Edge sharpness effects on computer-generated holograms

Rochester Institute of Technology

RIT Scholar Works

Theses

Thesis/Dissertation Collections

5-1-1981

Edge sharpness effects on computer-generated

holograms

Clinton S. Potter

Follow this and additional works at:

http://scholarworks.rit.edu/theses

This Thesis is brought to you for free and open access by the Thesis/Dissertation Collections at RIT Scholar Works. It has been accepted for inclusion in Theses by an authorized administrator of RIT Scholar Works. For more information, please [email protected].

Recommended Citation

E

DG E SHA

RPNESS

E

FFECTS

ON

COM

PUTE R

-GENE

RA TED H

OL OG RAMS

by

Cl

i nton

S

.

Potte

r

A th

e s

i

s s

u bmitte d in part

ial

fulf

illment

o

f

the

r

equir eme nt s fo

r

the

de gr ee o

f

Bac he lo

r

o

f

Sc

ie nce i

n

th

e Sc ho o l o

f

Pho t og

r

aph

i

c A

rts

an

d

Sc

i e nces in

t

he

Coll ege o

f Graphi c

Art

s

and Phot og

raphy

o

f th

e

R

och e s t er

I

nsti tut e o

f Te chnolo gy

Ma

y

,

1981

Sign ature of t

he

Aut

h or

Clinton S. Potter

.

Photog

ra ph

ic

Scie

nce

and I

nst r ume n t a tio n

Ce

rtifi

ed

Acc

ept e d

Ronald Jodoin

by

.

.. ...

..

..

.

... .. .

.

••

. • • . •

.

. ..

•..

... •..

. ....

The si s A

d v

iso r

J.F.

Carson

ROCHES

T

ER INS

Tl'l

.

rm O

F TECII NOLOCY

COLLEGE

O

F

G

RA P

HIC

ARTS

AND PIIOTOCRAPH Y

PERM

I

S

S

,

ON

FORM

Titl

e

of Th

esis

E

DGE SHA

RPNESS EF

FECTS

ON COM

PUTER -GENERATED

HO

LOGRAMS

I

C

linton

S

.

Po

t t e r

he

re by

g

ra nt

permis

sio n to

th

e W

a l l ace Mem

orial

Li

brary o

f

t

he Ro

che

s

t e r

I

ns t i t

ut

e o

f

Te

c hno lo

gy

t

o

repr

oduc e

my

the

sis

in

w

ho

l

e

o

r in

part.

Any

re

pr od uction wi

l l

not be for comme

rc i a l us e

o

r

r

r

o

f i t .

11

EDGE SHARPNESS EFFECTS ON

COMPUTER-GENERATED

HOLOGRAMS

by

Clinton

S.

Potter

Submitted to the

Photographic Science and Instrumentation Division in partial fulfillment of the requirements

for the Bachelor'

of Science degree

at the Rochester Institute of

Technology

ABSTRACT

Computer-generated holograms refer to a class

of holograms which are produced as a graphical output

from a computer. This paper investigates the relation.

ship between the resolution in the system used to re

duce the graphical artwork to a reasonable size for

diffracting

light and the _quality of the reconstructed

hologram. A number of reduction schemes are examined

including

a one-step reduction onto a general purpose

film,

a one-step reduction onto a high resolution film

and a

two-step

reduction scheme _using coherent

Ill

ACKNOWLEDGEMENTS

I am grateful for the

assistance,

_support, ideas

and occasional _prodding given

by

_my

advisor,

Dr. Ronald

Jodoin.

Also,

I am indebted to Professor John F. Carson

for his _assistance, ideas and words of wisdom. Without

the

help

of both these men and the others who gratefully

provided

ideas,

equipment,

and

support,

this work

IV

TABLE OF CONTENTS

Page

ABSTRACT ii

ACKNOWLEDGEMENTS iii

LIST OF FIGURES v

CHAPTER I: INTRODUCTION 1

CHAPTER II: METHODOLOGY k

General Description of Computer-Generated

Holograms k

Detour Phase : A Method to Encode Complex

Wavefronts 5

Technique for

Making

Computer-Generated

Holograms

7

Advantages of

Binary

Holograms 10

Calculation of the Hologram 11

Software Development 12

Target Design 14

Two Dimensional Fast Fourier Routine

Tests 18

The Photoreduction Steps' 18

Sign Conventions 20

Plotter Characteristics 20

Artwork Characteristics 21

Experimental Procedure for Photoreduction. ..21

Analysis of Spatial Differentiation

During

Reconstruction 31

CHAPTER III: SUMMARY AND CONCLUSIONS 34

CHAPTER IV: RECOMMENDATIONS 36

APPENDIX: SOFTWARE

37

REFERENCE LIST

V

LIST OF FIGURES

Figure _Page

1.0 Graphical Explanation of Detour Phase 6

2.0 Typical Hologram Cell

9

3.0 Optical System for Reconstruction of

Hologram

g

4.0 Data Matrix for Sampled Triangle Target 15

5.0 Data Matrix for Triangle Sampled as an

Even Function , . . , 16

6.0 Data Matrix for Triangle Sampled with

a Degraded Edge

17

7.0 Hologram Artwork for Figure 4.0 22

8 . 0 Hologram Artwork for Figure 5.0 23

9.0 Hologram Artwork for Figure 6.0 24

10.0 Optical Reconstruction of Figure 4.0

27

11.0 Optical Reconstruction of Figure 5. 0... 28

12.0 Optical Reconstruction of Figure 6.0

29

13.0 Spectral

Sensitivity

of Kodak High

Speed Holographic Film Type SO-253 32

14.0 Condenser Illumination Scheme for

CHAPTER I

INTRODUCTION

Through the use of _{computer-generated}

holograms,

it is possible to obtain images of objects that have

no physical _existence.

