D-Ai86 399 OVERESTIMATION OF ON-AXIS PONER DENSITY BY THE iFRAUNHOFER ('FAR-FIELD ) APPROXINATTON(U) MITRE CORPBEDFORD MA M M WEINER APR 87 MITRE MB7-37

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April 1987 M 87-37

AD-A186 399

M. M. Weiner Overestimation of/On-Axis Powr Density

IIby the Fraun: fer("Far-Field")

I, Approximation

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SCUARE APERTUREOF SlOE LENGTH 2b

A - NEAR FIELD FCOVAiLATlON

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0 0.5 1 1.5 2 2.5 3

FRESNEL NUtEER. MN./X

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11 TITLE (Include Security Claspfication)

Overestimation of On-Axis Power Density by the Fraunhofer ("Far-Field") Approximation

12 PERSONAL AUTHOR(S)Weiner, Melvin M.

13a TYPE OF REPORT 13b TIME COVERED 114 DATE OF REPORT (Yea, Month, Day) 15 PA E COuNTFinal I FROM _____TO ____I1987 April j 2

16 SUPPLEMENTARY NOTATION

17 COSATI CODES 10 SUJBJECT SERWS (Continue on revrse if necesary and identy by block number)fIELD GROUP SUB-GROUP tenna Aperture

Aperture Distribution of Uniform Intensity and PhaseCircular Aperture i. (continued)

UM~ASSIFIED

18. Constructive InterferenceDestructive Interference,Diffraction/Far-Field2FraunhofierFresne 1

SFresneloKirchhoff IntegralFresnel NumberNear-FieldOn-Axis Intensityon-Axis Power DensityOptical ApertureRectangular Aperture

A E

"9t-1

April 1987 M87-37

M. M. Weiner Overestimation ofOn-Axis Power Densityby the Fnnhofer

V T--rr, _ .,,, ("Far-Field")

OF Mt a AproximationJI I

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DEPT D6.

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MITRE Project Approv

R. B. Stevenson

The overestimation of power density by the Fraahofer("far-field") approximation to the more exact Fresnel (Onear-field")diffraction integral is determined as a function of Fresnel number

at arbitrary on-axis field points for rectangular and circular

aperture illuminations of uniform aplitude and phase. For square

and circular apertures, the overestimation is 2.8 percent and 1.3

percent, respectively, at a Fresmnel nmber of 1/S (corresponding to

the far-field boundary distance 2D2 /X) and 518 percent and 147

percent, respectively, at a Fresnel number of 1.

*This work was sponsored by the lectronic System Division, AirForce System Commnd, Hanscom Air Force Ease, Massachusetts.

ii

-.

ThBLK OF CONTENTS

SECTION PAGE

1. introduction . . . . . . . . . . . . . . . . . . . . . . . 1-1

3. Far-Field Formulation . . . . . . . . . . . . . . . . . . 3-1

4. Near-Field Formulation . . . . . . . . . . . . . . . . . . 4-1

5. Comparison of Far-Field and Near-Field Formulations . . . 5-1

List of eferences . . . . . . . . . . . . . . . . . . . . . 6-1

Appiendix A Derivation of the Fraunhofer and FresnelDifracti n ntal ra.. .. .. . . . . .. A-1

SECTII 1

INTRCUCTON

The Fraunhofer and Fresnel diffraction integrals are commonly

used analytical tools for the evaluation of fields originating from

illuminated apertures. These integrals are derived in appendix Afrom the Fresnel-Kirchhoff integral subject to small angleconditions. Whereas the exponential factor of the Fresnel integrand

retains both linear and quadratic phase terms (as a function of the

aperture coordinates) and is applicable to near-field points as wellas far-field points, the Fraunhofer integrand retains only linearphase terms and is applicable only to far-field points. TheFraunhofer "far-field" formilation is, therefore, an approximation

to the more exact Fresnel "near-field" formulation. (This statement

is not applicable at a lens focal point where both linear and

quadrature phase terms are zero.)

For on--xis field points and an aperture illumination ofuniform phase, the Fraunhofer phase term are zero, corresponding to

totally constructive interference. However, for the sameconditions, the Fresnel phase terms are non-zero, corresponding to

destructive interference among interfering rays. Consequently, the

far-field formulation yields a larger power density at an arbitraryon-axis field point than the more exact near-field formlation for

an aperture illumination of uniform phase.

