Montgomery self-imaging effect using computer-generated diffractive optical elements

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1 Otics Communications 225 (2003) Montgomery self-imaging effect using comuter-generated diffractive otical elements J urgen Jahns a, *, Hans Knuertz a, Adolf W. Lohmann b a FernUniversit at Hagen, Otische Nachrichtentechnik, Universit atsstr. 27/PRG, Hagen, Germany b Universit at Erlangen-N urnberg, Lehrstuhl f ur Nachrichtechnik, Cauerstr. 7, Erlangen, Germany Received 10 Aril 2003; received in revised form 16 July 2003; acceted 21 July 2003 Abstract The Montgomery self-imaging henomenon reresents a generalization of the well-known Talbot effect. Here, it is imlemented by using comuter-generated diffractive otical elements. A Montgomery interferometer consisting of two hase-comlementary elements is demonstrated and suggested as a device for temoral rocessing of otical signals in the s/fs-regime. Ó 2003 Elsevier B.V. All rights reserved. PACS: Bs; Jv Keywords: Diffraction; Otical information rocessing; Self-imaging; Talbot effect; Temoral rocessing 1. Introduction Self-imaging of an otical wavefield, i.e., its relication in the longitudinal direction without the use of a lens is an interesting henomenon for theoretical and exerimental reasons. Well-known is the case of self-imaging for eriodic objects first observed and described by Talbot [1]. Talbot selfimaging can be described in the following way: a (quasi-)monochromatic wavefield of wavelength k with lateral eriod 1=m 1 is also longitudinally eriodic. The longitudinal eriod z T often referred * Corresonding author. Tel.: ; fax: address: jahns@fernuni-hagen.de (J. Jahns). to as the Talbot-distance is given as z T ¼ 2=km 2 1. The Talbot effect has been widely studied (see, for examle, the articles by Winthro and Worthington [2] and Patorski [3]) and used for a number of alications such as interferometry [4], imaging [5] and beam slitting [5 7]. Recently, we suggested the use of the Talbot interferometer for the rocessing of temoral signals [8]. Talbot self-imaging occurs for wavefields with lateral eriodicity under the assumtion of araxial roagation. Montgomery [9] described a more general case and showed that lateral eriodicity is a sufficient but not a necessary condition for self-imaging. According to Montgomery, selfimaging occurs for wavefields whose angular sectrum is confined to satial frequencies that reresent concentric rings of well-defined radii. An /$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi: /j.otcom

2 14 J. Jahns et al. / Otics Communications 225 (2003) exact formula for the radius q n of the nth ring will be given in a later section, however, aroximately it is q n ffiffiffi n m 0. Such wavefields can be generated by 1- and 2-D objects. Furthermore, Montgomery self-imaging is not restricted to araxial wavefields. The Montgomery effect was indeendently studied by Lohmann [10] and Indebetouw [11,12]. In [12], we also find a first exerimental demonstration. There it was suggested that a Montgomery wavefield could be generated either by a diffracting mask or by a virtual object using a Fabry Perot etalon. The diffracting mask was imlemented as a binary amlitude Fresnel zone late which aroximates the frequency sectrum required for Montgomery self-imaging. Previous attemts to alied Montgomery objects include [13] and [14]. In both cases, satial filtering techniques were used. In [14], it is shown that under suitable conditions quasi-eriodic and aeriodic ring uils generate self-imaging wavefields. Here, a relationshi to Bessel or nondiffracting beams [15] is of interest: each ring in the uil is the source of a Bessel beam whose intensity distribution is invariant uon roagation. For Montgomery self-imaging to occur, the different Bessel beams, each characterized by a secific wave number, have to add u coherently at certain z-lanes. The earlier work just mentioned is based on analog exerimental techniques and thus offers limited design flexibility. This situation can be imroved by the use of comuter-generated diffractive otical elements (DOEs). The design and fabrication of DOEs is well established nowadays [16]. Phase-only DOEs can be custom-designed for secific tasks by using iterative Fourier transform algorithms and fabricated by lithograhic fabrication techniques. It is the goal of this aer to demonstrate this ossibility for Montgomery selfimaging with the urose to make this effect alicable in a similar way as the Talbot effect. For our demonstration exeriments, we use an interferometer consisting of two DOEs, called Montgomery interferometer analogous to the Talbot interferometer [4] (Section 3). As a otential alication and motivation for this work we describe in Section 4 the use of the Montgomery interferometer as an otical taed delay-line for the rocessing of temoral signals. We shall start, however, with a brief review of the self-imaging effect. 2. Theory of self-imaging Here, we consider only the 1-D case. Generalization to two dimensions (in either cartesian or cylindrical coordinates) is straightforward. The grating is described as Z G 1 ðxþ ¼ ~G 1 ðmþ exð2imxþdm ð1þ with ~G 1 ðmþ ¼ X A n dðm m n Þ: ð2þ n Proagation from the G 1 -location over a finite distance Dz is described by ~G 1 ðmþ! ~G 1 ðmþ ex 2iDz ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 k 2 m k 2 : ð3þ In the araxial case, one may aroximate this as ~G 1 ðmþ! G ~ 1 ðmþ ex½ ikdzm 2 Š: ð4þ For this case, a linear grating with m n ¼ nm 1 (n: integer) yields the Talbot effect, i.e., self-images aear at locations Dz ¼ Mz T ¼ Mð2=m 2 1kÞ with M ¼ 1; 2; 3;... Fractional Talbot images are observed for Dz ¼ðM=NÞz T where N is also an integer number. Rather than asking the question, how the wavefield looks like that is generated by a eriodic object, Montgomery [9] asked the question how an object has to look like in order to generate a wavefield with longitudinal eriodicity. If a wavefield is z-eriodic it must be reresentable in a Fourier series as uðx; zþ ¼uðx; z þ z T Þ ¼ X V m ðxþ exð2imz=z T Þ: ð5þ m Inserting this exression into the Helmholtz equation yields a differential equation for the V m ðxþ. Meaningful solutions result for satial frequencies [9,10]

