RESOLUTION LIMITS OF IN-LINE HOLOGRAPHIC ......RESOLUTION LIMITS OF IN-LINE HOLOGRAPHIC IMAGING...

RESOLUTION LIMITS OF IN-LINE HOLOGRAPHIC IMAGING SYSTEMS

by

Joseph Withers

A senior thesis submitted to the faculty of

Brigham Young University - Idaho

in partial fulfillment of the requirements for the degree of

Bachelor of Science

Department of Physics

Brigham Young University - Idaho

December 2013

BRIGHAM YOUNG UNIVERSITY - IDAHO

DEPARTMENT APPROVAL

of a senior thesis submitted by

Joseph Withers

This thesis has been reviewed by the research committee, senior thesis coor-dinator, and department chair and has been found to be satisfactory.

Date Todd Lines, Advisor

Date Evan Hansen, Senior Thesis Coordinator

Date Stephen Turcotte, Committee Member

Date Stephen McNeil, Dr.

ABSTRACT

RESOLUTION LIMITS OF IN-LINE HOLOGRAPHIC IMAGING SYSTEMS

Joseph Withers

Department of Physics

Bachelor of Science

Methods are developed and carried out to analytically and experimentally

determine the resolution limits for in-line holographic imaging systems. On-

axis resolution is limited by the lens numerical aperture, effective aperture and

effective pixel pitch. The strongest limit of these three constrains the entire

system. Experimental data shows strong agreement with these theoretical

limits. Off-axis resolution proves to be more complicated to model. It has the

same limitations as the on-axis resolution as well as limits from the detector

edge cutting off the airy disk. Thus, resolution at the edges will be less than

at the center. This degradation in resolution at the edges, increases with

reconstruction distance. With its fast recording and processing times and

three-dimensional ability, digital holography has become a popular method for

particle imaging. Understanding the resolution limits will enable scientists and

engineers to successfully design and build future holographic imaging systems.

ACKNOWLEDGMENTS

I would like to thank my mentor, Dr. Raymond Shaw, and those in the

Cloud Lab at Michigan Technological University, for their guidance and en-

couragement. I also thank the faculty at Brigham Young University-Idaho for

their helpful feedback throughout the semester. I give special thanks to my

wife, Meghan, for her continual patience and support.

Contents

Table of Contents vii

List of Figures ix

1 Introduction 11.1 Cloud Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Holographic Particle Imaging . . . . . . . . . . . . . . . . . . . . . . 11.3 In-line Digital Holography . . . . . . . . . . . . . . . . . . . . . . . . 21.4 The HOLODEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Resolution Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Theoretical Methods 52.1 Resolution Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Experimental Methods 113.1 Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Experimental Results 19

5 Conclusions 25

6 Future Research 276.1 Contrast Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . 276.2 Signal to noise ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.3 Roll-off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Bibliography 33

A 1951 USAF Test Chart Table 35

B CTF Calculator Manual 37

Index 43

vii

List of Figures

1.1 In-line holography setup . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Holodec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Numerical aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Effective aperture diagram . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Predicted resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.1 Holodec on optical bench . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Space between benches . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Holodec arms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4 Ruler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.5 Ruler and stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.6 Experiment setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.7 Reconstructed sample volume . . . . . . . . . . . . . . . . . . . . . . 163.8 USAF test chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.9 Coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.10 Test chart wheel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.1 Hologram compare . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Smallest element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.3 Predicted resolution with data . . . . . . . . . . . . . . . . . . . . . . 214.4 X axis resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.5 Y axis resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.6 Diagonal axis resolution . . . . . . . . . . . . . . . . . . . . . . . . . 24

6.1 Square wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.2 Gaussian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.3 Square wave with gaussian . . . . . . . . . . . . . . . . . . . . . . . . 286.4 Intensity with contrast . . . . . . . . . . . . . . . . . . . . . . . . . . 296.5 Contrast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.6 Intensity one . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.7 Intensity thirteen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

A.1 Resoltuion test chart table . . . . . . . . . . . . . . . . . . . . . . . . 35

ix

x LIST OF FIGURES

B.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38B.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38B.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39B.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39B.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39B.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40B.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40B.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41B.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Chapter 1

Introduction

1.1 Cloud Physics

Clouds in the atmosphere serve as a first order boundary between the earth and

the sun, which greatly affects climate and weather. The optical properties of clouds

cause the suns light to be reflected. These optical properties are dependent upon the

droplets from which the cloud is formed of. Cloud particle size and thermodynamic

phase affect simplified cloud and climate models. Knowing more about the number

density and size distributions of droplets in various clouds types enhances our un-

derstanding of the atmosphere and perhaps make our climate models more reliable.