In this paper, the overestimation of power density by the

far-field formlation is determined as a function of Fresnel numberat arbitrary on-axis field points for rectangular and circular

i 1-1

aperture illuminations of uniform amplitude and phase. For square

and circular apertures, the overestimation is found to be2.8 percent and 1.3 percent, respectively, at a Fresnel number of

1/8 (corresponding to the far-field boundary distance 2D2 A) and 518

percent and 147 percent, respectively, at a Fresnel number of 1.

The 1.3 percent result is compatible with a result reported

earlier(i).

The Fresnel number is defined in section II. Far-field and

near-field formulations are sumarized in sections III and IV,

respectively. A comparison of far-field and near-field formulations

is given in section V.

_JR

1-2

SECTION 2

DEFINITION OF FRESNEL NMMER

Consider a collimated beam, of uniform amplitude and phase withintensity I o (W/m2 ) at a wavelength X(m), which is incident on alimiting aperture. For a rectangular aperture, the clear aperture

is of width 2bx and height 2b (see figure la). For a circulary

aperture, the clear aperture is of radius b (see figure lb). Infigure 1, a collimated beam corresponds to a source point S at adistance d' .

4. For a circular aperture of radius b, the number N of Fresnel

zones, subtended by every point on the edge of the circular aperture

at a field point P(O,O,d) on the optical axis at a distance d fromthe aperture, is defined as

N -' d')k/ b2/AL; b<<d, b<<d' (1)

where

s (b2 + d2)1 /2 ' = (b2 + d'2)1 /2 (see figure ib)

L = [(l/d) + (1/d')] - 1

For a collimated beam, L - d and equation (1) reduces to

N = b2/d; b<<d, collimated beam (2)

For points P in the very near field, N a.. For points P in

the very far field, N 4 0. For a collimated beam, the boundary

distance between the near field and far field is conventionally

chosen as ( ' ) d - 2(2b)2/A which corresponds to a Fresnel number

N - b2/Xd - 1/8.

2-1

7 'I.

$4$

Sw -4w0 wS~

DO u

4.8 to.

144

00

2-2

4P

Consider now the rectangular aperture of figure la. Not every

point on the edge of the rectangular aperture subtends the same

number of Fresnel zones (except in the very far field).

Consequently, the effect of edge diffraction, on the intensity at a

near-field point, is less pronounced for a rectangular or square

aperture than for a circular aperture. Edge diffraction by arectangular aperture is characterized by the Fresnel numbers, Nx and

x - b2 AM); bx<<d, collimated beam (3a)

N- -4,- b /(Xd); b <<d, collimated beam (3b)Y Y

where2 2 1/2 2 2/sx W (bx + d2) , sy M(b 2 + d2) 1/ 2 (see figure la)

The Fresnel number Nx is the number of Fresnel zones subtended by

N every point on the edges x - + bx of the rectangular aperture at thefield point P if the point source S were replaced by a line source

of infinite extent parallel to the y axis and if the aperture were

of infinite extent in the y direction. A similar statement applies

to N Yif x and y are interchanged in the statement for N.

1%

2-3

' S" " .'" ' , '-. . o" ." " " '." ' " W ill . " " '.. .: ;'' '

SECTION 3

FAR-FIELD FORMIUATION

The on-axis intensity I (W/ ) at a far-field point

P(x,yz) - P(0,0,d) is found from the Fraunhofer diffraction

integral of appendix A to be (2)

I(OOd) " u(0O,,d) 12 - Io I dxody oI2X2dT A

IA2 PA Pg0 t - t o .1 -A , P collimated beam (4)

whereA m aperture area { 4bx by , rectangular aperture

Ae , circular aperture

Pt W total power radiated - 10 A4NA

g - aperture (antenna) gain *A

4n- V2bx)(V b ) 4 rectangular aperture

yj[n/(X/2b)]2 , circular aperture

It will be noted from equation (4) that the beam is no longer

collimated in the far field but appears to be spreading as thoughthe aperture were a point source whose radiation is confined

primarily to beam total angles of the order of (A/2b x ) in the x-z

plane and (V2b y) in the y-z plane.

3-1

The intensity I at a field point P(0,0,d) can be expressed in

terms of the number of Fresnel zones subtended by the aperture at

the field point. Noting from equation (2) and equation (3) that

A - nNd for a circular aperture and that A - 44N 4W Ad for axyrectangular aperture, equation (4) reduces to

I o 16 Nx N , rectangular aperture (Sa)

1 0 n 2 N2 circular aperture (5b)

collimiated beam, Nx , NY, N <_ 1/8

It will be noted from equation (4) or equation (5) that in the

far-field I 0 as 1/d 2 4 0 or equivalently I 0 as

N XN y(or N2) 0.

