3D self-imaging condition for finite aperture objects

15 November1996

OPTICS COMMUNICATIONS Optics Communications132 (1996)41-47

ELSEVIER

3D self-imaging condition for finite aperture objects R. M o i g n a r d i, J.L. de B o u g r e n e t de la T o c n a y e D~partement d'Optique, UMR CNRS 1329, Ecole Nationale Supdrieure des T~l~communicationsde Bretagne. BP 832, 29285 Brest Cedex. France

Received 10 June 1996;accepted 16 July 1996

Abslract The effect of object spatial truncation on self-imaging formation is investigated in the general case, resulting in the definition of a 3D self-imaging condition. Particular attention is paid to periodic objects for which a link between the selfimaging distance and the optimal 2D tiliag is demonstrated. We deduce the resulting minimum number of periods required to observe self-imaging. An experimental illustration is presented in a simple case e,nd the theoretical values compared to experimental results.

1. lntroduc~,~m The theory of self-imaging was investigated by Montgomery in 1967 [ 1], but this phenomenon remains attractive and still gives rise to many applications [2]. Although, some non-periodic objects can be found which self-image, periodic structures are of primary interest in physics (see Ref. [2] pp. 90-95). In particular, our analysis has been partly motivated by the recent attention paid to the implementation of programmable multi-level holographic elements using self- or multiple-imaging effects [3,4]. This leads us to consider the ease of finite aperture objects, one of whose critical points is related to the determination of the maximum number of independent elements which can be encoded within the transmittance support. In other words, this leads, in that case, to determine how many periods are necessary to self- or multipleimage accurately an object at a given location along the propagation axis. Our objective in this paper is to give the necessary condition for 3D self-imaging in i E-mail:[email protected].

the most general case. An extension of this result to multiple-imagingcan be achieved using the Winthrop and Worthington analysis proposed in Ref. [5]. In Section 2, we give a necessary condition for an object to self-image using the spectrum analysis proposed by Montgomery in Ref. [ 1 ]. The example of the zero order 2D Bessel function is treated in Section 3. In Section 4, we focus on 2D periodic objects, and we give the minimum distance of self-imaging in the general case. We end the paper by illustrating the different theoretical results with a simple example and experiment in Section 5.

2. Self-imaglng of finite aperture objects

Our analysis is based on Montgomery's paper [ 1]. We restrict ourselves to the so-called w e a k imaging which concerns objects whose spatial frequencies f x and fy verify a2(f~ +f~y) << 1, where a is the wavelength of the illuminatingmonochromatic source. According to Montgomery, self-imaging occurs at the distance z0, for an infinite aperture object o ( x , y ) il-

0030-4018/961512.00Copyright~) 1996ElsevierScienceB.V.All righuqreserved. PII S0030-40 i 8 (96) 00464-6

R. Moignard.J.L. de Bougrenetde la Tacnaye/OpticsCommunications132 fl996) 41.-47 luminated by a plane wave, if the spatial frequencies of the object are on circles centred on the origin and of radius Rk defined by

Rk = v~V~/A~0,

(1)

where k is an integer. If we now introduce the truncation function t ( x , y ) which limits the object o ( x , y ) to a finite support. For example, t(x, y) could be the rectangle function Rectr(x,y) or the circular function Circr(x,y) defined by

I I for Ix[ < T and [y[ < T, Rectr(x,y) =

0 otherwise,

(2)

and ! forx 2 + y 2 < T 2, 0 otherwise,

Circr(x,y) =

tance z0 for an object of frequency bandwidth B is then (3)

where T is real positive, then the spectrum of the truncated object becomes thus b( fx, fv) ® t'( f~, fy ), where ~ denotes the Fourier transform of u and ® the convolution operation. If t ( x , y ) ---RectT(x,y) then, sin(crf:~T) sin(TrfyT) ?(A, A) = ~'fxT erA. T ,

J, ~c r r ~ :

I/T<< Rk~o+i - Rk~o.

(6)

Assuming that z0 >> I/AB 2 (this condition is discussed in Appendix A) implies: T >> AzoB.