Particle imaging provides a way to measure both of these features.

1.2 Holographic Particle Imaging

With its fast recording and processing times and three-dimensional ability, digital

holography has become a popular method for particle imaging [6]. Three dimensional

length scales between particles can be visualized. Particle image velocimetry (PIV),

1

2 Chapter 1 Introduction

which requires the recording of many holograms, is now quite feasible with digital

processing. Thus holography is a strong candidate for imaging in situ cloud droplets.

1.3 In-line Digital Holography

While there are variations to holographic setups, e.g., reflection and transmission, the

basic principles are the same. An interference pattern, which contains information

about the phase and amplitude, is recorded. In figure 1.1, an in-line holography setup

is shown. A coherent laser source is expanded and collimated to produce plane waves.

The waves impinge upon the object, which scatters some of their light. The scattered

waves, or object beam, interact with the reference beam and create an interference

pattern that can be recorded.

Figure 1.1 An in-line holography setup is shown. The beam is expanded andcollimating to create plane waves. Object beam and reference beam interferewith each other and a pattern is recorded containing phase and amplitudeinformation.

Originally, interference patterns were recorded by photographic plates. Holograms

were then reconstructed by illuminating the recorded hologram with a coherent source.

As the wave interacted with the pattern it would interfere and form an image of the

object. The advances in computer technology in the past twenty years have en-

abled holographic particle imaging to go from film to digital recording [4]. Digital

holograms, recorded onto a charge-coupled device (CCD), can be numerically recon-

1.4 The HOLODEC 3

structed using Fourier transforms [1].

1.4 The HOLODEC

The Holographic Detector for Clouds (HOLODEC) is an in-line holographic instru-

ment that images cloud droplets and ice crystals. Cloud water droplet sizes range from

5 to 30 µm and ice crystals range from 50 to 2000 µm. The HOLODEC was designed

to image a range of particle types, e.g., liquid and ice; the smallest particle diameter

∼5 µm and a sample volume of ∼20 cm3. In order to know how small of particles

can be detected by the optics in the HOLODEC and correctly analyze data obtained

from it, a confident understanding of its resolution limits must be ascertained.

Figure 1.2 HOLODEC attached to wing of a C-130. HOLODEC is on rightwith the black tips.

Previously, the resolution of the HOLODEC was determined through a process of

recording holograms of a resolution test chart [5]. An algorithm was used to flag pixels

above a threshold intensity. The flagged pixels were set to a maximum intensity, while

all others were set to zero. By inspection, the smallest resolvable bar pattern on the

test target was identified, and the resolution limit was determined by looking up the

bar patterns width on a data table. While this method does provide a quantitative

measure of resolution, it does not tell us anything about the actual width of the

4 Chapter 1 Introduction

bar pattern compared with the detected width of the object being imaged. As the

resolvable limit drops below the size of an object, the image will begin to lose pixels.

Consequently, the object width detected is smaller than actual size. This resolution

test shows that the HOLODEC can detect objects as small as 6 µm at a distance

of 13 cm, it does not indicate the detected width. A resolution test that determines

the smallest resolvable bar width at its true size in the presence of noise will give an

accurate and operational resolution limit.

1.5 Resolution Limits

Digital holography comes with some strong resolution limits that must be taken into

account when designing imaging systems, namely pixel size, lens numerical aperture,

and reconstruction distance. The CCD introduces a finite pixel size that sets a limit

on particle imaging and sizing. As in all optical setups, the system is only as good

as the lens, thus the numerical aperture of the lens sets another limit. An interesting

feature of in-line holography is the ability to reconstruct down the z axis, the axis

going from the center of the lens on the camera side to the center of the lens on

the laser side (see figure 3.3). The greater the reconstruction distance, the smaller

the opening angle (θrec in figure 2.2) from the particle to the lens. Thus an effective

aperture is manifested, which further constrains the imaging system [3]. The strongest

resolution limit of these three constrains the entire system.