*3I

3-2

SECTION 4

Nin-rzW lVUAI W

The intensity I (W/%2 ) at uny c-axis field point P(0,0,d) of a

rectangular aperture with illi nation of uniform aplitude ad

phase, is found from the Freerel integral to be(3 )

I 40[C (v4 .2 /))IIC 2(A/-) + S2 (I , (6

colliimted berm, b,<(d, by<<d

where

V

C(v) - f cos[(w/2)t 2] dt - freame osine intqWal(4)

0

V

5 (W) - J &in[0vP2)t 2] dt, - Fremiel sine integral~4

0

Nx , NY - rresnel mbers defined by eoastion (3)

10- aperture intensity (%M )

The rreenl integrals hm the properties thmt ( 7 )

S(v) = - S(-v), C(V) - - C(-W) (7a)

S(O) a C(M) a 1 (7b)

S(0) - C(0) - 0 (7c)

4-1

ror Nx - Nm - equation (6) reduces to I - 1 i c h is to be

expected for points in the wry rear field. For M X My - O,

equation (6) reduces to I - 0 which is to be expected for points in

the very far field.

Equation (6) reduces to equation (5.) in the limit of mall

rresnel umbers. This result is demnstrated by noting that ( 5 )

C(W) - W - -2,,s + .... (8a)

S(W) - W~ 3 - V2 3 w + (8b)

C2(W) + 32(w) - -2 A Vp) 2 W6 + ... (6c)

2 2-a . j 4 21M)A-+S (94)

L-(02) 2(WY ay 4 IiV~2 2 1) 3 + (9b)

Substituting equations (9a) and (1b) with the conition (44 2/45)(N2

+ N) << 1, into equation (6), 1 - 10 16 Nx NY which is the result

given by equation 5(a).

For a circular aperture wich sutends a mall angle at a field

point P(OO,d), the on-axis intensity I is found from the rreenel

integral to be

4-2

ib- .}B .'T-im~i~&L r eu (isr 2/Xd) r dr

0

" o * " -39P (ib),2 " 1o {( - C-()2 .,. 2 ,))

- 4X sin2 (W/2); collivated beR, b<d (10)

viere

r - (x 2 + 7)1/2

N - b2/d.

For N - 1, 3, S, .... , 1 - 4X. Por N - 2o 4, 6, .... , 1-0.For N - , 1 - 1o . (8gotion (10) don not eutraplt to thisremlt bocmase equation (10) is not valid for spertures hi hultsd a large Wnle at the field point. See reference (6) forcircular apertures which sudtand a large angle at the field point. )With the mstitution eup (-i0) f I-iin for (my/12) << 1,equation (10) reduces to I m 1 d0 iRic is ceamtible with thefar-field remit given by equation S(b).

4-3

SRMIQ 5

The on-axis near-field intensities for square apertures of sidelength 2bx - 2b y- 2b and circular apertures of diameter D - 2b with

illumination of uniiformn ampitude aid phase, are compred in

figure 2 as a function of the Fresnel nmber N.- fy- N - b 2/M.The near-field formulations given by equations (6) aid (10) are alsocompared with those obtained from the far-field formulations givenby equations (5a) and (5b).

The peak intensities attained are 41 0 at Preanel nmbers Nol,3, 5, ... for circular apertures and 3.24 10at a Freanai nmberN - 3/4 for square apertures. The ainium intensities attained inthe near field are 0 10 at Fremnel w Irs N - 2, 4,6... forcircular apertures and 0.37 1 0 at a Fresnel, nmer N - 7/4 forsquare apertures. Te intensities obtained from tie far-fieldforulat ions exceed those obtained from the wnre exact near-fieldforinalations.

The ratios of them intensities are given in table 1 as afunction of the Freenel numer N. Fr space an circularapertures, the far-field foalation overestimmites the power densityby 2.8 percent ai 1.3 percent, respectively, at a Fresnel. medr of1/2 (corresponding to the far-field boundry distance 2D2A) and 516percent and 147 percent, respectively, at a Fremel, nvoir of 1.the overestimation 1(N0V) 2/sin2(MV2)J - 1 for circular apertures,has been reported earlier (l). Please note thve round-off error in

I

I I CWICULAR APERTURE

I I OF DAMETER 2bI I

II

3 I

z 2-

SQUARE APERTUREOF SIDE LENGTH

NEAR FIELD FORMULATIN

- - FARFIELD FORMULATlONO - 1 - L I-

0 05 1 15 2 25 3

FRESNEL NMtR, N I /), .

