, NATL INST. OF STAND & TECH

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^'"ffAU O* * NBS TECHNICAL NOTE 1084

U.S. DEPARTMENT OF COMMERCE / National Bureau of Standards
7he National Bureau of Standards' was established by an act of Congress on March 3, 1901. The

Bureau's overall goal is to strengthen and advance the nation's science and technology and facilitate their effective application for public benefit. To this end, the Bureau conducts research and provides: (1) a

basis for the nation's physical measurement system, (2) scientific and technological services for industry and

government, (3) a technical basis for equity in trade, and (4) technical services to promote public safety.

The Bureau's technical work is performed by the National Measurement Laboratory, the National

Engineering Laboratory, the Institute for Computer Sciences and Technology, and the Center for Materials

Science.

The National Measurement Laboratory

Provides the national systern of physical and chemical measurement;

coordinates the system with measurement systems of other nations and

furnishes essentia] services leading to accurate and uniform physical and

chemical measurement throughout the Nation's scientific community, in- dustry, and commerce; provides advisory and research services to other Government agencies; conducts physical and fhemical research; develops,

produces, and distributes Standard Reference 'Materials; and provides

calibration services. The Laboratory consists of the following centers:

• Basic Standards^

• Radiation Research

• Chemical Physics

• Analytical Chemistry

The National Engineering Laboratory

Provides technology and technical services to the public and private sectors to address national needs and to solve national problems; conducts research in engineering and applied science in support of these efforts; builds and maintains competence in the necessary disciplines required to carry out this research and technical service; develops engineering data and measurement

capabilities; provides engineering measurement traceability services; develops

test methods and proposes engineering standards and code changes; develops

and proposes new engineering practices; and develops and improves

mechanisms to transfer results of its research to the ultimate user. The

Laboratory consists of the following centers:

Applied Mathematics

Electronics and Electrical Engineering^

Manufacturing Engineering

Building Technology

Fire Research

Chemical Engineering'

The Institute for Computer Sciences and Technology

Conducts research and provides scientific and technical services to aid Federal agencies in the selection, acquisition, application, and use of computer technology to improve effectiveness and economy in Government

operations in accordance with Public Law 89-306 (40 U.S.C. 759), relevant Executive Orders, and other directives; carries out this mission by managing

the Federal Information Processing Standards Program, developing Federal ADP standards guidelines, and managing Federal participation in ADP

voluntary standardization activities; provides scientific and technological advisory services and assistance to Federal agencies; and provides the technical foundation for computer-related policies of the Federal Government. The Institute consists of the following centers:

• Programming Science and

Technology

• Computer Systems

Engineering

The Center for Materials Science

Conducts research and provides measurements, data, standards, reference

materials, quantitative understanding and other technical information fundamental to the processing, structure, properties and performance of materials; addresses the scientific basis for new advanced materials technologies; plans

research around cross-country scientific themes such as nondestructive

evaluation and phase diagram development; oversees Bureau-wide technical programs in nuclear reactor radiation research and nondestructive evaluation; and broadly disseminates generic technical information resulting from

its programs. The Center consists of the following Divisions:

• Inorganic Materials

• Fracture and Deformation

• Polymers

• Metallurgy

• Reactor Radiation

'Headquarters and [laboratories at Gaithersburg, MD, unless otherwise noted; mailing address Gaithersburg, MD 20899. ^Some divisions within the center are located at Boulder, CO 80303.
NBS

RESEARCH

INFORMATION CEMTER

Direct Measurement of the Electric Field ^lo^u'

of a Laser Pulse—Theory mj>

Eric G. Johnson, Jr.

Electromagnetic Technology Division

Center for Electronics and Electrical Engineering

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued August 1 985
National Bureau of Standards Technical Note 1084

Natl. Bur. Stand. (U.S.), Tech Note 1084, 60 pages (Aug. 1985)

CODEN:NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1985

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402
CONTENTS

Page

1. Introduction 1 2. Why Know the Details of a Laser Pulse? 4

3. Analytical Model Using Hermite Modes to Illustrate the

Basic Ideas of an Electric Field Measuring Apparatus 7 3.1 Overview 7 3.2 Beam Matching Telescopes 8

3.3 Optical Filter 11

3.4 Fourier Transform Lenses 12

3.5 Fourier Plane--What Is Present There? 14

3.6 Effects of Beam Clipping and Optical Axis Offset

between Fiber and Cross-Correlation Beam 20

3.7 Design of a High Precision Positioning Device to Center

the Optical Axis of Each Fiber to the Optical Axis of

the Cross-Correlation Process 23

3.8 The Mode Attenuator--The Tapered Optical Fiber 27

3.9 The Polarization Analysis Process--A Calcite Crystal 30

3.10 Detector Assembly 33

4. Realization of Nonlinear Filter Using Abstract Mathematics 34

4.1 Overview 34

4.2 Exact Equation Incorporating the Constraints of Fabrication 34

4.3 Evaluation of the dc Part 39

4.4 Evaluation of the Cell Positioning Error 39

4.5 Evaluation of the Antisymmetric Nonlinear Part 40

4.6 Evaluation of the Antisymmetric Quasilinear Part 43

4.7 Two Fabrication Strategies--Which One is Better? 47

4.8 Necessary Work to Realize the Apparatus 48

5. Conclusion 48

6. References 50

m

Direct Measurement of the Electric Field of a Laser Pulse--Theory

Eric G. Johnson, Jr.

Electromagnetic Technology Division

National Bureau of Standards

Boulder, Colorado 80303

We consider realizing an electric field measuring apparatus by

using optical processing, tapered optical fibers, and a pair of

detectors at the end of each optical fiber. Using an appropriate

computer-generated optical filter, we show it is possible to discriminate among a set of orthonormal modes used to represent the

spatial features of the electric field with a signal-to-noise ratio

of at least 100 to 1. If the positioning accuracies for various

parts of the apparatus are properly set up, it is expected that the

signal-to-noise ratio could be about 1000. The purpose of the

tapered and graded-index fiber is to select the fundamental propagating mode in a fiber and to attenuate the other modes. The

existence of this fiber allows the precise determination of the

strength of each of the orthonormal modes being used as the spatial

basis of the electric field before the optical processing. The

detectors then measure the strength of each fundamental mode.

There are 36 such modes for the apparatus under discussion--six for

the X dimension and six for the y dimension. Propagation of the

laser pulse is assumed to be along the z axis. We use two detectors for each mode so that the polarization is determined.

This paper presents the conflicts in the design and gives a solution. The complete evaluation requires assembly of the proposed apparatus to assess final accuracy.

Key words: beam profile measurements; cross-correlation; electric

field measurement; graded-index fibers; laser pulses; matched filters; mode analysis; mode filtering; numerical analysis; optical

fibers; optical filter; singular value decomposition; total beam

control

1. Introduction

We at NBS [1-4] have been developing an electric field measuring system

to measure the time and spatial features of a laser pulse with at least 1.0

percent accuracy. If certain critical elements can be set up, we expect the
accuracy to be 0.1 percent. We have found a number of conflicting constraints

on the design of the equipment. We address these constraints in this paper.

First, we show why the optical processing is desirable for proper electric

field measurements. Second, using simplified analytical models, we illustrate

the technical conflicts that result from using optical processing. They are:

(1) mode mixing due to clipping, various offsets in the optical axis, and

various mode mismatches at several locations; (2) mode mixing due to the

Fourier transform process; and (3) attenuating the undesired modes which

result from the cross-correlation process. From these illustrations and the

development of the computer-generated filter, which has nonlinear features, we

show how to get a resolution of the conflicts.

We get this resolution in three key technical stages:

1. The first stage uses a computer-generated spatial filter to act as a laser-basis mode to fundamental optical -fiber mode converter. In this

way we get a decomposition of the electric field of a laser pulse by an orthonormal mode analysis of the original laser pulse. Because there are fundamental constraints in the allowed form of the electric field at the

Fourier plane resulting from the matched filter cross-correlation process, it is necessary to attentuate the resulting undesired electric

field signals at the selected spatial locations in this Fourier plane.

2. The second stage uses a graded-index optical fiber at each of the

selected spatial locations in the Fourier plane. Each optical fiber

starts with 300 ym core diameter size and is adiabatically pinched to a single-mode core diameter around 10 ym. After the pinch, these fiber

cores return to the 300 ym diameter. This stage attenuates all but the

fundamental propagating modes in the optical fiber.

3. The third stage has a detector which measures the power from each fiber.

This step is done if there is no need for measuring the polarization or the carrier phase of the laser pulse. If there is a need for the latter,

then it is necessary to run the pulse from the optical fiber though a polarization separation device and finally to heterodyne at the detectors

the resulting signals with a selected reference beam from the same laser

pulse.
The paper discusses the necessary equipment stages to make a direct measurement of the electric field in the original laser pulse. These are: (1) a beamwidth, phase front, and an optical axis matching telescope; (2) the optical filter with beam steering; (3) a pair of lenses to perform the Fourier

transform and to allow precision adjustment of the focal length; (4) four 3-

by-3 assemblies of the fiber optic units with appropriate six degree of freedom adjustments; (5) the necessary elements for polarization assessment; and

finally (6) four 6-by-3 detector arrays with associated electronics for

recording the signals of interest. We are assuming that six orthonormal modes

for each dimension are sufficient. The critical parts in this apparatus are:

(1) the optical processing filter, (2) the positioning device for the individual optical fibers, and (3) the tapering process of the individual fibers.

Less critical parts are: (1) the detector assembly, (2) the mode matching

telescope, and (3) the Fourier transform lens assembly. The first needs proof

of final accuracy and the last just needs to be done in the context of the

entire apparatus.

The paper is structured at two mathematical levels. The first level uses

an analytical formulation of the problem with Hermite polynomials as the

orthonormal basis modes. The second level shows the more complex mathematical

steps which require a digital computer to generate the required optical filter. In this way all the technical issues for construction of the electric

field measuring apparatus should be clear.

Section 2 discusses why knowledge of the electric field is eventually

necessary for complete measurement of a laser pulse. Also this section gives

a brief functional diagram of the desired measuring unit so the reader has a visual sense of how the apparatus would work.

