SANDIA REPORT SAND951697 UC-910 Unlimited Release Printed November 1995

d ' Diffraction Efficiency Analysis for Multi-Level Diffractive Optical Elements

lreena A. Erteza

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SAND95-1697 Unlimited Release

Printed November 1995 Distribution Category UC-910

Diffraction Efficiency Analysis for Multi-Level Diffractive Optical Elements

1reena.A. Erteza Optical Systems and Image Processing Department

S andi a National Lab or at ories Albuquerque, NM 87185

. Abstract

PBssive optical components can be broken down into two main groups: refractive elements and diffractive elements. With recent advances in manufacturing technologies, diffractive optical elements are becoming increasingly more prevalent in optical systems. It is therefore important to be able to understand and model the behavior of these elements.

In this report, we present a thorough analysis of a completely general diffractive optical element (DOE). The main goal of the analysis is to understand the diffraction efficiency and power distribution of the various modes affected by the DOE. This is critical to understanding cross talk and power issues when these elements are used in actual systems.

As mentioned, the model is based on a completely general scenario for a DOE. This allows the user to specify the details to model a wide variety of diffractive elements. The analysis is implemented straightforwardly in Mathematica. [i'] This report includes the development of the analysis, the Mathematica implementation of the model and several examples using the Mathematical analysis tool. It is intended that this tool be a building block for more specialized analyses.

Contents 1 Introduction 1

1.1 General Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Diffractive Optical Elements . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Theory 2 2.1 Normal Incidence . . . . . . . ; . . . . . . . . . . . . . . . .’ . . . . . . . . . 2

2.1.1 Phase Function of Grating . . . . . . . . . . . . . . . . . . . . . . . 2 2.1.2 Diffraction Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Non-Normal Incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2.1 Phase Function of Grating . . . . . . . . . . . . . . . . . . . . . . . 4 2.2.2 Diffraction Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2.3. Focal Lengths for Various Orders . . . . . . . . . . . . . . . . . . . 7

2.3 Uniform Distribution Power Calculations . . . . . . . . . . . . . . . . . . 8 2.4 Gaussian Distribution Power Calculations . . . . . . . . . . . . . . . . . . 8

2.4.1 Complex Gaussian Parameter . . . . . . . . . . . . . . . . . . . . . . . 8 2.4.2 Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

c

3 Implementat ion in Mathemat ica 10 3.1 Mathematica Package (D0Eefficiency.m) . . . . . . . . . . . . . . . . . . . 10

3.3 Mathematica Scripts (maketab1e.m) . . . . . . . . . . . . . . . . . . . . . 15 3.2 Variables and Parameters (inputfi1e.m) . . . . . . . . . . . . . . . . . . . . 14

4 Examples ’ 15 4.1 Two Step Levels in Grating Period: “P2” . . . . . . . . . . . . . . . . . . 17 4.2 Three Step Levels in Grating Period: “P3” . . . . . . . . . . . . . . . . . 18 4.3 Four Step Levels in Grating Period: “P4” . . . . . . . . . . . . . . . . . . 19 4.4 Five Step Levels in Grating Period: “P5” . . . . . . . . . . . . . . . . . . 20 4.5 Six Step Levels in Grating Period: “P6” . . . . . . . . . . . . . . . . . . . 21 4.6 Eight Step Levels in Grating Period, 11 Modes Calculated: “P8” . . . . . 22 4.7 Eight Step Levels in Grating Period. 17 Modes Calculated: “P8-8modes” 23 4.8 Four Step Levels in Grating Period. Incident Angle = $: “P4thetaPi36” 24 4.9 Four Step Levels in Grating Period. Incident Angle = E: “P4thetaPi72” 26 4.10 Four Step Levels in Grating Period. Receiver Distance = 150 microns:

“recdistance” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5 Conclusion 28

List of Figures 1 General Scenario for DOE Analysis . . . . . . . . . . . . . . . . . . . . . . 1 2 Diffractive Grating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 Plane Wave Incident at Non-Normal Incidence . . . . . . . . . . . . . . . 5 4 Grating Geometry with Various Variables . . . . . . . . . . . . . . . . . . 6 5 Geometrical Picture of Extreme Ray of qth Order . . . . . . . . . . . . . 8

.. 11

1 Introduction

This analysis addresses the efficiency and power distribution between different modes in a diffractive optical element (DOE). This is useful in determining cross-talk issues among modes.

In this analysis, effort has been made t o make the analysis as general as possible. In doing so, there are many parameters available to the user to describe the exact situation to be analyzed.

We will begin by describing the general scenario. In the following section, the 'actual theory and equations used will be derived and presented. Then the implementation in Mathematica will be discussed, and some example calculations shown.

1.1 General Scenario

The general scenario is shown in Figure 1. Light is incident at angle 8 in medium of index p ~ . It hits a diffractive optical element of medium of index p. Because of the change in index of refraction, the light inside the DOE has an angle 8'. The thickness of the nth step in the diffractive optical element is d,. The medium underneath the DOE has index p ~ .

x

I I

Figure 1: General Scenario for DOE Analysis

From the analysis, we want to know if light is incident on the DOE, how much power goes into the various diffracted orders, and what is the power distribution for those different orders at various distances. Then, the power hitting the receiver or other locations can be calculated.