Computer-generated

holograms have a number of

applications which include 3-D image

display,

edge

enhancement,

image

deblurring,

optical data

processing,

matched

filtering,

interf erometry, optical shop

testing

i

and laser beam scanning.

The process of _synthesizing a hologram _generally

consists of four steps. First the propagation of the

complex amplitude from an object to the hologram plane

2

is computed.

The second

step,

still within the

computer,

is to encode the complex amplitude as a real non-nega

tive function from which the hologram artwork can be

generated on a graphic output device. The final steps

of this process are to make the artwork and reduce

3

it to a reasonable size for

diffracting

light.

Studies on computer-generated holograms can

be divided into three main categories:

1)

Coding

Tech

niques;

2)

Applications;

and

3)

Techniques for

improving

the _quality of computer-generated holograms. Most

models for

improving

quality

and applications of com

puter-generated

holograms.

Only

a few studies brief

ly

examine the photoreduction process _necessary to

reduce the artwork to a reasonable size for

diffracting

light.

Photoreduction is still an important _step in

an age of laser scanners. A scanner allows for the

direct

recording,

from the

computer,

of the hologram

artwork without the need of a reduction step. Unfortun

ately,

the raster size of the best scanners is _only

on the order of

12.5ym,

which is not _sufficiently small 5

for some applications.

Also,

the cost of the scanners

is prohibitive.

This paper examines the photoreduction _step

in computer-generated holograms and proposes a number

Footnotes for Chapter I

1

Wai-Han

Lee,

in Progress in

Optics,

Emil

Wolf,

ed., North Holland

Publishing

Co. _,

1978,

p. 173. 2

B. R. Brown and A. W.

Lohmann,

"Computer-gener

ated

Binary

Holograms," I.B.M. Journal of Research and

Development,

March 1

969

, p"^ 161.

3Ibid.

.

^Lee,

p. 122.

L. P. Yaroslavski and N. S.

Merzlynkov,

Methods

of Digital

Holography,

Dave

Parson,

trans.,

Consultants

CHAPTER II

METHODOLOGY

General Description of Computer-Generated Holograms

Computer-generated holograms refer to a class

of holograms which are produced as a graphical output

i

from a digital computer.

Basically,

the function

of a computer-generated hologram is to create an optical

wavefront from a set of computed data that are a proper

2 sampling of the complex wavefront amplitude.

In conventional off-axis reference beam holo

grams, the amplitude transmittance of a hologram re

corded under ideal conditions is proportional to:

t,

i =

|Rei2lT0CX

+ A, eq . 1.0

(x,y)

'

(x,y)

where

Rei27TOCX

represents the tilted reference wave

and

A(x

e1(p(x'y)

the object wave.

t(x

y) is the

resulting

intensity

variation of the interference pat

tern between the two waves.

In computer-generated

holograms,

the transmit

tance of the hologram and the object wave is not re

stricted to the relationship specified

by

eq. 10.

In

fact,

most of the work in computer-generated holo

wavefront amplitude for convenient production on com

puter graphic devices.

Coding

is the conversion of

a complex valued function into a

real,

non-negative

function in such a _way that the complex valued function

can be retrieved intact

by

optical means at a later

stage .

Detour Phase : A Method to Encode Complex Wavefronts

Brown and Lohmann

(1966)

used Lord Rayleigh's

idea of detour phase to code complex wavefronts. Slight

dislocations of some slits in a diffraction _grating

give rise to "ghosts" _{in the diffraction spectra.}

While the path difference for wavelets from adjacent

slits of a perfect grating in the first diffraction

order is _exactly one _wavelength, the path length differ

ence for wavelets from a dislocated slit and its neigh

bor will be greater or less than one wavelength (Figure

1

)

. The deviation from an integral

wavelength forms

the basis for _encoding the phase in Detour phase holo-4

grams.

There are a variety of methods for encoding

the modulus of the complex amplitude. In a manner

similar to that of the conventional hologram formed

by

interference,

the amplitude can be represented

by

the contrast of sinusoidal fringes. It is possible

Fig. 1.0 Graphical Explanation of the Detour Phase

The two lower grating slits are _slightly out of

position. _Therefore, the complex amplitudes in the slits

complete control over the amplitude

by

_encoding with

binary

patterns of _varying spatial density.

Allebach

(1981)

provides a good review of a

variety of

binary

_coding techniques and analyzes associ

ated representational errors.

Technique for

Making

Computer-Generated Holograms

To make a hologram

by

the detour phase

method,

the wavefront represented

by

the complex valued

function,

A, .^(x,y)

2>0

(x,y)

is first sampled at _equally spaced intervals sufficient

ly

small _according to the _sampling theorem. In _plotting

the

hologram,

the paper is divided into equally spaced

cells. Rectangular apertures are drawn inside each

cell. Each aperture is determined

by

three parameters:

its

height,

h _; its _{width, wm ;} and its center with '

nm ' nm'

respect to the _cell, c . Figure 2.0 shows one of

the cells in a detour phase hologram.

The parameters of the aperture are selected

as follows :

h -Advw = co c = co dx/2TrM _eq. 3.0

nm nm y nm nm ynm

A and co are the amplitude and phase of the sampled

nm Ynm

values taken at x =

ndx

and y =

mdy

are the sampling

8

of A is normalized to

1.0.

The parameter M determines the effective carrier

frequency

of the

hologram.

This

detour phase _coding procedure is then carried out fcr

all the cells in the

hologram.

When this

binary

pat

tern with _many small apertures is plotted and

photore-duced on photographic film it creates an amplitude

transmittance on the film given by:

t1

(x,y)

=

ERect[(x-ndx-cnm)/w]Rect

f(y-mdy)/hj

eq. 4.0

where

Rect(x)

=

1,

for x ^ 0.5

0,

otherwise.

This computer-generated hologram can be recon

structed

by

_placing it in the optical system illus trated in Figure 3.0.