Section 3 addresses the basic technical ideas in developing an electric

field measuring apparatus. Here we use Hermite modes so we can have analytical formulas to illustrate the ideas. This section contains the main body

of ideas about the electric field measuring apparatus. Each idea is discussed

to the detail necessary for the reader to understand what is so. There is a collection of figures which illustrate the basic physical construction ideas

at each stage. The critical dimensions are indicated in the text. The actual

chosen values are suggested so that the reader will have a concrete idea of
what should work. The final test will require construction of the unit. Discussion of that final test will be in a second paper after the unit is built

and tested. The purpose of this paper is to present the ideas so that such an apparatus could be built when it is economically desirable. The key ideas

are: (1) the beam matching by a pair of telescopes [5], (2) the cross- correlation process by an appropriately constructed optical filter [6], (3)

the Fourier transform lens, which has a variable focal length, (4) the tapered

fiber optic assembly which causes attenuation of all modes' of propagation

except the fundamental mode [7-10], (5) estimating and design for minimizing

the impact of various alignment and mode matching steps in the apparatus [11-

12], and in addition, (6) devising the necessary polarization [13] and detection techniques [14-17] for the resulting fundamental mode.

Section 4 addresses the problems in construction of the computergenerated filter [18]. Here we introduce the complex mathematics [19] necessary to get a filter which will allow optical processing with high precision.

Section 5 indicates what advances in construction techniques are necessary to physically realize this measurement apparatus.

2. Why Know the Details of a Laser Pulse?

The complete specification of how a laser pulse evolves in space requires

critical details of the electric field such as the electric field profile

transverse to the direction of propagation over the time period of the pulse.

Such measurements are not easy and, in fact, many would say impossible. With

current laser sources, mechanical adjustment parts, optical fibers, and electrochemical deposition techniques, we are beginning to get to the technical

capability where it is possible to measure the electric field of a laser

pulse. This paper describes how this can be done, the technical problems in

realizing that goal, and the degree the goal can be realized.

Why would anyone want to know the details of a laser pulse? The primary

reason is that we can then have better beam control of the spatial profile, of

the time dependence of the pulse, of the time and spatial dependence of the

polarization, and of the deviations from the carrier frequency. Better beam

control means we can do more precise operations with these laser pulses.
To illustrate, laser pulses are now being used as more and more precise

means for delivering energy of a specific form to a surface of a material to

modify that material in various ways. Examples range from simple welding or heat treatment, to the more esoteric uses such as chemical phase changes at

localized locations (read and write cycles of optical disks). In these illustrations, it becomes more and more appropriate to have a means to measure the

electric field of a laser pulse. The final applications may not need total

beam control. Ultimately there needs to be a precise measurement means to

verify the appropriate level of beam control in the various applications of

these laser pulses. Some new areas of technical endeavor where we suggest

that total beam control will be valuable are: (1) in the area of parallel

processing for pattern recognition of two-dimensional processes (for example,

time dependent checking of the heat distribution in a computer chip), (2) in

the area where a chemical reaction is taking place on a surface (here the

laser's time and spatial pattern may need to be specially controlled to drive

efficiently the chemical reaction in conjunction with the surface catalyst),

and (3) in the area of biological development in cells where it would be helpful to selectively cause a disturbance to the biological process with a specific pattern and with a specific pulse shape.

Current means for measuring the details of laser pulses are incomplete.

They can be designed to measure the flux (irradiance) (W/m^) or fluence (J/m^)

with a time resolution ranging from picoseconds to seconds. The details of

the phase front are usually only known by inference. This means that we have

lost at least half of the necessary information. Examples of such measurement

devices are: (1) single detectors which scan the laser beam and (2) twodimensional detector arrays [20-21].

Figure 1 shows a block diagram of an electric field measuring apparatus.

The numbers in this figure are references to later figures which show the

details of that block. Briefly, the steps in making an electric field meas- urement are:

Step A: get a sample of the laser pulse of interest by using a beamsplitter or a high spatial frequency grating.
Sampled

Laser

Pulse

E>

Beam Match

Teleecopes

A>-- B1 U 82

(••• fig- 2)

Optical Fourier

Filter Tr^nsfonn

A

c

Fourier X?^®^!?

Plane ^ptlcal

Fbers

V

(••• fig. 3)

Optical Axle"

(••• fig. 4)

(••• fig. 6) (••• fig. e-O)

Figure 1. Block diagram of the steps in the electric

field measuring apparatus.

Step B: modify this pulse so that it has a beamwidth, a phase front, and

an optical axis that are matched to an optical filter. Obviously this matching process will require high quality laser sources and high mechanical

stability in the beam sampling and beam modification. The telescope Bl and

the reverse telescope B2 with appropriate mechanical adjustments accomplish

this function.

Step C: use an optical filter which is coded with the following features, namely (1) unique beam steering for each orthonormal spatial decomposition of the electric field and (2) a proper intensity weighting for the orthonormal decomposition so that there is a cross-correlation pattern which has a nearly Gaussian profile of standard width for the desired correlation and that

is a linear combination of the higher order Hermite modes for the undesired

correlations.

Step D: use an adjustable Fourier transform lens to cause the resulting ~

pattern exiting from the optical filter to form the cross-correlation patterns

at the Fourier transform plane of this lens.

Step E: mount four blocks of 3-by-3 arrays of optical fibers. Each

fiber is located so that its optical axis is centered on the Gaussian profile

resulting from the desired cross-correlation process. The fundamental mode of

each graded-index fiber is Gaussian-like and hence is to be matched in phase

front, in beamwidth, and in beam center to the Gaussian profile resulting from

the cross correlation. To account for various errors in fabrication of these

blocks, they are mounted on positioning equipment which will allow 0.1 ym
positioning accuracy in all six degrees of freedom. The final errors in

fabrication may be reduced by proper use of the beam bending design in the

optical filter and by using piezoelectric units to center the optical axis of

each optical fiber relative to the optical axis of the desired cross- correlation mode.

Step F: each optical fiber has been modified so that its optical core changes slowly from 3U0 ym diameter to 10 vim diameter and back over a distance

of 3 cm. These optical fibers are immersed individually in an index matching

fluid so that we can get efficient attenuation of the modes not being sus- tained by the restricted core diameter. Each optical fiber is polished at the

front and back so that there is accurate mode coupling at the beginning and a good match to a calcite crystal at the end. The latter results in the separation of the two polarization states of that propagating single mode in the

optical fiber. The thickness of the crystal is long enough to allow separation of the polarization states and yet is short enough to minimize the beam

spreading of the fundamental mode in the crystal.

Step G: A detector is situated to measure the power in each polarized

mode. To allow heterodyne action, it is desirable to structure the detectors

so that there is some form of local oscillator signal with a carrier frequency

near that of the laser pulse. This step requires some form of integrated

optics to be associated with each detector as well as some form of integrated

electronics to record the time shape of the laser pulse. We do not discuss

how this is done.

3. Analytical Model Using Hermite Modes to Illustrate the Basic Ideas of an Electric Field Measuring Apparatus

3.1 Overview

In this section, there are nine subsections. Each subsection deals with

some aspect of processing the critical electric field information in the laser

pulse. Subsection 3.2 addresses the two beam matching telescopes. Subsection

3.3 addresses the optical filter from a construction point of view. Subsection 3.4 addresses how the Fourier transform lenses operate. Subsection 3.5

addresses the mechanical positioning of the optical fiber units and what is
present at the Fourier transform plane for the simplified analytical case. Subsection 3.6 addresses the errors in matching the optical axis of the

graded-index fiber to the optical axis of a particular cross correlation pro- cessor. Also it addresses what the finite size of the optical fiber does to

the information. At this stage it is appropriate to use these same formulas

to estimate the effects of the finite size apertures at other stages of this

apparatus. Subsection 3.7 shows how to construct the positioning equipment to

minimize the positioning errors. Subsection 3.8 addresses how the mode

attenuation process works within the tapered fiber. Subsection 3.9 addresses

the polarization separation process. Finally we close this section with the

brief description in subsection 3.10 of the detector process.

3.2 The Beam Matching Telescopes

Figure 2 shows the configuration of the four lenses used to set the laser

pulse to a standard beamwidth and to a flat phase front. There are two telescopes. One acts to create a local image and a flat phase front of the incident laser pulse. The second telescope acts as a beam expander to create a standard beamwidth and a flat phase front at the optical filter location indicated as q^y in figure 2. The f^^ are the focal lengths of the simple lenses.

The f^ and f„ are 2000 mm planoconvex lenses. The f, and f^ are -2000 mm

2 8*^ 4 6

planoconcave lenses. The distance between each critical optical stage is

indicated by d^. The critical optical stages are indicated by the complex

beam parameters for an ideal Gaussian beam. Thus we can use q^ = ibp + Zp,

where b^ = Wp^lVX), w^ is the beanv/idth at the nth location, and -z^ is the

relative distance to a point where the Gaussian beam has the width Wp [3,5].

We use the Gaussian beam as a means for design choice. The actual laser beam

need not be Gaussian. The final adjustment procedures would use the entire

apparatus to select the required standard beamwidth and the flat phase front.

How that is done will be described in the conclusions. For now we explain how

to do the crude adjustments of the beam matching telescopes.

There are two relationships which allow determination of position and

status of the laser beam at each optical stage.

Type 1: q , ^ ^n "^ ^n

^°'" ^ simple displacement with f^ = 0. (1)
Optical

Axis

B1 Part B2 Part

Laser

Pulse

"h ^3 <'4 <»5 ^6 "^7 "^B ^'t

Optical

Filter

•10

Figure 2. Beam matching telescope, step B in figure 1.

Type 2: q ^. = -f + f 2/(f _

"'^ q ) ^n+1 n n ^ n ^n^

for the action of the thin lens with d^ = 0. (2)

In a calculation sequence, we would assume that f^, b , z , b , and z are specified. We fix the d 19and d = 100 mm for the design conditions. We use the constraint parameter T = (d /d )i/2 = ^200 mm as a means for setting a convenient dynamic range for d and d of about 800 mm. We do the computation

by a self consistency sequence. First we compute the value of b using the

current values starting from optical stage q . Next we compute the value of

b using the current values starting from the optical stage q and going

backwards through the stages. The sequence is iterated until the computed b

in both directions are equal within the accuracy desired. For this iteration

process to work, we need to specify d and d for each iteration. We con- strain d = T^/d at each iteration. We get a new d m from the old d ° by the

relation

^sn-'s^^",/",'^'"- (3)
To start the iteration process we use the condition d = T/(M)^^2^ where

M = (bj^/b^p) ^''2. This M is a measure of the magnification change necessary to

match the input beamwidth and phase front at 1 to the standard output beamwidth and phase front at 10 where we assume w = 10 mm and a flat phase

front.

