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Generalized periodically focused beams and multiple on-axis lens imaging
Optics Communications 229 (2004) 39–57 www.elsevier.com/locate/optcom
Generalized periodically focused beams and multiple on-axis lens imaging Andrey G. Sedukhin
*
Russian Academy of Sciences, Institute of Automation and Electrometry, Prospekt Akademika Koptyuga 1, Novosibirsk 630090, Russia Received 31 July 2003; received in revised form 24 September 2003; accepted 20 October 2003
Abstract A class of periodically focused propagation-invariant monochromatic beams with sharp central peak is extended to the beams, of which the patterns of intensity at discrete transverse planes regularly spaced along the propagation axis may have an arbitrary, but the same, form. It is shown that the wave field of such beams in free space may be represented as a discrete set of nonuniform phase-distorted spherical wavelets converging to the vicinities of axial points at the indicated transverse planes and possessing the property of mutual similarity. High-order azimuth angular constituents of these wavelets are shown to have phase singularities appearing in the transverse planes as dark coaxial spots with bright rims. Hybrid refractive–diffractive systems are proposed, which are suited both for the direct high efficient production of the beams considered and for the multiple imaging of arbitrary plane objects. The results of numerical simulations are given for two cases of multiple reproduction of rings with large and small radii. Ó 2003 Elsevier B.V. All rights reserved. PACS: 41.85.Ct; 42.40.Jv; 42.60.Jf; 42.79.)e Keywords: Propagation invariance; Phase singularities; Diffractive multifocal lenses; Multiple imaging
1. Introduction In our previous paper [1] the existence of a new class of longitudinally periodic propagation-invariant monochromatic beams, referred to as periodically focused ones, was predicted heuristically and arrangements for production of these beams were proposed. An intensity pattern of such beams is periodically reproduced at the specified equidistant transverse planes and is centered about the propagation axis. The simplest rotationally symmetric beams with sharply defined central peak and low-intensity sidelobes have been demonstrated. It is characteristic that the field of these beams, considered just behind the output aperture plane of a beamforming system, can be decomposed into a discrete set of elementary fields which
*
Tel.: +7-3832-320-062; fax: +7-3832-333-863. E-mail address: [email protected] (A.G. Sedukhin).
0030-4018/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2003.10.026
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A.G. Sedukhin / Optics Communications 229 (2004) 39–57
correspond to the cuts of nonuniform strictly spherical wavelets converging to the regularly spaced coaxial focal points and having all the same energies and the amplitude distributions of angular spectra. The spacing between the focal points is chosen from the conditions which ensure the absence of significant interference of the wavelets in the vicinities of these points. In the present paper, we introduce more complicated beams, the total field of which in free space is represented as a superposition of mutually similar nonuniform phase-distorted converging spherical wavelets. We show that the nth wavelet considered just behind the output aperture plane of a beamforming system and conjugated with a vicinity of axial focal point zn can be represented in terms of two multiplicative constituents: the cut of a strictly spherical wavelet, converging exactly to the point zn (a fundamental constituent), and a suitable complex aperture-form function (a distortion constituent). The mutual similarity of the wavelets implies that there is the pairwise equality between the spherical projections of the amplitude and phase real-valued components of the complex aperture functions of two arbitrary wavelets onto unit spheres with centers in the axial points of convergence of the wavelets. Despite of certain restrictions, the two-dimensional (2-D) distributions of the amplitude and phase components of the aperture functions may have arbitrary, but the same, shapes. In particular, such shapes may lack the rotational symmetry and vary not only in the radial direction, but also in the azimuth angle. We derive the equations describing the exact forms of ideal infinite-extent generalized periodically focused beams in scalar approximation. The high-order angular constituents of the above complex apertureform function is shown to give rise to the phase singularities appearing as dark coaxial spots at regularly spaced transverse planes. Also, we propose optical systems for generation of real finite-extent beams, consider the direct association of these beams with multiple on-axis lens imaging, and make a brief comparison of these beams with those inherent in Talbot self-imaging effect. As it is shown, the proposed systems can perform either lens multiple Fourier imaging, when transforming the spectrum of an object, or lens multiple projection imaging, when reproducing the object itself. For numerical simulations of the finiteextent generalized periodically focused beams, we consider two examples of synthesizing the single-mode beams with ring focal patterns having quite large and minimum-permissible radii.
2. Exact scalar forms of generalized periodically focused beams First we present a mathematical description of ideal generalized periodically focused beams in scalar approximation. This description is important in methodological aspect and pertains, in essence, to the limiting forms of the beams generated by real finite-aperture systems, including the systems with high spatial resolution. Let us assume that in a sequence of the regularly spaced focal planes of the beam the nth position is defined as zn ¼ z1 þ ðn 1ÞZ, where z1 is the first position, Z ¼ znþ1 zn is the distance between adjacent planes and n ¼ 1; 2; 3; . . . is their current index. We now prove that the complex amplitude distribution of a generalized periodically focused propagation-invariant field at the output aperture plane of a beamforming system may be represented as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 X Dz exp ik R2 þ z2n þ iwn R pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi UðR; nR Þ ¼ A ; nR : ð1aÞ Rn R2 þ z2n n¼1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Here R ¼ x2 þ y 2 and nR ¼ arctanðy=xÞ are the radius and the azimuth angle at the output aperture plane z ¼ 0, Dz is a constant of reference distance, R R R A ; nR ¼ A ; nR exp iU ; nR ð1bÞ Rn Rn Rn
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is the complex aperture-form function with real-valued amplitude and phase components Aðm; nR Þ and Uðm; nR Þ (at m ¼ R=Rn and 0 6 m 6 1), and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1cÞ Rn ¼ zn ab = k 2 a2b is the boundary radius at the cross-section of the nth wavelet, with ab being the boundary radial spatial frequency. The last fractional term in Eq. (1a) describes the complex amplitude distribution at the crosssection of the nth uniform strictly spherical wavelet converging to the point zn and having the initial phase offset p < wn < p. As it follows from Eq. (1b), both the amplitude and the phase real-valued components of the complex aperture-form function – Aðm; nR Þ and Uðm; nR Þ – are double-normalized (in magnitude and in arguments nR and m) and are common to all the useful wavelets. It is well known that within scalar approximation the resultant field distribution in free space z > 0 can be calculated through the angular spectrum of plane waves [2]. To this end, we first expand the apertureform function in Fourier series in circular harmonics: 1 X R R A ; nR ¼ Am ð2Þ expðimnR Þ; Rn R n m¼1 with the Fourier coefficients determined as Z 2p R 1 R A ; nR expðimnR Þ dnR : ¼ Am Rn 2p 0 Rn The angular spectrum of plane waves, relative to the plane z ¼ 0, is then found as 1 1 X X m sða; nÞ ¼ FfUðR; nR Þg ¼ ðiÞ expðimnÞsmn ðaÞ;
ð3Þ
ð4aÞ
m¼1 n¼1
where a and n are, respectively, the radial spatial frequency and the azimuth angle in free space, F is the direct Fourier transform operator (see Appendix A), and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 R exp ik R2 þ z2n þ iwn pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi smn ðaÞ ¼ Dz Am ð4bÞ Jm ðaRÞR dR Rn R2 þ z2n 0 is the mth circular component of the angular spectrum for the nth phase-distorted spherical wavelet at the plane z ¼ 0, with Jm being the mth order Bessel function of the first kind. Substituting Eq. (1c) into Eq. (4b) and using the stationary-phase method [3] for asymptotic evaluation of the obtained integral expression under the condition kðz zn Þ ! 1, one can find the following approximate form: h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii mþ1 ðiÞ DzAm ða=ab Þ ðk 2 a2b Þ=ðk 2 a2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi smn ðaÞ ð5Þ expðizn k 2 a2 þ iwn Þ: k 2 a2 When deriving Eq. (5), the Bessel function Jm ðaRÞ was represented in complex form with separation of the useful diverging wave component from the well-known approximation of the modular and phase of a Bessel pffiffiffiffiffiffiffiffiffiffi ffi m function at large values of its argument: Jm ðaRÞ ðiÞ exp½iðaR p=4Þ= 2paR (this procedure has been described for the function J0 ðaRÞ in [4]). In fact, Eq. (5) is of the exact form, because the accepted condition kðz zn Þ ! 1 corresponds to consideration of a field in the far zone and is fully consistent with implicit definition of the angular spectrum of plane waves. To find the resultant complex amplitude of the field behind the output aperture, it is necessary to take the inverse Fourier transform of the angular spectrum of plane waves in free space: 1 1 n X X pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o Uðq; n; zÞ ¼ F1 sða; nÞ expðiz k 2 a2 Þ ¼ expðimnÞUmn ðq; zÞ; ð6aÞ m¼1 n¼1
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1 where q isp the radial ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi coordinate in free space, F is the inverse Fourier transform operator (see Appendix A), expðiz k 2 a2 Þ is the propagation transmission function of the angular spectrum in free space, and Z k pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi smn ðaÞ expðiz k 2 a2 ÞJm ðaqÞa da ð6bÞ Umn ðq; zÞ ¼ 0