(7)

Therefore we can deduce the following result: (4)

and if t(x, y) = Citer(x, y), then ¢rT2

Fig. I. Montgomery's circles and spectrum of a truncated

self-imagingobject,fx and fy are the spatialfrequencies.The approachedself.imagingphenomenonis restrictedto a limitedpart of the frequencyplaneby the circle of radius B.

A necessary condition for a finite aperture object to self-image over a distance zo is:

zo << T/AB,

(8)

)

(5)

where T is the truncation support size, A the wavelength and B is the spatial fi~luency bandwidth o f the object.

where ,/I is the first order Bessel function. The first zeros of tbese functions are respectively ! / T and 1.22/T. Thus, in both cases, the effect of the truncation results in a spre~ing of the spectral components proportional to lIT. If this spreading remains small compared to the difference Rk - Rk-I between two consecutive radii of the Montgomery's circles, we assume that an approached self-imaging occurs. This condition is stronger for high spatial frequencies as R k - Rk_ i tends towards 0 when k tends towards infinity (and thus Rk). To be verified, such a condition imposes a limitation on the spatial frequency bandwidth (denoted B) of tbe object (see Fig. i ). Therefore, at the observation distance z0, we denote k~ the integer part of B2Azo/2. The condition of approached self-imaging at the dis-

This relationwill now be illustratedby two examples.

~(A'f-")=

2

~-r~/-~+~

'

3. Non-diffracting Bessel beam A first well-known example of an object, for which self-imeging occurs, is the zero order 2D Bessel function Jo(x,(.~Z~_.In theory, as the Fourier transform of Jo(ax/x 2 +y2 ) (where a is a real positive number) is a circle of radius ~/(2~'), the Montgomery condition is automatically verified for all zo, and therefore this object self-images for any distance ar~ng z. The propagation properties of such beams have been rediscovered recently for the synthesis of non-spreading

R. Moignard. J.L. de Bm:grenet de la Tocnaye/ Optics Communications 132 (1996) 41-47

We consider a 2D periodic function F(x) where x = (x,y) with two linearly independent translation vectors (not necessary fundamental) at and a2 located in the plane z = 0 and illuminated by a monochromatic plane wave of wavelength A. One can expand F0(x) = F(x) in Fourier series: ,

Fo(x) =EEFn,.n2exp(2icrx.bn),

e

nl

Fig. 2. Example o f a 2D periodic object. The c o u p ~ ( a , b). (c, d) and ( e , f ) am fundamental an,.' the are~-t ef ttx~, parallelograms

definedby these couplesis A (from Eittel [71L beams [6]. Creating a Jo beam over an entire plane, requires an infinite amount of energy. Therefore, the above finite aperture consideration is particularly relevant in that case. The spatial ~ c y bandwidth of the Bessel function Jo(o~V/X2 + y2 ) being B = ot/(27r), by applying the result of formula (8), we find that this beam, when truncated, self-images (i.e., does not spread) along z up to a distance z~ given by z0 << 2,rT/A~.

(9)

This means that a spatial truncation of Jo results in a limitation of the self-imaging range along z. This upper bound can be compared with that obtained by the geometric approach proposed by Dumin .~n Ref. [6], assuming the approximation that z0 >> I/AB 2.

where n = (nt, n2) and bn = nlbl +n2b2. The vectors bl and b2 are the translation vectors of the reciprocal lattice and Fn~.n2are the Fourier coefficients of F(x). In the plane z = zo, the field repartition is given by the Fresnel diffraction theory for spatial frequencies fx and fy verifying A2(f 2 + f~y) << 1 (see Ref. [1] for instance):

Fzo(x) = exp (2i*rzo/A) x E E nl

Fn.... exp (-i,rAzo llbnll2) n2

x exp (2i~rx. bn).

( 11 )

The condition of self-imaging at the distance zo is [exp (-iTrAzo IIbnll2) = 1] for all nl and n2. This is equivalent to [" iTrAz0 /" nl2 n~ exp l - ~ [,-;--~,,2 + L sm 3'o \ liar II I]a2[]2

2nln2

)]

Ilalll Ila211COS3'a 4. 2D periodic objects A second interesting case concerns periodic objects. In the 2D case, the notion of period can be confusing, therefore we prefer to use the formalism of solid state physics to characterize a 2D lattice [ 7]. According to this formalism, we call a a translation vector of a lattice, if a translation by a does not modify the lattice. Any integer linear combination pal + qa2 of two translation vectors al and a2 with integer p and q, is a translation vector. A couple of translation vectors which can generate all of the translation vectors is said to be fundamental. The area of the parallelogram defined by two fundamental translation vectors is minimal but a couple of fundamental translation vectors is not unique (see Fig. 2).