FiRure 2. On-Axis Intensity for Square and Circular Apertures

of Uniform Phase and Amplitude Illumination

5-2

| n n n i n 4 n a.ni

I. It I 0 I IIII

I b

9639

4 S o W4 M lt

a I . . . . . . . ...

0.

5-3

II

reference (1] for a Fresnel number of 1/4 (G/Go - 0.95 and not

0.94). For Fresnel numbers corresponding to constructive or weakly

destructive interference, the overestimation is greater with square

apertures than with circular apertures. However, for Fresnel

numbers corresponding to totally or almost totally destructive

interference, the overestimation is greater with circular apertures

(see N-2 and N-2.25 in table 1). Constructive and destructive

interference are more pronounced for circular apertures and,

therefore, are in better and worse agreement, respectively, with the

totally constructive on-axis interference of the Fraunhofer

4integral.

5-

5-4

LIST OF REFERENCES

1. S. Silver, "Microwave Antenna Theory and Design" (McGraw-Hill,N.Y., 1949, MIT Reduction Lab Series, Vol. 12), p. 199.

2. M. Born and E. Wolf, "Principles of Optics" (Pergamon Press,Oxford, 1964), p. 386. Please not1 that equation (44) ofreference 2 omits the factor (l/d) (or equivalently, that theintensities in equation (44) are normalized to (l/d) 1.

3. G. R. Fowles, "Introduction to Modern Optics "(Bolt, Rinehartand Winston, Inc., New York, 1968, p. 131. See alsoreference 2, pp. 428-433.

4. M. Abramovitz and I. Stegun, "Handbook of MathematicalFunctions," U.S. Dept. of Commerce, National Bureau ofStandards, Applied Mathematics Series 55, tenth printing,Deceber 1972, p. 300, 7.3.1 and 7.3.2.

5. op. cit. 4, p. 301, 7.3.11 and 7.3.13.

6. op. cit. 2, pp. 370-375.

7. op. cit. 4, pp. 321-322

6-1

APPER4IX A

DERIVATION OF THE FRALIMMFE AND FRESNEL DIFFRACTION INTEGRALS

For a source point S(x',y'd'), arbitrary aperture point

Q(x01y0 ) with origin 0, and field point P(xy,d), the

Fresnel-Kirchhoff diffraction integral for the amlitude U(P)

(proportional to either the scalar electric or magnetic field

intensities) is given in the notation of figure 1 y)

Bi ff exD~ik(s'+s)] -*os4;,_co (44)d

UM(P - A (cos n s' n co (nsJxdyo (A-1)

where

s - 0, sf - SQ1

n M unit vector normbl to the aperture in the direction of the +zaxis

A - area of the clear aperture

B - complex constant - d I-- e-ikd' which satisfies the boundarycondition U(0,0,0) - /-10.0

(The aperture point Q(x0 , YO) is not restricted to the aperture

point Q shown in figure 1).

For the small angle conditions

44I 4 44 s %;M-8 A2

and the straight line condition

(0 M) - n radians (A-3)

(1) M. Born and E. Wolf, "Principles of Optics "(ftrgaon Press,Oxford 1964), pp. 382-383.

A-1

then

cos(in,') 1 cos(nit)) : -1, [conditions (A-2) and (A-3)) (A-4)

1/s's = 1/d'd [conditiun (A-2)] (A-5)

Expanding s' and s in a power series,

(X+ yy) X2 +2 xO+Y )2s'+s-d +d 0d + 02L Y 3 . .(A-6)

2d

exp [ik(s'+s)] exp [ik(d'+d)J exp (ikf) (A-7)

where

(XX + yy) 1 2 +2 (XX+ 2

d 2 L1Y 2d3 0

L 1 (1/d) + (l/d'))F'

Substituting equations (A-4) - (A-7) and the boundary condition

U(01010) - AF into equation (A-1l),0

U(P) - exp (ikd) jj exp(ikf) dxdyo (A-8)

If only the first term in the above expansion for f is retained,

then equation (A-8) reduces to

i/IffU(P) - ~~exp (ikd) ff ex [(-ik/d) (xxot w0.)] dxdyo (A-9)

A

A- 2

Equation (A-9) is known as the Fraunhofer diffraction integral.

If only the first two terms in the above expansion for f are

4 retained, then equation (A-8) reduces to

U( P) 0 s- exp (ikd)l f exp [(-ik/d)(xxt yy)]

exp [(ik2L)(x 2 y 2)] dxdy0) (A-10)

Equation (A-10) is known as the Fresnel diffraction integral.

A- 3

4.-A -4Z

p/tm E1) ;jjji