is the complex amplitude of the mth circular component of the nth wavelet. In Eq. (6b), we used the integration over the range 0 6 a < k to exclude the evanescent waves. Finally, on substituting Eqs. (5) and (6b) into Eq. (6a), we arrive at the following expression for the resultant wave field: h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 1 1 Z ab Am ða=ab Þ ðk 2 a2b Þ=ðk 2 a2 Þ X X pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Uðq; n; zÞ ¼ Dz ðiÞmþ1 expðimnÞ k 2 a2 0 m¼1 n¼1 h i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp iðz zn Þ k 2 a2 þ iwn Jm ðaqÞa da; ð7Þ with the particular distributions of the dominant nth constituents at the transverse planes z ¼ zn represented as h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii Z ab Am ða=ab Þ ðk 2 a2 Þ=ðk 2 a2 Þ 1 X b mþ1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Un ðq; n; zn Þ ¼ Dz expðiwn Þ ðiÞ expðimnÞ Jm ðaqÞa da: 2 2 k a 0 m¼1 ð8Þ From Eq. (8), one can see that however complicated the complex amplitude distributions of the different nth constituents of a beam at their own planes z ¼ z1 , z ¼ z2 , z ¼ z3 ; . . . ; z ¼ zn would be, they are fully propagation-invariant accurate to the phase offsets wn . This result obviously to give proof to our representation regarding the form of a generalized periodically focused beam at the output aperture plane of a beamforming system, which is given by Eqs. (1). As can be seen from these equations, each nonuniform phase-distorted spherical wavelet of the beam possesses the property of similarity to other wavelets, since there is the pairwise equality between the spherical projections of both the amplitude and the phase realvalued components of the complex aperture functions of two arbitrary wavelets onto unit spheres centered at the axial points of convergence of the wavelets. Let us now analyze the resultant solution for the wave field of an ideal generalized periodically focused beam governed by Eq. (7). All the beamlets corresponding to the constituents of this equation with different indices m and n, as well as the superposition of these beamlets describing the total wave field, are the exact solutions of the nonparaxial Helmholtz wave equation with a marginal condition on the cutoff frequency ab . The field compositions defined as superpositions of the beamlets with different index n, but common index m, can be referred to as single-mode generalized periodically focused beams. According to Eq. (7), the intensity distribution of the wave field of ideal single-mode beams is rotationally symmetric. Touching on a beam of fundamental mode with m ¼ 0, we point out the following. As would be expected, for all spherical wavelets with m ¼ 0 and n P 1 the field distribution near the focal points zn (the nth 3-D point spread function) will fully be consistent with the well-established and rigorous integral representation of Debye, see Ref. [5, Eqs. (2.6) and (2.7)], if one will make change for the parameters we adopted (ks? ¼ a), will take account of the origin of the coordinate system (z ! z zn ), and will make generalization for the case with a nonuniform phase-distorted aperture function [A0 ðaÞ 6¼ const: at 0 6 a 6 ab ]. This fact reinforce, in part, our previous statement on the exact form of expression (5). The high-order modes with m ¼ 1; 2; . . . constitute the periodically focused beams which should be related to those having the so-called phase singularities (also known as wavefront screw dislocations or optical vortices) with topological charge m (see a review on the wave fields with phase singularities in [6]). The distinctive feature of the latter beams for the case with rotational symmetry is that they have helical wavefronts, whose phase at the arbitrary transverse plane is varied over 2pm on a closed path around the
A.G. Sedukhin / Optics Communications 229 (2004) 39–57
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propagation axis with the on-axis amplitude of the field being dropping to zero. The fact that the above high-order modes belong to precisely those beams is evidenced by occurrence of a separable azimuthdependent phase term expðimnÞ in Eqs. (7) and (8) and by nullification of the integrands in these equations due to the identity Jm ðaqÞjm6¼0;q¼0 ¼ 0. The relevant annulus-like intensity patterns may be called mth-order doughnut modes. It is worth noting also that, as it follows from Eqs. (4a) and (5), the angular spectrum of plane waves for any nth useful wavelet of a generalized periodically focused beam may be conceived as a nonuniform phasedistorted spherical wave diverging from the respective axial point zn and having the same complex amplitude distribution within a solid angle subtended by the boundary angle hb ¼ arcsinðab =kÞ: 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ðiÞm expðimnÞsmn ðaÞ sn ða; nÞ ¼ expðiz k 2 a2 Þ h
m¼1
i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Sða; nÞ exp iðz zn Þ k 2 a2 ip=2 þ iwn ; h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 1 Am ða=ab Þ ðk 2 a2b Þ=ðk 2 a2 Þ X m pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Sða; nÞ ¼ Dz ð1Þ expðimnÞ : k 2 a2 m¼1
ð9aÞ ð9bÞ
Here Sða; nÞ is the complex amplitude of the angular spectrum, which is common to all wavelets and is varied, in the general case, both in spatial frequency a and in azimuth angle n. The last exponential term in Eq. (9a) is the nth strictly spherical wavelet converging to the point zn . Using Eq. (9b), it can easily be found that each nth useful wavelet will possess the same finite energy.
3. Implementation As can be seen from Eqs. (1) of the present paper and from [1, Eq. (4)], the only differences between the complex amplitude distribution at the output aperture plane inherent to a generalized periodically focused beam and that peculiar to a periodically focused beam with stigmatic ray convergence are the complex character of the output aperture functions of the respective wavelets and the possible lack of rotational symmetry of these functions. Furthermore, the front of the nth wavelet, Wn , associated with the nth complex aperture function and being fixed with respect to the output plane, may conveniently be subdivided into two parts: Wn ¼ Wn0 þ Wn00 ; Wn0
ð10Þ Wn00
where is the front of the strictly spherical wavelet converging to the point zn and is the wavefront deformation peculiar to a specific pattern to be generated at the transverse focal plane z ¼ zn . Given equidistant foci z1 ; z2 ; z3 ; . . . ; zN (with subscript N denoting their total number), the amplitude aperture-form function, Aðm; nR Þ, and also the above wavefront deformation which defines the phase component of the complex aperture-form function, Uðm; nR Þ, one can obviously reconstruct the requisite real-valued amplitude components A1 ðRÞ; A2 ðRÞ; A3 ðRÞ; . . . ; AN ðRÞ of the aperture functions and the fronts W1 ; W2 ; W3 ; . . . ; WN of wavelets relative to the output aperture plane. Fig. 1 is an illustration of such the reconstruction for the simple paraxial case with rotational symmetry, when the amplitude aperture function of the nth wavelet has the appearance of a circular function Dz Dz R An ðRÞ ¼ circ ; ð11Þ zn zn DR and the nth wavefront deformation Wn00 has the conical form, such that the wavefront phase delay varies as
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Fig. 1. Torus-like fronts of wavelets, Wn , and the circular real-valued amplitude components of their aperture functions, An ðRÞ, at the output plane of a rotationally symmetric beamforming system producing invariable ring-like amplitude patterns at three regularly spaced planes in free space. The focal points ðqring ; zn Þ are marked with bullets. The fronts denoted as Wn0 and Wn00 are, respectively, the spherical and conical parts of the nth wavelet front Wn . The distance constant is assumed to be equal Dz ¼ z1 .
Un ðRÞ ¼ kqring R=zn ;
ð12Þ
where circðrÞ ¼ 1 for 0 6 r 6 1 and 0 otherwise, with DR ¼ Dzab =k and qring a1 b being constant values. As is easy to see, the multiple focusing to a ring of radius qring is performed in this case. Rotationally symmetric beamforming systems for production of periodically focused beams with no the wavefront deformation of spherical wavelets (with Wn00 ¼ 0) have already been considered in [1]. A key component of these systems is a diffractive multifocal (DMF) lens with uniform beam splitting and a naturally spaced on-axis array of foci. This DMF lens is characterized by a transmittance function T 0 ðrÞ and in conjunction with a positive refractive lens L (a focal distance apart from the DMF lens) is capable of generating a complex beam consisting of the plurality of strictly spherical wavelets with fronts Wn0 . Shown in Fig. 2 is a beamforming system with collimated illumination of the DMF lens. Below, we will prove that for production of a generalized periodically focused field it will suffice to insert into the system an auxiliary spatial-frequency filter having a transmittance T 00 ðr; nr Þ and located, e.g., in the immediate vicinity of the DMF lens (here nr stands for the azimuth angle at the plane of the DMF lens). The role of this filter is just in the correction for necessary wavefront deformations of wavelets, Wn00 . However, for better understanding and convenience of further presentation, we first carry out the resumptive consideration of a system with pure stigmatic multiple focusing. As was shown [1], the transmittance of a planar phase DMF lens in the parabolic approximation must be periodic in r2 and may thus be represented as a Fourier series: " 2 # 0 2 1 1 X X r ikn r 0 0 0 T ðrÞ ¼ exp½iu ðrÞ ¼ an0 exp i2pn an0 exp ¼ ; ð13Þ Kr 2F n0 ¼1 n0 ¼1
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Fig. 2. Modifying the arrangement of a periodically focused propagation-invariant beam generator from the simplest one, with forming a uniform array of coaxial spots at multiple equidistant focal planes [with no the filter with transmittance T 00 ðr; nr Þ], to the generalized one, with forming a uniform array of coaxial patterns of an arbitrary form at the same focal planes [with the filter with transmittance T 00 ðr; nr Þ being added].