Chapter 2

Theoretical Methods

2.1 Resolution Derivations

As previously noted, assuming all other aberrations are negligible, the resolution of

a holographic imaging system is limited by three factors: the numerical aperture

NAlens, imposed by the lens itself (which is a property of the lens design), the reso-

lution limit from diffraction and reconstruction aspects of in-line holography NArec,

and the effective pixel size Dpixel,rec. Each of these limits are analytically determined.

Calculating the resolution of a diffraction limited system begins with the Airy

disk, or Poisson’s spot. In Fresnel diffraction, constructive interference occurs at the

center of the interference pattern from a circular aperture. At a minimum, the airy

disk needs to be detected. The Rayleigh criterion says the smallest resolvable angle

in a diffraction limited system is

sin θ ≈ 1.22λ

D, (2.1)

where D is the diameter of a circular aperture, λ is the wavelength, 1.22 comes

from the first zero of the first order Bessel function, and θ is the angular resolution.

Recall that the numerical aperture, which is a measure of the acceptance angle of a

5

6 Chapter 2 Theoretical Methods

lens, is given by

NA = n sin θ, (2.2)

where θ is half of the opening angle. Since the medium is air, n ≈ 1, thus

NA ≈ sin θ, (2.3)

and

NA ≈ 1.22λ

D. (2.4)

Solving for D we get

D ≈ 1.22λ

NA. (2.5)

This equation determines the smallest resolvable diameter as a function of the

wavelength and the lens numerical aperture [3]. The HOLODEC has a numerical

aperture of 0.087 and a magnification of 2.5. Numerical apertures typically range

from 0.07 for low-power lenses to 1.4 or so for high-power (100X) ones [2].

Figure 2.1 Numerical aperture for a lens is a function of the half angle fromthe from the focal length.

The lens numerical aperture is defined from the focal length (see Figure 2.1). In

holography, objects can be imaged from a range of distances from the lens. Imaging

particles at varying distances from the lens imposes an effective aperture, NArec, due

2.1 Resolution Derivations 7

to the varying opening angle. For far field diffraction

sin θ ≈ θ, (2.6)

and thus

NArec ≈ θrec. (2.7)

θrec arises from a geometrical derivation, and is

Figure 2.2 Deff , shown in the figure, arises from the size of the cameradetector and magnification of the system. The half angle from the axis tothe edge of the detector is a further constraint from the numerical apertureof the lens. The wider angle in the figure is for the numerical aperture. Thesmaller angle is for the effective aperture. As particles are imaged at variousdistances from the camera in the sample volume, the opening angle, θrec,changes. And thus the effective aperture changes.

θrec = arctanDeff/2

z. (2.8)

The series expansion for the inverse tangent is

arctanx = x− x3

3+

x5

5− x7

7+ ... (2.9)

Because z >> D/2, a first order approximation is sufficient.

(2.10)


and thus

NArec ≈Deff

2zrec. (2.11)

Deff is the geometrical mean of the diameter of the effective sensor size. The

effective sensor size is obtained by dividing the actual dimensions of the sensor by the

magnification. For the HOLODEC

Deff =√9.66mm× 14.4mm. (2.12)

Magnification effectively makes the pixels smaller in order to make the object

larger. According to the Nyquist sampling theorem, at least two pixels are needed to

resolve the smallest feature. This means that the pixel size on the object side must be

at most half of the optical resolution to avoid further constraining the resolution [3].

The limit from the effective pixel size is found by dividing the pixel size on the object

side by the Magnification multiplied by a factor of 2:

Dres,pixel =2Dpixel,obj

M. (2.13)

The pixel pitch on the object side for the HOLODEC II is 7.4 µm, thus Dres,pixel =

2.96 µm.

A plot of these resolution constraints on the HOLODEC is shown in figure 2.3.

The resolution is based on objects being on the z axis. The resolution of the entire

system will be limited to the greatest resolution constraint. At distances z < 80 mm,

the resolution is limited to Dres,pixel. At z > 80 mm, the resolution is limited to

Dres,rec, which is linear in z.