The hologram is illuminated

by

a collimated

laser beam. The Fourier transform of the desired

wave-front occurs in an off-axis region on the Fourier

trans-form (back

focal)

plane of Lens L. . The aperture mask

passes _only one diffracted wave through the optical system. Lens

L?

performs the inverse Fourier transform

to produce the wavefront

,icp(x,y) eq. 2.0

(x,y)

at the back focal plane of lens

L2

. The aperture mask

Fig

2.0

A

Typical Hologram Cell

h is proportional to the

nm amplitude of the

complex wavefront

fc

1

c is proportional to the phase of the

nm complex wavefront

Source

Fig. 3- 0 Optical System for Reconstruction _o$

Hologj

am

Lens 1

/

&

Hologram

Planr

Lens 2

Image Plane

>T

>^

>^

>

10

The wavefront at the back focal plane of the lens

L?

is the bandpassed output of the diffracted wave from

o

the hologram.

A more rigorous analysis of the Detour phase

coding procedure is outlined

by

Lohmann and Paris

(1967)

Advantages of

Binary

Holograms

Binary

holograms offer a number of features

that make them _particularly attractive.

The ease and _accuracy with which

binary

patterns

can be _{photographically} reproduced compared with _grey

patterns are well known and form the basis for the

halftone _printing process. Like halftone

photographs,

the

binary

hologram is quite insensitive to non-linear

photographic _effects; thus much less control over ex

posure and development is needed

during

the reduction

9

process .

Binary

holograms are as efficient as _ordinary

thin emulsion holograms can be. The _only difference

is that

binary

holograms are similar to square wave

gratings, whereas _ordinary holograms are similar to

sinusoidal gratings.

Another advantage of the

binary

hologram over

the _grey hologram is that it directs more light to

the reconstructed image. A square wave _representing

2

11

11

into the first diffraction order as compared with a

fully

modulated sine wave _{grating representing the}

brightest possible _ordinary thin emulsion

hologram.

The reconstruction of a

binary

hologram also

yields less noise from light scattered

by

photographic

grain structure than an _{ordinary grey}

hologram.

At

transmittance values near 1.0 there are few grains

to cause

scattering,

whereas near 0.0 transmittance

several layers of silver grains make the emulsion reli

ably opaque . The transmittance of a conventional grey

hologram fluctuates near a value of 0.5 where grain

scattering is most severe 12

Calculation of the Hologram

The procedure in the preparation of a

binary

Fourier hologram reduces to the

following

operations.

First a resolution element of the object Ax is selected

so as to _satisfy the requirements of _quality of the

reconstructed image. This automatically defines the

hologram size D, _, where

D^

= co Xf

h max

eq. 5.0

The input to the computer is in the form of a matrix

of complex amplitudes

f(nAx,mAy) = f

nm

12

2 ?A

which consists of N complex

numbers,

where N = /Ax

is equal to the number of resolvable elements _along

the X axis. The next _step is to find the function

f f

nm nm n

v = = _eq. 7.0

sinc[cAx(xQ+nAx)

]

sinc[c(M+g)]

Since such qualities of the image as _uniformity of

the field and its brightness are function of c and

M,

the permissible values of these parameters have

to be found subject to the requirements stated above.

If M =

1,

c =

I,

we find that co = 1 and the bright

ness of the image is high at the _center, but it falls

to zero at the edges. If M =

2,

c =

l

, co = 2/tt

and the quality of the image improves but its bright

ness decreases

by

a factor of 20.

The machine Fourier transformation

V *

V.,

(j5co,k<5co)

=

|V.,

|el(cpjk/27T)

eq. 8.0

nm jk ' Jk'

is performed most economically on a computer with the

aid of the

Cooky-Tukey

algorithm, known as the Fast

Fourier

Transform,

which reduces the computing time

by

the factor of

(41og2N)/N2

compared with that needed

13 in the conventional Fourier transform program.

Software Development

A Xerox/Honeywell Sigma

9

computer and a Zeta

13

available for computation and

plotting

of the holograms.

Software was designed around the capabilities of the

hardware and was written in Fortran IV.

The program consists of the

following

major

routines :

1)

Read Data

Matrix.

This routine reads a

64 x 64 element _array from a target file.

2)

Division

by

a weighted sine function accord

ing

to _eq. 7.0.

3)

Two Dimensional Fast Fourier Transform.

This routine computes the Fast Fourier Transform of

the sine divided real matrix. It uses the _separability

property of two dimensional Fourier transforms. The

routine computes the row

transforms,

multiplies

by

the number of elements in the row and then computes

the column transform.

4)

System of Equations. This routine determines

the height and placement of the rectangular aperture

in each hologram cell. It performs the

following

opera tions :

a) Converts matrix data from the Cartesian

Complex form a+ib to the polar complex form

AeicP.

b)

Normalizes A and cp.

c) Converts normalized A and _cp to scaled

plotting parameters

Wnffl

and

Pnm,

where

14

the height of the rectangle to the center of

the hologram cell.

5)

Plot Routine. Plots W and P as a

rec-nm nm

tangle in each hologram cell. The plot routine allows

the artwork to be _easily scaled

by

_manipulating the

value of the SCALE variable.

The program can be found in Appendix A. It

uses _{approximately} 16K words of variables (1 complex

64 x 64 element _array, 2 real 64 x 64 element _arrays,

and 1 complex 64 element array). Computation time

is dependent on object _complexity but generally requires

less than two minutes. Eighty-five granules of disc

storage are required to plot the hologram.

Target Design

Holograms were computed and plotted for a number

of targets.