The remaining technical points about the beam matching telescope are:

aberrations, extraneous light due to reflections and the positioning equipment

necessary for the above adjustments, and to fix the optical axis. We address

briefly these points. The aberrations are expected to be small for the choice

of F number and field of view. If it is desirable to make them even smaller,

then it is desirable to either use two lenses in place of each lens to make

the adjustment for minimum aberration over a selected range of magnifications

and phase fronts. In the construction of the prototype apparatus, we suggest

that this correction be ignored. Reflections from the surfaces can be dealt

with two ways, namely use an anti reflection coating on each lens for the wavelength range of interest or use off-axis mirrors. Again for the first apparatus, we suggest transmission optics with anti reflection coating set for

1.0 ym. There remains one final point, the selection of positioning equipment

for these lenses. We suggest that an optical rail with a means to position

each lens on the rail to an accuracy of 1 ym is necessary. Each lens unit

would have a means to assure that its optical axis matches the reference optical axis with an accuracy of 1 ym. Because the total distance from stage q

to stage q will change for a generic laser pulse, it is desirable to design

a means to eliminate the influence of this total distance change on the rest

of the apparatus. One way to do this is to use a pair of 45° prisms with

anti reflecting coating on the surfaces where the laser pulse enters and exits.

Under these circumstances the change in distance is perpendicular to the rest

of the apparatus and would not affect their relative locations. As always

there is a price, it is necessary to correct for the change in the phase front

and polarization state of the laser pulse due to these beam turning operations

which can be done with appropriate use of Hermite modes. This completes the

discussion of this subsection except to remind the reader that there is a need

to specify the aperture size of each lens. One criterion for selection of the

aperture will be discussed in section 3.6.

10
Detail of C|f^ cell Xm=m(d)±s

yn 3nd±s

DetaH of 0^^, spot

i"i

fn, «Yn + nid,

Figure 3. Optical filter, step C in figure 1 (front view).

3.3 Optical Filter

In this section we address the physical construction of the optical filter. The particular construction parameters are specific to what can be

fabricated at one particular facility. If a different optical-filter-generation facility is used then the construction parameters may be different. Figure 3 shows the sequence. First there is a glass plate with W = 70 mm. Each

cell indicated by c^ ^ has d = 10 ym with the range of n and m from 1 to 35.

The positioning accuracy of each cell has s = 0.15 ym. Within each cell are 400 spots with their corner locations at the coordinates e^^ ,f^ . Each spot D has a size given by an a and b which are uniquely specified for each cell.

This implies that 98.0 million data values must be specified for a given optical filter.

11
We assume that the radiation is 100 percent transmitted through the

D"^'"^^, ^. spot where the area is clear and percent transmitted through the

shaded area of that same spot. This cell simply samples the electric field

illuminating the filter and passes that part within each shaded region. This

fact means there is no true gray scale for the optical filter. There is some sampling noise due to this binary sampling process and thus a restriction in

dynamic range of the action of the filter on the laser pulse. Beyond these

constraints, there is a second form of sampling error because the allowed

values of a are in discrete steps of 0.3 urn and because the allowed values of

b are in discrete steps of 0.25 pm. The range of a is from 2 ym to 8 ym. The

range of b is to 10 ym. These allowed conditions are unique to the optical

fabrication facility. Other facilities would change these conditions. Section 4 will address the consequences of these sampling constraints. We assume

for simplicity in this section that we have a gray scale filter with no restrictions on dynamical range. By this step we can use analytical expressions for the discussion. It will be necessary to mount the filter perpendicular to the optical axis and to make sure there is no rotation of this filter relative to the axis.

3.4 Fourier Transform Lenses

Figure 4 shows the nominal structure of the pair of lenses used to define

the Fourier transform segment of this apparatus. The anodized tube is used to

reduce the spurious radiation on the optical fibers in the Fourier plane.

This tube is only representational. The proper shielding may be different.

We use the same design formulas as were used in subsection 3.2. Here we use two lenses and we know the beam parameters at q , to be w = 10.0 mm and the

phase front is flat. A critical adjustment of these two lenses will make the

effective focal length at stage q^ show that Xfg^^ = 1 mm^ is true for the

wavelength of interest. For our discussion we assume X = 1.00 ym. Incidentally, this wavelength change will be important in the adjustment of the beammatch stage just as it is important in this Fourier transform stage. The

design conditions are to have q^ with a beamwidth w^ = ( X/iT)fg|:^/w^ and

z^ = 0. We assume for this discussion that the two lenses are plano-convex

with f^ = 2200 mm and f^ = 1800 mm. The distance d^ is controlled by the

12
Optical

Axis

Tube of black anodized Al

Optical

Filter

04 Qe

Fourier

Plane

Figure 4. Fourier transform lenses, step U in figure 1.

mounting hardware for the filter, by the thickness of the lenses, and by the

changes in d and d . We assume d is about 100 mm. The distance d = 982 mm

and is set by requiring f^^^ = lOOU mm. The distance d is adjusted to

satisfy the constraints at q . The key approximate formulas are:

5

eff = ^.if. - dj/f,.. (4)

and

^3 = da ^'(^ - d^)/(b^)2. (5)

with the definition

d^ = f + f

a 2 4 f,^/(% -d,). (6)

These are derived byJ noting^ the size of b^ and b, relative to f^ and f in the

6 1 2 If exact formulas. We get d = 39 mm for our example. There needs to be a precision adjustment of 1 um accuracy for stages q and q . The value of d will

change because the distance between q and q is fixed. The distances change

13
with wavelength to keep ^fgff constant. This constant keeps the Fourier

transform plane unchanged relative to the optical filter. We first use the

approximate equations to set starting values and then use a self-consistency

iteration procedure with the exact equations to finally get the correct d and

d values, b and b are constant in this iteration process.

Again precision adjustment equipment is necessary,

3,5 Fourier Plane--What Is Present There?

The Fourier plane contains the results of the cross-correlation from the

optical filter and the Fourier transform lenses acting on the laser pulse.

Figure 5 shows four physical structures in this plane. Each structure is held

by a three-axis positioner with accuracy of 0.1 \m in each dimension, a single

rotation unit of accuracy 1 mrad, and by two double tilt positioners with

accuracy of 1 mrad. Each structure is made of a 3-by-3 array of optical

fibers mounted in a square pattern indicated by the dimensions in figure 5,

The approximate dimensions of the structure set d = 20 mm. The hole diameter

is d = 350 ym, where it is assumed that each optical fiber has a diameter

d^ = 325 ym including the protective sheath and an graded-index core diameter

b of 300 ym. The actual placement of the holes in the electroform sheet, as indicated by the distance d , can be either periodic for each 3-by-3 array or uniquely located for each structure. If there are no significant ghosts due

to the cross-correlation process in the computer generated filter, then a periodic structure is adequate. If there are significant ghosts, then a nonperiodic structure is necessary. In our discussion we assume that the

periodic structure is adequate. This point will be further discussed in sec- tion 4, The remaining dimension is the absolute positioning of the four

structures as indicated by D^ and D^, Using coordinate notation (0^,0^), we

have four number pairs with the values in millimeters each of (6,28, 6.28),

(6,28, -8,7), (-8,7, 6,28), and (-8,7, -8,7), These coordinates locate where

the corner optical fiber nearest the optical axis has its center relative to

that optical axis. There is a correction to these coordinates to get the d^

and d shown in figure 5. This depends on the final construction details of

the 3-by-3 structures. That correction is expected to be about 5 mm. The

final dimension of the spacing between the optical fibers has d = 5 mm for

the periodic case.

14
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O o o o o

1

o o <5lL o o o

o o

^ '^

-^cu

Electroform sheet of

nickel with holes

Figure 5. Fourier plane, step E in figure 1 (front view).

The physical dimensions are now defined, and we are ready to look at the

analytical formulas to show the kind of cross-correlation pattern produced in

the Fourier plane.

We represent the electric field just before the optical filter by the

form in eq (7)

n,m=0

where the following definitions are noted:

(7)

H^(/2x/w) = (^)"exp[-z2 + 2/2 x z/w]

W(x) = exp[-(x/w)2],

z=0

Z(x,Zi,w) E exp [-Zi^ + 2 xZi/2/w] W(x)

C^ = 2" n!//^.

z,=0'

15
C 6" = ^

/ dx H (/2" x/w) H (/2x/w)[W(x)]2,

J- _oo ••

H (-/2x/w) = (-1)" H (/2 x/w).

and

Z = X or y polarizations. (8)

Here the w is the Gaussian width of the beam. Z represents the two polarization states and the n and m represent the mode label for x and y coordinates

of Hermite functions. C^ and C^ are the necessary normalizations for the

orthonormal expansions. I^r\ m Z ^^^ ^^^ strengths of each mode in the electric

field. Z represents the generator of the Hermitian modes, z is set zero after the differentation process. The symmetries of the Hermitian modes are also indicated.

We assume that the optical filter can be represented as

T(x,y) = T(x) T(y). (9)

This means we can separate the two dimensions and do the analysis with one dimension only. With this assumption we can write the electric field after

the filter as

where we have the one-dimensional field mode as

E,(x) = (^)" Z(x,Zi,w) T(x). (11)

The definition of the filter function is

5 T(x) =

I B.H.(/?x/w)cos[u(j)x + v(j)]W(x)/Wl(x)C ). (12)

j=0 J J ^

16
Here we require the filter function to be real and do not worry about the possibility it can be negative. We also assume only the first six Hermitian

modes are of interest. We are implicitly assuming that most of the energy for

each dimension of the electric field is contained within these first 36 modes —6 for each dimension. If that is not true, then it is necessary to use a different set of orthonormal modes so we can use continue to use 6 modes per

dimension. In eq (12) we define a weighting function Wl as,

Wl(x) = exp[-(x/wj2]. (13)

Its purpose is to allow the cross-correlation process to have exact orthogonality for the fundamental propagating mode at the Fourier transform plane.

We use v(j) to set the phase symmetry of T(-x) = -T(x).

v(j) = V2 j = 0, 2, 4 (14)

=0 j = 1, 3, 5.

This is done because the actual filter must have a dc offset to prevent negative values. This dc offset is important only to make sure that a dc signal

peak in the Fourier plane does not match the signal peaks for the cross- correlation process. We ignore the dc term in the optical filter. The positioning of the cross-correlation processes is controlled by the u(j) which is

set by eq (15)

u(j) =|f x^"(j). (15)

x°f(j) are specified locations in the Fourier-transform plane relative to the

optical axis. The strength of each cross-correlation process is controlled by

the Bj terms.