1.2 Diffractive Optical Elements

A diffractive lens emulates the phase function of a refractive lens, using a phase grating. The phase grating imparts the same phase delay as a refractive lens, but modulo 2w. The modulo 2w grating is called a blaze.

If the ,phase function has continuous phase levels, then 100% efficiency can be obtained. Typically, however, the continuous valued phase function is approximated by discrete values, giving rise to “Binary Diffractive Optics”. The analysis to’be presented is directed toward binary diffractive optical elements.

The phase function of a diffractive lens is periodic in $, with period A. Therefore, the lens transmittance can be expanded into a Fourier series in the variable $ or &. The blaze as a function of $ or & is a stepped ramp. Fourier decomposition describes this

integer. Recall that the phase of a spherical wave is exp(-iF$) for a wave with focal length f. Therefore, the various orders have focus at i. stepped ramp as the weighted sum of terms exp(-i2wqz), where z = 2 f ~ , rz and q is an

If the medium has index 7, then a plane’ wave incident on a phase plate with phase exp(-iqF:$) results in a wave identical to a converging spherical wave in media 7, converging at a focus, d.

9

To summarize, the focal power of a lens depends onZy on what kind of phase delay is imparted to the wave exiting the lens. For a refractive lens, the phase delay distribution will be of the form: $ ( r ) = bo(f - d m ) . An incident plane wave then becomes exp(i(yt - $ ( T ) ) . If f >> T , then $(r ) x 7. -zr2

The Fourier analysis of a diffractive lens is based on the following statements:

1. The lens is a phase plate.

2. Taking the Fourier series of the phase decomposes an incoming plane wave into a

3. The focal lengths of these waves is dependent on the periodicity of the phase‘plate.

series of converging and diverging spherical waves.

2 Theory

2.1 Normal Incidence

Consider the problem of a plane wave illuminating the grating shown in Figure 2.

2.1.1 Phase Function of Grating

The effect of the grating is to impart a phase function onto the incoming light. For an incident plane wave, the phase function after passing through the grating can be given by exp[-i$(f, 41.

2

Figure 2: Diffractive Grating

If surrounding medium is air, and

P = number of step levels in the grating period

A = one grating period

0, d l , 2dl,3dl.. . ( P - l ) d l = geometrical heights of the steps

d , = ndl f o r n = 0 , 1 , 2 ... P - 1 , o r p A < t < n - n+ l A P

p(A) = index of refraction of step material

8 = angle of incidence

then $,(A) = n@l(A)

The phase steps $n(X) form a stepped phase function $(<, A)

n = 0 , 1 , 2 . . . P - l .

For maximum efficiency on a blaze, $, = $2r, 0 5 n < P - 1, resulting in

where p = p(A0) and A0 is the design wavelength.

3

2.1.2 Diffraction Efficiency

If a grating is illuminated by a plane wave, the light is split into several diffraction orders. The power into the different orders can be found by taking the square of the coefficients of the Fourier series.

Let uq(X) be the fraction of radiant energy at wavelength X diffracted into order q by the grating.

sin r A where [ = A - 2. Then using C,";: exp [*Am] = a=,

2.2 N.on-Normal Incidence

A more general scenario includes the possibility of the light coming in at an angle, as in Figure 1. For non-normal incidence, the phase delay imparted by the grating must be calculated as a function of the incident angle. In addition, the phase function of the incoming plane wave is no longer constant along the direction; it is a function of [.

2.2.1 Phase Function of Grating

The total phase delay for normal incidence is given by:

(b = k/& + kPR(dp - dn)

4 = LPRdp + k(P - PR)dn, where k = 2n/X and dp is the height of the thickest step.

From Snell's law, if the incident angle is 0, p ~ s i n ( 0 ) = psin(0'). The total phase delay is then:

dP P PR (b = ~ P R - + k(- - -)dn, cos 0 cos0' cos0

So the phase delay in going through the nth step is

(8) 2n 1 x cos0'cos0 + n = - (P COS 0 - /.LR COS 0')dn

If the DOE is designed for maximum efficiency for non-normal incidence, then

n2n 2n 1 ?jn=-=- (P COS 0 - /LR COS 0')dn, P x cos0'cos0 (9)

4

resulting in

where dl = d,/n.

2.2.2 Diffraction Efficiency

If the light incident on the grating is not normal t o the grating, then the phase of the wave is not constant over the grating. The phase after the grating is the product of the phase incident on the grating and the transmittance function of the grating. To find the diffraction efficiency of the various orders of the grating, we need to find the Fourier decomposition of the product of the incoming phase and the phase function of the grating.

To describe the incoming phase distribution from a non-normal plane wave, consider Figure 3. If X is the wavelength of the illumination, then the phase function of the plane wave is given by

Figure 3: Plane Wave Incident at Non-Normal Incidence

The fraction of radiant energy at wavelength X diffracted into order Q by the grating, uq(X), is then

where

and

n = 0 , 1 , 2 . . . P - 1 .