Early

experimentation was unsuccessful

because the target was

insufficiently

sampled or the

target was particularly complicated. A large triangle

was used for the work in this thesis because the geo-i-/-.

metric shape is _easy to distinguish from background

noise

during

reconstruction. The triangle target was

sampled

by

three different methods. The triangle was

sampled in the middle of the data matrix as in Figure

4.0,

sampled as an even function in one dimension

(Fig

" "-"-'_<J<J<J

'.'J<JWV,-"JVWJ'.>WWVJWWW

OOOOUUOUUUUUUUUOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

15

0 OOOO 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0000 0 0 0 0 0 0 0 0 0 0 0 0 00 0O O O0 0 0000 0 0000

OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 100000000000000000000000000000000

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 00_{0 0 0 0}00 0000 0o on ooooo 0

00000000000000000000000000000

11111

000000000000000000000000000000

00000000000000000000000000001

1 1 1 1 1100000000000000000000000000000

0000000000000000000000000001

1 11111 1 _{10000000000000000000000000000}

000000000000000000000000001

111111111 _{10000000000O000000000O000000}

00000000000000000000000001 1111111111 1 100000000000000000000000000 0000000000000000000000001111111111111 110000000000000000000000000

000000000000000000000001 111111111111111 1000000000000000000000000 000000000000000000000011 1111111111111111 100000000000000000000000

0000000000000000000001 1111111111111111111 10000000000000000000000

000000000000000000001

lllllllllllllillllll

11000000000000000000000

00000000000000000001 1

111111111111111111111

1 100000000000000000000

000000000000000000

lllllllllllllllllllllllllll

_{0000000000000000000}

000000000000000001

llllllllllllllllllllllllllll

_{000000000000000000}

00000000000000001 1_{lllllllllllllllllllllllllll1 100000000000000000}

000000000000000

lllllllllllllllllllllllllllllllll

0000000000000000

00000000000000

lllllllllllllllllllllllllllllllllll

ooooooooooooooo

00000000000001

llllllllllllllllllllllllllllllllllll

oooooooooooooo

0000000000001 1 11111111111111111111111111111111111 1 10000000000000

00000000000

11111111111111111111111111111111111111111

000000000000

0000000000111111111111111111111111111111111111111111100000000000

000000001 1111111111111111111111111111111111111111111 1 10000000000

0000000111111111111111111111111111111111111111111111 1 10000000000

00000011 11111111111111111111111111111111111111111111111000000000

0000000000000000000000000000000000000000000000000000000000000000

0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000

0000000000000000000000000000000000000000000000000000000000000000

0 0 00 0 0 0 0 0 0 0 0 0 000 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0000 0 0 0

OOOOOi 300000000 OOOOOi 300000000 OOOOOi 300000000 OOOOOl 300000000 OOOOOi 300000000 OOOOOi. 300000000

00000< F1s-

k.o

300000000

OOOOOl 300000000

00000< Data Matrix for Sampled 300000000

00000< 300000000

00000< Triangle Target 300000000

OOOOOC 300000000 OOOOOC 300000000 OOOOOi. 300000000 OOOOOC 300000000 OOOOOC 300000000 OOOOOC 300000000 OOOOOOuvvyyvuuwyyuyyyyyciyuyywuyuuyyuuiiyULiUUUUUUWUUUOOOOOOOOOO 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0000 0 0 0 0 0 000000000000000000 0 0 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000

0000000000000000000000000000000000000000000000000000000000000000

00uv^guMmimiMiiKMi_{y 00 y 0 V 000 00 0 0}

00 0 0 00 00 000 0 0 0 0 0 0 0 0 0 000 0000 0

000000000 0 0 000 000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0000 0 000 0 0 0 0 0 0 0000 0 0 0 0

ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo

000000000000000000000000000000000000000000000000000000000000000

100000000000000000000000000000000000000000000000000000000000000

110000000000000000000000000000000000000000000000000000000000000

111000000000000000000000000000000000000000000000000000000000001

111100000000000000000000000000000000000000000000000000000000011

1 1 1 1

100000000000000000000000000000000000000000000000000000001

1 1

11111

1000000000000000000000000000000000000000000000000000001

_{1 1 1}

111111

10000000000000000000000000000000000000000000000000001

₁₁₁₁

1 1 1 1 1 1 1

100000000000000000000000000000000000000000000000001

₁₁₁₁₁

11111111

1000000000000000000000000000000000000000000000001

_{1 11111}

1111111111

0000000000000000000000000000000000000000000001

_{1 111111}

11111111111

00000000000000000000000000000000000000000001

_{1 1111111}

11111111111

1000000000000000000000000000000000000000001

_111111111

11111111111 1

10000000000000000000000000000000000000001

1 111111111

1111111111111_{100000000000000000000000000000000000001}1 1111111111

1111111111111 11000000000000000000000000000000000001111111111111

11111111111111110000000000000000000000000000000001 1 111111111111

1111111111111111100000000000000000000000000000001 1 1111111111111

111111111111111111000000000000000000000000000001111111111111111

111111111111111111100000000000000000000000000011111111111111111

111111111111111111 110000000000000000000000000111111111111111111

111111111111111111111000000000000000000000001111111111111111111

111111111111111111111 10000000000000000000001 1 111111111111111111

111111111111111111111 1 100000000000000000001 11111111111111111111

11111111111111111111111 1000000000000000001 111111111111111111111 11111111111111111111111110000000000000001 1 111111111111111111111

000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 16 Fig. 5.0

Data Matrix for Triangle

Sampled as an Even Function

AAAAAliCuiAAAAAAAAAAAAAAAAAAAAAAAAj-u-iJ-iA^AAAAAiV^wi, 000001 O00001 000001 OOOOOl OOOOOi OOOOOl 00000( 00000< 00000( 00000< OOOOOC OOOOOC OOOOOC OOOOOC OOOOOC OOOOOC OOOOOC OOOOOCvvvvwyvvvvwvvyuwuvvyvyyuwyywyvyyyyvywyyyjyy^OOOOOOO 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000090000000000000000000000000

11a n

(, .,,.tr_-^_;^r:_:^___r,...