The action of the Fourier-transform lenses changes the one-dimensional

field mode indicated by eq (11) to a Fourier-transformed field mode indicated

in eq (16):

S (u)

=—

/ dx e^'^^T(x) H (/2x/w)W(x). (16)

" /Xf -»

"

17
Equation (16) has a constant phase term missing which is of no importance in our discussion. If we insert the form for T(x) we get the transformed field

mode

5 S (u) = I R {exp[iv(j)] S^ .(u)}/C., (17)

-j_Q "= "»J J

where we have the cross-correlation field mode defined in an orthonormal

expansion of Hermite functions. Thus

oo S .(u) =

I G . H (v7u/w.) W2(u)/C , (18) n,j' j^fg n,j,m m' ' f' ^ '' m' ^ '

with each mode coefficient given as

Here we have a fundamental generator function defined as

D E^^ / du dx e^"^ Z(x,Zi,w) Z(x,Z2,w)Zl(u,Z3,w.)/Wl(x) , (20)

w^/xf ^

where

W2(u) = exp[-(u/w^)2] (21)

and where Zl is the same as Z with u replacing x and Wf replacing w.

We can integrate analytically over x and u in eq (20). First we make a set of definitions

1 2 1 q^ W2 W^2»

1 _ 1 , q^

W^2 ' 4 '

and

18
w^ = wb. (22)

The generator D becomes

D

=

/2tt -^qp exp

-z2-z2-z2+2^z|+2-P^'^^

W 2-3--w2^2(Zi+Z2)

+ 12-^—^Z, Z,+ Z2 WW ^ ^ ^ (23)

This equation form shows us what the cross-correlation process is doing. The

z = case corresponds to the fundamental mode in eq (18), which corresponds

to m = 0. The desired cross-correlation process has only correlation between

the n = j terms. This can only happen when the relation

is true. This implies

pq = WW,

/2

w^ = 2/Wi,

p

= w^(l - (ww^) 2/8)^/2

(24)

and

q = w/[/2(l - (ww^) 2/8)^/2], (25)

Under these conditions D can now be written as

D = — exp [2ZiZ2 + z^(l - l/b2) + i2 Z3(z i+Z2)/b].

/Xf

(26)

When b = 00, we have the ideal cross correlation, a delta function. Only

z = case has no mixing between n ^ j terms. Notice that the mode series

are infinite. This implies a clipping error when the beam is injected into

19
the optical fiber. The value b = 1.0 corresponds to the situation that the

m = mode has a width equal to the Fourier transform of the original Gaussian

beam before the optical filter. The best we can do is to get a nominal minimum in the strength of the higher order modes at the Fourier plane. We assume

that this is done. In the optical fiber subsection we find that the value of

the width is 26.68 ym. This implies b = 1.19.

There remains only one more step in this subsection, to explicitly differentiate the generator D to find G in eq (19). We not do this since the

expressions are complex and all the details of interest have been shown and

discussed except to note that the m = 1 case is 90° out of phase relative to

the m = case. This situation allows a precise alignment strategy for the

optical fibers. This is true because we have set up the apparatus with flat

phase fronts at each of the critical locations, namely the optical filter and

the Fourier-transform plane. Other locations may not allow this situation to

be true. A careful analysis of other situations would be necessary to check

this point.

3.6 Effects of Beam Clipping and Optical Axis Offset between Fiber

and Cross-Correlation Beam

Here we look at the mode mixing effects due to clipping and to a shift in

the optical axis of the cross-corrrelation mode compared to the optical axis

of the optical fiber. Figure 6 illustrates the points. The value d is the

radius of the optical fiber which accepts the laser radiation, and e is the

amount of relative shift of the optical axis. The critical parameter of scale

is the width of the fundamental mode. We use w = 26.68 ym as that width. The

value of d is 150 ym. This implies a distance of 5.62 times the characteristic width of the fundamental mode. This will imply a very small level of

clipping when the number of modes present in the cross-correlation process is

sufficiently small. This is the case when the value of b discussed in the

previous subsection is near enough to one. If we require a position adjustment accuracy of 0.5 ym for the optical axis, then we would have sufficiently

small error in the offset between the two optical axes. It would be about 0.1

percent or less depending on the stability of the technique for adjusting the

optical axis. We look at this point in the next subsection. We now get the

analytical expression for the effects of clipping and offset.

20
Later

Beam

- d-+€ -

Optical

Axis of

Cross-correlation

di-€ —

Optical

Axis of

Graded Index

Fiber

Figure 6. One-dimensional version of clipping and offset effects due to

the graded-index fiber acting on a cross-correlation structure.

The one-dimensional electric field for one polarization state can be

written as

^(^'^) = ^ ^m^^) Z(x+e,z,w)/C^,

m=0

(27)

where x is the coordinate perpendicular to the optical axis of the optical

fiber, e is the offset of the optical axis of the cross-correlation process, w

is the width of the fundamental Hermite mode, A^ is the strength of the mth

mode, and C^ is defined in eq (8). The Z is defined in eq (8) also.

Once the radiation is launched within the optical fiber, we would have

the following form for the electric field, namely

21
m=0 ^

The only change between eqs (27) and (28) is the strength coefficients. We are particularly interested in the B term. It provides all information about

the cross-correlation process and about the particular orthonormal mode. The

ideal situation is no offset error and no error due to clipping. B is given

as

00 ^0 =

I A i^f' D2(z,0,d,,B)/C (29)

m^=0 ""i ^^

'"i

where D is defined as

/2 r^^

D2(ZiZi,di,e) = — j dx Z(x+e,z,w) Z(x,Zi,w) (30)

If we make the definitions

a E /2e/w, b = /2 d^/w, u = /2x/w,

and

c = z - a/2, (31)

then D is given as --^

D2(z,0,di,e) = exp[-a2/4 - az] D3(b,c) (32)

where D is defined as

b-c

D3(b,c) = / exp[-u2] du. (33)

-b-c

If we make the approximation that b is very large and a is small, then the

relationship between B and A is

Bo = Ao - aAi/2. (34)

22
The key fact to notice here is that an error in the optical axis can be significant because the strength of A^ is comparable to A . First we note the fact

that A^ is 90° out of phase with A^. This means the magnitude of B is proportional to a^. If a = 0.03 we would have a 0.1 percent error due to alignment. This implies a positioning accuracy of 0.5 ym. Subsection 3.7

describes how this can be achieved. If we make the approximation that a is zero and b is large, then the relationship between B and A is

00 Bo = Ao(l - e-^^/b) -e-^^ I A^^ [Hjb) - H^(-b)]/C . (35)

m=0

Here we simply notice that b needs to be chosen so that the highest A^ modes

are less than 1 percent of the A mode. The choice of a graded-index optical

fiber of 150 urn radius and w = 26.68 ym allows b = 5.62. As long as m is less

than 30 modes we have no significant clipping effect.

3.7 Design of a High Precision Positioning Device to Center the Optical Axis

of Each Fiber to the Optical Axis of the Cross-Correlation Process

Figure 7(a) shows the nominal mounting of the optical fibers shown as C so that the piezoelectric material can be used to get precision adjustment of

each fiber's optic axis. The piezolelectric material is indicated by A and

B. They each have the dimensions of 0.1 mm thick, 1.0 mm wide, and 20 mm

long. There are six such units for each 3-by-3 array of optical fibers. To

support these units, there is a ceramic frame with notches to allow the piezoelectric material to lengthen at least 50 ym. The outside pair of notches are placed with center of ±5 mm relative to the center notch. The width of the

notches is 0.12 mm. The depth of the notches is 0.2 mm. The notches run from

the front edge to a depth of 2.2 mm. The frame has the dimensions, d = 1.0

mm, d = 30 mm and d = d, = 21.9 mm. This frame is constructed so that front

and back edges are strictly parallel. This frame is rigidly attached to the

electroform in the back. The piezoelectric material is placed in the frame.

The remaining big step is to get precision in the placement of the optical

axes of the nine fibers. We need the coordinates to be accurate to ±1.0 ym to

the design positions. If we can get that accuracy, then it will be possible

to use the piezoelectric material to place the optical axes to ±0.2 ym accuracy, which would eliminate the offset error.

23
A,B - Piezoetectric material C - Optical fiber with polished ends

Indtvldual wire leada for poling voltages In each segment v

^"CZ

Edge c

Bottom P

Metal Film Ground Wire

(a) (b)

Figure 7. (a) Mounting of piezoelectric material to allow two-dimensional

position adjustment of the optical fibers, (b) Detail of a piezoelectric material.

We presume a jig to allow placement of the nine optical fibers to x and y

axis coordinates of ±5 ym for each fiber relative to the center fiber of the

3-by-3 array of fibers. The accuracy of this placement must be at least

±0.5 ym. We suggest that the jig can be constructed using a set of holes con- structed by electroformed material with the nine hole sizes made 1 ym greater

(301 ym) than the outside diameter for the uncoated part of the fiber. We

believe the fiber can be constructed with a consistent diameter within 1 ym.

Assuming this is true, the jig is used to align the 3-by-3 array of fibers.

The back electroformed unit becomes a second jig with one modification, namely

that the hole size is 350 ym. It becomes a part of the apparatus. The precision jig is temporary. It would be located on the front side of the 3-by-3

24
array at a distance of 40 mm from the back jig. It is also attached by using

a solvent to dissolve the sheath of the nine fibers and then inserting the

fibers into the precision jig. The conical shape of the hole in this jig

would expand toward its back. A number of precision spacers of 1 ym difference in length is used to align the front jig relative to the back jig and to

render the two jigs rigid to each other. The thickness of both jigs would be

about 0.5 mm so they are sufficiently rigid. Notice there is a 10 mm separation between the frame and the front jig. This should be enough distance to

allow the gluing of the piezoelectric materials to the optical fibers and to

the frame. The glue or epoxy must provide a rigid mechanical bond for the

optical fibers and for one side of each piezoelectric slab. To make the unit

insensitive to mechanical vibration and just allow the slab to lengthen or shrink freely, the other side of the slab needs to be attached just on the

edge of the notch with a rubber type of material —perhaps silicone rubber.

There remains one final construction detail, namely how to cut the fibers

to a proper length and polish them. We use a new jig made like the cover

plate indicated in figure 5 where the holes are 350 ym in diameter and where

the conical shape of the holes expands toward the front. We remove the jig

with the 301 ym holes and rigidly attach the new jig to the frame. We now have optical fibers sticking out about 10 mm above the new jig. The front

part of the jig is covered with paraffin wax. This will make the unit sufficiently rigid to allow the cutting and polishing of the excess optical fibers.