5

With the help of Figure 4, we can rewrite this as

a m = 1112 (12)

(13) -i2n I = 1' exp [+A - z)pRsin e] exp[-i+(z, A)] exp[2niqzldz

(14! n n + l - 5 X S - P P - 4 = n?ltl(x>,

3 -i2n P-1 I = /n exp [T(A - z ) p ~ sin 6 exp[-in$l(X)] exp[2aiqz]dz ' (15)

n=O B

... 1 - x

A - 5

0 1IP ZIP 3/P

0 Y P U P W P ...

Figure 4: Grating Geometry with Various Variables

First, let's do the integral, integrating only those things that are a function of z.

Substituting this result in the equation for I ,

Looking at just the summation and recalling

sin[nA] N-1

m=O Am] = sin[nA/N]

6

yields

(q- +pR sin e)n

P ])I. (18) A sin [. (+ -

P 2n

Therefore we have:

uq(X) = 1112 = II*, therefore

where $1 = $l(X).

2.2.3 Focal Lengths for Various Orders

The actual focal length of a diffractive lens depends only on the periodicity of the phase function it imparts, and on the medium after the lens.

.*

Recall that a blazed grating is periodic in the quantity & with period 1. This means that if light of wavelength A is incident on this element and then passes through air, that the first order focus will be at F. We use the symbols A and F to emphasize that these are "design" parameters.

If wavelength of light X is incident on a grating that is periodic in & with period 1, and then passes through air, the first order focus will be at F. This is because from the Fourier decomposition, we have a phase function broken into components exp[-i2nqz]. If this wave is propagating in air, then

If this wave is propagating in a medium with index q, then

meaning that the first order focus in medium of index q is y . Note, for a DOE, the only fixed physical parameter is the groove spacing. If you take a DOE, you will find that it is periodic in T ~ . Let that measured period be designated T,. For light of wavelength X i incident on this DOE, the first order focus will be at %.

7

2.3 Uniform Distribution Power Calculations

The paraxial focus of the Qth diffracted order is at a distance f awz from the 1 ns. Using geometrical optics, one can calculate the radius, x;, at plaie z where the extreme ray from the edge of the zone plate which passes through the qth order focus strikes. See Figure 5. This quantity, z;, is only meaningful for distances greater than 0. ( z = 0 + the plane of the DOE.) . .

Figure 5: Geometrical Picture of Extreme Ray of qth Order

I x q = wo, q = o

f z < - Q . f z 2 - Q f

q > o

q > O

q < o

=!l x; = wo(1- -),

xq = wg( - - l),

xq = wo(1- -),

f I zq

I zq f

-. !l f

Assuming light is uniformly distributed over the circle of radius xk, the percentage of light due to mode q hitting the receiver at a distance z is

2.4 Gaussian Distribution Power Calculations

Because lasers are often used to illuminate diffractive optical elements, it is useful to incorporate gaussian beams into the analysis.

2.4.1 Complex Gaussian Parameter

Recall the complex gaussian parameter, q(z) is defined as:

1 . A a -- 1 --

q(z) - R(z ) nnw2(z)'

The gaussian field is then given by

where

and k = F. n is the refractive index in which the gaussian beam is propagating. Power is the integral of the intensity squared, and I = EE*. E is normalized, using

~ 2 * ~ - e ~ ~ [ - 2 ~ 2 / w 2 ] ~ d ~ d e = -w2. 'IT

- 2

Since power is conserved, the total power in the beam is ,302. To be consistent with our notation, a = E:.

In the complex gaussian parameter q, R is the radius of curvature, and w is the beam waist. The evolution of the complex gaussian parameter as it propagates through a sequence of elements (propagation, lenses, interfaces) can be followed by applying the ABCD law as follows.

4 1 + B Cqi t D' q2 =

where A, B , C, D are the elements of the ray matrix which describes the relation of a ray in plane 1 to a ray in plane 2. If there is a sequence of elements, then the total ray matrix is the matrix product of the individual element ray matrices. The order of the product is important, with the last element's matrix occurring at the left of the product.

The ray matrix for propagation of a distance d in a homogeneous medium is

and the ray matrix for a lens of focal length f , (f > 0 converging, f < 0 diverging) is

1 0

I-t 11

For the case of a beam propagating through a lens with effective focal length propagating a distance d, the total ray matrix (hence the subscript T ) is

and then

Let's assume the gaussian beam hitting the DOE is a plane wave, with waist, 200. Then the complex gaussian parameter is

inawi q1 = - x

If the DOE is not at the laser waist, then q1 could have a real component.

The complex gaussian parameter at a distance d away from the DOE, can be found by applying Equations 26 and 29. Then the waist can be calculated from Equation 22.

9

2.4.2 Power

Once the waist at the distance z is found, then the relative power hitting a receiver of radius Rreceiver centered on the optical axis can be calculated using

aq [ 1 -exp [ -%yr]] To find the relative power hitting a more general location at a distance z from the DOE, the following two equations may be used:

1 2 R relative power = a - [Bend - Ostarf

- (33)

and

The first equation has the area of interest specified by polar coordinates, Tinner, Touter,

Bstart, Bend. The second equation has the area of interest specified by z and y coordinates, 51,129 Y1, Y2.

