atkinson, Clifford)

Exhibit 8

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Spatial Frequency: Optipedia, Free optics information from SPIE

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EXCERPT FROM

Modulation Transfer Function in
Optical and Electro-Optical

Systems

Spatial Frequency

Author(s): Glenn D. Boreman

Excerpt from Modulation Transfer Function in Optical and Electro-Optical Systems

An object- or image-plane irradiance distribution is composed of "spatial

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frequencies" in the same way that a time-domain electrical signal is composed
of various frequencies: by means of a Fourier analysis. As.seen in Fig. 1.3, a

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given profile across an irradiance distribution (object or image) is composed of
constituent spatial frequencies. By taking a one-dimensional profie across a
two-dimensional irradiance distribution, we obtain an irradiance-vs-position
waveform, which can be Fourier decomposed in exactly the same manner as if
the waveform was in the more familar form of volts vs time. A Fourier

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decomposition answers the question of what frequencies are contained in the
waveform in terms of spatial frequencies with units of cycles (cy) per unit
distance, analogous to temporal frequencies in cy/s for a time-domain
waveform. Typically for optical systems, the spatial frequency is in cy/mm.
An example of one basis function for the one-dimensional waveform of Fig. 1.3
is shown in Fig. 1.4. The spatial period X (crest-to-crest repetition distance) of

the waveform can be inverted to find the x-domain spatial frequency denoted by
E == l/X.

v'

Two-dimensional

".1,

image distribution

~

t:Y;d)

g(x¡i Jj')

Xi

u:.

One-imensional
profile g(x.) ,.1,
y.')
r

.~'¡

Figure 1.3 Definition of a spatial-domain irradiance waveform.

Irrdiance

x

.~X-Spatial period
Figure 1.4 One-dimensional spatial frequency.

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Fourier analysis of optical systems is more general than that of timedomain
systems because objects and images are inherently two-dimensional, and thus
the basis set of component sinusoids is also two-dimensional. Figure 1.5

illustrates a two-dimensional sinusoid of irradiance. The sinusoid has a spatial
period along both the x and y directions, X and Y respectively. If we invert these

spatial periods we find the two spatial frequency components that describe this
waveform: E = 1/X and ri = l/Y. Two pieces of information are required for
specification of the two-dimensional spatial frequency. An alternate

representation is possible using polar coordinates, the minimum crest-to-crest
distance, and the orientation of the minimum crest-to-crest distance with

respect to the x and y axes.

y

Figure 1.5 Two-dimensional spatial frequency.
Angular spatial frequency is typically encountered in the specification of imaging

systems designed to observe a target at a long distance. If the target is far
enough away to be in focus for all distances of interest, then it is convenient to
specify system performance in angular units, that is, without having to specify a
particular range distance. Angular spatial frequency is most often specified in

cy/mrad. It can initially be a troublesome concept because both cycles and
milliradians are dimensionless quantities but, with reference to Fig. 1.6, we find
that the angular spatial frequency Eang is simply the range R multiplied by the
target spatial frequency E. For a periodic target of spatial period X, we define an
angular period e == X/R, an angle over which the object waveform repeats itself.
The angular period is in radians if X and R are in the same units. Inverting this
angular period gives angular spatial frequency Eang= R/X. Given the resolution
of optical systems, often X is in meters and R is in kilometers, for which the
ratio R/X is then in cy/mrad.

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Spatial Frequency: Optipedia, Free optics information from SPIE

Page 3 of 3

IIII ç = ¡IX. Spal ~ency (eylm)
! i "":~.:,... i; · RJ
· X' " " Angular spatil frequency
S~tiaipeod (mì....~\. (eyclesradian)

.... "4:9 \ An peod
=X/R
t....

(radians)

"'I _ '.

R\,"'""'",-"
..'\\,
rage (m)'..\....
....

..

......."..

.,..'....

...\.
'. \.

;',\.

\,....
'''''\ft

','
..:..

-:.
..:~
....

Obseatio;.......

point ...

Figure 1.6 Angular spatial frequency.

Citation:

EXCERPT FROM

G. D. Boreman, Modulation Transfer

Optical and Electro-Optical Systems

Modulation Transfer Function in

Function in Optical and E1ectro-

Optical Systems, SPIE Press,

Author(s): Glenn D. Boreman

Bellingham, WA (2001).

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d"M¡lSl

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