where u0 ðrÞ is the phase function of the DMF lens, F is the focal length for the first converging diffraction pffiffiffiffiffiffiffiffi ffi order n0 ¼ 1, Kr ¼ 2kF is a constant equal to the radius of the first Fresnel zone for the principal focus the distance F apart from the DMF lens, and the Fourier coefficients an0 are dependent on the shape of the phase function within Fresnel zones and are determined as " 2 # Z Kr 2 r an0 ¼ 2 T 0 ðrÞ exp i2pn0 r dr: ð14Þ K Kr 0 r For a well-designed DMF lens, the values jan0 j2 , accounting for energy percentage in the n0 th diffraction 2 orders, are practically equal within an indexing set of useful orders, N0 (that is jan0 j jn0 2N0 ’ const:). Besides, 0 the efficiency of the DMF lens, is close to unity P sum of 2these values over the set N , defining the light 2 2 ( n0 2N0 jan0 j ’ 1, so that for any n0 2 N0 and j 62 N0 jan0 j jaj j ). For the DMF lens with odd or even 0 number of useful diffraction orders, N , the set N includes, respectively, the integers n0 ¼ 0; 1; 2; . . . ; ðN 1Þ=2 or n0 ¼ 1; 3; 5; . . . ; ðN 1Þ. Under the normal collimated illumination of such DMF lens, the field with an array of real and virtual coaxial foci is generated, the positions of these foci being nonlinearly distributed at distances F =n0 away from the DMF lens. Placing the DMF lens at the front focal plane of a positive refractive lens L with focal length f and meeting the restrictions given below, enables one to transform such the field to that possessing the propagation invariance within a discrete finite set of equidistant transverse planes behind the refractive lens. In what follows we will go from the indexing system with integers n0 , defined within the set N0 , to the indexing system with natural numbers defined within the set N ¼ f1; 2; . . . ; N g. The relation between indices n and n0 will then be defined as ð2n N 1Þ=2 for odd N ; n0 ðnÞ ¼ ð15Þ 2n N 1 for even N : On applying for the system of Fig. 2 the ordinary lens equation (relating the positions of foci of the DMF lens and their images behind the refractive lens), 1 1 1 þ ¼ ; 0 f þ F =n ðnÞ zn f
ð16Þ
the nth positions of the coaxial foci of a generated beam, zn , relative to the output aperture plane, can be determined as ð17Þ zn ¼ z1 þ ðn 1ÞZ; n ¼ 1; 2; . . . ; N ;
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where z1 ¼ f ðN 1ÞZ=2 is the position of the first focus and 2 f =F for odd N ; Z ¼ znþ1 zn ¼ 2f 2 =F for even N
ð18Þ
ð19Þ
is the distance between adjacent foci. Let us now place in the immediate vicinity of the DMF lens an auxiliary spatial-frequency filter with transmittance T 00 ðr; nr Þ proportional to the inverse Fourier transform of a desired propagation-invariant distribution at the equidistant focal planes: T 00 ðr; nr Þ / F1 fUn ðkq=f ; n; zn Þg:
ð20Þ
00
We shall think of the function T ðr; nr Þ as being pure phase one and expand it in a Fourier series in terms of circular harmonics: T 00 ðr; nr Þ ¼ exp½iu00 ðr; nr Þ ¼
1 X
bm ðrÞ expðimnr Þ;
ð21Þ
m¼1
where u00 ðr; nr Þ is the phase function of the filter and the Fourier coefficients bm ðrÞ are defined as Z 2p 1 T 00 ðr; nr Þ expðimnr Þ dnr : bm ðrÞ ¼ 2p 0
ð22Þ
To find successively the complex amplitude distributions at the plane of the lens L and in free space behind it, let us write down the appropriate expressions on the basis of Fresnel diffraction integral in the cylindrical coordinate system: Z 1 2 k R2 ikr 0 00 exp ik f þ UðR; nR Þ ¼ AðrÞT ðrÞT ðr; nr Þ exp r i2pf 2f 2f 0 Z 2p ikRr cosðnR nr Þ dnr dr; exp f 0 Z Rb k q2 ikR2 ikR2 exp ik z þ þ exp Uðq; n; zÞ ¼ R i2pz 2z 2f 2z 0 Z 2p ikRq cosðn nR Þ dnR dR; UðR; nR Þ exp z 0
ð23Þ
ð24Þ
where AðrÞ is the input real-valued aperture function. Next we suppose that the aperture of the lens L does not truncate practically the beams of useful diffraction orders, so that the condition Rb ! 1 can be considered as being meeting. Then, on inserting Eq. (23) into Eq. (24) and using Eqs. (13), (21) and the formulas [7] Z Z
2p m
0 1
exp ½imf ix cosðf vÞ df ¼ ðiÞ 2p expðimvÞJm ðxÞ;
ð25Þ
imþ1 bc iðb2 þ c2 Þ Jm expðiaf ÞJm ðbfÞJm ðcfÞf df ¼ exp ; 2a 4a 2a
ð26Þ
2
0
we obtain the following expression for the resultant field:
A.G. Sedukhin / Optics Communications 229 (2004) 39–57
Uðq; n; zÞ ¼
1 X k exp ½ik ð f þ zÞ ðiÞmþ1 expðimnÞ f m¼1 2 Z 1 1 X ikr 1 z n0 krq an 0 bm ðrÞAðrÞ exp þ r dr: Jm f f2 F f 2 0 n0 ¼1
47
ð27Þ
Going again to the system with indexing in n [see Eq. (15)], we obtain that, without significant mutual interference of the wavelets near the points zn ¼ z1 þ ðn 1ÞZ [where the argument of the exponential function in the integrand of Eq. (27) vanishes], the complex amplitude distribution of the field at the transverse plane z ¼ zn takes the form of the nth constituent: Z 1 1 X k krq mþ1 ðiÞ expðimnÞ bm ðrÞAðrÞJm r dr; ð28Þ Un ðq; n; zn Þ ¼ exp ½ikðf þ zn Þan f f 0 m¼1 where an an0 ðnÞ and the subscript n0 ðnÞ is defined by Eq. (15). Alternatively, Eq. (28) can be written in terms of the following Fourier transform: Un ðkq=f ; n; zn Þ ¼ cn FfAðrÞT 00 ðr; nr Þgja¼kq=f ;
ð29aÞ
where cn ¼
kan exp ½ik ð f þ zn Þ: if
ð29bÞ
Finally, from Eq. (29a) it follows that T 00 ðr; nr Þ ¼ F1 fUn ðkq=f ; n; zn Þg=½cn AðrÞ:
ð30Þ
Eq. (30) confirms that, with a correction for the input aperture function AðrÞ, the transmittance T 00 ðr; nr Þ is indeed proportional to the inverse Fourier transform of the desired propagation-invariant distribution at the equidistant focal planes [see Eq. (20)]. Furthermore, analyzing Eqs. (28) and (29) along with Eq. (21), one can conclude that the intensity distributions of a generated beam at the equidistant focal planes well-separated along the propagation direction is really much close to a desired propagation-invariant pattern provided that jan j2 ’ const. In fact, the indicated equations describe in paraxial approximation the focal-plane complex amplitude distributions of the same generalized beam as that has been analyzed earlier in nonparaxial scalar treatment [see Eq. (8)]. It is easy to show that the complex aperture-form function for the nth nonuniform phase-distorted spherical wavelet of that beam is related, in paraxial approximation, to the beamforming-systemÕs characteristics and the complex amplitude distribution at the corresponding focal plane as: R R R f A ; nR ¼ an A rb T 00 rb ; nr ¼ F1 fUn ðkq=f ; n; zn Þg; ð31Þ Rn Rn Rn k where rb is the boundary aperture radius of the DMF lens and of the auxiliary filter with transmittance T 00 ðr; nr Þ. The last expression was obtained by comparing the components of Eq. (28) and the paraxial form of Eq. (8) with taking into account Eqs. (2), (21) and (29) choosing Dz ¼ f , and omitting unessential phase constants. Clearly, a pattern of intensity distributions at the transverse focal planes is strongly dependent on the Fourier coefficients bm ðrÞ of the function T 00 ðr; nr Þ. In general case, the pattern is azimuth-modulated and may have an arbitrary form. However, of special interest are single-mode beams characterized by a rotationally symmetric pattern with only one coefficient bm ðrÞ not being zero. The beams of zero-order mode with b0 ðrÞ being real and nonzero relate to the simplest form of periodically focused beams with multiple stigmatic ray convergence and have already been demonstrated (mainly in paraxial approximation) in [1].
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4. Multiple on-axis lens imaging Let us now discuss the issue of resemblance of generalized periodically focused wave fields and the wave fields inherent in Talbot self-imaging effect. Under collimated monochromatic illumination with light of the wavelength k, periodic objects with sufficiently large transverse dimensions are known to be self-imaged at regularly spaced transverse planes. For example, the pattern of a periodic grating is reconstructed (without transverse shift) at multiple transverse planes separated by the Talbot distance [8] ZT ¼ 2K2u =k;
ð32Þ
where Ku is the period of the grating. This effect is observed in the near field without the aid of lenses or other optical accessories and may be therefore referred to as lensless multiple Fresnel imaging. In using the beamforming system of Fig. 2, the minimum distance between the regularly spaced transverse planes, at which the pattern of an object can be reproduced (with N being odd), is defined as 2
Z ¼ f 2 =F ¼ 2ðf =Kr Þ k:
ð33Þ
Eqs. (32) and (33) disclose that the distances ZT and Z have quite different dependence on the wavelength k. Besides, the important feature of the latter system is the fact that the reconstructed pattern is directly associated with not the object itself but with its Fourier image. In Fig. 2, the Fourier image is represented by an objectÕs mask, which has the transmittance T 00 ðr; nr Þ and is placed at the plane of the DMF lens; the reproduction being realized by means of an auxiliary Fourier transform lens L. In this connection, such reproduction may be called lens multiple Fourier imaging. If the multiple imaging of an object itself is desirable, there have to be obvious modifications in the optical system. To associate the object with its intermediate Fourier image at the plane of the DMF lens, one more Fourier transform refractive lens may be introduced, for example. In this way, we arrive at a classical two-Fourier-lens imaging system (coherent optical processor) [2] considered yet in extended treatment as a system with plurality of output focal planes and employing the DMF lens as a spatialfrequency filter (see Fig. 3). The complex amplitude distribution at the plane of the object to be reproduced may now be related to the complex amplitude distribution at the plane situated just before the DMF lens through the suitable inverse Fourier transform. Thus, the transmittance of a new amplitude objectÕs mask t00 ðq0 ; nq0 Þ turns out to be also proportional to the inverse Fourier transform of the transmittance T 00 ðr; nr Þ of the auxiliary filter used in the previous system of Fig. 2:
Fig. 3. Multiple imaging system with a filter DMF lens having the phase transmittance T 0 ðrÞ. At the equidistant transverse planes z ¼ z1 ; z ¼ z2 ; . . . ; z ¼ z7 , the system provides the invariant projection imaging of an arbitrary object represented by a mask with the amplitude transmittance t00 ðq0 ; nq0 Þ.