3 J. Jahns et al. / Otics Communications 225 (2003) Fig D satial frequencies allowed for Montgomery objects according to Eq. (6). The little black dots mark the outermost frequency with index m ¼ 0orn¼m max, resectively, whereas the little white dots mark the ring with index n ¼ 0orm ¼ m max, resectively. m 2 m ¼ 1 2 m 2 m ¼ 0; 1; 2; 3;...; m max: k z T ð6þ It should be noted that this result could have been also derived directly from Eq. (3) which reresents a eriodic function rovided that Eq. (6) is valid. For the 2-D case, we relace m 2! m 2 x þ m2 y.in that case, the allowed frequencies for a Montgomery object reresent concentric rings in the frequency lane (Fig. 1). Note, that the outermost ring with the highest satial frequency occurs for m ¼ 0. The maximum index, m max, occurs for the smallest satial frequency and is given as the largest integer smaller than z T =k: m max 6 z T =k. Instead of m we may also use the index n ¼ m max m which increases with the satial frequency. For small values of n, one may write:m n ffiffiffi n m 0 with m 0 ¼ m n¼0 which is the araxial limit to the Montgomery criterion [10]. In this case, one finds that z T 2=km 02. to the Talbot interferometer [4]. It consists of two diffractive elements, G 1 and G 2, searated by a multile of the longitudinal eriod z T (Fig. 2). For the exeriments, G 1 and G 2 were imlemented as hase-only comuter-generated DOEs calculated by an iterative Fourier transform algorithm. Reminiscent of the rings occurring in the Montgomery theory, G 1 and G 2 were designed with a frequency sectrum consisting of seven rings as shown in Fig. 3(a). In order to kee fabrication and the exeriment simle we chose the design arameters such that the satial frequencies turned out to be relatively large. In our exeriment, the longitudinal eriod was chosen to be z T ¼ 10,000k and k ¼ 632:8 nm. Each DOE consists of ixels with a ixel size of 8 8 ðlmþ 2 and four discrete hase levels (Fig. 3(b)). The far-field diffraction attern of a single DOE is shown in Fig. 3(c). The diffraction efficiency was calculated to be 50.6%. This value is lower than tyical values known for hase-only DOEs. We attribute this to the fact that in our demonstration exeriment a wavefield with radial symmetry is generated by a DOE based on a cartesian grid. This mismatch leads to increased satial quantization errors and thus to losses. It can be exected, however, that for other situations better otimization results and thus higher efficiencies can be achieved. Self-imaging is now demonstrated by lacing a second DOE at a distance z T behind the first. We used two hase-comlementary objects, i.e., G 2 ¼ G 1, so that the wavefront behind G 2 should ideally be a lane wave that can be focused to a shar sot in the Fourier lane. (This feature may 3. Exerimental demonstration In order to verify Montgomery self-imaging with DOEs, we use an exerimental setu similar Fig. 2. Otical setu with two gratings at a distance Dz. The second grating is designed and ositioned such that it is hasecomlementary to the wavefield generated by G 1.