3 Implement at ion in Mat hemat ica

The analysis presented in the previous sections has been implemented in Mathematica. The package D0Eefficiency.m should be read in to set up the needed definitions and functions.

The variables and parameters for the specific problem to be analyzed can be read in from a file. An example of this is shown in the section Variables and Parameters. The parameters and variables can be changed at any time during a Mathematica session.

Finally, a Mathematica script called maketab1e.m is presented. Each time it is read in, it acts like a command to produce a table of pertinent results. The variables NumModes and RecPZane are used by maketab1e.m to determine the number of modes in the table, and the distance of the receiver plane. The output of maketab1e.m is saved in a file specified by the variable outputfile. This variable is a string, and can be changed at any time during a Mathematica session.

Maketab1e.m is just an example of how the package D0Eefficiency.m can be used. Other scripts can be made to produce custom outputs.

3.1 Mathematica Package (D0Eefficiency.m)

BeginPackage ["DOEef f iciency ' "1

(* PARAMETERS F.ROM ACTUAL DOE SPECIFICATIONS

10

rReceiver = rad ius of rece iver aper ture D i a m = diameter of l e n s

(Only need 2 out of t h r e e of t h e following, s ince they a r e r e l a t ed : ) TO= per iod of g ra t ing i n r squared lambda0 = design wavelength f O = design f o c a l length (TO = 2 lambda0 f0 ) (If you're comparing designs, then values f o r lambda0 and f O are known. and lambda0, t o ca l cu la t e f O . )

If you have an ac tua l DOE, then you can use TO (measured)

muL = index of r e f r a c t i o n i n subs t r a t e mu = index of r e f r a c t i o n of g ra t ing muR = index of r e f r a c t i o n of media above g ra t ing

P = number of s t e p l e v e l s i n t h e optimthetao i s True i f d l is optimized f o r inc ident angle the ta0 ;

g ra t ing per iod

i s False i f d l is optimized f o r normal incidence '

waist0 = input waist the t a0 = angle of incidence f o r which d l i s optimized lambda = wavelength, ca lcu la ted as lambdaO/lambdaratio t h e t a = angle of incidence upon g ra t ing lambdaratio = r a t i o of lambda0 t o lambda *I

(* CALCULATED QUANTITIES (Intermediate) :

f a = ac tua l f o c a l length

thetaprime = angle ins ide g ra t ing thetaL = angle i n subs t r a t e

d l = thickness of each s t e p (can be optimized f o r normal o r oblique incidence)

P h i l = t h e phase delay from one s t e p *I

(* CALCULATED QUANTITIES (Final) : a[q] = r e l a t i v e power i n order q xprime[z,q] = rad ius of spot a t d is tance z, due t o order q z=O ==> l oca t ion of t h e g ra t ing ro[z,ql = r a t i o of area of order q t o rece iver relpower[z,q] = r e l a t i v e percentage of l i g h t , due t o order q,

cgp[z,ql = complex gaussian parameter a t d is tance z, f o r order q a t d i s t z h i t t i n g recvr

11

waist[z,q] = waist f o r gaussian beam a t d is tance z, f o r order q gaussrelpower[z,ql = r e l a t i v e power h i t t i n g t h e r ec i eve r , due t o order q, a t d is tance z, assuming gaussian beams relpouerpolaruindow[z,q,rl,r2,thi,th2] = t h e amount of r e l a t i v e power i n a window between r ad i i rl and r2 , and angles t h l and th2 , at a d is tance z f o r order q, assuming gaussian beams relpouerwindow[z,q,xl~x2,yl,y21 = t h e amount of r e l a t i v e power i n a window between x i and x2, and y l and y2, a t a d is tance z f o r order q , assuming gaussian beams *> lambda : = lambdaO/lambdaratio

fa::usage = " f a is t h e ac tua l f o c a l length of t h e DOE, with muL and muR t h e surrounding ind ices of r e f r ac t ion" f aair : = lambdarat io*f 0 f a := faair*muL

thetaprime::usage = "thetaprime is t h e angle in s ide t h e DOE" thetaprime := ArcSin[muR/mu * SinCthetal] thetaoprime : = ArcSin[muR/mu * Sin [theta01 I

(* t he t a0 = angle of incidence f o r which d l is optimized *) (*optimthetaO is t r u e i f g ra t ing optimized f o r incident angle of t h e t a *> optimthetao = False Phi1::usage = "Phi l is t h e phase delay through one s t e p l e v e l of t h e DOE" P h i l := 2*Pi*lambdaratio/P /; !optimthetaO $8 theta==O P h i l := 2*Pi*lambdaratio/P * (mu*Cos [theta] - muR*Cos [thetaprime] ) / (mu*Cos [theta01 - muR*Cos [thetaprime01 ) * (Cos [theta01 * Cos [thetaprime01 ) /(Cos [theta] * Cos [thetaprimel /; optimthetao P h i l := 2 * P i * lambdaratio * (mu*Cos[theta] - muR*Cos[thetaprime])/ (P * CosCtheta] * Cos[thetaprime] * (mu - muR) )

a::usage = "a[order] gives t h e r e l a t i v e power i n a

a[orderval-I: = N [ L i m i t [ ( Sin[Pi/P* (order - lambdaratio*muR*Sin [theta] )I / (Pi/P*(order - lambdaratio*muR*Sin[thetal > > 1-2 * ( Sin[P*(Phil/2 -(order (P*Sin[Phil/2 -(order - lambdaratio*muR*Sin[theta] )*Pi/P] 1-2, order->orderval] 1

given d i f f r a c t e d order"

