Far Field Diffraction Patterns of Circular Gratings

Andre Fedotowsky and Kurt Lehovec

The fine structure of the annular images of circular gratings is analyzed in terms of diffraction patterns ofaxicon pairs. Each image arises substantially from only one converging and one diverging axicon of equaldeflection angle. Single main lobe, symmetric double main lobe, and various intermediate asymmetricdouble lobe structures are obtained depending on the phase relationship and strengths of the two axiconbeams, which in turn depend on the design of the circular grating. Approximate expressions are derivedfor the intensity distribution and energy content of the symmetric single and double lobe images.

1. Introduction

It is well known that circular gratings generate aset of far field diffraction patterns that can be fo-cused into annular images by a lens. Tichenor andBracewell 1 published photographs of ring images inthe Fraunhofer region of a circular grating consistingof narrow annular slits that confirm previous theoret-ical results.2 It was found that the ring images wereasymmetric and not centered at the diffraction an-gles predicted by the familiar relations involvingwavelength and grating period of an ordinary one-dimensional grating. Axial diffraction patterns ofcircular gratings in the Fresnel region have beenstudied by Dyson.3 The purpose of this paper is toanalyze the fine structure of the far field diffractionpatterns of circular gratings.

In Sec. II we discuss a general representation of thecircular grating by an infinite set of axicons, and weexpress the intensities of images of various orders asthose of axicon pairs. McLeod,4 Fujiwaru,5 and Litand Brennan 6 considered the axial image or focus ofaxicons in the Fresnel region. We are interested inthe diffraction patterns of axicons in the Fraunhoferregion.

In Sec. III we give approximate analytical expres-sions for the intensity distribution and energy con-tent of symmetric single lobe and double lobe images.Sec. IV discusses the image shapes of selected binarycircular phase gratings.

The authors are with the University of Southern California, LosAngeles, California 90007.

Received 14 November 1973.

11. Expansion of a Radial Periodic Filter Into a Seriesof Axicons

We shall use the dimensionless variables z and r,which are related to the physical coordinates shownin Fig. 1 by

r = p/a,z = 2aR/xf, (1)

where f is the focal length, and the wavelength ofthe incident coherent radiation.

Any radial periodic filter function can be expandedin the series

imz r -imz rF(r) = bo + (bme0 + bme 0)

m>Q(2)

The trigonometric terms exp(+imzr) are the com-plex transmissions of converging and diverging axi-cons of small deflection angle am = mz0 X/(27ra) (Fig.2). Converging and diverging axicons with the samediffraction angle produce ring images of the same ra-dius Rm = fam.

The normalized far field diffraction pattern of anaxicon

IU ..(z) = 2 e arJ (z r)rdr, (3)

with U(O) = 1, the axial field arising from a clear ap-erture, and Uo(z), the Airy disk 2J1 (z)/z. It is shownin the Appendix that Eq. (3) can be approximated by

U,,, a- I I "(1 + i)E*(z)/I2,

for a > O and Ua = U* a for a < Owhere

z = z - aI =z - mz 0,and

E = C + iSj,

with

(4)

(5)

(6)

2638 APPLIED OPTICS / Vol. 13, No. 11 / Noivember 1974

INCIDENT ,0-PLANEMONOCHROMATIC i fWAVE I

CIRCULAR IMAGEGRATING LENS PLANE

Fig. 1. Coordinates in the aperture and image planes.

2 - 2V ~

Vm(z) = bmUm + bmU*Z

= e"' 4 (bmE - b mEi)(mzo)'/2

el/4 [(bm - ibm)CI - i(bm + ib m)Si](mzO)'1" 2, (10)

and by neglecting the interference arising from theother ring images. For all circular binary gratings(i.e., radial periodic, filters that have only two valuesof the complex transmission) the strength of bothbeams is equal: I bl = I b-mf. The shape of the dif-fraction pattern depends then solely on the phasedifference y)m defined by

iv.e = bm/b m,

and Eq. (10) becomes

Vm(Z) = 2bmle m (mz )-1/2-i /2

-[Ce m cos(y,,/2 + 7r/4) +S fieiy /1

(11)

cos(y,,,/2 - T/4)].

(12)b, 0 R/f1

bl}=~~~~~~~~~~~~~~~~V =J_ R Uifj

b2 '= Rv

U V=IU+ UaJ2

-2R /f

Fig. 2. Equivalent axicons of radial periodic filters. The con-verging and diverging axicons of strength b and b combine togenerate the mth order image I V,,,l2 = Um + Ualj 2 located at

the deflection angle am = mRO/f = ma.