4 16 J. Jahns et al. / Otics Communications 225 (2003) Fig. 3. (a) Designed diffraction attern showing the rings in the Fourier lane according to MontgomeryÕs theory. (b) Calculated hase grating consisting of ixels, each grey level reresents a different hase value. (c) Exerimental diffraction attern of the grating shown in (b). eventually turn out to be useful to collect the light with a oint detector.) As Fig. 4 shows, a shar focal sot is indeed observed in the outut lane as exected. Since each DOE generates unwanted diffraction orders and stray light as well, there is a certain amount of background illumination, which, however, is diluted over the outut lane and, therefore, low in intensity. The efficiency of the interferometer was obtained by measuring the relative efficiency of the zeroth order. The measured value of aroximately 19% was in close agreement with the calculated efficiency for the interferometer which is aroximately 21%. The latter value is determined by the square of the efficiency of a single DOE (theoretical value aroximately 25% in our case) and the reflection losses ð0:96 4 0:85Þ. Fig. 4. Exerimental diffraction attern of two-grating setu as shown in Fig. 2. The DOEs were the same as those used for Fig. 3 searated by z T. 4. Montgomery interferometer as a taed delayline filter In [8], the Talbot interferometer was suggested for imlementing a temoral taed-delay line filter. As exlained there, the temoral imulse resonse of the Talbot setu consists of a series of delayed delta-like imulses. Different grou time delays result for different diffraction orders. More general, the temoral imulse resonse of an otical system is directly associated with its angular sectrum exressed in terms of the satial frequencies m x and m y. The grou time delay s of a wave acket roagating under an angle a relative to the otical axis (here the z-axis) from a lane z to another lane z þ Dz is Dz= cosðaþ sðaþ ¼ : ð7þ c The directional cosine cosðaþ ¼km z can be exressed by the satial frequencies m x and m y to yield Dz sðm x ; m y Þ¼ r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð8þ c 1 k 2 m 2 x þ m2 y Consequently, any otical setu with an extended angular sectrum (1- or 2-D) has a temoral imulse resonse of finite extension and may, therefore, be otentially useful as a temoral filter. The imulse resonse may be continuous or discrete. A discrete imulse resonse occurs, for examle, when elements are used that generate a

5 J. Jahns et al. / Otics Communications 225 (2003) discrete angular sectrum. The use of DOEs is of articular interest because of the design freedom they offer. Hence our interest in the Talbot and Montgomery interferometer as filtering devices. In the Talbot case, laterally eriodic gratings are used which is equivalent to an equidistant sacing of satial frequencies: m n ¼ nm 1. This choice of satial frequencies, however, results in a quadratic increase of the time delay of the nth diffraction order Dz s n ðm x Þ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c 1 ðkm n Þ 2 ðdz=cþ½1 þð1=2þðnkm 1 Þ 2 Š: ð9þ In the case of the Montgomery interferometer, however, the satial frequencies are given by Eq. (6). Inserting this into Eq. (8) and using Dz ¼ M z T we now obtain a time delay s m that increases linearly with the index m of the diffraction order s m ¼ mm k c : ð10þ Note, that this is an exact result. Alternatively, we could have used the aroximate, yet more intuitive, equation for the satial frequencies of a Montgomery object consisting mostly of coarse structures for which m n ffiffi n m 0. With this and by using the same aroximation as in Eq. (9), we also arrive at the linear deendency between the time delay and the diffraction order. To summarize the contents of this section: by going from linear gratings with equidistant sacing of the satial frequencies as in the Talbot case to a Montgomery object with nonlinear (aroximately square-root-like) sacing of the satial frequencies, we can build an interferometric device that imlements linear time shifts. This may be of interest to build a taed-delay-line filter with alications for s/fs-ulses. The use of comuter generated diffractive elements allows one the flexible design of secific filter resonses. 5. Conclusion We have demonstrated Montgomery self-imaging with comuter-generated diffractive otical elements. For the exerimental demonstration, the Montgomery interferometer was used. The flexibility in the design of DOEs oens u the ossibility to aly the Montgomery effect to different tasks in otical information rocessing, similar to the Talbot interferometer. As a secific examle we suggest the use of the Montgomery interferometer as a temoral filter for Terahertz and otical frequencies. The use of 1-D gratings aears attractive for multilexing uroses. More work is currently ongoing. This includes theoretical and exerimental work on the temoral roerties of the Montgomery interferometer and considerations on the otential and limitations for temoral signal rocessing as well as ractical imlementations. References [1] H.F. Talbot, Philos. Mag. 9 (1836) 401. [2] J.T. Winthro, C.R. Worthington, J. Ot. Soc. Am. 55 (1956) 373. [3] K. Patorski, Progr. Ot. 27 (1989) 1. [4] A.W. Lohmann, D.E. Silva, Ot. Commun. 2 (1971) 413. [5] O. Bryngdahl, J. Ot. Soc. Am. 63 (1973) 416. [6] R. Ulrich, T. Kamiya, J. Ot. Soc. Am. 68 (1978) 583. [7] A.W. Lohmann, Otik 79 (1988) 41. [8] J. Jahns, E. ElJoudi, D. Hagedorn, S. Kinne, Otik 112 (2001) 295. [9] W.D. Montgomery, J. Ot. Soc. Am. 57 (1967) 772. [10] A.W. Lohmann, Otical Information Processing, Erlangen, 1978 (Chater 18). [11] G. Indebetouw, Ot. Acta 30 (1983) [12] G. Indebetouw, Ot. Acta 35 (1988) 243. [13] J. Ojeda-Casta~neda, P. Andres, E. Teıchin, Ot. Lett. 11 (1986) 551. [14] G. Indebetouw, J. Ot. Soc. Am. A 9 (1992) 549. [15] J. Durnin, J. Ot. Soc. Am. A 4 (1987) 651. [16] S. Sinzinger, J. Jahns, Microotics, second ed., Wiley- VCH, Weinheim, 2003.