A.G. Sedukhin / Optics Communications 229 (2004) 39–57
t00 ðq0 ; nq0 Þ / F1 fT 00 ðkr=f ; nr Þg;
49
ð34Þ
0
where q and nq0 are, respectively, the radial and azimuth coordinates at the plane of the object. The overall process of monochromatic multiple image replication, as applied to an imaging system of Fig. 3, can also conveniently be described within the framework of Fourier optics, if the following simplifications are met: (i) both refractive lenses do not introduce the essential confinement in the passage of useful spectral components and (ii) the image planes are well separated from one another to avoid overlapping of adjacent on-axis point spread functions. In this case, the complex amplitude distribution just behind the plane of DMF lens will be proportional to the product FfUðq; n; 3f Þgja¼kr=f ;n¼nr T 0 ðrÞ, with Uðq; n; 3f Þ Uðq0 ; nq0 Þ denoting the complex amplitude distribution just behind the objectÕs mask. Taking into consideration Eq. (29a), the complex amplitude distribution at the nth equidistant image plane z ¼ zn may be expressible as 9 #> kr n ðnÞ = Un ðq; n; zn Þ / F FfUðq; n; 3f Þgj a¼kr=f T 0 ðrÞ exp i > > 2F n¼nr ; : a¼kq=f ( ) kr2 n0 ðnÞ 0 ¼ F FfUðq; n; 3f Þgj a¼kr=f F T ðrÞ exp i 2F n¼nr a¼kq=f a¼kq=f Uðq; n; 3f Þ an dðqÞ ¼ an Uðq; n; 3f Þ; 8 > <
"
2 0
ð35Þ
where is the 2-D convolution operator, dðqÞ is the Dirac delta function, the dependence n0 ðnÞ is defined by Eq. (15), and the quadratic exponential term accounts for the axial focal shift of the nth image. When deriving Eq. (35), we assume that for the nth complex beamlet generating the nth image, FfT 0 ðrÞ exp ½ikr2 n0 ðnÞ=ð2F Þgja¼kq=f ¼ PSFn ðq; zn Þ an dðqÞ, with PSFn ðq; zn Þ denoting the amplitude point spread function of the imaging system for the nth beamlet. The change of sign at the argument q in Eq. (35) implies that the generated image will be inverted with respect to the objectÕs image. With double Fourier transform processing performed in the system by refractive lenses, the reproduction in view may be called lens multiple projection imaging. In common with conventional non-aberrated imaging systems with a single output focal plane, the quality of reproduction may be characterized in the last instance by the relationship between the maximum spatial frequency of an object and the boundary spatial frequency of the imaging system. An additional point to be emphasized is that the multiple projection imaging can be realized not only with two-Fourierlens systems but also with single-lens ones. For example, a single-lens system presented in [1, Fig. 4] is well suited for this purpose, if one places a reconstructed object at the plane of a point source S. Alternatively, a filter approximating the Fourier transform of the object can likewise be placed at the front focal plane of the output refractive lens L. An apparent advantage of the lens multiple imaging over the lensless Talbot multiple imaging is the fact that a pattern to be reproduced may have an arbitrary form and may also be imaged as both scaled up and down with respect to the objectÕs sizes. The beamforming systems based on DMF and refractive lenses and having the on-axis character of image replication are, in fact, complementary to the well-known in-plane multiple imaging systems based on Dammann gratings [9,10]. It is notable also that a closely similar system with on-axis multiple imaging has recently been proposed and investigated in [11–14]. A distinctive key component of that system is a Fabri– Pero interferometer acting as a comb-like spatial-frequency filter and originating behind the output refractive lens a collection of interfering zero-order Bessel beams. However, the narrow-band nonlinear character of the transmittance of the filter gives rise to the reduced light efficiency of the imaging system as
50
A.G. Sedukhin / Optics Communications 229 (2004) 39–57
well as to the limitation of the full depth of a generated field and to the moderate invariance of relevant images. The imaging properties of such the system are confined due to the fact that its point spread functions possess marked transverse sidelobes (for systems with rotational symmetry these functions are close to the zero-order Bessel function), the main intensity maxima of these functions having approximately linear growth with propagation distance. The advantages of imaging systems with DMF lenses are thus the enhanced light efficiency, the large full depth of field, and the high quality of image replication due to high linearity of transmitting the spatial frequencies. 5. Numerical simulations Let us now consider examples of synthesizing some finite-extent periodically focused beams which possess a high degree of propagation invariance, but lack stigmatic on-axis ray convergence. For a rotationally symmetric system of Fig. 2, one can take a ring-like intensity distribution as being the simplest suitable pattern reproduced at the equidistant focal planes. A mathematically idealized desired distribution of the complex amplitude at the nth plane may then be represented in the form: Uðq; n; zn Þ ¼ dðq qring Þ exp½iWðnÞ; ð36Þ where WðnÞ is the arbitrary phase function which can be expanded in a Fourier series in circular harmonics. For simplicity of further derivations, consider the single-mode cases with WðnÞ ¼ mn. Additionally, suppose that the incident wave is the plane one and has a constant amplitude within input aperture, so that AðrÞ ¼ circðr=rb Þ. Using Eqs. (30) and (36) and omitting the amplitude and phase constants, we then get that the transmittance T 00 ðr; nr Þ must be consistent with the mth order Bessel function: T 00 ðr; nr Þ / Jm kqring r=f expðimnr Þ: ð37Þ When using the further asymptotic expansion of the Bessel function Jm ðkqring r=f Þ at large values of its argument and going to the complex representation with dropping the slowly varying amplitude factor, we find the following approximate expression for the transmittance function: T 00 ðr; nr Þ exp½iðkqring r=f mnr mp=2 p=4Þ þ exp½iðkqring r=f þ mnr mp=2 p=4Þ:
ð38Þ
It is easy to see that the first and second phase-only components of this function represent, respectively, the converging and diverging pure-conical (m ¼ 0) or helical-conical (m 6¼ 0) waves. Being passed through the output refractive lens, each wave is reshaped and provides, at the equidistant focal planes, the in-phase beam focusing to a ring of radius qring . Because of this, either or both components of Eq. (38) are suitable for the desired focusing. Besides, numerical simulations reveal that the shape of the focal-plane intensity distributions in the vicinity of radius qring is scarcely affected by the quantity m, if only qring a1 b or, differently, if qring Wring , where Wring is the full width of the ring at half intensity maximum. On this basis, one may set m ¼ 0, when synthesizing the beams with large-ring intensity patterns. Then, the simplest transmittance function generating, e.g., a converging pure-conical wave will be that of a refractive axicon: T100 ðrÞ ¼ b0 ðrÞ ¼ exp½iu001 ðrÞ ¼ exp ikqring r=f : ð39Þ In the back focal plane of a positive refractive lens the axicon is known indeed to form a ring-like intensity distribution with rather weak transverse diffraction sidelobes [15]. The plots of the amplitude components of the aperture functions and the front shapes of relevant wavelets, which have been demonstrated in Fig. 1 earlier, represent just the same asymptotic curves as those related to a periodically focused beam produced by the beamforming system with an auxiliary filter having the transmittance according to Eq. (39). Neglecting the beam vignetting by the positive refractive lens L and using Eqs. (39) and (28), the ring-like intensity distribution at the nth output focal plane and the full width of this distribution at half intensity maximum can be assessed, respectively, as
A.G. Sedukhin / Optics Communications 229 (2004) 39–57
51
2 Z rb k kqring r kqr In ðq; n; zn Þ ¼ an exp i r dr ; J0 f f f 0
ð40Þ
Wring ¼ 6:223=ab ;
ð41Þ
with ab krb =f . In the first numerical example, suppose that the transmittance of the auxiliary filter is given by Eq. (39) with a desired ring distribution, to be formed at the equidistant focal planes, having the maximum intensity radius qring ¼ 0:5 mm and the full width of the ring at half intensity maximum Wring ¼ 10 lm (when the condition qring Wring holds). From Eq. (41), we readily find the preliminary value of the required boundary spatial frequency: ab ¼ 622:3 mm1 . To characterize the beam, it is essential also to specify such its parameters as the design wavelength k, the number of useful foci N , the axial position of a nearest useful focus z1 , the maximum propagation distance zN , and the distance between adjacent useful foci Z. For the sake of convenience, we set these parameters to be consistent with those of a fundamental-mode periodically focused beam, which has been investigated in [1]. Namely, we assume that k ¼ 0:6328 lm, zN ¼ 1 m, N ¼ 9, Z ¼ 100 mm, and z1 ¼ 2Z ¼ 200 mm. Using a calculation procedure presented in the reference 2 mentioned, pffiffiffiffiffiffiffiffiffi one can immediately find that f ¼ ðzN þ z1 Þ=2 ¼ 0:6 m, F ¼ f =Z ¼ 3:6 m, and Kr ¼ 2kF ¼ 2:135 mm. Let the DMF lens be of continuous-relief profile and has the following phase function: " 2 # 4 X e Þþ2 e 0 Þ cos kr n0 ; e u0 ðrÞ ¼ arg e a 0 expði w a n0 expði w ð42Þ 0 n 2F n0 ¼1 e ;...; w e g ¼ f0:906; 2:966; 1:601; 1:015; where fe a0; . . . ; e a 4 g ¼ f1:002; 0:976; 0:969; 0:936; 1:030g and f w 0 4 0:488g. We derived the last function from the generating phase function of a 1-D fan-out grating which, in turn, was designed by the virtual-source approach [16] for the fan-out N ¼ 9 [17]. The theoretical light efficiency of the DMF lens is equal to 99.28%. In Figs. 4(a) and (b), the different views of the central portion of the phase function are shown. pIn final ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi stage, we find the values of other quantities: P ¼ Round½Za2b =ð4pkÞ ¼ 310, a0b ¼ 4pkP =Z ¼ 621:9 mm1 , rb ¼ f a0b =k ¼ 37:582 mm, and Rb ¼ zN a0b =k qring þ pBzN =ðf a0b ÞjB¼14 ¼ 62:3 mm. Using Eqs. (23) and (24), the calculation of a generated wave field was carried out. The entire longitudinal intensity profile of this field at maximum intensity radius qring and the superimposed entire transverse intensity profiles at equidistant focal planes z ¼ z1 ; z ¼ z2 ; . . . ; z ¼ z9 are shown, respectively, in Figs. 5(a) and (b). The superimposed local longitudinal intensity profiles I1 ¼ Iðqring ; z z1 Þ; . . . ; I9 ¼ Iðqring ; z z9 Þ and the superimposed local transverse intensity profiles I1 ¼ Iðq qring ; z1 Þ; . . . ; I9 ¼ Iðq qring ; z9 Þ are represented, respectively, in Figs. 5(c) and (d). The last plots characterize the degree of scattering of the useful intensity point spread functions of the beamforming system both in magnitude and also in longitudinal and transverse displacements. As another example, which might be useful, e.g., for precise alignment, we now consider the special case of generating a singular longitudinally periodic beam with strictly zero intensity on the optical axis and ring-like intensity distributions at the focal planes, with a rather small maximum intensity radius qring . In this case, the condition qring Wring may does not hold and the transmittance function T200 ðr; nr Þ, coming from either component of Eq. (38) at qring ! 0 and m 6¼ 0, has the form T200 ðr; nr Þ ¼ bm ðrÞm6¼0 ¼ exp½iu002 ðrÞ ¼ exp ðimnr Þ:
ð43Þ
As can be seen, the filter with such transmittance originates an mth order helical wavefront. From Eqs. (28) and (43) it follows that, in the idealized case of the total absence of beam vignetting by the positive refractive lens L, the intensity distribution at the nth output focal plane will take the form:
52
A.G. Sedukhin / Optics Communications 229 (2004) 39–57
Fig. 4. Halftone representation of the central portion of the phase function of a continuous-relief zero-order DMF lens (a) and its radial section (b). The first three Fresnel zones are shown only. The design parameters are as follows: N ¼ 9, Kr ¼ 2:135 mm, and P ¼ 310 (with rb ¼ 37:582 mm).