2.1 Resolution Derivations 9

Figure 2.3 Plot of predicted resolution limits as a function of reconstructiondistance z. Limits from the numerical aperture Dres,lens and the pixel sizeDres,pixel are constant in z and constrain the system up to z = 80 mm. Theeffective aperture from reconstruction linearly limits the system in z and isthe stronger constraint past z = 80 mm.

Chapter 3

Experimental Methods

3.1 Alignment

Reconstructing many holograms can be quite tedious. This is especially true for large

sample volumes, so knowing how far to reconstruct saves time. An alignment method

was therefore created. A three-dimensional coordinate system is used between the

HOLODEC arms. As previously noted, the z axis is the line from the camera lens to

the laser lens, and z = 0 at the camera lens outer surface. The plane perpendicular

to the z axis is the x, y plane. Due to the sizes of optical benches that were available,

the HOLODEC and the optical mount are on two different optical benches. This

presents a complication in ensuring proper alignment. There is a slider between the

HOLODEC arms that needs to be parallel to the z axis of our coordinate system.

The slider is fixed to the optical bench not containing the HOLODEC. Thus the

optical bench needs to be aligned with the HOLODEC. The HOLODEC is supported

by a mount that is bolted to the optical bench. It was assumed the precision of this

mount was sufficient for alignment purposes and that the HOLODEC is therefore

in-line with the optical bench. Thus in theory, only the optical benches need to be

11

12 Chapter 3 Experimental Methods

aligned with each other, and then the HOLODEC and optical mount will be aligned.

Once the optical benches were aligned, further tuning was accomplished by recording

holograms at each lens, reconstructing, and then using holoViewer to locate the object

positions on the x, y plane. Once the object has the same x, y position at both lenses,

the mount and slider are aligned.

Figure 3.1 HOLODEC shown mounted to optical bench.

Once the mount and HOLODEC are aligned there needs to be a way to know

approximately where the object is along the z axis. One way to do this is to simply

measure it. This is tedious to do every time, so a systematic way was created. An

object was placed 5 cm from the lens on the camera side. This distance was measured

with a ruler and a caliper.

Once the position of the object was determined with confidence, a ruler was at-

tached to the optical bench next to the slider (slider and mount shown in figure 3.5).

Then a piece of thin polyimide tubing was attached to the mount so that it would

move with the mount along the z axis and mark the z position.

Once the hologram has been reconstructed to the estimated position based on the

system above, the hologram can be fine-tuned for better focus.

The off-axis, meaning off the z axis, or on the x, y plane perpendicular to the z

axis, alignment of the object is based on positioning noted on axes in holoViewer (a

3.2 Data Acquisition 13

Figure 3.2 Alignment between optical benches aligns HOLODEC andmount. Dual stage mount is shown on slider between HOLODEC arms.

hologram reconstruction software).

3.2 Data Acquisition

Reconstructed sample volumes taken by the HOLODEC show a trend of decreasing

detected number of particles near the outer edges, especially the farther down the z

axis they are. This is particularly true of the corners as shown in figure 3.7

It is assumed that the decrease in detected particle count is due to a decrease in

resolution, i.e., the actual particle density does not decrease. There is an uncertainty

that what is being detected are actual particles and what their sizes are. The laser

intensity is greatest in the center, or on the z axis, of the sample volume, and drops

radially outward from it. The resolution may be lower from this decreasing intensity

of light hitting the CCD at the outer areas. Particles on the outer edges of the sample


Figure 3.3 The z axis shown between the Arms of HOLODEC. The laserand camera are in the left and right arms, respectively.

Figure 3.4 Ruler used to calibrate z axis measurement. Object is measured5 cm from lens on camera side.

volume will have parts of their diffraction rings cut off. Theoretically this would lower

resolution at the edges. This would be especially true as particles are imaged farther

from the camera lens as their diffraction patterns would be larger.

Resolution data was obtained using a 1951 USAF resolution test chart. The USAF

chart contains bar patterns organized by size into groups and elements. This test chart

is ideal due to its range of bar widths, from 2.19-125 µm. The resolution test chart

was placed in the sample volume of the HOLODEC along the z axis and individual

holograms were recorded at z = 10, 20, 30,130 µm. To have good representative data,

this process was repeated five times.