(!0)

n2

= 1

(12)

for all nl and n2. This condition holds if and only if the translation vectors verify the Loseliani condition [8]: Ilalll2 [[a2]]2 and ~Ilatll co s 3'a are rational numbers,

(13)

where Ya is the angle between ai and a2. We now introduce considerations on the observation distances. Let us consider 2D periodic objects verifying condition (13), we know that they self-image at discrete locations along the propagation axis. The first plane where the object self-images is at a distance Z-r called the Talbot distance. It is very important to evaluate this distance since the truncation window size T is linearly dependent on the observation distance (see Eq. (7)). In most applications, the smaller the object is, the easier it is to implement. So, we can write:

44

R. Moignard. J.L. de Bougrenetde ia Tocnaye/Optics Communications 132 (1996) 41-.47

[[al [j2 = ma and [[a, [[ m" j{a:[Iz n~ i ~ ' ~ cosy. = n-~'~'

(14)

(A is also the area of the basic pattern of the periodic object). We have:

where ma and no (respectively m" and n ' ) are two integers with no common factor. In the case of an orthogonal lattice (i.e. "/ = ~'/2), we take m" = 0 and n~ = na. With these notations and according to (12), one can find that the minimum distance for selfimaging associated with the translation vectors a, and

A = jla, II × Ila211 x I sinYol,

a2 is:

Applying the result of Eq. (7) implies that the truncation support T depends not only on the space bandwidth product A x B but also on the way the information is spatially arranged (i.e. it depends on no, n ' , pa, ma, m'o and Ya).

2

2n"

2

,

~ . 2 = "~nallalH -~a sin2"~a = ~ma a2

,,/

gsin2Va, (~5)

where Pa is the greatest common divider of no and n~. One can notice that this distance is a function of the u~mslation vectors describing the 2D periodic lattice. However, a Talbot distance exists physically and must be independent of the chosen translation vectors. Thus, among all the possible couples of translation vectors (a~,a2), we have to find the one which minimbes the distance d~.2. Intuitively, one can assume that this minimum is reached for a fundamental couple of translmion vectors. For a mathematical demonstmtion, see Appendix B. The following rule can be stated: The Talbot aismnc¢ o f a 2D periodic object is the miaimum distance o f s~lf-imaging associated with any f u n d a m e n t a l couple of a'anslation vectors. If a! and a2, making an angle 7a form such a fundamental couple verifying: 2l

[a~!l

__m~

[la2!i 2 = ~':a

llal II

m"

and ,-r---~.cosy~ = - lla2ll n'a'

(16)

where ma and na (respectively m~ and n") are two integers with no common factor, then the Talbot distance ZT is given by

= 2 malJa2JJ2 ~a sin2,,/a '

ZT -- i V nama - - sinya x A. Pa

(19)

5. Experimental verification 5.1. Illustration In order to illustrate the above analysis, we consider a simple case. For 2D periodic pixelated objects, the trnr:cation is expressed in terms of numbers of periods. Therefore, if we now consider the case of optical implementation, the number of pixels per period (if we use a spatial light modulator (SLM)), or the object spatial resolution (if we use a fixed transparency ), become the relevant parameters in terms of information encoding capacity. ~his feature becomes very critical when dealing, for instance, with holographic element synthesis, because it indirectly influences the number of degrees of freedom. For instance, if we assume that a square pattern of size d x d is replicated N x N times, it yields T = Nd. According to the Talbot distance which is, in this case, ZT = 2d2/A, if we want to use the properties of the periodic object at a fractional Talbot distance r/s ZT where r and s are integers and for which multipleimaging occurs [5], condition (7) becomes: N >> 2

ZT = dla.2 = 2nai[aliJ2 n---~sin2 Va A Po

(18)

giving:

rdB.