- lambdaratio*muR*Sin[thetal )*Pi /P)I /

12

xprime::usage = "xprime is t h e rad ius of t h e d i f f r a c t e d beam of a given order a t a given d is tance , assuming plane waves of uniform u n i t amplitude. xprime is only meaningful f o r d i s tance g r e a t e r than 0. xprime Cdist, ,order,] := N C w a i s t O * (order/f a*dist -113 / ; ((order) (d i s t>= (f a/order) ) )

k%

xprimeCdist,,order,l := N[waistO*(l-dist*order/fa)] /; (order>O)&% ( d i s t < ( f a/order)&&(dist>=O)) xprime Cdist, ,order,] : = N [waistO* (-order/f a*dis t +1)3 / ; (orderC0) && ( d i s t >=O) xprime Cdist, ,order,] : = N C w a i s t O l / ; order==O

ro::usage = "the f r a c t i o n of power of a given order a t a given d is tance h i t t i n g t h e rece iver of rad ius rReceiver, assuming plane waves of uniform u n i t amplitude" r o [dist , , order,] : = l/ ; rReceiver > xprime [d is t , order] r o [dist , , order,] : = l/ ; xprime [d is t ,order] == 0 r o [dis t - , order,] : = N C(rReceiver/xprime [ d i s t , order] 1-23

relpower: :usage = "the r e l a t i v e power of a given order a t a given d is tance h i t t i n g t h e rece iver ,

relpower [d is t - , order,] : = r o [ d i s t ,order] *a [order] assuming plane waves of uniform u n i t amplitude"

cgpl := (I*muR P i waist0^2)/lambda (* cpgl could have a r e a l p a r t i f DOE i s n ' t a t t h e laser waist *)

cgp: :usage = "cgp is t h e complex Gaussian parameter of t h e beam f o r a given order a t a given dis tance" cgp [dis t - ,order-] : = (cgpl*(l-dist*order/f a) + d i s t ) / (cgpl* (-order/f a) + 1)

waist: :usage = "waist is t h e gaussian beam waist f o r a given order a t a given d is tance . Waist i s only meaningful f o r d i s tance g r e a t e r than 0." waist Cdist, ,order,] : = N [ Sqr t [lambda/

(ImCl/cgp Cdist ,order1 I *muL*Pi* (-1) 1 1

1 /; d i s t >= 0

gaussre1power::usage = "the r e l a t i v e power of a given order at a given d is tance h i t t i n g t h e rece iver , assuming Gaussian beams" gaussrelpower [dist , ,order,] : = a [order] * (1 - Exp [-2*rReceiver-2/ (waist [ d i s t ,order] -211 )

relpowerpo1arwindow::usage = "relpowerpolarwindow gives t h e

13

r e l a t i v e power i n a window denoted by inner

f o r a given order and dis tance" relpowerpolarwindow [dist , ,order, , r in , ,rout, , t h s t a r t , ,thend,l :=

(Exp [-2*rin-2/ (waist [d i s t ,order1 -211 - Exp [-2*rout^2/ (waist Cdist ,order1 -211

and outer rad ius , and t h e t a start and t h e t a end

l/ (2*Pi) *a[orderl * ( thend-thstar t ) * .__.

re1powerwindow::usage = "relpowerwindow gives t h e r e l a t i v e power i n a window denoted by xl ,x2, and y l , y2 f o r a given order and dis tance" relpowerwindow [dist , ,order, ,xi, ,x2, ,yl, ,y2,1: = a [order] * NIntegrate C2/ (Pi*waist Cdist ,order] -2) * Exp C-2* (x-2+ y-'2) / (waist [d i s t ,order] -211,

d i f f ef f Cabsorder,] : = <x 9x1 , ~ 2 3 , <y , y1 ,y231

Table [a [i] ,(i , -absorder , absorder)]

outputpwr [numorders,] : =

EndPackage n Table[relpower[i*lO^-6,j] ,(i ,O,numorders),(j, 1 ,numorders)l

3.2 Variables and Parameters (inputfi1e.m)

rReceiver = 100 10--6 D i a m = 100 10--6 lambda0 = 980 10--9 lambdaratio = 1 f O = 100 10--6 muL = 1.0 mu = 3.5 muR = 1.0 P = 4 t h e t a = 0 uaist0=30 10--6 optimthetao = False the t a0 = 0 (*if optimtheta = True, then need t o specify thetaO*)

NumModes = 5 (* w i l l s tudy modes from - NumModes t o + NumModes *) RecPlane = f a (* t h e d is tance of t h e rece iver plane *)

outputf i l e = "outputf i l e "