Each of the holograms were then reconstructed to the object distance, where


Figure 3.5 The slider is what is attached to the optical bench. The dualstage mount is on top of the slider. A is attached to the optical bench toprovide an approximate reference for the object. Polyimide tubing is used asa marker to indicate the approximate z distance of the object in the hologram.

the image had the optimal focus. The resolution of a given image was determined by

identifying the smallest complete bar pattern. The vertical and horizontal components

needed to be resolvable and complete. Each of the five resolution values for a given

reconstruction distance were averaged.

The resolution perpedicular to the z axis can be obtained using the same 1951

USAF resolution test chart. Holograms of the test chart were recorded at the same z

distances aforementioned. For each z distance, roughly 20 holograms were recorded

along the x axis, 30 along the y axis, and 20-30 radially outward following the diagonal

line in figure 3.9.

An optical mount was built to hold the test chart. The optical mount is on a

slide, which can move laterally (along z) the entire length of the sample volume. The

mount itself has two stages allowing for more accurate lateral movement along z as

well as perpendicular movement (along x). The final piece to the mount is a variable

vertical post which allows for accurate vertical adjustments.


Figure 3.6 These images show the HOLODEC, optical benches, and opticalmount configuration.

Figure 3.7 Reconstructed sample volume from HOLODEC. Particle numberdensity decreases at the edges. This decreases appears to be stronger at largerz.


Figure 3.8 1951 USAF Resolution Test Chart. Bar pattern are organizedinto groups and elements. Groups are in columns and individual elements ina group are in the rows. Each trio of bars is an element [7].

Figure 3.9 This diagram shows a side view of the leading edge of theHOLODEC. Coordinate system perpendicular to the z axis is shown. The zaxis goes into the page. Holograms were recorded along the x, y and diagonalaxes.


Figure 3.10 1951 USAF resolution test chart wheel that was used.

Chapter 4

Experimental Results

Once the necessary holograms were recorded along the z axis and radially outward

along the x and y axes, they were reconstructed to determine the resolution. A

MATLAB program was used to view and reconstruct the holograms. The code,

HOLOSUITE [1], utilizes a user interface that allows one to pick a desired hologram

and reconstruction distance. The appropriate reconstruction distance is found when

the objects in the hologram are at the greatest focus. Selecting the distance of best

focus is up to the viewers discretion. Objects will generally have the smallest size and

sharpest edges at their best focus.

Figure 4.1 On the left is a raw unconstructed hologram. On the right isthe same hologram reconstructed to the distance the object appears in focus,i.e., the object distance.

19

20 Chapter 4 Experimental Results

Once the hologram has been reconstructed to the desired distance, the resolution

is found from the USAF resolution chart [3]. The smallest complete bar pattern

in the chart gives the smallest resolvable diameter. In order for a bar pattern to be

complete, the horizontal and vertical bars must be relatively whole. The actual width

of bars is found by looking up the corresponding group and element numbers on a

table (see appendix A.1). Figure 4.2 shows a circle around the smallest complete bar

pattern. Its group and element numbers are 5 and 4 respectively. The width of this

element is 11.04 µm.

Figure 4.2 Smallest complete element is circled.

This process was repeated for each recorded hologram. Figure 4.3 shows a plot

of the predicted resolutions as well as the USAF test chart resolution data along the

z axis. Each resolution data point is taken from an average of 5 separate holograms

each recorded at the same distance.

The experimental data shows agreement with the predicted resolutions. That is,

the resolution is limited by the numerical aperture of the lens, the effective aperture

introduced from reconstruction and the effective pixel size. The data also agrees with

the most limiting factor being the limit of the entire system.

Figures 4.5, 4.6 and 4.7 show the results of radial resolution. These plots show

the resolution limit versus distance from the hologram center for a given z. Figure

4.5 shows resolution along the x axis. Data for z = 1, 7 and 13 cm are shown. As

21

Figure 4.3 Plot of predicted resolution limits with USAF resolution testchart data along the z axis. Each data point represents an average takenfrom 5 separate holograms at that reconstruction distance. Experimentaldata shows agreement with predicted resolution.