2 Z rb k kqr Iðq; n; zn Þ an Jm r dr : f f 0
ð44Þ
Of particular interest is a maximum-compressed ring distribution corresponding to the first-order modes with m ¼ 1. In accordance with Eq. (44), the full width of the ring zone of this distribution at half intensity maximum amounts to Wring ¼ 2:623=ab :
ð45Þ
If need be, the other sizes can likewise be determined. For instance, the radius of the central dark spot at half intensity maximum and the maximum-intensity radius are found to be qdark spot ¼ 1:205=ab and qring ¼ 2:452=ab , respectively. Using the binary representation for the structure of the DMF lens introduced in [1], it makes sense to combine the elements with transmittances T 0 ðrÞ and T200 ðr; nr Þ into a single element with transmittance T ðr; nr Þ ¼ T 0 ðrÞT200 ðr; nr Þ ¼ exp½iuðr; nr Þ;
ð46aÞ
where uðr; nr Þ ¼
2 Q P Y pY kr sgn þ mnr 2pðuq þ p 1Þ 2 p¼1 q¼1 2F
ð46bÞ
is the binary phase function of the element, P is the number of Fresnel zones within the aperture of the element relative to its principal focus the distance F apart, Q is the number of transition points within one
A.G. Sedukhin / Optics Communications 229 (2004) 39–57
53
Fig. 5. Intensity patterns of the output field for a beamforming system corresponding to Fig. 2, designed for illumination with a uniform collimated input beam, and consisting of a DMF lens with the phase function of Fig. 4, a refractive axicon with the phase function u001 ðrÞ ¼ kqring r=F ¼ 0:00524r=k, and a refractive lens with f ¼ 600 mm and Rb ¼ 62:3 mm [the values of other design parameters are as follows: k ¼ 0:6328 lm, Z ¼ 100 mm, z1 ¼ 200 mm, zN ¼ 1 m, F ¼ 3:6 m, Wcirc ¼ 10 lm, and a0b ¼ 621:9 mm1 ]: the entire longitudinal intensity profile at the maximum intensity radius qring (a), the superimposed entire transverse intensity profiles at the equidistant focal planes z ¼ z1 ; z ¼ z2 ; . . . ; z ¼ z9 (b), the superimposed local longitudinal intensity profiles I1 ¼ Iðqring ; z z1 Þ; . . . ; I9 ¼ Iðqring ; z z9 Þ (c), and the superimposed local transverse intensity profiles I1 ¼ Iðq qring ; z1 Þ; . . . ; I9 ¼ Iðq qring ; z9 Þ (d).
period of the generating phase function of a 1-D fan-out grating at which the phase function undergoes abrupt changes, and 0 < uq 6 1 is the normalized position of the qth transition point within one grating period. We assume here that the arguments of the function uðr; nr Þ are varied in the ranges 0 6 r < rb and 0 6 nr < 2p only. The integrated element with transmittance of Eq. (46) may be referred to as an mth-order singular DMF lens. In the numerical simulation, the full width of the ring zone at half intensity maximum is chosen to be equal Wring ¼ 20 lm at m ¼ 1. The values of the main parameters of the beam to be synthesized and of the focal lengths derivable from them – k, z1 , zN , Z, N , f , and F – are assumed to be the same as in the previous case, when synthesizing the beam with large-ring focal patterns. The preliminary value of the required boundary spatial frequency can immediately be determined from Eq. (45) as ab ¼ 2:623=Wring ¼ 131:2 mm1 . From [18], we adopt the following normalized ordered set of transition points at which the generating phase function of a 1-D binary fan-out diffractive grating undergoes abrupt changes within a period: fu1 ; . . . ; u6 g ¼ f0:06668; 0:12871; 0:28589; 0:45666; 0:59090; 1:00000g (the theoretical light efficiency of this structure is equal to 72.49%). The values of other parameters are finally determined as: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P ¼ Round½Za2b =ð4pkÞ ¼ 14, a0b ¼ 4pkP =Z ¼ 132:17 mm1 , rb ¼ f a0b =k ¼ 7:987 mm, and Rb ¼ zN a0b =k þ pBzN =ðf a0b ÞjB¼14 ¼ 13:9 mm. Fig. 6 shows the central portion of the phase function of the calculated DMF element. Fig. 7(a) displays the map of intensity contours of the generated first-order singular beam in the meridional plane. The longitudinal intensity profile at radius qring ¼ 18:5 lm, where the main intensity maxima are observed, and the superimposed transverse intensity profiles at the equidistant focal planes z ¼ z1 ; z ¼ z2 ; . . . ; z ¼ z9 are shown, respectively, in Figs. 7(b) and (c).
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A.G. Sedukhin / Optics Communications 229 (2004) 39–57
ϕBF (rad)
(a)
π _ 2
0
_ –π 2 –3
(b)
–2
–1
0
1
2
3
Radius r (mm)
Fig. 6. Graphic representation of the central portion of the phase function of a binary-phase first-order DMF lens (a) and its radial section (b). The first three Fresnel zones are shown only. The design parameters are as follows: N ¼ 9, Kr ¼ 2:135 mm, and P ¼ 14 (with rb ¼ 7:987 mm).
6. Conclusions We have shown that, apart from sharply peaked monochromatic periodically focused beams with multiple stigmatic on-axis ray convergence, there can exist periodically focused beams having an arbitrary, but the same, form of a desired propagation-invariant pattern reproduced at the equidistant focal planes. In the general case such the beams are represented as a discrete set of nonuniform phase-distorted spherical wavelets which possess the property of mutual similarity of their complex aperture functions. The last property means that the spherical projections of the real-valued amplitude and phase components of the complex aperture functions of the wavelets onto unit spheres, centered relative to the respective regularly spaced coaxial focal points, are pairwise the same for all the wavelets. In alternative treatment, the angular spectrum of plane waves for each useful wavelet may be conceived as a nonuniform phase-distorted spherical wave diverging from the respective on-axis focal point and having a changeless complex amplitude distribution. At the output aperture plane of a beamforming system the field of any wavelet can be decomposed into two multiplicative constituents: the complex amplitude at the cut of a strictly spherical wavelet, converging to a respective on-axis focal point, and a complex aperture-form function proportional, in paraxial approximation, to the inverse Fourier transform of a desired pattern. The high-order angular constituents of the aperture-form function give rise, in turn, to phase singularities appearing at the equidistant focal planes as dark coaxial spots with bright rims. The simplest optical arrangement for generation of a finite-extent generalized periodically focused beam can be implemented in the form of a positive refractive lens and the combination of two spatial-frequency phase filters placed in the front focal plane of the lens. The transmittance of the first filter corresponds to that of a diffractive multifocal lens and is associated with the fields of the above strictly spherical wavelets. The transmittance of the second filter is associated with the above complex aperture-form function and is defined by the inverse Fourier transform of the desired pattern. Both filters can be integrated in a single diffractive element.