;^,-j (;)q(-j(;)0^_{0 q ^ q ^ q q 0 0 0 0 ^}

0^ ^ ^ ^ ^ ^ ^ ^0^ ^ ^ 0 ^ ^ Q Q Q QA

0000000000000000000000000000000000000000000000000000000000000000

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 0 0 0 0 0 0 0 0000 0 0 0 0 0 0 0 0 0 0 0 0000 0 0 0 0 00000000 0

0000000000000000000000000000000100000000000000000000000000000000

0000000000000000000000000000001210000000000000000000000000000000

000000000000000000000000000001222 1 000000000000000000000000000000

0000000000000000000000000000122222100000000000000000000000000000

0000000000000000000000000001222222210000000000000000000000000000

0000000000000000000000000012222222221000000000000000000000000000

0000000000000000000000000122222222222100000000000000000000000000

0000000000000000000000001222222222222210000000000000000000000000

0000000000000000000000012222222222222221000000000000000000000000

0000000000000000000000122222222222222222100000000000000000000000

0000000000000000000001222222222222222222210000000000000000000000

0000000000000000000012222222222222222222221000000000000000000000

0000000000000000000122222222222222222222222100000000000000000000

0000000000000000001222222222222222222222222210000000000000000000

000000000000000001222222222222222222222222222

1

000000000000000000

0000000000000000122222222222222222222222222222100000000000000000 0000000000000001222222222222222222222222222222210000000000000000 00000000000000122222222222222222222222222222222210000000(30000000

0000000000000122222222222222222222222222222222222100000000000000 0000000000001222222222222222222222222222222222222210000000000000 00000000000 12222222222222222222222222222222222222221000000000000 0000000000122222222222222222222222222222222222222222100000000000

0000000001222222222222222222222222222222222222222222210000000000

AA AAA AAA 1 ??77???9?'????'7?????'???'?9?t1?'???'????^????''?7??7 1 A A A A A A A A A

0000000 1111111111111111111111111111111111111111111111111 00000000

0000000000000000000000000000000000000000000000000000000000000000

0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000

00000000000000000000 0 00 000 0 0 0 0 0 0 0 0000 000 00000 0 000 00 0 00000 0 00000 0 000000000 00 OO00O Qn r>Q Q Q'"> <"<i">'">rt'-* '">'"'_&Qaa a a a a a a a a a a a aa a a a a a axiclqaaa

y 00 0 0

17

000000 000000 000000 000000' 0000001 000000' 000000' 000000' 000000' 000000' 000000'

000 0001

000 0001

000000' 000000' 0000001 0000001 OOOOOOL Fig.

6.0

Data Matrix for Triangle Sampled

with a Degraded Edge

vvV*V'"^*"**

18

Two Dimensional Fast Fourier Routine Tests

A number of fast Fourier transform routines

were tested for speed and _accuracy with 8 and 16 point

known transforms. It was determined that International

Mathematical and Statistical Libraries

(

IMSL

)

subroutine

FFTRC was most applicable.

FFTRC computes the fast Fourier transform of

a real vector of length N where N is _any positive even

integer. The output vector X is defined mathematically

as

N-1 0 . .. /M

X,

. =

JA.

2TTijk/N

where k = 0 N-1

k+1

jTo

J

FFTRC factors N into its prime factors and applies

1 A

Cooley-Tukey

techniques for each prime factor.

The Photoreduction Steps

The _plotting, photoreduction and reconstruction

of a computer-generated hologram transform mathematical

models and

theory

to real physical existence. To re

alize the noise and efficiency advantages of

binary

holograms it is essential that the resolution of the

photoreduction be sufficient to ensure a step edged

binary

pattern for all aperture sizes. The contrast

ratio of clear apertures to background should be greater

19

should remain uniform _{throughout the reduced}

hologram,

regardless of feature size. The edge transition region

should be an order of magnitude smaller than the aper

ture size.

A number of factors _influence the _quality of

the reduced

hologram.

_High

quality artwork should

be used and it should be illuminated to provide maximum

contrast. The lens used for reductions must be con

sidered. Factors such as _resolving power and field

coverage and the effects of

aberrations,

distortion

and curvature of the plane of

focus,

must be considered.

Factors _relating to the photographic material also

play a large role in the _quality of the reduced holo

gram. Such factors as light

scatter,

granularity,

contrast and the method of _processing influence image

quality.

Finally,

there is the camera that makes the

reduction.

Vibration,

poor

focus,

misalignment of

focal plane position or failure to

keep

the axis of

the lens _exactly perpendicular to the film plane can

15

degrade the reduced hologram.

The reduction in contrast of fine detail due

to lens performance has important consequences in the

recording of the image

by

a photographic emulsion.

The hologram artwork has two basic brightness levels

- the _aperture brightness and the background bright

20

a _variety of feature sizes will have a _variety of il

luminance

levels,

_{ranging from that of the}

coarsest

lines to that of the background.16

In an ideal

binary

reduction of the hologram

exposure is adjusted to

keep

the apertures clear.

As a result the smaller apertures will be recorded

on the toe of the characteristic curve and the contrast

of the hologram will be further reduced.1^

Sign Conventions

For the purpose of this paper the original art

work with a white background and dark rectangular aper

tures will be considered to have a positive sign.

A reduction of the artwork onto a negative _working

film system will be considered to have a negative sign.

For reconstruction the hologram must have a negative

sign; that

is,

a dark background with clear apertures.

Plotter Characteristics

The theoretical resolution of the Zeta plotter

is _.06mm,

but,

_practically, because of ink _spreading

into the paper _surface, the resolution of the plotter

system is more on the order of .3mm. A black Pentel

pen is used for plotting on plain translucent 32" x

32" _{paper. A}

clipping procedure has been employed

in the software to _clip all apertures with widths

21

Artwork

Characteristics

The 16 inch square artwork contains 4096 cells

in a 64 x 64 array.

Plotting

time was

typically

30

minutes.

