Pg. 1

Application of Linear Systems

Analysis to 2-D Optical Images

(Fourier Optics)

An Intuitive Approach

Andrew Josephson

ajosephson@comcast.net

Pg. 2

Fourier Transforms of Electrical Signals

• As electrical engineers, we conceptualize the

Fourier transformation and Fourier synthesis of

voltages and currents frequently in circuit analysis

• Why do we use the Fourier Transform?

• There are other transforms

– Hilbert

– Hankel

– Abel

– Radon

• What makes the Fourier Transform special?

Pg. 3

Linear Time Invariant Systems

)(tx )()()( txthty Linear Time

Invariant System

)(th

)( dtx )()()( dtxthdty Linear Time

Invariant System

)(th

)()()()()( 21 thtxthtxty )()( 21 txtx

Linear Time

Invariant System

)(thLinearity

Time Invariance

Pg. 4

Fourier Analysis of LTI Systems

)()( Xtx

)()()()( HXYty

Linear Time

Invariant System

)()( Hth

When a system can be

classified as LTI, we can

analyze it easily with

Fourier Analysis…why?

Pg. 5

• The Eigen function of an LTI system is a

mathematical function of time, that when applied as

a system input, results in a system output of

identical mathematical from

– The output equals the input scaled by a constant „A‟

– Delayed in time by „d‟

Eigen Function of LTI Systems

)(t )()()()( dtAtthty Linear Time

Invariant System

)(th

Pg. 6

Eigen Function of LTI Systems

• Complex exponentials are the Eigen functions of LTI

systems

• Complex exponentials are also the Kernel of the

Fourier Integral

tjet )(

dtetxX tj

)(

2

1)(

Pg. 7

Fourier Analysis of LTI Systems Revisited

...)( 21

210 tjtj ececctx

...)( )(

22

)(

110021 dtjdtj ecKecKcKty

Linear Time

Invariant System

)(th

• Conceptually, when we analyze an LTI system, we represent the input signal x(t) as a summation of linearly scaled Eigen functions (complex exponentials)

– Fourier Decomposition

– The signal‟s “spectrum”

• We can then run each complex exponential through the system easily because they give rise to a linearly scaled output that is delayed in time

• The output y(t) is the summation of these scaled and delayed complex exponentials

Pg. 8

Answer to the Million Dollar Question

• We use the Fourier Transform to analyze LTI systems because the Eigen function of an LTI system IS the Kernel of the Fourier Integral

• When we do not have an LTI system, we usually assume it is closely approximated by one, or force it to operate in a well-behaved region – Linearization

• The complex exponentials we deal with in circuits are single variable functions with independent variable „t‟

• Where else in electrical engineering do we use complex exponentials?

Pg. 9

Plane Waves

• Complex exponentials are also used to describe

plane waves

– The plane defines a surface of constant phase

• These are multivariate functions with independent

variables R = (x,y,z)

xy

zk

R

RjkeERE 0)(

Pg. 10

Plane Waves as Eigen Functions

• Plane waves are Eigen functions of certain system

types as well

– Since the independent variable is no longer time, we

aren‟t interested in Linear Time Invariant Systems

• We are now interested in the more general Linear

Shift Invariant System

– Shift refers to spatial movement since the independent

variables now describe position instead of time

Pg. 11

Fourier Optics

• Fourier Optics is the application of linear shift

invariant system theory to optical systems

• The plane wave in an optical system is represented

by the multivariate complex exponential just like the

sine wave in an LTI system is represented by the

single variable

• Just like an electrical signal can be represented as

summation of sine waves, an optical image can be

represented as a summation of plane waves

– Angular Plane Wave Spectrum

tje

Pg. 12

Concept of Spatial Frequency

• Assume a plane wave propagates in z-direction

(down the optical axis)

• The image plane (x-y) is normal to the optical axis

• The projection of lines of constant phase onto the x-

y plane is zero (spatial frequency equivalent to DC)

kx

z

yjkzeEzyxE 0),,(

Pg. 13

• Deflecting the wave-vector at an angle other than

zero gives a projection of the plane wave intensity

into the image plane (max and min)