C Z) = (2) r COS (r),rl 2drSt z 7;: J 0 sin

= ZL 2) (,z) T IC (Z) ]

and

C.(z) ( 2)i/2 fl Cog zr)r-'/ 2drS,, (z) IT ) rO sin

tabulated Fresnel integrals.7 The function

Ej 2 = C2 + S2

(7)

(8)

(9)

which describes the energy distribution generated byan axicon, is shown in Fig. 3.

An excellent approximation to the intensity of themth image (m 2 1) of a circular phase grating is ob-tained by combining the field strengths of the corre-sponding axicons,

The same type of approximation has been found byBracewell and Thompson2 for the case of a circulargrating consisting of narrow, equally spaced slits.Our derivation is simpler and more direct, and is ap-plicable to all circular gratings.

The phase difference 'Ym depends on the particulargrating used. Four cases of special interest are listedin Table I and are illustrated in Fig. 3. For ym =-r/2, there is a symmetrical main lobe and in thecase of -y = +7r/2, there are two main lobes of equalstrength. The lobe structure can be varied contin-uously between these two extremes, e.g., for oy 0,an asymmetric double lobe structure is obtained thatis the mirror image of that for m = shown in Fig. 3.The intensities have been normalized to provide anenergy content of unity under each curve.

C (z).3 ---- s2(Cz)

/2 (C '+S')

F32 I 0 1 2

Z /77r -

Fig. 3. The function (C12 + S12)/2 characterizes the ring image ofa single axicon. The function (C - S2/2 describes the shape ofall images (m = 1,3 .. ) of a circular binary phase grating of innerdisk radius r equal to the zone width Ar = 7r/z,,. The functionsC12 and S12 are asymptotic forms to the ring images m = 1,5,9 .. .and m = 3,7,11 .. , respectively, of a circular binary phase grating

of inner disk radius r = 3/4 Ar.

November 194 / Vol. 13, No. 1 1 / APPLIED OPTICS 2639

bo

Table I. Fine Structure of the Diffraction Images of BinaryCircular Gratings for Four Special Values of the Phase Dif-

ference -ir < y < r of Corresponding Axicon Pairs

,Y. I Vm(2)|2mZo/41bmj2

-7r/2 C12+ 7r/2 S,2

0 (Cl + S,)2/27r (Cl - S)2/2

.6 _ H5(Y)I / //

.2 /

//

0 I 2 3T /7r -

Fig. 4. Percentage energy flux H(2) within an annulus of width 22

with respect to the total energy in the image order under consider-ation. The function Hc applies to images of intensity distributionC1

2 , and H, applies to the intensity distribution S 12 shown in

Fig. 3.

111. Intensity Distribution and Energy Content ofSymmetric Single Lobe and Double Lobe Images

The intensity distributions for the axicon pair ofphase angle ym = -7X/2 or 7r/2, respectively, i.e., thesingle lobe and symmetric double lobe images, are

V=(z) I 2 = 4 bm 12(mz)-lC12 (Z) (13)

The peak intensities of the single lobe images are

IVm(0)12 = 32!bml2 [927ma]J1 = 11141 b.l 2(MZo)-', (14

and those of the double lobe images are

IVm(± 0.7r) 2 = 0.781bm 2 (r")-I. (15'

The energy fraction with respect to the incident fluxthat is contained within the range +zh of the mthimage is

Hm( ) 2 m I VmI(Z)2ZdZ. (16)

Substituting Eq. (13) and approximating the factor zin the integrand by mzo, one obtains

Hc (z)

H,(Z) |SI

The functions HC and H8 are plotted in Fig. 4.total energy contained in the mth image is

Hm(O) = 2bm12.

(18)

The

(19)

IV. Special Cases of Binary Circular Phase Gratings

A. Binary Phase Grating sgn sinz0 (r - r) with 0 < r0

< r/z.This grating consists of zones of equal width Ar =

lr/z0 and of alternating phases, except for the inner-most zone of width r. The grating period is 2r.The coefficients of the Fourier expansion Eq. (2) are8

= ± 2/(m7r), m = 1,3...

b~m= 0 , m = 0,2,4...; (20)

and their phase difference for odd m is

Y = 27(1 - 2mr/tr). (21)

Thus by selecting appropriate values for r, any oneof the shapes listed in Table I, as well as those for in-termediate cases, can be given to an odd-order ringimage. However, only for r0 /Ar = 0 or 1/2 do all oddimages have the same shape, since ym = 7r or 0, re-spectively. Thus it is impossible to have all odd-order ring images of the symmetric single lobe typefor the grating considered in Sec. IV. A.