Depending

on the

complexity

of the

target,

approximately

20-30%

of the cells are filled with rec

tangular _apertures. Feature sizes range from .4mm

to

approximately

1 cm and the average hologram cell

spacing is .6 cm. The rectangles _are filled

by

hand-coloring with a black Pentel pen. The contrast ratio

of the artwork is _{approximately} 10 to 1. The hologram

artwork for each of the triangle targets is illustrated

in Figures

7.0,

8.0 and 9.0.

Experimental Procedure for Photoreduction

Kodak Technical pan film was used for initial

investigations. The artwork was mounted on a wall

and

diffusely

illuminated with two 1000 watt tungsten

sources. A 100x photoreduction onto Technical pan

film was made in one _step using a Nikon 50mm macro

camera objective at f/5.6. An exposure series was

made at one-stop intervals and the film was processed

to a gamma of 2.8.

The _resulting holograms were 4mm square with

an average aperture spacing of .06mm.

On examination under a high _quality Olympus

I I

22

23

m I

Zk

25

capabilities of Technical pan film are

extremely

poor.

Smaller features were in some cases not recorded and

edge transition regions of the apertures approached

the size of the medium features

(1-2mm).

The grain

size of the film approached the smallest feature sizes.

An attempt was made to reconstruct the hologram

by

placing it in the optical system described earlier

(Figure 3.0). A series of _spatially differentiated

triangles was evident in the first and second diffrac

tion orders. The _quality of the reconstruction was

poor and cluttered with background noise caused

by

the large grain structure in the hologram.

Kodak High Speed Holographic film type SO-253

was used for further investigations. The theoretical

resolving power is 1250

p/mm.

A 100x _one-step reduc

tion of the artwork was made in the same manner as

stated previously. An exposure series was made at

1 _stop intervals and the film was processed to a gamma

of 6.C.

On examination under the

microscope,

it was

determined that all aperture sizes were recorded but

the transmittance of the smaller apertures was signifi

cantly less than the larger apertures. Grain size

of the SO-253 was at least an order of magnitude smaller

than the smallest feature sizes. The edge transition

region was an order of magnitude smaller than the aper

26

The holograms were _{reconstructed} and _spatially

differentiated triangles were _{clearly visible} in the

7th diffraction order.

It was thought that a better _quality reduction

and an increase in lens performance could be achieved

if a filter was used over the objective. Artwork was

reduced 100x in one _step onto Kodak High Speed Holo

graphic film SO-253 using a

red,

green,

and blue filter

(Wratten number

25,58,47)

over the objective. An ex

posure series was made at one _stop intervals and was

processed as stated previously.

Under microscopic

examination,

the holograms

resembled those made earlier onto SO-253 without the

filters. The exposure series did not compensate for

the varying filter factors and the spectral _sensitivity

of the film.

Consequently,

the maximum

density

of

the holograms varied and it was difficult to make sub

jective comparisons _among the filtered reductions.

On _{reconstruction,} the holograms yielded approx

imately

the same results as without the filters. Spa

tially

differentiated triangles were evident beyond

the 5th diffraction order.

It was discovered that proper exposure

during

reduction is an essential factor for the quality of

reconstruction. The next experiment optimized the

28

riEtrre-6re-30

Because of the spectral

sensitivity

of SO-253

(see

Figure

13.0),

further experimentation was done

with a red filter

(Wratten

#25)

over the objective.

Thirteen exposures were made over a range of two stops.

This was accomplished

by

_placing a 1.2 N.D. filter

over the objective and _making exposures _varying between

4 and 16 seconds at

f/5.6.

The film was processed

as before to a gamma of

6.0.

On

reconstruction,

the reduction with the best

image _quality and brightest

image,

compared to back

ground

noise,

was reduced _using an exposure time of

12 seconds. The integrated diffuse

density

of this

hologram,

measured at the center of the hologram with

a 2mm

aperture,

was 1.95.

Experimentation was started _using a two-stage

reduction scheme. A high _quality 10x reduction was

made of the artwork onto Kodalith film _using a 4 x 5

camera and a high _quality process camera lens at f/ 11.

The quality of the reduction was excellent and managed

to capture the intricacies of the pen movement. The

reduction was essentially

binary

with step edged pat

terns for all aperture sizes. The contrast ratio was

on the order of 10,000 to 1.

A condenser illumination scheme was set up on

an optical bench (Figure 14.0) and the uniformity of

31

to be adequate. The Kodalith intermediate was placed

in the system and photographed with a Nikon 50mm macro

at f/5.6 onto

SO-253

film.

A focus and exposure series

was run. The film was processed and examined under

a microscope. The hologram was found to be out of

focus. Depth of focus was calculated as 80u.

Reconstruction was not attempted because the

hologram was the _wrong sign. No further experimentation

was attempted.

Analysis of Spatial Differentiation

During

Reconstruction

During

reconstruction the triangles are subject

to spatial differentiation. This high pass

filtering

effect can be _clearly observed in the reconstructed

image "because Fourier spectra of the original object

are not taken to be _uniformly distributed over the

entire hologram plane." "Multiplication

by

a random

phase factor on the object is effective for _removing

this effect since the addition of the random phase

factor to the object causes the Fourier transform values

19

o.o

-t-l

o

09 co

CD

0)

ct

32

ij-oo

500

'600

xtfavelengfch

(nm)

700

Fig. 13-0

SpecggiigSijaltiftJj

5jp80jgk2^gh Speed

Transparency

Final

Image

J

Filtered Source

\

i

Objective Condenser

Fig. 14.0 Condenser Illumination Scheme for $wo-

33

Footnotes for Chapter II

1

Wai-Hon

Lee,

in Progress in

Optics,

Emil

Wolf,

ed.,

North Holland

Publishing Co.,

1978,

p. 121. 2

B. R. Brown and A. W. Lohmann_, "Computer-Gener ated

Binary

Holograms,"

I.B.M. Journal of Research

and

Development,

March

1969,

p. 161 .