• This example deflects k into the y-direction creating

a nonzero spatial frequency in the y-direction

Concept of Spatial Frequency

kx

z

y

Pg. 14

Concept of Spatial Frequency

• Increasing the angle of deflection increases the

spatial frequency of intensity maximum/minimum

kx

z

y

k

x

z

y

Pg. 15

Concept of Spatial Frequency

Pg. 16

Concept of Spatial Frequency

• Spatial frequency in the y-direction can be denoted

as and has units 1/cm yf

yf

1

Pg. 17

Angular Plane Wave Spectrum

• An arbitrary 2-D field distribution (image) can be decomposed

into a spectrum of plane waves

– Assuming monochromatic light

• Just like the Fourier Transform of an electrical signal

represents the magnitude and phase of each sinusoid in the

signal spectrum, the 2-D Fourier Transform of an image

represents the magnitude and phase of each plane wave in

the image spectrum

• High spatial frequency -> plane wave at large angles

• Low spatial frequency -> plane wave at small angles

Pg. 18

2-D Fourier Transform

• Consider an arbitrary 2-D black and white image in

the XY-Plane

• The image can be described mathematically by

some function U(x,y,z=0)

– „U‟ is optical intensity versus position

– Optical intensity is just proportional to

• The angular plane wave spectrum of the image is

related to the 2-D Fourier Transform

2E

YXYXYX dfdfyfxfjzyxUffA

2exp)0,,(),(

Pg. 19

LSI Optical Systems

• In Linear Shift Invariant optical systems, we can use

Fourier analysis to decompose an image into its

spectrum, multiply the spectrum by the optical

transfer function(s), and inverse transform to get the

resulting output

– Note: Free space propagation of optical images can be

modeled as an LSI system

– This technique correctly models diffraction

– This technique produces identical results to the full

Rayleigh-Sommerfeld solutions

Pg. 20

Thin Lens

• A thin lens can be modeled as a phase shifting

device

– Assumes that no optical power is absorbed

– Using the refractive index, n, and the radius of curvature,

a mathematical transfer function can be calculated

D1 D2

y x y x

R1 R2

n

Pg. 21

Thin Lens

• To determine the optical transfer function of the

simple lens system

– Free space propagate D1

– Multiple by lens transfer function

– Free space propagate D2

D1 D2

y x y x

R1 R2

n

Pg. 22

Fourier Transforming Lenses

• A special value of D1 exists where many terms in

the optical transfer function simplify

– This special value is called the focal length

– When the input image is placed one focal length away, the

optical transfer function at one focal length after the lens

becomes a 2-D Fourier Transformation of the input image

– Most of us already kinda knew that…

21

111

1

1

RRn

D

Pg. 23

Fourier Transforming Lenses

• A delta function and sine wave (complex

exponential) form a Fourier Transform pair

• What is the image equivalent of a delta function?

– A point of light

• We know that a plane wave is a complex

exponential

– A point of light and a plane wave form a Fourier Transform

Pair

• This is exactly what happens when we place a point

of light one focal length away from a lens

Pg. 24

Fourier Transforming Lenses

• An optical delta function placed one focal length

away is transformed into a plane wave one focal

length away (and always)

• This point source has been „collimated‟

F

y x y x

F

Pg. 25

Spatial Filtering – A simple 2 Lens System

• With two lenses, we can construct a system that

produces the Fourier transform of the input image

and then transform this again to create the original

image

F

y x y x

F F

y x

F

Fourier Transform

of Input Image

Pg. 26

Spatial Filtering – A simple 2 Lens System

• We now have direct access to the image spectra and can

filter it physically with apertures

y x

• The low frequency components (small

angular deflection from optical axis)

are contained within the center of the

image spectrum

• Using a circular aperture and blocking

out a portion of spectrum re-creates

the image with the higher frequency

components blocked – low pass filter

Pg. 27

Spatial Filtering – A simple 2 Lens System

• http://micro.magnet.fsu.edu/primer/java/digitalimaging/processing/fouriertr

ansform/index.html

• The link above gives many interactive images and spatial filtering

examples

– High pass

• Block out the image spectra around origin

– See high resolution portion of image remain unchanged

– Low pass

• Allow low frequency planes waves (small angles) to pass through the aperture

– Blurs image by removing high frequency plane waves

– Can be used to balance versus higher frequency image noise

– Band Reject

• Find an input image with a periodic grating (Black Knot Fungus)

• Image spectra is periodic

• Band reject the aliases

• Recreate image without grating presents