expected, the resolution decreases with reconstruction distance. The first 2500 µm

from the center shows relatively constant resolution. This likely because the pixel

limit, which is constant, is the strongest resolution limit in this region. The closer to

the edge of the detector the object is, the closer the airy disk will be as well. When

the airy disk starts to get cut off by the edge of the detector resolution decreases. On

the x axis, for z = 1 cm, the cutoff appears to begin around 4000 µm from the center,

or 500 µm from the edge. As the object moves farther from the detector, its airy

disk will be larger. This should cause the resolution to degrade sooner, i.e., closer to

the center. The larger airy disk will reach the edge sooner. On the x axis plot, for

z = 13 cm, the resolution starts degrading just after 2000 µm from the center. The

plot for the x axis shows the decrease in resolution happens closer to the center of

the hologram at greater reconstruction distances.

Figures 4.6 and 4.7, resolution plots for the y and diagonal axes, show a similar


trend as observed in figure 4.5 for the x axis. Resoultion decreases radially outward,

and with greater z.

Figure 4.4 Plot of resolution versus distance from the center of the hologramalong the x axis. Resolution decreases with distance from center. The furtherin z the test charts were, the faster the resolution decreased.

23

Figure 4.5 Plot of resolution versus distance from the center of the hologramalong the y axis. Resolution decreases with distance from center. The furtherin z the test charts were, the faster the resolution decreased.


Figure 4.6 Plot of resolution versus distance from the center of the hologramalong the diagonal axis. Resolution decreases with distance from center. Thefurther in z the test charts were, the faster the resolution decreased.

Chapter 5

Conclusions

Methods were developed to analytically and experimentally establish the resolution

limits of the HOLODEC. On-axis resolution is limited by the lens numerical aperture,

the effective aperture from reconstruction, and the effective pixel pitch. Experimental

data from the HOLODEC shows strong agreement with these limits. These three

limits must be taken into account in the design of in-line holographic imaging systems.

Off-axis resolution of the HOLODEC proves to be more complicated. It has the

same limitations as the on-axis resolution as well as limits from the detector edge

cutting off the airy disk. Due to limits at the edges, the HOLODEC, or any other

in-line holographic system, will have less resolution at the edges than at the center.

This degradation in resolution at the edges increases the further in z (further from the

camera side) the objects are, thus the cone shaped profile of particle number density

observed in figure 3.7.

The attained resolution data will serve as a guide when analyzing in situ particle

data. Effectively determining the resolution along the z axis allows a way to com-

pare experimental to predicted resolution. Experimental agreement with predicted

resolution demonstrates the utility of the three aforementioned resolution limits.

25

26 Chapter 5 Conclusions

Chapter 6

Future Research

6.1 Contrast Transfer Function

A common method for determining the resolution of an optical system is to measure

its contrast transfer function. As the light from an object passes through a lens, the

edges of the image are blurred. As an example, take the initial intensity profile of a

single bar from the 1951 USAF resolution test chart to be figure 6.1.

Figure 6.1 Initial intensity profile of a single bar form a 1951 USAF resolu-tion test chart

After the light from the bar passes through a lens the intensity profile looks like

figure 6.2.

27

28 Chapter 6 Future Research

Figure 6.2 Intensity profile of a single bar after passing through a lens.

If the before and after profiles are superimposed the result would be something

similar to figure 6.3.

Figure 6.3 . Initial and final intensity profiles are superimposed.

The contrast, or “roll-of” of the image is defined as

Contrast =max−min

max+min, (6.1)

where the max and min values are the upper and lower bounds of the intensity.

Contrast, or the depth of the intensity well, is defined as the ratio of the intensity

amplitude over the level of bias. A useful way to use the contrast is to plot the

contrast as a function of the number of lines per unit length. It is expected that

the contrast would decrease as the number of line pairs per unit length increases.

Contrast as a function of frequency is called the contrast transfer function (CTF). A

common measure of the performance of an optics system is to plot the normalized

6.1 Contrast Transfer Function 29

CTF verse frequency. To normalize it you simply divide the equation by the largest

value of contrast.

Figure 6.4 shows an example of the intensity, contrast and CTF for a bar pattern

of increasing frequency.

Figure 6.4 Intensity, contrast and CTF for a bar pattern of increasing fre-quency [8].

In the plot in figure 6.4, it can be seen that the contrast decreases with higher fre-

quencies, or smaller objects. The resolution is determined by identifying the frequency

associated with the intersection of the CTF curve and the average noise contrast.