(20)

$

(17)

where Pa is the greatest common divider of no and n'. Let S be the area of the parallelogram defined by a couple of fundamental translation vectors (al, a2)

This gives the minimum number of basic pattern repetitions.In the case of a pixelated object, and if the pixel size is t x t,the overall number of pixels S of the device used to display the periodic object must verify: s>>

2s t

/

2stB

.

(21)

R. Moignard. J.L. de Bougrenet de la Tocnaye/Optics Communications 132 (1996) 41-47

Llli LIBK [am

lNmllH ann!

lull Im~lll aluminum IIllllllll [ l l l l l l l

a)

Illlil

amid

b)

NNINNI cl Fig. 3. Basic pattern of the u.~d object (a) and theoretical field repatlition at half (b) and quarter (c) Talbot distances. For instance, if we consider the quarter Talbot distance (for such an application, see for instance Ref. [ 3 ] ) for a pixelated basic pattern of 8 x 8 pixels, and according to the bandwidth B which is equal in this case to I/t, we must use a device with an overall number of pixels S verifying S >> 32 x 32.

Fig. 4. Experimentalfieldr~panifionat the quarterTalbotdistance for differentperiodsnumber, (a) N = 8, (b) N = 16, (c) N -- 24 and (d) N=32.

5.Z Experiment The above condition on the minimum number of periods is, in practice, too severe. To obtain a more precise limit we introduce additional considerations, like the accuracy of the object reconstruction. Such a criterion has been introduced, for instance, in array illuminator design. We prefer here not to enter into such considerations which make the self-imaging condition more dependent on the objects. However, in order to assess the approximation of our analysis, we propose to compare the deviation between theory and experiment with a concrete example. For the sake of simplicity, we have considered a binary amplitude transparency whose basic pattern is shown in Fig. 3a: this is a 8 x 16 pixel rectangle. The pixel size is t = 25.4 /~m, and thus the basic pattern is a 203.2/.tin x 406.4 /zm rectangle. The Talbot distance is, in this case, Z'r = 52.8 cm at A = 633 nm. We have observed the field repartition at the half and the quarter Talbot distances (the theoretical field repartitions at these distances are shown in Figs. 3b and 3c). Assuming that the bandwidth is B = i/f, and applying (20), one can find that we need N x N replications of the basic pattern with N >> 8 to obtain an approached multipleimaging at ZT/4. We show in Fig. 4, four pictures cot-

Fig. 5. Experimcnlalfield repatlition for N = 24 for different observation d~s~mces,(a) ZT. (b) ZT/2 and (c) ZT/4. responding to four different number of periods ( N " 8,16, 24 and 32). One can observe that with 8 replications, the image is completely bluffed, but with 24 replications, the resuh is already quiet good since we can see details of pixei size, In the same way, we can assess the influence oftbe distance by comparison, for a given value of N, oftbe reconstruction for ZT, Z-r/2 and ZT/4 (see Fig. 5). In this particular case, and using a subjective criterion (for a more precise image reconstruction criterion, see, for instance Ref. [9] ), we can see that there is only a factor 3 between the theoretical lower bound of (20) and the experimental result. However, we can consider that condition (20) is always verified for an order of magnitude of the lower bound.

R. Moignard.J.L. de Bougrenetde la TacnayelOpticsCommunications132 (1996)41-47 !

A~kno~=age~m~

~2 m

Special thanks to H. H~unara and ,I.N. Provost for valuable discussions and Y. Defosse and K. Heggarty for supplying the gratings.

Hence

MI.IM2,2 - MI.2M2,1 [-M2,lbl + Ml.!a2] • (B.4)

Ila! II2

[[a2ll2=

A. Verification of the condition

M 2.2mbn~ ÷ M!2.2nb%' - 2M2,2ML2m~nb M21mbnlb -{- M21nbntb -- 2M2.1M2,2mlbnb

Ila!ll i!a21"----/COSVa = [ M2,2M2.lmbn~ + MI.2Ml.lnbn~ We have to verify the assumption made in Section (2), which is zo :~ I/AB 2. We consider a pixelated 2D periodic object which has a square basic pattern. One can assume fllat B = ! / L At the fractional Talbot distance distarce zo =- r/sZT with Zr = 2d2/A, ~ i s condition becomes N 2 >> s/(2r). This condition is sLrongcr when the fractional order is smaller. For inslance, at the quarter Talbot disUmce, the condition N 2 :~ 2 which is, almost always, verified.