14

I

3.3 Mathematica Scripts (maketab1e.m)

defficiencies = diffeffCNumModes1 dradii = Table [xprime [RecPlane , il , (i , -NumModes ,NumModes)l dwaists = Table [waist [RecPlane , i] ,(i, -NumModes ,NumModes)] drpwr= Table [relpower [RecPlane, il , Ci , -NumModes ,NumModes)] dgausspwr= Table Cgaussrelpower CRecPlane , il ,(i, -NumModes ,NumModes)l

titles=Table [Mode [il , Ci , -NumModes ,NumModes)] PutAppendr OutputForm I: TableForm[(defficiencies,dradii,dwaistsadrpwradgausspwr~, TableDirections->(Row , Column, Row ,Column ,Row), TableSpacing->C3,13, TableHeadings->(C"ef f ic" , "radii", "waists", llrpwrll, "gausspwr"), titles)]] , outputf ilel PutAppend [" II , outputf ilel PutAppend ["P = I f , P , outputf ilel PutAppend ["Receiver plane at d = PutAppend["******************************************************** #I,

outputf ilel PutAppend [I1 I 1 3

outputf ile]

,RecPlane , outputf ilel

4 Examples

In this section we show some sample output using the files DOEefficiency-m, inputfi1el.m and maketab1e.m. The files were generated using the following commands.

<<DOEefficiency.m <<inputfilel.m

outputfile = "P2" P = 2 <<maketable .m

outputf ile = "P3" P = 3 <<maketable .m

outputf ile = "P4" P = 4 <<maketable .m

outputfile = "P5" P = 5

<<maketable .m

outputfile = "P6" P = 6 <<maketable .m

outputf i l e = "P8" P = 8 <<maketable .m

outputf i l e = "P8-8modes'' NumModes = 8 <<maketable .m

outputf i l e = "P4thetaPi36" theta = Pi/36 P = 4 <<maketable .m

outputf i l e = "P4thetaPi72" theta = Pi/72 NumModes = 5 <<maketable .m

outputf i l e = "recdistance" theta = 0 RecPlane = 1 .5 10--4 <<maketable .m

Quit

The files generated are as follows.

4.1 Two Step Levels in Grating Period: “P2”

eff i c r a d i i waists rpur gaus spur

Mode[-51 0.0162114 0.00018 0.000180003 0.00500352 0.00746666

Mode[-41 0 0.00015 0.000150004 0 0

. Mode [-SI 0.0450316 0.00012 0.000120005 0.031272 0.0338017

Mode[-21 0 0.00009 0.000090006 0 0

Mode [-11 0.405285 0.00006 0.000060009 0.405285 0.403715

ModeCOl 0 0.00003 0.000030018 0 0

-6 1.03981 10 0.405285 0.405285 ModeClI 0.405285 0.

ModeC21 0 0.00003 0.000030018 0 0

Mode[3] 0.0450316 0.00006 0.000060009 0.0450316 0.0448573

ModeC41 0 0.00009 0.000090006 0 0

ModeC51. 0.0162114 0.00012 0.000120005 0.0112579 0.0121686

1tp = II

2 ”Receiver plane a t d = 0.0001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ‘I

II I t

11 11

4.2 Three Step Levels in Grating Period: “P3”

eff i c

Mode [-53 0.0273567

Mode[-41 0

Mode[-33 0

r a d i i waists

0.00018 0.000180003

0.00015 0.000150004

0.00012 0.000120005

Mode C-21 0.170979 .. 0.00009 0.000090006

Mode[-11 0 0.00006 0.000060009

ModeCOl 0 0.00003 0.000030018

Mode E11 0.683918

ModeC21 0

ModeC3I 0

Mode [41 0.0427449

0. -6

1.03981 10

0.00003 0.000030018

0.00006 0.000060009

0.00009 0.000090006

0.00012 0.000120005

rpwr gausspwr

0.00844343 0.0126

0 0

0 0

0.170979

0

0

0.683918

0

0

0.0427449

0

0.1565

0 *

0

ModeC51 0

sp = I 1

3 “Receiver plane a t d = 0.0001

I t

It* * * * * * ** * * ** * ***** * * * * * * ** * * * * * ** * * * * * ** * * * * * * ** * * * * * * * * It

I 1 I t

0.683918

0

0

0.039125

0

4.3 Four Step Levels in Grating Period: “P4”

Mode C-51

Mode [-43

Mode 1-33

Mode C-21

Mode [-11

Mode [O]

Mode [l]

Mode [2]

Mode [31

Mode C41

Mode [SI

Ilp = II

4

I1

ef f ic

0

0

0.0900633

0

0

0

0.810569

0

0

0

0.0324228

radii

0.00018

0.00015

0.00012

0.00009

0.00006

0.00003

0.

0.00003

0.00006

0.00009

0.00012

waists

0.000180003

0.000150004

0.000120005

0.000090006

0.000060009

0.000030018

-6 1.03981 10

0.000030018

0.000060009

0.000090006

0.000120005

rpwr

0

0

0.0625439

0

0

0

0.810569

0

0

0

0.0225158

gausspwr

0

0

0.0676035

0

0

0

0.810569

0

0

0

0.0243372 II

“Receiver plane at d = 0.0001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1 11

19

. ._

4.4 Five Step Levels in Grating Period: "P5"

, e f f i c

Mode C-51- 0

Mode C-41 0.0546963

Mode[-31 0

Mode[-21 0

Mode[-11 0

Mode101 0

Mode C11 0.87514

Mode121 0

ModeC31 0

Mode141 0

r a d i i

0.00018

0.00015

0.00012

0.00009

0.00006

0.00003

0.