3Lee,

p. 121.

Brown and

Lohmann,

1969,

p. 161.

5Ibid.

Lee,

p. 127.

7Ibid.

, p. 128.

8Ibid.

, pp. 127-128.

Q

Brown and

Lohmann,

1969,

p. 161. 1 0

A. W. Lohmann and D. P.

Paris,

"Binary

Fraun-hofer Holograms," _{Applied Optics}

, . Ootober

1967,

p. 1739.

1 1

B. R. Brown and A. W.

Lohmann,

Complex Spatial

Filtering

with

Binary

Masks," Applied

Optics,

June

1966,

p. 968. 1 1

C. M.

Soroko,

Holography

and Coherent

Optics,

Albin

Tybulewicz,

trans.,

Plenum

Press,

1980,

pp. 498- i

499.

UIbid.

, p. 499.

15IMSL

Library

1,

Volumes

I,

-II and III,. Edition

7,

Houston,

Texas,

July

1979-Microphotography

, Eastman Kodak Data Book

P-52,

1976,

p. 3.

17Ibid.

18Ibid.

1^K.

Nagashima and T.

Asakura,

"Simple

34

CHAPTER III

SUMMARY

AND

CONCLUSIONS

To produce

binary

_{step edged apertures on the}

order of 5pm requires a _{microphotographic} system with

extreme _{resolving capabilities. If the apertures are}

considered as

bars,

to just resolve a ₅ micron bar

would require a system that is capable of _resolving

200 lp/mm. To resolve the 5 micron aperture as a

binary

step-edged pattern would require a system that is _cap

able of _resolving 1000-2000 lp/nm.

Very

few systems

have these _resolving capabilities.

It was demonstrated that the holograms can be

reduced in one _step onto a high

contrast,

general photo

graphic film (Technical pan

film)

_using a good _quality

35mm camera and still yield

barely

acceptable recon

structions.

Better _quality reconstructions are obtainable

by

using a high resolution film in a one _step reduction.

Exposures should be bracketed at least at _stop inter

vals to yield the hologram best suitable for recon

struction.

Best reductions could _possibly be made

by

_using

a 2 stage reduction process. A positive intermediate

35

using a condenser illumination scheme. The intermedi

ate could be photographed with careful attention to

focus . A reduced hologram made in this manner would

theoretically

have a higher contrast ratio

and,

_perhaps,

may yield a better reconstruction.

The two-stage reduction process requires an

additional generation compared to a direct reduction.

The extra effort _may not be acceptable for a _slightly

36

CHAPTER IV

RECOMMENDATIONS

Future work in this area should perhaps investi

gate more

quantitatively

the _relationship between varia

tions in the reduction's image _quality on reconstruc

tion. The

two-step

reduction process should be more

carefully examined and optimized so that high _quality

reductions could more _easily be performed. Software

should be modified to include random phase _coding of

the image matrix to rid the system of spatial differen

37

38

****** INPUT IMAGE

MATRIX*****

rnMpr^D^?IM(S4'G4)'IWK(7)'AMP^4,G4)

COMPLEX

A ( G4, 64 ) ,F C G4 )

DO 30 1=1,G4

READ (105,

10)AIM(I,J),

_J=l,64 00010 FORMAT (64F1.0)

00030 CONTINUE

N=₆₄

M=1.5

C=.25

*****

DIVISION

_{BY SINC*****}

CALL

ASINC(AIM,M,C,N)

DO 36 1=1 ,64

DO

35,J=l,G4

A(I,

J)=AIM(I,

_J)

00035

CONTINUE

00036 CONTINUE

**#**2D FFT*****

***** ROW TRANSFORMS*****

DO 60 1=1,64

DO 40 J=l,64

F( J)=A(I, J)

00040 CONTINUE

CALL FFT2CCF,6,IWK)

DO 50 J=l,64

ACI, J)=F( J)*N

00050 CONTINUE

00060 CONTINUE

##*## COLUMN TRANSFORMS*****

DO 90 J=1,G4

DO 70 1=1,64

F(I)=A(I, J)

00070 CONTINUE

CALL FFT2C(F,6,IWK) DO 80 1=1,64

ACI, J)=F(I) 00080 CONTINUE 00090 CONTINUE *#***SCALING SEQUENCE***** APHAMAX=0 AMPMAX=0

DO 120 1=1,64

DO 110 J=l,64

AMPCI, J)=AMDSZ(A(I, J) ) AIMCI, J)=PHA(A(I, J) )

Routine for Calculation

of the Hologram

0\3110 00120 IFCAMPCI IFCAIMCI; CONTINUE CONTINUE

DO 180 1=

DO 170 J=

AMPCI, J)=

AIMCI, J) =

J) .GT.AMPMAX) AMPMAX=AMP(I, J)

J) .GT.APHAMAX) APHAMAX=AIM ( I,J )

1 ,64 1,64 AMPCI AIMCI J3/AMPMAX J3/APHAMAX 00160 00170 00180

WRITE (108, 160) AMPCI, J) ,AIM(I

FORMAT ( F 1 0. G,T20,F 1 0. 6 3 CONTINUE

CONTINUE

39

SUBROUTINE

ASINC ( U,M,C, N )

* ADDITIONAL

SUBROUTINES REQUIRED!