-- (MI.IM2.2 + Ml,2M2.1)m~nb]

x [M 2,1mbn~ + M!.lnbnb2

' -- 2M2.!M2,2m~nb]-!.

With these notatiops, we obtain:

,~

d 2 --

• 2Yb, nullblll 2 --~ sm

-

(B

s)

where Pb is the greatest common divider of nb and n~. In (B.5), we can simplify the two ratios by Pb, giving

IL S d f - l m ~ g i n g minimum distance

Let ( a l , a 2 ) he a fundamental couple of translation vectc~rs of a 2D periodic object. Let (hi,b2) he a couple of linearly independent translation vectors. We can ~Tite:

Ila!ll2

~ama

Ha2l]2

=" 13ana'

Iladl

_ ~m"

[[-'~21lc o s y o - ~'na''}"

(B.6)

where (aa, a'0,13a, 13'o) are non-zero integers and

bl = Ml,lal -[- ML2a2~

b2 = M2.tat + M2.2a2,

(B.I)

where (Mt,b ML2, M2a, M2.2) are integers, We denote A the outer product. One can show:

Hbl A b2H = [MLtM2.2 - M2A, ML2[ Hal A a2[]. (B.2)

13ana = ~'0n'0 = M2.lmb-~h + M21 -- 2M2.1M2,2mlb~ b .

Using qa to denote the greatest, common divider of [3ana and 13"n'0,we can demonst~,~atethat qa <--Pa13a13'~. We thus denote:

Let Vb he the ang|e hetween bl and b2. From Eq. (13), we can write

2 [i2 ~pa aP sin2ya, d = ~(pana)llaJ qa

~[i~aii 2 ~ mb and Ilbmll

giving:

n'-~

~

_ m£

cOS~'b -- t~b*

(B.3)

where mb and nb (respectively m~ and n~) are two integers with no common factor. From (B.I), we have |

aD = Mi.uM2.2- Ml.2M2.;

[g2,2b! - Ml,2b2] ,

(B.7)

I ~ t nt......~

t 2 H2~a sin2Ta" d >_ "~nollal

(B.g)

(B.9)

One can deduce that d >_ d~'2. However, from Eqs. (B.5) and (B.9) and noticing that q~ = 13ana, we can write:

41

R. Moigoord,J.L. de Bougrener de lo Tocnoye/OpricsCommunicodons132 (19%) 41-47 2

~efe~n~

2 ni&

Ms,,mb-n: + M,,, -

d =i

Pb

-

2M2.1~2.2~:~)

= $%.IMI.Z

-

?‘b

IlUl II2

[ I] W.D. Montgomery, J. Opt. Sot. Am. 57 ( 1967) 772.

[2)

,n2 * ya

K. Patorski, Progress in Optics, Vol. XXVII,

ed. E. Wolf

(Yorth-Holland. Amsterdam, 1989).

[31 H. Hamam and J.L. de Bougrenet, Appl. Iaio.

M,*&.t)2

Optics 35 (1996)

(41 V. Aniz6n and 1. Ojeda-Castaiieda, Appl. Optics 33 (1994) 5925.

(B.10)

151 J.T. Winthrop and C.R. Worthington, J. Opt. Sot. Am. 55 (1965) 373. 161 J. Dumin, J. Opt. Sot. Am. A 4 (1987) 651.

From (B.2), we have:

171 C. Kittel. Introduction to solid state physics (Wiley, New York.

llat~1211u2112~~t,t~2.2

-

= llbl 11211~2112 sin2Ybv

Mt,2M2,t12sin2x

(B.ll)

which allows us to conclude that d = dFa2.We have therefore established the following result:

df.22 4%

1976).

IS] R.E. LuseGni. Opt. Spectroscopy 55 (1984)

(B.12)

191 AR Smimov. Opt. Spectroscopy 44 (1978)

544. 208.