The 3-by-3 array unit is mounted in a lathe with the rotation axis parallel to

the optical fiber axes. Using a thin one-sided wedge angle tool, we cut as close as posible to the front surface of the jig. We then polish the unit so that the nine optical fibers are optically flat relative to each other to at

least 1 ym. Once this step is done, then the remaining wax is removed by a solvent and by ultrasonic cleaning. The new jig is now removed. Figure 7(b)

shows the detail for the metal films and wire leads on the piezoelectric

material. The bottom side is covered with a metal film to act as the ground

plane. A ground wire is attached with a conducting epoxy. The top side is

covered with metal film except for three regions of width d^ = 1 mm where

there is no film. The intervals between these regions are d = 5 mm. There

are four individual wires attached with conducting epoxy to each film segment.

The five wires pass through the frame of figure 7(a) through notches (not

25
A.C - QIasi suport for flb«r B - QIass support for tapered position of fiber D - Electroform of nlcl^el with holes to all fibers to be held

E - Electroform with holes to shield out extraneous radiation F - Polished ends of optical fiber Index Matching Fluid

Qraded Index

Fiber with Taper

Figure 8. Side view of one optical fiber showing the individual supports.

shown in the figure) in the front surface of this ceramic frame. These wires

provide the means for getting a dc electric field onto each segment of the

piezoelectric material.

There remain two primary points of discussion, namely the reason for the

choice of 30 mm for the length of the fiber and the dynamic range of the movement of the fiber due to the piezoelectric effect. We discuss the effects of

four potential voltages on each piezoelectric strip. To first approximation,

we can expect a sum of linear shifts. A more exact analysis would produce a two-dimensional shift pattern. To understand this requires assembly of the

unit and testing of the unit. We now address the primary point.

Figure 8 shows how a single fiber is mounted to the support base. Figure 7(a) shows how the array of fibers are mounted. The key point to note is

that this is a cantilevered beam configuration. This means that we can look

up the formulas for the displacement of the end of the beam under two

conditions--gravity and external force at the end. The gravity force is a distributed force. Its effect, which is proportional to the fourth power of

the length, depends on the flexibility of the fiber. A 30 mm length implies a sag of 5 ym at the end. Under these conditions a force of 1 gram will imply a displacement proportional to the third power of the length, namely 310 pm. In

26
contrast, a 20 mm length implies as sag of 1 urn and a corresponding displacement of 92 ym. The former results assume no significant resistance to movement of the end. The pair of piezoelectrics act to support the end of the

fiber and to control its position with no important resistance to the two

transverse motions.

To address the final primary point, we need to specify the piezoelectric

material. With our thickness of 0.1 mm and with the application of 100 V, we

can develop an electric field of 1 MV/m. If we have a length of piezoelectric

material of 5 mm, a mechanical electric coupling efficiency of 0.6, and piezoelectric constant of 270 x 10'^^ m/V, we can get a displacement of 0.8 ym.

This implies a range of adjustment of ±0.4 ym which is adequate for fine tuning the optical axis of each fiber. It will be necessary to temperature control this 3-by-3 array so that the temperature effects in the piezoelectric

material become unimportant. The best way would be to temperature control the

entire unit beyond the Fourier plane to some fixed value such as 30°C with an accuracy of 0.1°C. In this way we would guarentee the dimensional stability

of all the parts including the piezoelectric material- In addition we would

control the temperature dependence of the responses of all the detectors.

3.8 The Mode Attenuator--The Tapered Optical Fiber

Figure 8 shows the structure of the tapered optical fiber. We address

first the structural details, next the construction details, and finally the

mode stripping process.

We look at the structure from the left to right. E is the cover plate at

the Fourier-transform plane. G is the graded-index fiber. It goes from the cover plate to the back support indicated by C. D is the support, which acts

as the rigid locator for the optical fiber. The glass supports A and C are used to hold the fiber and are used to prevent the index matching fluid within

support B from leaking out. The optical fiber is tapered from the 150 ym

radius to about 5 ym and back to 150 ym within the length of the support B.

The 5 ym location is near the center of support B. Both ends of the fiber are polished. The nominal dimensions are d = 30 mm, d = 10 rmi, d = 50 mm, and

d. = 10 mm. The diameter of the supports A, B, and C must be less than 9 mm

with a capillary of about 400 ym diameter.

27
The suggested construction details are to first mount a 3U0 mm long fiber

inside support A with 40 mm in front of the support. Use an index matching

glue to do the mounting. Insert the fiber into structure B and suspend the

unit with support A holding up the fiber unit. To prevent support B from falling off, clamp the bottom end of the fiber with a metal clamp. The unit

should now have a space of about 180 mm between mount A and the support B where we can see the optical fiber. Using a CO laser, we heat a segment of

fiber. The weight of the support B and the metal clamp will stretch the

optical fiber to form a taper. The length and pith of the taper is controlled

by how far the stretch can take place. By having a jig which will stop the

fall of support B and which will make sure there is no twisting or turning of

the support B, we can control the final size of the taper. This process is

reported until the fiber is the correct size. After the glass fiber is so contructed, we put a drop of index matching fluid at the top of the support

where the optical fiber enters, then we slide the support B up to support A

and glue them together. This action is necessary to protect the taper and to

support the optical fiber. It is desirable to make the start of the melt as close as possible to the A unit. During this assembly it is necessary to make

sure that the index matching fluid fills the support B. After this step is

completed, the support C is added and glued to the support B and to the

optical fiber. At this stage the optical fiber is now in a rigid mount and

has been protected. The next step in construction is to polish the optical

fiber at the exit end.

Within this construction process, there are several points to notice:

(1) there are nine of these units for each 3-by-3 array and there are four

such arrays, (2) the unit A is glued to the support structure D of the array.

At this point, the optical fiber is placed 40 mm in front of the unit D. Figure 8 shows the optical fiber terminating before the cover plate. The con- struction of the piezoelectric unit left the optical fiber flush with the

front of the cover plate. To make the fiber terminate before the cover plate,

we add a spacer of the thickness of the cover plate. The cover plate is

coated with a radiation absorbing material so that there is minimum stray

radiation getting to the optical fiber. Also it is desirable to orient the

cover plate E so that the conical shape of each hole in the electroform

expands toward the optical fiber.

28
We now have some analytical points to make. The mode-count number V in

one-dimension for optical fibers is

V = nbTT(2A) /^A, (36)

where A = (An)/n, b is the radius of the optical fiber, n is the index of

refraction of the glass and X is the wavelength of laser radiation. If we assume A = 0.001, b = 4.75 ym, n = 1.5, and X = 1.0 ym, we find that V = 1,

which means that we have only one propagating mode present in the optical

fiber for each polarization state. In contrast, if we assume b = 150 ym, we find that V = 32. This means that there are about 2000 modes in the fiber.

Assuming at the start a uniform rod of length 1 , and at the end a uniform taper, both in shrinking and expansion phase, the new length, 1, of the

tapered optical fiber compared to the old length, 1 , is given as

1^ = 1(1 - a + a2/3),(37)

where a is the ratio of the difference between the initial radius of the fiber

to the final radius divided by the initial radius. If we use for the intial

length 1 = 10 nm and a = 145/150, we get 1 = 29.0 (im or a change of length of

19 mm. The taper angle is 2r/l = 2 deg. Equation (37) is derived from the

assumption that glass material is conserved.

It is convenient at this time to add one more formula to this list,

y the one that

graded-index fiber.

namely the one that specifies the width w of the fundamental mode of the

Wy = (X a/nrr) V2(l/2A)^/4, (38)

Using the values given for eq (36) we get w^ = 26.68 ym for a = 150 ym and

w = 4.75 ym for a = 4.75 ym. A change in the wavelength of 10 percent causes

a 5 percent change in the width of the fundamental mode. To get highest

accuracy in the electric field measuring apparatus, it is necessary to match

the optical filter cross-correlation width w at the particular wavelength. A

calculation just like the beam clipping would show the mode mixing error due

29
to beamwidth mismatch of 1 percent would be 2 percent error. This is not to

be ignored. This fact basically implies that each optical filter will be unique for each wavelength.

The ideal procedure is to have the support glass made of a radiation

absorbing material so we can be sure all modes that do not satisfy the support

condition given by the radius of the graded-index fiber will be absorbed as quickly as possible and the propagating modes will not be absorbed. The delicate point is to get the neck of the taper long enough to allow the modes

above the fundamental mode to be attenuated and yet not cause the structure to

break during the fabrication process. The above construction procedure is

expected to do the job. What is very critical is to make sure that the

stretching process keeps the graded-index structure cylindrically symmetric

over the entire length of the taper. In this way we can be sure the two fundamental modes are not significantly attenuated. It should be obvious to the

reader that this taper method for attenuation of modes depends intimately on the form of the modes. The primary loss is the energy exterior to the core.

By having a Gaussian fundamental mode, we tend to place the energy of the

undesired modes exterior to the fundamental modes and hence can get mode

attenuation with minimal mode mixing. One final point of this subsection. It

is not necessary to get the neck to 5 ym radius. What is necessary is to get

the neck small enough that the fundamental mode still propagates and the other

modes are attenuated because they are not being sustained by the core. This

condition happens as long as the radius a is less than 9.49 pm. We suggest

that the best range has a equal to about 6 \im. In this way we do not have the

fiber too small and yet make the neck long enough to attenuate undesired

modes.

3.9 The Polarization Analysis Process--A Calcite Crystal

If the optical fiber and the tapering process do not cause more than 10

percent mixing between the two polarization states, then it is possible to

structure the apparatus so that polarization can be accurately measured for

each of the original laser modes. A calibration of the apparatus is necessary

to determine the level of the polarization mixing process. For illustration

we assume that the original 300 ym graded-index fiber has a polarization mixing rate of 0.1 percent per centimeter for the fundamental mode in the fiber.

30
We assume the taper process increases this mixing to 1 percent per centimeter.

Under these conditions, the actual mixing would be 4 percent which would be

tolerable. The actual way to attain this situation will require that the

graded-index fiber be made as strain-free as possible and as cylindrically

symmetric as possible. One way to estimate how well this can be done is to

note that current single mode fibers with such cylindrical symmetry can be

made with a birefrigence of 5° per meter. This implies a mixing of 0.02 percent per centimeter which is well within our calculation.

If we have a calcite crystal which has been cleaved and polished with the

principal section normal to the cleaved surfaces, we can get a separation of

the ordinary and extraordinary polarizations. The separation angle is 6.2°.

A 5 mm thick slab will separate a laser beam into two beams 0.54 mm apart.

The original 27 m width beam will expand to 65 ym width over this distance.

Thus the beam separation is 8.3 beamwidths which is adequate for separation.