0.00003

0.00006

0.00009

waists

0.000180003

0.000150004

0.000120005

0.000090006

0.000060009

0.000030018

-6 1.03.981 10

0.000030018

0.000060009

0.000090006

rpwr gausspwr

0 0

0.0243095 0.032209

0 0

0 0

0 0

0 0

0.87514 0.87514

0 0

0 0

0 0

ModeCSl 0 0.00012 0.000120005 0 . 0

up = II

5 "Receiver plane a t d = 0.0001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

II II

II II

20

4.5 Six Step Levels in Grating Period: "P6"

eff i c r a d i i waists rpwr gausspwr

Mode [-SI 0.0364756 0.00018 0.000180003 0.0112579 0.0168

Mode[-41 0 0.00015 0.000150004 0 0

Mode[-31 0 0.00012 0.000120005 0 0

Mode[-21 0 0.00009 0.000090006 0 0

Mode[-11 0 0.00006 0.000060009 0 0

ModeCO] 0 0.00003 0.000030018 0 - 0

-6 ModeCll 0.911891 0. 1.03981 10 0.911891 0.911891

ModeC23 0 0.00003 0.000030018 0 0

ModeC33 0 0.00006 0.000060009 0 0

ModeC41 0 0.00009 0.000090006 0 0

ModeC51 0

Ilp = I 1

I 1

0.00012 0.000120005 0 0 11

6 "Receiver plane a t d = I'

0.0001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1

I1 11

21

4.6 culated: "P8"

Eight Step Levels in Grating Period, 11 Modes Cal-

ef f ic

Mode[-53 0

Mode[-41 0

Mode[-31 0

Mode[-21 0

Mode[-11 0

ModeCOl 0

r a d i i wais ts rpur

0.00018 0.000180003

0.00015 0.000150004

0.00012 0.000120005

0

0

0

0.00009 0.000090006 0 :-*

0.00006 0.000060009

0.00003 0.000030018

Mode111 0.949641 0.

Mode121 0

ModeC3l 0

ModeC41 0'

0.00003

0.00006

0.00009

ModeC51 0 0.00012 I t

up = I1

8 "Receiver plane a t d = 0.0001

-6 1.03981 10

0.000030018

0.000060009

0.000090006

0.000120005

0

0

gaus spur

0

0

0

0.949641 0.949641

0 .-

0

0

0 11

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I 1 11

22

4.7 culated: "P8-8modes"

Eight Step Levels in Grating Period, 17 Modes Cal-

e f f i c r a d i i wais ts rpur gaus spur

0 0

0.00336466 0.00568519

Mode[-81 0 0.00027 0.000270002

Mode [-71 0.0193804

Mode[-61 0

0.00024 - 0.000240002

0.00021 0.000210003

0.00018 0.000180003

0.00015 0.000150004

0.00012 0.000120005

0.00009 0.000090006

0 0 -..

Mode[-51 0

Mode[-41 0 r' ..

Mode[-31 0

Mode[-21 0

Mode[-11 0

ModeCOl 0

0.00006 0.000060009 0

0

0

0 0.00003 0.000030018

-6 1.03981 10 0.949641 ModeCll 0.949641 0. 0.949641

ModeC21 0

Mode[3] 0

ModeC41 0

ModeC51 0

0.00003 0.000030018

0.00006 0.000060009

0.00009 0.000090006

0.00012 0.000120005

0 0

ModeC6) 0 0.00015 0.000150004 0

ModeC73 0 0.00018 0.000180003

0.00021 0.000210003

0

0

0

ModeC81 0

Ilp = II

8 "Receiver plane a t d = 'I

0.0001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II II

II

0 II

23

4.8 = L. “P4t het aP i36”

Four Step Levels in Grating Period, Incident Angle 36’

eff i c r a d i i waists rpwr gausspwr

-7 Mode[-81 9.1125 10

-7 1.25 10

-7 2.18636 10 0.00027 0.000270002

0.00024 0.000240002

0.00021 0.000210003

0.00018 0.000180003

Mode C-71 0.0136199 0.00236456 0.00399535

Mode C-63 0.000448798 0.000101768 0.000163633

Mode [-51 0.000159434 0.0000492081 0.0000734324

-6 Mode[-41 3.56769 10

. - 6 . 0.00015 0.000150004 . 1.58564 10

- 6. 2.1009 10

Mode C-31 0.0717795 0.00012 0.000120005 0.0498469 0.0538793

Mode [-23 0.00381742

Mode [-11 0.00349098

Mode [O] 0.0078458

0.00009 0.000090006 0.00381742 0.00349414

0.00006 0.000060009 0.00349098 0.00347747

0.00003 0.000030018 0.0078458 0.0078458

-6 1.03981 10 0.820964 Mode E11 0.820964

Mode 121 0.00454486

0. 0.820964

0.00454486 0.00003 0.000030018 0.00454486

Mode C3l 0.000486292 0.00006 0.000060009 0.000486292 0.000484409

-6 ’ ‘.