A.G. Sedukhin / Optics Communications 229 (2004) 39–57
55
300 Radius ρ (µm)
200 100 0 100 200 300 0.1
(a)
0.2
< 0.005
0.3
0.4
0.005 0.01
0.5
0.6 0.7 0.8 Propagation distance z (m)
0.01 0.02
0.02 0.05
ρring = 0.7 18.55 µm 0.8 0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.1
0.2
(b)
0.05 0.1 Intensities I1, ... , I9 (arb. units)
Intensity I (arb. units)
1 0.9
0.3
0.4
0.5
0.6
0.7
Propagation distance
0.8
z (m)
0.9
1
1.1
0.9
1.2
1.0
1.1
0.1 0.2
1.2
1.3
0.2 0.5
1.4
0.5 1.0
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
120 100 80
(c)
60
40
20
0
20
40
60
80 100 120
Radius ρ (µm)
Fig. 7. Intensity patterns of the output field for a beamforming system corresponding to Fig. 2, designed for illumination with a uniform collimated input beam, and consisting of a DMF lens with the phase function of Fig. 6 and a refractive lens with f ¼ 600 mm and Rb ¼ 13:87 mm [the values of other design parameters are as follows: k ¼ 0:6328 lm, Z ¼ 100 mm, z1 ¼ 200 mm, zN ¼ 1 m, F ¼ 3:6 m, Wcirc ¼ 20 lm, and a0b ¼ 132:2 mm1 ]: the map of the intensity contours of the generated field in the meridional plane (a), the longitudinal intensity profile at the maximum-intensity radius qring (b), and the superimposed transverse intensity profiles at the equidistant focal planes z ¼ z1 ; z ¼ z2 ; . . . ; z ¼ z9 (c).
When using projection optical systems with diffractive multifocal lenses, the wave field of a generated finite-extent periodically focused beam is defined by structure of a planar object, for which the patterns reproduced at the equidistant focal planes of the beam are its own images. The effect of multiple projection imaging realized by these systems might be useful in numerous applications. As in the case of a conventional monochromatic imaging system with no spatial-frequency filters, the process of image formation at each equidistant focal plane is shown to be representable through the squared convolution of the complex amplitude distribution of an image to be reproduced and the point spread function of a multiple-imaging system, with only difference that the last function should be taken for the respective diffraction order.
Acknowledgement The author thanks the reviewer for its constructive comments.
Appendix A Consider the derivation of the direct and inverse Fourier transforms, when one is using the cylindrical coordinate system and the mathematical forms adopted in the present paper. By definition, the direct
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A.G. Sedukhin / Optics Communications 229 (2004) 39–57
Fourier transform of a function gðu; vÞ in the Cartesian system ðu; vÞ and the inverse Fourier transform of a function Gðxu ; xv Þ in the Cartesian system ðxu ; xv Þ can be represented as (see e.g. [19]): Z Z 1 1 gðu; vÞ exp ð ixu u ixv vÞ du dv; ðA:1Þ Ffgðu; vÞg ¼ Gðxu ; xv Þ ¼ 2p 1 Z Z 1 1 F1 fGðxu ; xv Þg ¼ gðu; vÞ ¼ Gðxu ; xv Þ exp ðixu u þ ixv vÞ dxu dxv ; ðA:2Þ 2p 1 with xu and xv being the angular frequencies. pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Going from the Cartesian system (u; v) to the polar coordinate system (r; nr ), such that r ¼ u2 þ v2 and nr ¼ arctanðv=uÞ, and going, in the frequency domain, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi from the Cartesian system (xu ; xv ) to the polar system (a; n), such that a ¼ x2u þ x2v and n ¼ arctanðxv =xu Þ, we rewrite Eqs. (A.1) and (A.2) as Z 2p Z 1 1 gðr; nr Þ exp ½ iar cosðnr nÞr dr dnr ; ðA:3Þ Gða; nÞ ¼ 2p 0 0 Z 2p Z 1 1 gðr; nr Þ ¼ Gða; nÞ exp ½iar cosðn nr Þa da dn: ðA:4Þ 2p 0 0 Using the identity Z 2p exp ½imf ix cosðf vÞ df ¼ ðiÞm 2p expðimvÞJm ðxÞ
ðA:5Þ
0
and expanding the functions Gða; nÞ and gðr; nr Þ in Fourier series 1 X Gða; nÞ ¼ Gm ðaÞ expðimnÞ
ðA:6Þ
m¼1
and
1 X
gðr; nr Þ ¼
gm ðrÞ expðimnr Þ;
ðA:7Þ
m¼1
with the following substitution into Eqs. (A.3) and (A.4) and changing the order of integration and summation, we get the following expressions for the direct and inverse Fourier transforms in the polar systems: Z 1 1 X m Ffgðr; nr Þg ¼ Gða; nÞ ¼ ðiÞ expðimnÞ gm ðrÞJm ðarÞr dr; ðA:8Þ 0
m¼1
F1 fGða; nÞg ¼ gðr; nr Þ ¼
im expðimnr Þ
m¼1
where gm ðrÞ ¼
1 X
1 2p
Gm ðaÞ ¼
1 2p
Z
Z
1
Gm ðaÞJm ðarÞa da;
ðA:9Þ
0
2p
gðr; nr Þ expðimnr Þ dnr ;
ðA:10Þ
0
Z
2p
Gða; nÞ expðimnÞ dn:
ðA:11Þ
0
References [1] A.G. Sedukhin, Periodically focused propagation-invariant beams with sharp central peak, Opt. Commun. 228 (2003) 231. [2] J.W. Goodman, Introduction to Fourier Optics, first ed., McGraw-Hill, New York, 1968 (Chapter 3).
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Generalized periodically focused beams and multiple on-axis lens imaging Andrey G. Sedukhin
*
Russian Academy of Sciences, Institute of Automation and Electrometry, Prospekt Akademika Koptyuga 1, Novosibirsk 630090, Russia Received 31 July 2003; received in revised form 24 September 2003; accepted 20 October 2003
Abstract A class of periodically focused propagation-invariant monochromatic beams with sharp central peak is extended to the beams, of which the patterns of intensity at discrete transverse planes regularly spaced along the propagation axis may have an arbitrary, but the same, form. It is shown that the wave field of such beams in free space may be represented as a discrete set of nonuniform phase-distorted spherical wavelets converging to the vicinities of axial points at the indicated transverse planes and possessing the property of mutual similarity. High-order azimuth angular constituents of these wavelets are shown to have phase singularities appearing in the transverse planes as dark coaxial spots with bright rims. Hybrid refractive–diffractive systems are proposed, which are suited both for the direct high efficient production of the beams considered and for the multiple imaging of arbitrary plane objects. The results of numerical simulations are given for two cases of multiple reproduction of rings with large and small radii. Ó 2003 Elsevier B.V. All rights reserved. PACS: 41.85.Ct; 42.40.Jv; 42.60.Jf; 42.79.)e Keywords: Propagation invariance; Phase singularities; Diffractive multifocal lenses; Multiple imaging
1. Introduction In our previous paper [1] the existence of a new class of longitudinally periodic propagation-invariant monochromatic beams, referred to as periodically focused ones, was predicted heuristically and arrangements for production of these beams were proposed. An intensity pattern of such beams is periodically reproduced at the specified equidistant transverse planes and is centered about the propagation axis. The simplest rotationally symmetric beams with sharply defined central peak and low-intensity sidelobes have been demonstrated. It is characteristic that the field of these beams, considered just behind the output aperture plane of a beamforming system, can be decomposed into a discrete set of elementary fields which
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Tel.: +7-3832-320-062; fax: +7-3832-333-863. E-mail address: [email protected] (A.G. Sedukhin).
0030-4018/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2003.10.026
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A.G. Sedukhin / Optics Communications 229 (2004) 39–57
correspond to the cuts of nonuniform strictly spherical wavelets converging to the regularly spaced coaxial focal points and having all the same energies and the amplitude distributions of angular spectra. The spacing between the focal points is chosen from the conditions which ensure the absence of significant interference of the wavelets in the vicinities of these points. In the present paper, we introduce more complicated beams, the total field of which in free space is represented as a superposition of mutually similar nonuniform phase-distorted converging spherical wavelets. We show that the nth wavelet considered just behind the output aperture plane of a beamforming system and conjugated with a vicinity of axial focal point zn can be represented in terms of two multiplicative constituents: the cut of a strictly spherical wavelet, converging exactly to the point zn (a fundamental constituent), and a suitable complex aperture-form function (a distortion constituent). The mutual similarity of the wavelets implies that there is the pairwise equality between the spherical projections of the amplitude and phase real-valued components of the complex aperture functions of two arbitrary wavelets onto unit spheres with centers in the axial points of convergence of the wavelets. Despite of certain restrictions, the two-dimensional (2-D) distributions of the amplitude and phase components of the aperture functions may have arbitrary, but the same, shapes. In particular, such shapes may lack the rotational symmetry and vary not only in the radial direction, but also in the azimuth angle. We derive the equations describing the exact forms of ideal infinite-extent generalized periodically focused beams in scalar approximation. The high-order angular constituents of the above complex apertureform function is shown to give rise to the phase singularities appearing as dark coaxial spots at regularly spaced transverse planes. Also, we propose optical systems for generation of real finite-extent beams, consider the direct association of these beams with multiple on-axis lens imaging, and make a brief comparison of these beams with those inherent in Talbot self-imaging effect. As it is shown, the proposed systems can perform either lens multiple Fourier imaging, when transforming the spectrum of an object, or lens multiple projection imaging, when reproducing the object itself. For numerical simulations of the finiteextent generalized periodically focused beams, we consider two examples of synthesizing the single-mode beams with ring focal patterns having quite large and minimum-permissible radii.