* _SINC

DIMENSION _U(64,G4)

DO 20 1=0,63

DO 10 J=0,63

X=C*CM+(I/N)

)

Y=SINC(X)

J)/Y IF_{(Y.LT. 0.001) UCI, J)=l} .0

IFCY.GE.O. 001) U(I , J) =U(I

00010 CONTINUE 00020 CONTINUE RETURN END FUNCTION SINCCX) PI=3. 14159

IFCX.LT. .015) GOTO 10

SINC=(SIN(X*PI) )/(X*PI)

GOTO 20

00010 SINC=1.0

00020 RETURN

END

****** THIS SUBROUTINE COMPUTES THE PHASE OF A COMPLEX

* NUMBER FUNCTION PHACX) COMPLEX X Y=REAL(X) Z=AIMAG(X) PHA=ATAN(Z/Y) RETURN END

****** THIS SUBROUTINE COMPUTES THE MODULUS OF A COMPLEX

****** NUMBER

FUNCTION AMDSZ(X)

COMPLEX X

Y=REAL(X) Z=AIMAG(X)

AMDSZ=_{( ( Y**2 )}+( Z**2 ) )**. 50

***## _THIS

C=.50

PROGRAM PLOTS 64*64 CGH******

^0

00010 00020 00030 00040 00050 00060 00100 00010 00020 00100

1 ,

1234,2111

₎

C, P,ArAINC,SCALE,CELLS )

CP, A, A IMC, SCALE, CELLS)

SCALE=1S

AINC=.005

CELLS=S4

CALL BEGPT(2,34,

DO 30 J=40,71

DO 10 1=40,71

READ(105,100)A,P

CALL RECPLOT(I,J1

CONTINUE

DO 20 1=8,39

READC105, 100)A,P

CALL RECPLOT(I,J,

CONTINUE CONTINUE

DO 60 J=8,39

DO 40 1=40,71

READ(105,100)A,P

CALL RECPLOT(I,J,

CONTINUE

DO 50 1=8,39

READ(105,100)A,P

CALL RECPLOT (I,J,C,P,A,AINC, SCALE,CELLS )

CONTINUE CONTINUE

FORMAT ( F 1 0. 6,T20,F 1 0.

CALL FINPT

END

SUBROUTINE RECPLOTd,

(I/CELLS)*SCALE

C, P, A,A INC, SCALE,CELLS )

6)

J, CP, A,AINC,SCALE, CELLS)

TO 100 Y 3) YY,2) YY+CC2) A Y=C J/CELLS)*SCALE CC=(C/CELLS)*SCALE WW=(A/CELLS)*SCALE

IF ( WW.LT. 0.005) GO

PP=(P/CELLS)*SCALE

XX=X-(WW/2)

YY=(Y+PP)-(CC/2)

CALL PLOT(XX

CALL

CALL

CALL

CALL

Y3=_YY

DO 100 M=l

Y3=Y3+AINC

IF(Y3.GT.YY+CC) GO TO 100

CALL PL0T(XX,Y3,3)

S=_M/2

T=INT(S)

IF(S.EQ.T) GOTO 10

REFERENCE LIST

Allebach,

J. P.

"Representation-Related

Errors in

Binary

Digital Holograms: A Unified Analysis."

Applied Optics

(1981),

Vol.

20,

#2,

p. 290.

Brown,

B. R. and

Lohmann,

_{A. W. "Complex Spatial Fil}

tering

with

Binary

Masks." _Applied

Optics,

Vol.

5,

#6,

p. 967.

Brown,

B. R. and

Lohmann,

_{A. W.} "Computer Generated

Binary

Holograms."

I. B.M. Journal of Research

and _Development

(

1969)

, Vol.

13,

p. 160.

IMSL

Library

1,

Volumes

I,

II and

III,

Edition 7.

Houston,

. Texas

,

July

1979.

Lee,

Wai-Hon. "Computer-Generated Holograms: Techniques

and Applications. In Emil

Wolf,

ed.,

Progress

in

Optics,

Volume XVI. Amsterdam: North-Holland

Publishing

Company,

1978.

Levi,

Leo (ed.). Applied Optics: A Guide to Optical

System

Design,

Vol . 2~. New York: John

Wiley

and

Sons,

T"98"0

.

Lohmann,

A. W. and

Paris,

D. P.

"Binary

Fraunhofer

Holograms,

Generated

by

Computer." _Applied

Optics

(1967),

Vol.

6,

#10,

p. 1739.

Microphotography

. Eastman Kodak Data Book

P-52,

1976.

Nagashima,

K. and Asakura , T. "Simple Computer-Generated

Holograms Displayed

by

an X-Y Plotter." _Optics

and Laser

Technology

(1978),

Vol.

10,

#6,

p. 310.

Stevens,

G. W. W.

Microphotography

. New York: John

Wiley

and

Sons,

1968.

Yaroslovski,

L. P. and

Merzlyakov,

N. S. Methods of

Digital

Holography

(Dave

Parsons,

Trans.

)

.

BIBLIOGRAPHY

Cathey,

W. T. Optical Information

Processing

and Holo

graphy. New York: John

Wiley

and

Sons,

1974.

Collier,

R.

, iBUrkhardty --C. _, and

Lin,

L. Optical Holo

graphy. New York: Academic

Press,

1971.

Lee,

Wai-Hon. "Effect of Film-Grain Noise on Perfor

mance of Holographic Memory."

Journal of Optical

Society

of America

(1972),

Volume

62,

#6,

p. 797.

Lohmann,

A. W. and

Paris,

D. P.

"Binary

Image Holograms.'

Journal of Optical

Society

of

America,

Vol.

56,

M,

p.

537.

Loomis,

J. S. "Computer-Generated

Holography

and Optical

Testing." _Optical

Engineering

(1980),

Vol.

19,

#5,

p. 6T9T

Meyer,

A. J. and

Hickling,

R. "Holograms Synthesized

on a Computer-Operated

Cathode-Ray

Tube." _Jour

nal of the Optical

Society

of America (196 7) _,

Vol.

57,

#11,

p. 1388.

Smith,

Howard M. Principles of Holography. New York:

John

Wiley

and

Sons,

1975.

Soroko,

G. M.

Holography

and Coherent Optics .(Albin

Trans .

)

. New York: Plenum

Press,

1980.

Waters,

J. P. "Holographic Image Synthesis

Utilizing

Theoretical Methods." _{Applied Physics Letters}