The binary phase filter sgn cos(zr - 7r/4) thatarises for rd/Ar = 3/4 has single lobe images of typeC12 for m = 1,5,9 .. .; and symmetric double lobe im-ages of type SI2 for m = 3,7,11 .... This is illus-trated in Fig. 5 for this binary filter with eight filterzones, i.e., for Ar = 1/8 = w/z0 . The intensity distri-bution I V(z)l2 was calculated by the formulas for thediffraction pattern associated with filters consistingof annular zones of constant transmission given byBoivin. 9 Images are located at z = mz, where m isodd. The main lobe of the images of order m = 1and 5 has a single peak, while the images of order m= 3 and 7 are double-peaked. The asymmetry of thedouble peak structure m = 7 results from the inter-ference from lower order images. This is reduced byincreasing the number of zones, since the distancebetween images increases.

H(-) 2 b 12 Ha (z)Hm(Z) 2~~H b,, ) (17)

where

2640 APPLIED OPTICS / Vol. 13, No. 11 / ftvember 1974

Fig. 5. The first four ring images of the eight-zone binary circular

grating sgn cos(zr - 7r/4).

-- - ' HC (7)

- lv(i)I2

z/? -

Fig. 6. First and third order images, V(z)12 , of the ten-zone circu-lar binary grating sgn cos(zr - r/4) and the asymptotic forms C1

2

and S12. Abscissa is 2 =z - mz where mz0 is the center of the mth

ring image.

The intensity distribution V(z) 2 in the vicinity ofthe images m 1 and 3 is shown in more detail inFig. 6 for the case of a filter having 10 zones, usingthe deviation = z - mz0 from the center of theimage as abscissa. The dotted line is the contribu-tion of only the axicon pairs m = 1 or 3, respectively.The error of neglecting the contributions from axi-cons n d m in the vicinity of the image located atmz0 is obviously quite small.

The energy flux contained in the first order imageis H1 = 87r-2Hc, e.g., 46% of the incident flux is locat-ed within 2 = 0.4ir (half width) and 56% within+0.87r (full lobe). This comes very close to the opti-mal concentration of incident radiation into a narrowannulus. 10

The sum Hm adds up to unity, not withstandingthat the functions Vm(z) of Eq. (13) are not strictlyorthogonal to each other. However, they do notoverlap appreciably for sufficiently large z0; other ap-proximations made in deriving Eqs. (13) and (17)happen to compensate for the small error due to non-orthogonality.

B. Grating of Tichenor and Bracewell 1

This grating consists of equidistant (2Ap) circularslits of width E. Let po be the radius of the first slit.Assuming thatc << 2Ap(m7r)-, one has

|bmi. = IrE/2Ap (22)

and

ym =-47rmp 0/2Ap (23)

for all integers m. The two values po/2Ap = 0 and 1/2result in y}m = 0(mod 2 ir), i.e., all images exhibit theasymmetric double lobe structures (C1 + Sj2/2.

No binary circular grating whose period consists oftwo zones can have all images of the single lobe typeC1

2. When chosing po/2Ap = 1/8, the first four ymbecome -r/2, x, w/2, 0; and this order is repeated ingroups of four for higher m values. All the lobes list-ed in Table I appear in the ring images of this grat-ing.

The reason why linear gratings can have all imagesof the single lobe type and circular gratings cannot isas follows: the linear analog of the circular grating isnot the linear grating, but the bigrating with complextransmission or reflectance Fix), with F(x) a period-ic function for x > 0. An example for a bigrating is asequence of alternating absorbing and transmittingstrips of equal width, except for the central strip,which has a smaller width. Analogous to the repre-sentation of a circular grating as a sum of axicons, thelinear bigrating can be represented as the sum of con-verging and diverging biprisms. As with the circulargrating, linear bigratings have diffraction images con-sisting of a single lobe, a double lobe, or asymmetricstructures in between. A symmetric double lobe dif-fraction pattern might be of some interest for spec-troscopic measurements, since locating the minimumis often more precise than measuring the position of asingle lobe maximum.

V. Summary

The filter function of any radical period filter canbe developed into a Fourier series, the terms of whichrepresent filter functions of converging and divergingaxicons. A pair of such axicons of equal (small) de-flection angle generates two beams that combine intoone of the diffraction images of the circular grating.Both beams are symmetric and are located at theclassical diffraction angle for a linear grating. In thecase of circular binary phase gratings, these twobeams have equal strength; and the shape of the dif-fraction image then depends solely on the phaseangle between them. Depending on the magnitudeof this phase angle, single lobe, symmetric doublelobe, and various intermediate cases, such as theasymmetric double lobe structures photographed byTichenor and Bracewell,1 can be obtained.