ModeC41 3.89264 10 0.00009 0.000090006 -6

3.89264 10 -6

3.56299 10

Mode [51 0.0283434 0.00012 0.000120005 0.0196829 0.0212752

Mode E61 0.000475649 0.00015 0.000150004 0.00021 14 0.000280095

Mode 171 0.0000863412 0.00018 0.000180003 0.0000266485 0.0000397671

-7 ModeC81 9.5184 10

Ilp = I I

4 “Receiver plane a t d = 0.0001

I t

-7 2.15837 10

II

-7 3.47043 10 0.00021 0.000210003

24

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 'I

25

4.9 - vr) “P4thetaPi72”

Four Step Levels in Grating Period, Incident Angle - E.

Mode C-51

Mode [-41

Mode C-31

Mode C-23

Mode C- 13

Mode [O]

Mode 113

Mode C23

Mode C31

Mode C41

Mode C5l

Ilp = 11

4

I 1

ef f i c

0.0000392628

-7 2.50903 10

0.0810384

0.000958416

0.000917026

0.00215619

0.820748

0.0010458

0.000114273

-7 2.62089 10

0.0305592

“Receiver plane at d = ‘I

0.0001

r a d i i

0.00018

0.00015

0.00012

0.00009

0.00006

0.00003

0.

0.00003

0.00006

0.00009

0.00012

waists

0.000180003

0.000150004

0.000120005

0.000090006

0.000060009

0.000030018

-6 1.03981 10

0.000030018

0.000060009

0.000090006

0.000120005

rpwr

0.0000121182

-7 1.11512 10

0.0562767

0.000958416

0.000917026

0.00215619

0.820748

0.0010458

0.000114273

-7 2.62089 10

0.0212217 11

gausspwr

0.0000180837

-7 1.47749 10

0.0608292

0.000877252

0.000913475

0.00215619

0.820748

0.0010458

0.000113831

-7 2.39894 10

0.0229384

2G

4.10 tance = 150 microns: "recdistance"

Four Step Levels in Grating Period, Receiver Dis-

e f f i c r a d i i waists rpwr gausspwr

Mode[-51 0 0.000255 0.000255005

Mode[-41 0 0.00021 0.000210006

Mode [-SI 0.0900633 0.000165 0.000165007

Mode[r2] 0 0.00012 0.00012001

Mode[-11 0 0.000075 0.0000750162

0 0

0 0

0.0330811 0.0468581

--. 0 0

0 0

ModeCOl 0 0.00003 0.0000300405 0 0

ModeCll 0.810569 0.000015 0.0000150809 0.810569 0.810569

ModeC21 0 0.00006 0.0000600203 0 0

ModeC31 0 0.000105 0.000105012 0 0

ModeC41 0 0.00015 0.000150008 0 0

ModeC51 0.0324228 0.000195 0.000195006 0.0085267 0.0132609 II II

Ilp = II

4 "Receiver plane at d = 0.00015 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I'

II II

5 Conclusion

We have presented the analysis of a general diffractive lens, addressing the issues of efficiency and power distribution of the various modes. The analysis described in the first part of this report is implemented using Mathematica. The package D0Eefficiency.m contains the basic functions necessary to calculate a variety of quantities of interest in this problem. We have presented representative scripts which set values for the parameters of the problem and which can be used to produce output. Once D0Eefficiency.m is read in, any Mathematica commands may be used, and other scripts can be written to produce custom output.

z

References [l] Kai Chang, editor. Handbook of Microwave and Optical Components, volume 3. J.

[2] H. Dammann. Blazed synthetic phase-only holograms. Optik, 31:95-104,1970. Wiley and Sons, 1990.

[3J H. Dammann. Spectral characteristics of stepped-phase gratings. Optik, 53(5):409- 417, 1979.

[4] H. Nishihara and T. Suhara. Micro fresnel lenses. Progress in Optics XXIV, pages 1-37,1987. Elsevier Science Publishers B.V.

[5] G. Swanson. Binary optics technology: The theory and design of multi-level diffrac- tive optical elements. Technical Report T R 854, MIT Lincoln Labs, August 1989. Prepared for DARPA.

[6] G. Swanson. Binary optics technology: Theoretical limits on the diffraction efficiency of multilevel diffractive optical elements. Technical Report T R 914, MIT Lincoln Lab, March 1991. Prepared for DARPA.

[7] Stephen Wolfram. Mathematica: A System for Doing hfathematics by Computer.

[8] A. Yariv. Optical Electronics. Holt, Itinehart and Winston, 1985. Addison-Wesley, 2nd edition, 1991.

28

DISTRIBUTION:

MS 9018 MS 0899 Technical Library, 4414 MS 0619 Print Media, 12615 MS 0100 Document Processing, 7613-2

For DOE/OSTI

Central Technical Files, 8523-2

29

Org. Bldgi Name Rec'd by low. Bldg. Name Rec'd by