There is one concern in the use of calcite: it is brittle. The assembly pro- cess will need to account for this fact. It may be necessary to make the C

structure nine separate calcite crystals mounted in a plastic frame which is

then mounted as shown in figure 9.

Figure 9 shows the structure of the assembly of the calcite crystal as well as the other parts of the 3-by-3 array. There are six points to notice:

1. The calcite crystal is either 20 by 20 mm or is nine crystals of 4 by

4 mm in si ze.

2. The glass rods and calcite are encased by two nickel electroforms and a rectangular aluminum frame of length d

= 75 mm and cross section of

d = 25 mm.

3. The ends of each glass rod are coated with a light absorbing paint except

at the optical fiber.

4. The calcite is matched to the optical fiber by index matching fluid such

as mineral oil.

5. The back piece of electroform, F, has nine circular holes of 2 mm diameter. These are needed to allow the detectors to see the laser radiation

from each beam.

6. To eliminate mixing of radiation, the space between the glass rods is

filled with a light-absorbing material. If the unit works as intended,

31
A - Nickel etoctroform for support

of glass rods

B - Glass rods

C - Calcite crystal assembly to separate polarization

D - Polarization beam (extra ordinary)

E - Polarization beam (ordinary)

F - Nickel electroform for support of glass rods, and crystal, and detectors

Q - Aiumimim case to create rigid frame

H - Space filled with light absorbing material

Hole to allow beams

to reach detectors

Figure 9. Polarization separation and support structure.

then it is desirable to use epoxy for mechanical rigidity. For testing

purposes, it may prove better to use a liquid such as water or oil with a dye in it. The liquid has the advantage that it can be removed if the

unit needs to be repaired. A high specific heat material such as water

will attenuate significantly the temperature fluctuations of the environment acting on the 3-by-3 array.

If the polarization mixing is greater than 10 percent, then it will be

necessary to do the polarization analysis just after the optical filter with a very good quality polarizer. The effects of the phase shift will need to be

considered. In this case, the polarization measurement would be done by

repeating the measurement with the orientation of a linear polarizer shifted

by 90° between each measurement.

32
3.10 Detector Assembly

We now need to measure the strength of each Gaussian beam and its phase

function as a function of time.

If it is desirable to measure the phase, then it will be necessary to use these 72 beams and mix them with a reference beam which has phase dependence

that is known in some sense to produce a signal that allows determination of

the relative phase of these 72 beams to the reference beam. The reference

beam could come from a sampled beam obtained at the optical axis of the apparatus with a single mode fiber. This sample is gotten at the Fourier plane

with due consideration of various propagation distances.

For now, we assume the phase measurement process is not necessary. Under

this assumption, figure 10 shows a possible configuration for the detectors

used to measure the strength of a Gaussian beam. We see the calcite assembly

designated as C, the holder for the detectors designated as B, and the holder

for the assembly and the glass rods designated as A. The detectors are located in the holes made in A. The detectors have a width of 0.5 mm and are

A - Electroform support with holes

B - Electronic circuit board to hold detectors

C - Calcite crystal assembly

F - Index matching fluid between

calcite and detector

. DJD^ ^^° detector assemblies

^ to measure laser power

G - Optical fiber • - Extraordinary beam

o - Ordinary beam

Figure 10. The case where only laser power is measured,

33
centered for each beam. These detectors are matched to the calcite with index

matching fluid so that there is minimal reflection. The key reflection is the

one from the surface of each detector. It propagates back to the front sur- face of the calcite assembly where it is absorbed by the paint on the back

surface of the glass rods. This process should minimize the effects of the

reflected beams. The detectors would be either silicon for the visible light

range or pyroelectric detectors if the power levels are high enough.

Once the power is measured, the current or voltage is then captured by some form of sample and hold circuitry for the A/D processing. Selection of

that circuitry is dependent on the time resolution desired. Because the

detectors are small, we can expect resolution of signals at faster than nano- second rates.

4. Realization of Nonlinear Filter Using Abstract Mathematics

4.1 Overview

In this section we develop the mathematical expressions necessary for the

computer programs. These programs are used to generate the optical filter.

Subsection 4.2 generates the exact equation and subsequent approximations

using all the various constraints due to the fabrication process. Subsection

4.3 examines the part of T(x) which has the spatial dc characteristic of the

fabrication process. Subsection 4.4 examines what happens due to the 0.15 ym

error in positioning the cell. Subsection 4.5 examines the antisymmetric part

that is nonlinear. Subsection 4.6 examines the antisymmetric part that is

quasilinear. Finally, subsection 4.7 discusses two strategies for fabrication

of the two-dimensional optical filter to establish the most accurate one.

4.2 Exact Equation Incorporating the Constraints of Fabrication

We assume that the separability into one-dimensional structures is sufficient. Under these circumstances the resulting pattern of a single mode at

the Fourier-transform plane is given as

4

S(n,u) =

I S(n,u,r) (39)

r=l

34
Here

L

1 2 S(n,u,r) = -!- / dx H (x) T[x,a(r) ,b(r)]exp(iux) . (40)

/Xf L " "2

We have formed four types of radiation patterns due to the structure of the

holes in the optical filter. T(x,a(r) ,b(r) ) gives the pattern for each type.

Equation (42) and others below will define this function. For now notice the

Hp(x), this represents the n mode of the orginal orthonormal decomposition of

the electric field at the optical filter. This n mode is like the Hermite

mode but does not have to be exactly the same. We assume that n ranges from 1 to 6. Also notice that the range in integration is -(L/2) to +(L/2). This is

due to the finite aperture of the optical filter which has L = 70 mm.

Because we are only interested in the fundamental fiber mode, we multiply

the expression in eq (40) by that in eq (41), namely

^^^exp H(u - u(r))/wJ2}, (41)

and integrate over u to get the strength of each radiation pattern in the fundamental mode. There is an approximation in this expression which is less

than 0.2 percent. We get

Sl(n,k,r) = /-^ I H^(y^)B(z^)exp[iu(k)y^ - (w^^/2)2] (42)

We have a large number of definitions to note:

First we give the form of T, namely

N T(x,a(r),b(r)] =

I [9(x - x. - a.(r)) - e(x - x - b (r))], (43)

where we have below the step function, the centroid coordinate, the half width

of each hole in the optical filter, and the nominal stepping interval between

the holes.

e(x) =1 X > e(x) =0 X < 0,

35
y^^ = x^ + (a^(r) + b^(r))/2.

z^ = (b^(r) - a^(r))/2, and

x^ = D4 (2£ - 1). (44)

The width of the fiber in u coordinates is related to the actual fiber mode

width in position coordinates of the Fourier plane as

^m = ^^^fibep/^^ = ^-^^^ ('^'"^'^' (45)

Some values that are specified by the conditions of design are:

u(k) = N(k) Au

Xf = 1.0 (mm)2

AU = 2Tr Ax/Xf = 0.0898 (mm)""^

Au^g^^ = 2Tr Ax^g^^/Xf = 31.42 (mm)'-^

Ax = Xf/L = Xf/(4ND4) = (1/70) mm

Ax^ n = Xf/ Ax T1 = 5 mm

cell cell

Ax^g^^ = 200 ym. (46)

The cell set is defined by the 200 ym cell size. This implies a spacing in

the Fourier plane of 5 mm. The second spacing is defined by the finite aperture.

Some further definitions or consequences of constraints in the design

are:

D4 = DO/2 = 5 ym.

36
DO = 10 ym,

N = 3500,

and

W... = 26.68 for X = 1.0 ym. (47) fiber

DO is the fundamental maximum spot size of the optical filter and N is the

total number of spots available in a 70 mm width filter. Wf^^^g^ will change

according to eq (38) as the wavelength is changed. For maximum accuracy, each

new wavelength will require a new spatial filter. B(z^) is given and approximated to

B(z^) = 2 sin[u(k)zp/u(k)

= 2z^ [1 - (u(k)z^)2/6]. (48)

The term that is cubic in z^ can be a 6 percent correction to the filter so it

must be retained.

The individual strengths are summed to produce the net strength of the

fundamental mode, namely

4

Sl(n,k) = y Sl(n,k,r). (49)

r=l

At this point, we specify the a^(r), b^(r), z^^, and y^. We use table 1

to specify these quantities. The TYPE label notes the following: DC is the

necessary offset to prevent signal at the hole from becoming negative, AC is

the desired signal which can be negative. El is due to the error in cell

placement affecting the offset, and E2 is due to the error in cell placement

affecting the desired signal.

37
Table 1.

TYPE r ^£(^) b,(r) ^z ^r^j

DC 1 -D4 04/

2

-04/

2

AC 2 H g,/2 g,/2

El 3 e,-D4 -D4 -e,/2 C/2-D4

E2 4 ^a H-'H s,/2 9^,^-/2

There are some symmetries to note relative to i and -a + 1, namely

^-£+1 '^e

^-i+l "^£'

e„ < 0.15 ym = 0.03 D4,

|gj < D4. (50)

The above also shows the restrictions in size of the error e and the range of

the desi red signal .

The beam turning is specified by the N(k). It has the following values

and properties. Here k ranges from -6 to +6.

N(-k) = -N(k)

N(0) = N(k) = 440,615,790,965,1140, and 1315 (51)

for k = 1 to 6. These values are chosen to allow the necessary physical spacing of the optical fibers and their associated equipment as well as avoiding

resonances due to the cell placement errors. They occur at multiples of 350.

The first mode correlation, N(l), is placed far enough so that the signal on the optical axis has no significant contribution to that correlation. N(2) is

spaced far enough from N(l) so that there is less than 1 percent mixing

between the N(2) and N(l) terms.

The remaining symmetry relationships are shown in table 2.

38
Table 2.

D4/2

-9ji/2

-e-£+l/2

^-£+1/2

-y,- D4

This completes the general specification of the mathematics for the

expected field strengths due to the four possible radiation patterns. We are

now ready to look at those individual patterns to assess what they will do to

the signal at each fiber.

4.3 Evaluation of the dc Part

The dc part is symmetric to better than 0.1 percent. Thus we can simplify the dc strength to the form in eq (52) where the y^^ and z^ are defined in

table 1 with r = 1.

. /2

N Sl(n,k,l) =^ y H (y )B(z )[e

/Xf il=l " ^ *

iu(k)y , -iu(k)y

^ + (-1)"^ e ^]

exp[-(w^^/2)2], (52)

The key point to note about this expression is that the signal strength at

each k location is the Fourier transform of the nth mode. For k not equal

zero, this strength is less than 0.1 percent of the peak for the signal of

interest and hence is ignored.