2. Exact scalar forms of generalized periodically focused beams First we present a mathematical description of ideal generalized periodically focused beams in scalar approximation. This description is important in methodological aspect and pertains, in essence, to the limiting forms of the beams generated by real finite-aperture systems, including the systems with high spatial resolution. Let us assume that in a sequence of the regularly spaced focal planes of the beam the nth position is defined as zn ¼ z1 þ ðn 1ÞZ, where z1 is the first position, Z ¼ znþ1 zn is the distance between adjacent planes and n ¼ 1; 2; 3; . . . is their current index. We now prove that the complex amplitude distribution of a generalized periodically focused propagation-invariant field at the output aperture plane of a beamforming system may be represented as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 X Dz exp ik R2 þ z2n þ iwn R pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi UðR; nR Þ ¼ A ; nR : ð1aÞ Rn R2 þ z2n n¼1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Here R ¼ x2 þ y 2 and nR ¼ arctanðy=xÞ are the radius and the azimuth angle at the output aperture plane z ¼ 0, Dz is a constant of reference distance, R R R A ; nR ¼ A ; nR exp iU ; nR ð1bÞ Rn Rn Rn
A.G. Sedukhin / Optics Communications 229 (2004) 39–57
41
is the complex aperture-form function with real-valued amplitude and phase components Aðm; nR Þ and Uðm; nR Þ (at m ¼ R=Rn and 0 6 m 6 1), and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1cÞ Rn ¼ zn ab = k 2 a2b is the boundary radius at the cross-section of the nth wavelet, with ab being the boundary radial spatial frequency. The last fractional term in Eq. (1a) describes the complex amplitude distribution at the crosssection of the nth uniform strictly spherical wavelet converging to the point zn and having the initial phase offset p < wn < p. As it follows from Eq. (1b), both the amplitude and the phase real-valued components of the complex aperture-form function – Aðm; nR Þ and Uðm; nR Þ – are double-normalized (in magnitude and in arguments nR and m) and are common to all the useful wavelets. It is well known that within scalar approximation the resultant field distribution in free space z > 0 can be calculated through the angular spectrum of plane waves [2]. To this end, we first expand the apertureform function in Fourier series in circular harmonics: 1 X R R A ; nR ¼ Am ð2Þ expðimnR Þ; Rn R n m¼1 with the Fourier coefficients determined as Z 2p R 1 R A ; nR expðimnR Þ dnR : ¼ Am Rn 2p 0 Rn The angular spectrum of plane waves, relative to the plane z ¼ 0, is then found as 1 1 X X m sða; nÞ ¼ FfUðR; nR Þg ¼ ðiÞ expðimnÞsmn ðaÞ;
ð3Þ
ð4aÞ
m¼1 n¼1
where a and n are, respectively, the radial spatial frequency and the azimuth angle in free space, F is the direct Fourier transform operator (see Appendix A), and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 R exp ik R2 þ z2n þ iwn pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi smn ðaÞ ¼ Dz Am ð4bÞ Jm ðaRÞR dR Rn R2 þ z2n 0 is the mth circular component of the angular spectrum for the nth phase-distorted spherical wavelet at the plane z ¼ 0, with Jm being the mth order Bessel function of the first kind. Substituting Eq. (1c) into Eq. (4b) and using the stationary-phase method [3] for asymptotic evaluation of the obtained integral expression under the condition kðz zn Þ ! 1, one can find the following approximate form: h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii mþ1 ðiÞ DzAm ða=ab Þ ðk 2 a2b Þ=ðk 2 a2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi smn ðaÞ ð5Þ expðizn k 2 a2 þ iwn Þ: k 2 a2 When deriving Eq. (5), the Bessel function Jm ðaRÞ was represented in complex form with separation of the useful diverging wave component from the well-known approximation of the modular and phase of a Bessel pffiffiffiffiffiffiffiffiffiffi ffi m function at large values of its argument: Jm ðaRÞ ðiÞ exp½iðaR p=4Þ= 2paR (this procedure has been described for the function J0 ðaRÞ in [4]). In fact, Eq. (5) is of the exact form, because the accepted condition kðz zn Þ ! 1 corresponds to consideration of a field in the far zone and is fully consistent with implicit definition of the angular spectrum of plane waves. To find the resultant complex amplitude of the field behind the output aperture, it is necessary to take the inverse Fourier transform of the angular spectrum of plane waves in free space: 1 1 n X X pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o Uðq; n; zÞ ¼ F1 sða; nÞ expðiz k 2 a2 Þ ¼ expðimnÞUmn ðq; zÞ; ð6aÞ m¼1 n¼1
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A.G. Sedukhin / Optics Communications 229 (2004) 39–57
1 where q isp the radial ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi coordinate in free space, F is the inverse Fourier transform operator (see Appendix A), expðiz k 2 a2 Þ is the propagation transmission function of the angular spectrum in free space, and Z k pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi smn ðaÞ expðiz k 2 a2 ÞJm ðaqÞa da ð6bÞ Umn ðq; zÞ ¼ 0
is the complex amplitude of the mth circular component of the nth wavelet. In Eq. (6b), we used the integration over the range 0 6 a < k to exclude the evanescent waves. Finally, on substituting Eqs. (5) and (6b) into Eq. (6a), we arrive at the following expression for the resultant wave field: h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 1 1 Z ab Am ða=ab Þ ðk 2 a2b Þ=ðk 2 a2 Þ X X pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Uðq; n; zÞ ¼ Dz ðiÞmþ1 expðimnÞ k 2 a2 0 m¼1 n¼1 h i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp iðz zn Þ k 2 a2 þ iwn Jm ðaqÞa da; ð7Þ with the particular distributions of the dominant nth constituents at the transverse planes z ¼ zn represented as h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii Z ab Am ða=ab Þ ðk 2 a2 Þ=ðk 2 a2 Þ 1 X b mþ1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Un ðq; n; zn Þ ¼ Dz expðiwn Þ ðiÞ expðimnÞ Jm ðaqÞa da: 2 2 k a 0 m¼1 ð8Þ From Eq. (8), one can see that however complicated the complex amplitude distributions of the different nth constituents of a beam at their own planes z ¼ z1 , z ¼ z2 , z ¼ z3 ; . . . ; z ¼ zn would be, they are fully propagation-invariant accurate to the phase offsets wn . This result obviously to give proof to our representation regarding the form of a generalized periodically focused beam at the output aperture plane of a beamforming system, which is given by Eqs. (1). As can be seen from these equations, each nonuniform phase-distorted spherical wavelet of the beam possesses the property of similarity to other wavelets, since there is the pairwise equality between the spherical projections of both the amplitude and the phase realvalued components of the complex aperture functions of two arbitrary wavelets onto unit spheres centered at the axial points of convergence of the wavelets. Let us now analyze the resultant solution for the wave field of an ideal generalized periodically focused beam governed by Eq. (7). All the beamlets corresponding to the constituents of this equation with different indices m and n, as well as the superposition of these beamlets describing the total wave field, are the exact solutions of the nonparaxial Helmholtz wave equation with a marginal condition on the cutoff frequency ab . The field compositions defined as superpositions of the beamlets with different index n, but common index m, can be referred to as single-mode generalized periodically focused beams. According to Eq. (7), the intensity distribution of the wave field of ideal single-mode beams is rotationally symmetric. Touching on a beam of fundamental mode with m ¼ 0, we point out the following. As would be expected, for all spherical wavelets with m ¼ 0 and n P 1 the field distribution near the focal points zn (the nth 3-D point spread function) will fully be consistent with the well-established and rigorous integral representation of Debye, see Ref. [5, Eqs. (2.6) and (2.7)], if one will make change for the parameters we adopted (ks? ¼ a), will take account of the origin of the coordinate system (z ! z zn ), and will make generalization for the case with a nonuniform phase-distorted aperture function [A0 ðaÞ 6¼ const: at 0 6 a 6 ab ]. This fact reinforce, in part, our previous statement on the exact form of expression (5). The high-order modes with m ¼ 1; 2; . . . constitute the periodically focused beams which should be related to those having the so-called phase singularities (also known as wavefront screw dislocations or optical vortices) with topological charge m (see a review on the wave fields with phase singularities in [6]). The distinctive feature of the latter beams for the case with rotational symmetry is that they have helical wavefronts, whose phase at the arbitrary transverse plane is varied over 2pm on a closed path around the
A.G. Sedukhin / Optics Communications 229 (2004) 39–57
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propagation axis with the on-axis amplitude of the field being dropping to zero. The fact that the above high-order modes belong to precisely those beams is evidenced by occurrence of a separable azimuthdependent phase term expðimnÞ in Eqs. (7) and (8) and by nullification of the integrands in these equations due to the identity Jm ðaqÞjm6¼0;q¼0 ¼ 0. The relevant annulus-like intensity patterns may be called mth-order doughnut modes. It is worth noting also that, as it follows from Eqs. (4a) and (5), the angular spectrum of plane waves for any nth useful wavelet of a generalized periodically focused beam may be conceived as a nonuniform phasedistorted spherical wave diverging from the respective axial point zn and having the same complex amplitude distribution within a solid angle subtended by the boundary angle hb ¼ arcsinðab =kÞ: 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ðiÞm expðimnÞsmn ðaÞ sn ða; nÞ ¼ expðiz k 2 a2 Þ h
m¼1
i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Sða; nÞ exp iðz zn Þ k 2 a2 ip=2 þ iwn ; h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 1 Am ða=ab Þ ðk 2 a2b Þ=ðk 2 a2 Þ X m pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Sða; nÞ ¼ Dz ð1Þ expðimnÞ : k 2 a2 m¼1
ð9aÞ ð9bÞ
Here Sða; nÞ is the complex amplitude of the angular spectrum, which is common to all wavelets and is varied, in the general case, both in spatial frequency a and in azimuth angle n. The last exponential term in Eq. (9a) is the nth strictly spherical wavelet converging to the point zn . Using Eq. (9b), it can easily be found that each nth useful wavelet will possess the same finite energy.