The intensity distribution of the first order imageof the filter sgn cos(zr - 7r/4) is a single lobe struc-ture that is well approximated by the analytical func-tion C1

2 describing the intensity distribution result-ing from a converging and a diverging axicon of equaldeflection angle z and phase difference -ir/2. Inthis image, 56% of the incident energy is concentrat-ed within the annulus of radius z and width Az =1.67r.

This work was supported by the Joint ServicesElectronics Program through the Air Force Office ofScientific Research under Contract F44620-71-C-0067.

November 1974 / Vol. 13, No. 11 / APPLIED OPTICS 2641

Appendix: Derivation of Eq. (4)

Using the large argument asymptotic form of J,Eq. (3) becomes

U, (z) 2 (-) "f eiarcos(zr - 1T/4)r 112dr,

which transforms into

Ua(z) = (2z)-t/2[(1 + i)Ej'(z) + (1 - i)E1(z + 2a)]

using the identity

e i"r cos(zr - 7/4) = [(1 - i)ei(2arzr)

+ (1 + i)efzr]/(2vI)

E1 is defined by Eqs. (6) and (7). For 2 = z - a << a,one has z 1 /2 -_ a-1/ 2. The second term in bracketscan be neglected since El(z) z- 1 .

References1. D. Tichenor and R. N. Bracewell, J. Opt. Soc. Am. 63, 1620

(1973).2. R. N. Bracewell and A. R. Thompson, Astrophys. J. 182, 77

(1973).3. J. Dyson, Proc. R. Soc. Lond. A248, 93 (1958).4. J. K. McLeod, J. Opt. Soc. Am. 44, 592 (1959).5. S. Fujiwaru, J. Opt. Soc. Am. 57, 287 (1962).6. J. W. Lit and F. Brennan, J. Opt. Soc. Am. 60, 370 (1970).7. M. Abramovitz and L. A. Stegun, Eds., Handbook of Mathe-

matical Functions (National Bureau of Standards, Appl.Math. Series 55, Washington, D.C., 1964), p. 321.

8. The Fourier coefficients for gratings consisting of alternatingopaque and transparent zones are bm = (m7r)' for odd m;b,m = 0 for even m > 2; and bo = 1/2.

9. A. Boivin, Theorie et Calcul des Figures de Diffraction deRevolution (Gauthier Villars, Les Presses de l'Universit6 LavalQuebec, 1964).

10. A Fedotowsky and K. Lehovec, Appl. Opt. 13, Dec. (1974).

Fundamental Physical ConstantsNew Valves

A landmark review of the measurements bearing on thefundamental physical constants appears in the Journal ofPhysical and Chemical Reference Data, Vol. 2, issue 4, pre-

pared by E. Richard Cohen and Barry N. Taylor. Both havebeen active for a number of years in precision measurementsand their analysis. Cohen is presently at the Science Centerof Rockwell International and Taylor is Chief of the ElectricityDivision of NBS.

Included in the review are a detailed analysis of each perti-nent measurement, an assessment of sources of error ineach technique employed and a comparison of values ob-tained from different laboratories throughout the world. Thedata was subjected to a least-squares fit in order to arrive atrecommended values and limits of error. The final result is arecommended set of some fifty fundamental physical con-stants and combinations thereof.

Selections of recommended values of the fundamentalconstants have been carried out at various times in the past,the most recently by Taylor, Parker, and Langenberg in 1969.Since then a number of developments have occurred whichnecessitated a new analysis. Among these are routine mea-surements of the Josephson effect to very high accuracy inseveral laboratories, a major improvement in the accuracy

with which the speed of light is known, and improved mea-surements of the magnetic moment of the proton in terms ofnuclear magneton. The new set of recommended constantscontains significant changes, relative to the 1969 set, in theAvogadro constant, the Faraday and the magnetic moment ofthe proton. In addition, the limits of error have been substan-tially reduced in many cases.

The work was supported in part by the NBS Office of Stan-dard Reference Data and by Rockwell International and wasreviewed by the Committee on Fundamental Constants of theNumerical Data Advisory Board, National Academy of Sci-ences-National Research Council. It formed the basis for the1973 report of the Task Group on Fundamental Constants ofCODATA, the Committee on Data for Science and Technolo-gy of the International Council of Scientific Unions, of whichCohen is chairman. The set of fundamental constants in thepaper has been recommended by CODATA for internationaluse, and is expected to be adopted by the international scien-tific unions and standardizing bodies.

Copies of The 1973 Least Squares Adjustment of theFundamental Constants, may be obtained for $5.00 fromJPCRD Reprint Service, Room 604, American Chemical Soci-ety, 1155 Sixteenth Street, N.W., Washington, D.C. 20036.

2642 APPLIED OPTICS / Vol. 13, No. 11 / November 1974