4.4 Evaluation of the Cell Positioning Error

The two cell positioning error terms are shown in eq (53) and (54) below

with r = 3 and r = 4, respectively.

iu(k)y

Sl(n,k,3) i /? N

/xf £=1 " ^ ^

- (-1) e ^-ii+l'

* exp [-(wj^/2)2] (53)

39
Sl(n,k,3) =^ I_ H^(y^)[V

^

(-1^ ^ VmJ

exp [-(wj/2)2] (54)

We notice here the error is 3 percent of the basic term and is further reduced

since both terms are basically the Fourier transform of the fundamental mode.

This is very weak at the locations where k ;^ 0. The resonance due to cell

positioning error is handled by having N(k) not located at those cell induced

resonances. Thus we neglect these error terms.

4.5 Evaluation of the Antisymmetric Nonlinear Part

The strength that has the desired signal is the r = 2 case of table 1 and

it has the foi^m given by eq (55).

Sl(n,k,2) =

I H (y ) [e - (-1) e J /Xf il=l " *

* B(z^) exp C-(w^^/2)2]. (55)

This expression contains all the details for the optical filter. It is useful

to rewrite this expression to get it into a real-function form and to establish the form for the gj^.

Fi rst we define

using

Si(n,k,2) =il^IiOA(n,k)(i)^(^) (56)

/Xf

WA(y^) = exp[-(w^^/2)2]

H

= 9/2,

and

40
g^ = [aD4 h^]q. (57)

Here the [ ]q represents the fact that the spot size adjustments are discrete

rather than continuous. We note the constraints

U < a < 1 (58)

Because of the way a spot is generated we have certain conditions on the

values of a. For now, a is equal to 0.6 for the x-dimension pattern and to

0.7 for the y-dimension pattern. The discrete conditions are in steps of 1 for a range of ±30 units for the x dimension and ±20 for the y dimension.

This latter condition means that the y dimension has more quantization error. This fact can affect the final strategy for development of the optical filter.

This point will be discussed in more detail in subsection 4.7. With the above

definitions A(n,k) becomes

A(n,k) =

I H^(y^)B(z^)WA(y^) [u(k)y^]/DO (59)

£=1 cos

with

r(k) = 1 for k = 1,3,5 use sin

= for k = 2,4,6 use cos.

(60)

and with

W(y^) = l/WA(y^). (61)

Notice that A(n,-k) = -A(n,k) for k odd and A(n,-k) = A(n,k) for k even. This means it is necessary to calculate only the positive k values. We define four

functions for convenience. Namely

Gl(s,kl,£) = W(y^)H^(y^) cos [u(s)y^] sin [kl Au y^.

41
G2(s,kl,?,) E W(y^)H^(y^) cos [u(s)y^] cos [kl Au y^],

G3(s,kl,P0 = W(y^)H^(y^) sin [u(s)y^] sin [kl Au y^],

and

G4(s,kl,£) = W(y^)H^(y^) sin Cu(s)yj,] cos [kl Au y^]. (62)

Using these definitions and noting the symmetry for g^, we can further define

in terms of some coefficients B(s,kl), C(s,kl) the composite functions. For

s = 1,3,5

N2

U(s,y.) = I [B(s,kl)G4(s,kl,^) + C(s,kl)Gl(s,kl , £)] (63)

^ kl=0

and for s = 2,4,6

N 2 U(s,y ) E ); [B(s,kl)G2(s,kl,£) + C(s,kl) G3(s,kl,£)] (64)

kl=0

(note N^ < 87).

The h^ is finally defined as

6

h^ = P a I U(s,yp (65)

s=l

where the P is adjusted to make contraint in eq (58) true.

A look at these equations shows that the relationships are nonlinear in

B(s,kl) and C(s,kl). This means the process for determining the values of B and C requires an iteration process to finally get the proper values of B and

C. The sequence is to get the value of A(n,k) to be as small as possible for

the cases n ^^ k and to get the values of A(n,n) to be approximately the same and as large as possible consistant with the constraints of the q.

The computation sequence is to start with a selected set of 6(s,kl) and

C(s,kl) and compute g^ with choice of contraints in a and the q operation in an interative manner until the values of g. are unchanged. For the range of

42
values of y^, this takes no more than three iterations. Once g^ is determined, then it is possible to compute A(n,s). The computation sequence

described here is strai yhtforward. The hard part is developing a means for

improving the selected set of B(s,kl) and C(s,kl) to get the best solution in

as efficient a way as possible. The next subsection addresses this problem.

4.6 Evaluation of the Antisymmetric Quasilinear Part

It is necessary to develop an iterative strategy to determine eventually

the proper B and C values in some least squares sense. This subsection shows

the necessary formulas and discusses the approximations and concepts necessary

to get these formulas. In summary, the process is to get a quasilinear relationship between the changes in B and C and relate it to the difference

between the current matrix A(n,k) and the desired form for A. We now step

through the formulation.

The change in B is designated as Bl(s,kl). The change in C is designated

as Cl(s,kl). The equation relating the error to the desired form is written

as

6 87 Al(n,k,s,kl)Bl(s,kl)

s=l kl=l A2(n,k,s,kl)Cl(s,kl)

Here, A^^'^'"^'^^(n ,k ) corresponds to the terms computed in eq (59) after the g^

terms have been stabilized to self consistancy.

The new B and new C are obtained from the old 6 and old C by

B'^(s,kl) = BO(s,kl) + Bl(s,kl)

C^s,kl) = CO(s,kl) + Cl(s,kl) (67)

w ith the constraint that the k. = case has no Bl and CI terms. Instead B

£ and C here are restricted to

B(s,0) =1 1 < s < 6 (68)

C(s,U) =

43
This constraint guarantees that the computer search for the ideal solution

will not cause the B and C to be zero.

If we replace y^ by x^^ and drops the (zj,)^ term in B(zj^), we get a term

linear in B(s,kl) and C(s,kl). As long as the primary source of mixing

between the different modes is dominated by the linear term we have a good

approximation. Since this is so, we can find the Al and A2 matrices. If

necessary, we could use the old g^ and replace Xp^ by y^ to get the quasilinear

form for the Al and A2 matrices. In this case, the computer analysis would

involve another level of computation. Fortunately, it is not necessary to do

this complex process. We now make a number of definitions. First, we have

the Fourier transform of the cross-correlation product written as

N QI[n,s,kx] = ); H^(x^)H^(Xj,)cos(kx.Au-x^)

N Ql[n,s,kx] =

I H^(x^)H^(x^)cos(kx.Au.x^) (69)

Next, we note the symmetries as

Ql[n,s,-kx] = Ql[n,s,kx]

Q2[n,s,-kx] = -Q2[n,s,kx] ^^ (70)

Also, there are four spatial frequency terms to be defined, namely

ka»Aus kl •Au + u(k)+u(s)

kb»AuE-kl»Au + u(k)+u(s)

kC'AuE kl •Au + u(k)-u(s)

kd»Au=-kl'Au + u(k)-u(s). (71)

Finally, a projection operator for k and s can be defined as

44
P^ (k) =|(1 ± (-1)^). (72)

Using these definitions, we can write out the form of Al and A2 where the top

sign corresponds to the Al case and the bottom sign corresponds to case.

Al(n,k,s,kl) = -^ [Ql(n,s,ka) [-P^

[Ql(n,s,kb) [+P^

[Ql(n,s,kc) [±P

[Ql(n,s,kd) [ P

[Q2(n,s,ka) [±P^

[Q2(n,s,kb) [ P^

[Q2(n,s,kc) [ P^

[Q2(n,s,kd) [±P^

k)P,

k)P

k)P

k)P

k)P

k)P

k)P

k)P

+ P (k)P

+ P (k)P

+ P (k)P

+ P (k)P

+ P_(k)P^

+ P (k)P

T P_(k)P^

- P_(k)P^

s)

s)

s)

s)

s)

s)

s)

he A2

(73;

We are finally able to set up the proper numerical analysis problem.

First we group the n, k labels into a single label and we group the s, kl,

with Bl or CI into a single label, thus

Map n Tt k into 1 < K3 < 30

Map s,Kl,B,C into 1 < K4 < 1044, (74)

We next construct a rectangular matrix M and two vectors T and E as follows

M[K4,K3] = [Al ® A2] ,

T(K4) = [Bl © CI],

45
and

E(K3) = -A^'''^"^n,k) n t k. (75)

It is now possible to write eq (66) in matrix form as

TM = E. (76)

We now do a singular value decomposition of M into two orthonormal matrices U and V and into a diagonal matrix D [19]. Thus,

M = UDV^. (77)

The solution in a least squares sense is given by

T = EVDlu"^. (78)

Here the requirement is made that Dl is inverted in a constrained way. Thus,

Dl = 1/D for D(£,) > lu""^ D (79) ^ '' max ^

'

or

= U for D(£,) < 10""^

^ ^' max'

where

1 < £^ < 30.

(BO)

We select this level of cutoff because the analysis is carried to the 0.1 percent level. To do better would require consideration of the dc and error

terms. We suggest that it is premature to work to a higher accuracy than this

0.1 percent.

There remains only one step in the determination of the B and C matrices.

It is necessary to determine when the error in the cross-correlation process

is sufficiently small. We define the error as

ERROR = ); |A(n,k)|/30. (81)

n^k

46
Define the signal as

SIGNAL =

I |A(n,k)|/6. (82)

n=k

Finally, accept the computer results as correct when

ERROR < lO"-^ SIGNAL. (83)

4.7 Two Fabrication Strategies--Which One Is Better?

We define briefly how the filter is constructed. The sequence starts

with the construction of a spot. A spot has up to 40 lines written in the y

dimension, and each of these lines has a length ranging from 2 to 8 ym in

length. The construction sequence is to generate a cell of 20 by 20 spots and

then to go to the next cell along the x dimension for a total of 3bO cells.

The of cells are generated in the y direction one row at a time until there

are 350 such rows of cells.

There are two basic construction strategies: The first is to have every

spot written with all 40 lines and to vary the length of the lines according

to the gj^ pattern given by the theory in this paper with a = 0.6. This pattern is generated for every spot in a cell and for every row of cells. The

resulting pattern would then look like a one-dimensional pattern with variable

width lines in the x dimension and running in the y dimension the full 70 mm. This master plate is then copied by contact printing to generate the secondary

plates. Two secondary plates are put together with one plate rotated 90°

relative to the other and placed so that the two light absorbing parts of each

plate are next to each other. In this way we have constructed a twodimensional pattern with maximum accuracy relative to the q operation. The

problems are to make sure the optical axis of each plate is exactly the same

and that the rotation angle is truly 90°.