This method of determining the resolution would be a viable and useful for the

HOLODEC. The CTF method does not have the user bias that the method outlined

in this paper has; however, there are some difficulties. It turns out the HOLODEC’s

laser profile is not uniform and the CTF method depends on a uniform background.

As the contrast is a function of the minimum and maximum intensities, a varying

background will alter the contrast. Figure 6.5 shows 2 CTF curves for the HOLODEC,

at a reconstruction distance of 1 cm and 13 cm. Figure 6.6 and 6.7 shows the intensity

pattern of the bar pattern at the two reconstruction distances.


Figure 6.5 Two CTF curves shown for the HOLODEC at z = 1 cm andz = 13 cm. CTF curves are normalized and the units of the x axis are inµm.

Figure 6.6 Bar pattern intensity at z = 1 cm.

Observing the intensity pattern, one can readily see the effects of a uninform

background. In order for the CTF method to be used on the HOLODEC, the laser

profile must be made uniform, or a way to compensate for the uneven background is

needed.

6.2 Signal to noise ratio

Another method to determine the resolution is to calculate the signal to noise ratio.

Similar to the contrast yransfer function, the signal to noise ratio reflects the contrast

between the signal and noise of the imaging system.

The noise can be calculated by averaging the intensity of the hologram at the

current reconstruction distance. A ratio would then be taken between the intensity

6.3 Roll-off 31

Figure 6.7 Bar pattern intensity at z = 13 cm.

of the bar pattern and the averaged noise. The greater the ratio the greater the

contrast.

6.3 Roll-off

The optics and electronics of an imaging system transfer the sharp edges of a square

signal into a “roll-off” as seen in figure 6.3. For large objects, i.e., greater than 40

µm, the “roll-of” accounts for less than 5% of their width. For small objects, i.e., less

than 10 µm, which is typical for cloud particles, this “roll-off” accounts for significant

portion of their width. It becomes difficult to accurately determine the sizes of small

objects. Researching a method to quantify the size of an object in the presence of a

“roll-of” would improve cloud particle sizing.

Bibliography

[1] Fugal, Jacob P., Timothy J. Schulz, and Raymond A. Shaw. “Practical Methods

for Automated Reconstruction and Characterization of Particles in Digital In-line

Holograms,” Meas. Sci. Technol., Vol. 20 (No. 7), 2009.

[2] Hecht, E., Optics. (Addison-Wesley, 2002), p. 100000

[3] Henneberger, J., Fugal, J. P., Stetzer, O., and Lohmann, U. “HOLIMO II: a

digital holographic instrument for ground-based in situ observations of micro-

physical properties of mixed-phase clouds,” Meas. Sci. Technol. 6, 2975-2987,

2013.

[4] Meng, H., Pan, G., Pu, Y., and Woodward, S. “Holographic particle image

velocimetry: from film to digital recording,” Atmos. Meas. Tech. Vol. 15 (No.

4), 2004.

[5] Spuler, S., Fugal, J., “Design of an in-line, digital holographic imaging system

for airborne measurement of clouds,” Appl. Opt. 50, 1405-1412 (2011).

[6] Yang, W., Kostinski, A., Shaw, R., “Phase signature for particle detection with

digital in-line holography,” Opt. Lett. 31, 1399-1401 (2006).

[7] 1951 USAF resolution test chart image.

http://www.thorlabs.com/images/TabImages/R1DS1N closeup 780px.jpg.

33

34 BIBLIOGRAPHY

[8] Contrast transfer function curves.

http://spie.org/Images/Graphics/Publications/TT52 P9-12 f1.11.jpg.

Appendix A

1951 USAF Test Chart Table

Figure A.1 Table shows width of bar patterns from the 1951 USAF resolu-tion test chart. Bar widths are in µm.

35

36 Chapter A 1951 USAF Test Chart Table

Appendix B

CTF Calculator Manual

A 1951 USAF resolution test chart is typically used in calculating the contrast transfer

function (CTF) of an electro-optical imaging system. The purpose of this manual is

to instruct the user on how the ctf.m code works and how to use it in MATLAB.