The second strategy is to generate the y pattern as well as the x pattern

on a single master plate. The secondary plates are then generated from this

master plate by contact printing. Here the errors of positioning the plates

47
are eliminated. The trade-off is a reduced accuracy in the q operation along

the y dimension.

Both techniques need to be tested to determine which is more appropriate.

At this time, we think the first strategy will be the more accurate as well as more flexible. The key question will be how well the transfer process works

to create secondary plates. If it works well, then the cost of fabrication of

plates will be substantually reduced. If it does not work well, then it may

be necessary to generate each secondary plate just as the master was done.

This path will be expensive.

4.8 Necessary Work to Realize the Apparatus

The theoretical work has been completed and summarized in this paper.

The computer codes are written and are believed to be debugged. What needs to

be done now are:

1. Construction of the optical filter via the two strategies and testing

those units to confirm that there are no bugs in the computer code and to

check the accuracy of construction of the optical filter.

2. Construction of the tapered fiber with the specifications indicated in

this paper.

3. Construction of the 3-by-3 tapered fiber units with the piezoelectric

material. Testing of these units to verify they work as intended.

Issues that will need to be addressed are the time stability of the units

and hysteresis effects of the piezoelectric material.

4. Construction of the two optics, the telescopes, and Fourier transform

pair of lens, of this apparatus and testing them.

5. Finally the complete construction of the electric field apparatus and its

testing for accuracy and stability.

5. Conclusion

We now summarize this paper. There is the theoretical analysis of an electric field measuring apparatus. The key mathematical formulas for each

stage are shown and a concrete set of values are selected so that you can have

a picture of the apparatus.

48
The paper was structured to show the basic ideas and then to show the

complex ideas for construction of the optical filter.

In the basic ideas section, you saw first that there is need for a pair

of telescopes to match the beamwidth and phase front of the incident laser

beam to that needed at the optical filter. Second, you saw how the optical

filter is constructed and how its effect on the laser beam is transformed by a pair of lenses. Third, you learned that there are selected locations where

there are cross-correlation structures which contain information on the

strength, phase, and polarization of the modes used to represent the transverse spatial features of the electric field. Fourth, you learned that the

key step in measuring the electric field requires an array of tapered optical

fibers to capture the fundamental cross-correlation mode and to make it the

fundamental propagation mode of the fibers. You learned how this capture can

be performed and were given estimates of the errors involved in this capture.

Finally, you were shown how the beams from the fibers were separated according

to polarization and how the detectors were set up to measure the resulting

beams. At this point, you have the potential of 72 time series signals which

characterized the strength of each orthonormal mode in the original electric

field. In this paper, we assumed that the power is measured for each mode.

In the construction of the optical filter section, you saw the mathematics necessary for its construction and estimates of the errors involved in

making approximations to these mathematics. You learned that the basic con- struction of the optical filter requires solution to nonlinear equations and some very complex mathematics.

We finish this conclusion with two comments:

1. Once the apparatus is constructed, it will be possible to do a calibration process with a single well characterized cw beam to evaluate the

unit and to use some form of matrix analysis to reduce substantially some

of the remaining errors. This calibration process will define precisely

along the optical axis where we have made the electric field measurement.

Other locations would have their electric fields defined by using either

a new calibration scan or by using the theory to propagate the laser beam

to the new locations.

49
2. We have emphasized Hermite modes in this discussion. In order to have

good separation between the cross-correlation structures in the Fourier

plane, it is necessary to stay away from sharp cut-off structures for

representing the orthonormal set. For example, a plane wave illuminating

the optical filter would imply an Airy pattern where the effective range

of the electric field is 44 times greater than that implied by Hermite

modes. This means that beam steering of the cross-correlation process

would have to be incorporated to account for this kind of smearing of the

cross-correlation structure. We suggest it is best to avoid this problem

by using carefully smoothed corners in the .orthonormal mode set so that

this ringing in the Fourier plane is minimal [22].

In a complex work of this sort there are a large number of people and

groups who have helped me at various stages in the work. Matt Young, Aaron

Sanders, Don Jennings, Norris Nahman, Bruce Danielson, Douglas Franzen, Gordon

Day, Alvin Rasmussen, Robert Gallawa all helped in giving me their technical

advice and thoughts. The members of the Scientific Computing Division, namely

John Gary, Roland Sweet, Michael Maish, James Williams, Lloyd Fosdick, Robert

Schnabel , and David Lovering have all helped by giving me their thoughts on the mathematics and by making sure the computer would work for me. Gary and

Schnabel were especially helpful in developing the appropriate Singular Value

Decomposition Strategies. The excellent drafting was done by Nick Sanchez.

The manuscript preparation was handled by Edie DeWeese and secretarial activities were handled by Susie Rivera. The Combined Calibration Group (CCG) of

the Department of Defense has partially funded the background necessary to get

this work where it is today.

6. References

[1] Johnson, Eric G. Jr. Laser beam profile measurements using spatial

sampling, Fourier optics, and holography. Nat. Bur. Stand. (U.S.) Tech,

Note 10U9; 1979 January.

50
[2] Johnson, Eric G. Jr. Design of a reflection apparatus for laser beam

profile measurements Nat. Bur. Stand. (U.S.) Tech. Note 101b; 1979

July.

[3] Johnson, Eric 6. Jr. Beam-profile measurements of laser pulses using a spatial filter to sample the Hermite modes for a string of pulses. Nat.

Bur. Stand. (U.S.) Tech. Note 1057; 1982 September.

[4] Johnson, Eric G. Jr. Using optical processing to find the beam profile

of a laser pulse (theory). Proc. SPIE 499:75-88; 1985.

[5] Kogelnik, H.; Li, T. Laser beams and resonators. Proc. IEEE, 54:1312-

1329; 1966 October.

[6] Cathey, W. Thomas. Optical information processing and holography. John

W. Wiley & Sons, pp. 1-200; 1974.

[7] Allan, W. R. Fiber optics, theory and practice. Plenum Press, Chapter

9, pp. 177-197; 1973

[8] Lindgren, L.; Vilhelmsson, K. Mode excitation in graded-index optical

fibers. J. Lightwave Tech. LT-2(2) :559-563; 1984 August.

[9] Ozeki, Takeshi; Ito, Takao; Tamura, Toshifumi. Tapered section of multimode cladded fibers as mode filter and mode analyzers. Appl. Phys. Lett.

26(7):386-388; 1975.

[10] Montalti, F.; Valli, M.; Vannucci , R. Optical fiber technology--

analysis of tapered graded-index fibers. Elect. Commun. 56(4) :431-434;

1981.

[11] Modern examples of piezoelectric coefficients are shown in report PD-9247

Modern Piezoelectric Ceramics, Vernitron Corporation. No endorsement is

implied by using this example.

[12] Vidosic, J. P. Mechanics of material. Chapt. 5 in Standard handbook for

mechanical engineers, 8th Ed. T. Baumeister, E. A. Avalline, and

T. Baumeister III, eds. Macgraw Hill; 1978. pp. 5-38

[13] Hecht, Eugene; Zajac, Alfred. Optics. Addi son-Wesley; 1979. p. 234.

[14] Barnoski, M. K. Introduction to integrated optics. Plenum; 1974.

[15] Teich, M. C. Infrared heterodyne detection. Proc. IEEE 56:37-46; 1968.

[16] Sze, S. M. Physics of semiconductor devices. Wi ley-Interscience,

pp. 625-686; 1969.

[17] Day, G. W.; Hamilton, C. A.; Gruzensky, P. M.; Phelan, R. J., Jr. Performance and characteristics of polyvinylidene fluoride pyroelectric

detectors. Ferroelect. 10:99-102; 1976.

51
[18] National Computer Holography Facility at Aerodyne Research, Inc., see discussion in Laser Focus; November 1982 on pages 42-46. The design specifications where chosen since they have been well specified for use by

the general public.

[19] Nash, J. C. Compact numerical methods for computers: linear algebra and

function minimisation. John Wiley & Sons; 1979. Of particular interest

here is the discussion in chapter 3 on the singular value decomposition

procedure.

[20] Wilson, Scott D.; Reed, Timothy. Beam and spot profiles measurement

methods for optical storage systems. SPIE 499:65-74; 1984.

[21] Frank, J. D. Beam profile measurement for target designators. SPIE 499;

56-64; 1984.

[22] Priestley, M, B. Probability and mathematical statistics. Spectral

Analysis and Time Series, Vol. 1, Univariate Series, Academic Press;

1981. Here are some examples of cutoff techniques to avoid ringing.

52
NBS-114A (REV. 2-ec) U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO. NBS TN-10.84

2. Performing Organ. Report No. 3. Publication Date

August 1985

4. TITLE AND SUBTITLE

Direct Measurement of the Electric Field of a Laser Pulse—Theory

5. AUTHOR(S)

Eric G. Johnson, Jr.

6. PERFORMING ORGANIZATION (If joint or other thon NBS, see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP)

10. SUPPLEMENTARY NOTES

I I Document describes a computer program; SF-185, PIPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bi bliography or literature survey, mention it here) We consider realizing an electric field measuring apparatus by using optical processing, tapered optical fibers, and a pair of detectors at the end of each optical fiber,

Using an appropriate computer-generated optical filter, we show it is possible to discriminate among a set of orthonormal modes used to represent the spatial features of

the electric field with a signal-to-noise ratio of at least 100 to 1. If the positioning accuracies for various parts of the apparatus are properly set up, it is expected

that the signal-to-noise ratio could be about 1000. The purpose of the tapered and

graded-index fiber is to select the fundamental propagating mode in a fiber and to

attenuate the other modes. The existence of this fiber allows the precise determination of the strength of each of the orthonormal modes being used as the spatial basis

of the electric field before the optical processing. The detectors then measure the

strength of each fundamental mode. There are 36 such modes for the apparatus under

discussion--six for the x dimension and six for the y dimension. Propagation of the

laser pulse is assumed to be along the z axis. We use two detectors for each mode so that the polarization is determined.

This paper presents the conflicts in the design and gives a solution. The complete

evaluation requires assembly of the proposed apparatus to assess final accuracy.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon s) beam profile measurements; cross-correlation; electric field measurement; graded-index

fibers; laser pulses; matched filters; mode analysis; mode filtering; numerical analysis; optical fibers; optical filter; singular value decomposition; total beam control

13. AVAILABILITY

[V| Unlimited

I I For Official Distribution. Do Not Release to NTIS

rX~l Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C

20402.

I I Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

60

15. Price

* U.S. GOVERNMENT PRINTING OFFICE:1985— 580-369 I 25361 SCOMM-DC 6043-P80



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