1. Place the image file and ctf.m code in the current folder in MATLAB.

2. Call function from command window. The inputs for the function are the

filename and the association name of the variables the function will save in the

workspace. Generally the name of the variable association name is one which

ties the variables to the image. If you are working with the image testchart.png

and you want the variables association name to be testchart, you would type

>> ctf(testchart.png, testchart)

Once the function executes and the figures open it will look like figure B.1.

3. The red box defines what pixels are being plotted and averaged. Plot 1 in figure

B.1 shows the intensity averaged over the columns within the red box. Plot 2

shows the intensity averaged over the rows within the red box. The red box

37

38 Chapter B CTF Calculator Manual

Figure B.1 Plot 1 is the top right, and plot 2 is just below it.

will be set at some default location and will need to be moved to the desired

location using the sliders and/or edit text boxes shown in figure B.2.

Figure B.2

There are two ways to make sure the red box is in the correct location: 1)

look at the image, and 2), look at the intensity plots. If you want to analyze a

vertical bar pattern you will want the left and right edges to be just beyond the

outer edge of the left and right hand bars. You will want the top and bottom

edges of the red box to be a pixel or two inside the bar pattern. For looking

at horizontal bar patterns, its just the opposite. You want the top and bottom

just outside the upper and lower bar edges and the left and right edges just

inside the left and right edge of the bar pattern.

4. Once you have the red box in the desired place, you will select the group and

39

Figure B.3

element number of that bar pattern.

Figure B.4

An outline of the actual bar widths will appear in the intensity plots as in figure

B.5.

Figure B.5

5. The data points within the dashed bars will be averaged to find the maximum

intensity, Imax, in the contrast equation. The data points in between the dashed

bars will be averaged to find the Imin. Sometimes there will be data points on

the boundaries that may be undesirable to include in the calculations, since

its value will go into the average. A way around this is to shrink the dashed

lines using the line cut edit box. Each value entered represents how many data

points on each end of each bar is cutoff. The dashed bar pattern may need to


be shifted to the left or right. Use the vertical line shift slider and edit box,

shown in figure B.2, if you are using the vertical intensity plot, and horizontal

if otherwise.

Figure B.6

Select the desired number of data points to cut out. Although it will not appear

on the plot, the data points between the dashed lines will be cut on each side

by the amount the user specified.

6. Once you know the correct data points will be used for the CTF calculation you

need to add the vertical or horizontal vector to the CTF vector. The vertical

and horizontal vectors are are the data points shown in plots 1 and 2 in figure

B.8. The CTF vector is the vector containing the contrast and corresponding

frequency data points. This button will cause the contrast to be calculated

based off the averages of the low and high data points in the plot. It will then

store this data with the corresponding frequency from the group and element

number.

Figure B.7

When you add the vector, the portion you added will show up in the third plot

41

of figure two. This plot builds as the user adds more data.

Figure B.8

7. Once the contrast has been calculated for the given region in the red box, the

user must move the red box as noted in step 3 to the next bar pattern. And

the steps 4-6 are repeated.

8. When the user has stored all the contrast data they want, they must click

“Calculate CTF.” This will plot the CTF verses line frequency as well as store

the contrast, minimums, maxes, and frequencies in a matrix in the workspace.

CTF plot shown in figure B.9.


Figure B.9

Index

Index1951 USAF resolution test chart, 14Airy disk, 5Alignment, 11Climate models, 1Contrast transfer fuction, 27Contrast transfer fuction, calculation,

28Contrast transfer fuction, calculator

manual, 37Coordinate system, z axis, 14Coordinate system, off-axis, 17Data aquisition, 13Effective aperture, 7HOLODEC, 3In-line digital holography, 2Magnification of the HOLODEC, 6Numerical aperture, 5, 6Particle imaging velocimetry, 1Pixel resolution, 8Rayleigh criterion, 5Reconstruction code, 12, 19Resolution derivations, 5Resolution, radial, 15

43

RESOLUTION LIMITS OF IN-LINE HOLOGRAPHIC ......RESOLUTION LIMITS OF IN-LINE HOLOGRAPHIC IMAGING...

Documents

Transcript of RESOLUTION LIMITS OF IN-LINE HOLOGRAPHIC ......RESOLUTION LIMITS OF IN-LINE HOLOGRAPHIC IMAGING...