3. Implementation As can be seen from Eqs. (1) of the present paper and from [1, Eq. (4)], the only differences between the complex amplitude distribution at the output aperture plane inherent to a generalized periodically focused beam and that peculiar to a periodically focused beam with stigmatic ray convergence are the complex character of the output aperture functions of the respective wavelets and the possible lack of rotational symmetry of these functions. Furthermore, the front of the nth wavelet, Wn , associated with the nth complex aperture function and being fixed with respect to the output plane, may conveniently be subdivided into two parts: Wn ¼ Wn0 þ Wn00 ; Wn0
ð10Þ Wn00
where is the front of the strictly spherical wavelet converging to the point zn and is the wavefront deformation peculiar to a specific pattern to be generated at the transverse focal plane z ¼ zn . Given equidistant foci z1 ; z2 ; z3 ; . . . ; zN (with subscript N denoting their total number), the amplitude aperture-form function, Aðm; nR Þ, and also the above wavefront deformation which defines the phase component of the complex aperture-form function, Uðm; nR Þ, one can obviously reconstruct the requisite real-valued amplitude components A1 ðRÞ; A2 ðRÞ; A3 ðRÞ; . . . ; AN ðRÞ of the aperture functions and the fronts W1 ; W2 ; W3 ; . . . ; WN of wavelets relative to the output aperture plane. Fig. 1 is an illustration of such the reconstruction for the simple paraxial case with rotational symmetry, when the amplitude aperture function of the nth wavelet has the appearance of a circular function Dz Dz R An ðRÞ ¼ circ ; ð11Þ zn zn DR and the nth wavefront deformation Wn00 has the conical form, such that the wavefront phase delay varies as
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A.G. Sedukhin / Optics Communications 229 (2004) 39–57
Fig. 1. Torus-like fronts of wavelets, Wn , and the circular real-valued amplitude components of their aperture functions, An ðRÞ, at the output plane of a rotationally symmetric beamforming system producing invariable ring-like amplitude patterns at three regularly spaced planes in free space. The focal points ðqring ; zn Þ are marked with bullets. The fronts denoted as Wn0 and Wn00 are, respectively, the spherical and conical parts of the nth wavelet front Wn . The distance constant is assumed to be equal Dz ¼ z1 .
Un ðRÞ ¼ kqring R=zn ;
ð12Þ
where circðrÞ ¼ 1 for 0 6 r 6 1 and 0 otherwise, with DR ¼ Dzab =k and qring a1 b being constant values. As is easy to see, the multiple focusing to a ring of radius qring is performed in this case. Rotationally symmetric beamforming systems for production of periodically focused beams with no the wavefront deformation of spherical wavelets (with Wn00 ¼ 0) have already been considered in [1]. A key component of these systems is a diffractive multifocal (DMF) lens with uniform beam splitting and a naturally spaced on-axis array of foci. This DMF lens is characterized by a transmittance function T 0 ðrÞ and in conjunction with a positive refractive lens L (a focal distance apart from the DMF lens) is capable of generating a complex beam consisting of the plurality of strictly spherical wavelets with fronts Wn0 . Shown in Fig. 2 is a beamforming system with collimated illumination of the DMF lens. Below, we will prove that for production of a generalized periodically focused field it will suffice to insert into the system an auxiliary spatial-frequency filter having a transmittance T 00 ðr; nr Þ and located, e.g., in the immediate vicinity of the DMF lens (here nr stands for the azimuth angle at the plane of the DMF lens). The role of this filter is just in the correction for necessary wavefront deformations of wavelets, Wn00 . However, for better understanding and convenience of further presentation, we first carry out the resumptive consideration of a system with pure stigmatic multiple focusing. As was shown [1], the transmittance of a planar phase DMF lens in the parabolic approximation must be periodic in r2 and may thus be represented as a Fourier series: " 2 # 0 2 1 1 X X r ikn r 0 0 0 T ðrÞ ¼ exp½iu ðrÞ ¼ an0 exp i2pn an0 exp ¼ ; ð13Þ Kr 2F n0 ¼1 n0 ¼1
A.G. Sedukhin / Optics Communications 229 (2004) 39–57
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Fig. 2. Modifying the arrangement of a periodically focused propagation-invariant beam generator from the simplest one, with forming a uniform array of coaxial spots at multiple equidistant focal planes [with no the filter with transmittance T 00 ðr; nr Þ], to the generalized one, with forming a uniform array of coaxial patterns of an arbitrary form at the same focal planes [with the filter with transmittance T 00 ðr; nr Þ being added].
where u0 ðrÞ is the phase function of the DMF lens, F is the focal length for the first converging diffraction pffiffiffiffiffiffiffiffi ffi order n0 ¼ 1, Kr ¼ 2kF is a constant equal to the radius of the first Fresnel zone for the principal focus the distance F apart from the DMF lens, and the Fourier coefficients an0 are dependent on the shape of the phase function within Fresnel zones and are determined as " 2 # Z Kr 2 r an0 ¼ 2 T 0 ðrÞ exp i2pn0 r dr: ð14Þ K Kr 0 r For a well-designed DMF lens, the values jan0 j2 , accounting for energy percentage in the n0 th diffraction 2 orders, are practically equal within an indexing set of useful orders, N0 (that is jan0 j jn0 2N0 ’ const:). Besides, 0 the efficiency of the DMF lens, is close to unity P sum of 2these values over the set N , defining the light 2 2 ( n0 2N0 jan0 j ’ 1, so that for any n0 2 N0 and j 62 N0 jan0 j jaj j ). For the DMF lens with odd or even 0 number of useful diffraction orders, N , the set N includes, respectively, the integers n0 ¼ 0; 1; 2; . . . ; ðN 1Þ=2 or n0 ¼ 1; 3; 5; . . . ; ðN 1Þ. Under the normal collimated illumination of such DMF lens, the field with an array of real and virtual coaxial foci is generated, the positions of these foci being nonlinearly distributed at distances F =n0 away from the DMF lens. Placing the DMF lens at the front focal plane of a positive refractive lens L with focal length f and meeting the restrictions given below, enables one to transform such the field to that possessing the propagation invariance within a discrete finite set of equidistant transverse planes behind the refractive lens. In what follows we will go from the indexing system with integers n0 , defined within the set N0 , to the indexing system with natural numbers defined within the set N ¼ f1; 2; . . . ; N g. The relation between indices n and n0 will then be defined as ð2n N 1Þ=2 for odd N ; n0 ðnÞ ¼ ð15Þ 2n N 1 for even N : On applying for the system of Fig. 2 the ordinary lens equation (relating the positions of foci of the DMF lens and their images behind the refractive lens), 1 1 1 þ ¼ ; 0 f þ F =n ðnÞ zn f
ð16Þ
the nth positions of the coaxial foci of a generated beam, zn , relative to the output aperture plane, can be determined as ð17Þ zn ¼ z1 þ ðn 1ÞZ; n ¼ 1; 2; . . . ; N ;
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A.G. Sedukhin / Optics Communications 229 (2004) 39–57
where z1 ¼ f ðN 1ÞZ=2 is the position of the first focus and 2 f =F for odd N ; Z ¼ znþ1 zn ¼ 2f 2 =F for even N
ð18Þ
ð19Þ
is the distance between adjacent foci. Let us now place in the immediate vicinity of the DMF lens an auxiliary spatial-frequency filter with transmittance T 00 ðr; nr Þ proportional to the inverse Fourier transform of a desired propagation-invariant distribution at the equidistant focal planes: T 00 ðr; nr Þ / F1 fUn ðkq=f ; n; zn Þg:
ð20Þ
00
We shall think of the function T ðr; nr Þ as being pure phase one and expand it in a Fourier series in terms of circular harmonics: T 00 ðr; nr Þ ¼ exp½iu00 ðr; nr Þ ¼
1 X
bm ðrÞ expðimnr Þ;
ð21Þ
m¼1
where u00 ðr; nr Þ is the phase function of the filter and the Fourier coefficients bm ðrÞ are defined as Z 2p 1 T 00 ðr; nr Þ expðimnr Þ dnr : bm ðrÞ ¼ 2p 0
ð22Þ
To find successively the complex amplitude distributions at the plane of the lens L and in free space behind it, let us write down the appropriate expressions on the basis of Fresnel diffraction integral in the cylindrical coordinate system: Z 1 2 k R2 ikr 0 00 exp ik f þ UðR; nR Þ ¼ AðrÞT ðrÞT ðr; nr Þ exp r i2pf 2f 2f 0 Z 2p ikRr cosðnR nr Þ dnr dr; exp f 0 Z Rb k q2 ikR2 ikR2 exp ik z þ þ exp Uðq; n; zÞ ¼ R i2pz 2z 2f 2z 0 Z 2p ikRq cosðn nR Þ dnR dR; UðR; nR Þ exp z 0
ð23Þ
ð24Þ
where AðrÞ is the input real-valued aperture function. Next we suppose that the aperture of the lens L does not truncate practically the beams of useful diffraction orders, so that the condition Rb ! 1 can be considered as being meeting. Then, on inserting Eq. (23) into Eq. (24) and using Eqs. (13), (21) and the formulas [7] Z Z
2p m
0 1
exp ½imf ix cosðf vÞ df ¼ ðiÞ 2p expðimvÞJm ðxÞ;
ð25Þ
imþ1 bc iðb2 þ c2 Þ Jm expðiaf ÞJm ðbfÞJm ðcfÞf df ¼ exp ; 2a 4a 2a
ð26Þ
2
0
we obtain the following expression for the resultant field:
A.G. Sedukhin / Optics Communications 229 (2004) 39–57
Uðq; n; zÞ ¼
1 X k exp ½ik ð f þ zÞ ðiÞmþ1 expðimnÞ f m¼1 2 Z 1 1 X ikr 1 z n0 krq an 0 bm ðrÞAðrÞ exp þ r dr: Jm f f2 F f 2 0 n0 ¼1
47
ð27Þ
Going again to the system with indexing in n [see Eq. (15)], we obtain that, without significant mutual interference of the wavelets near the points zn ¼ z1 þ ðn 1ÞZ [where the argument of the exponential function in the integrand of Eq. (27) vanishes], the complex amplitude distribution of the field at the transverse plane z ¼ zn takes the form of the nth constituent: Z 1 1 X k krq mþ1 ðiÞ expðimnÞ bm ðrÞAðrÞJm r dr; ð28Þ Un ðq; n; zn Þ ¼ exp ½ikðf þ zn Þan f f 0 m¼1 where an an0 ðnÞ and the subscript n0 ðnÞ is defined by Eq. (15). Alternatively, Eq. (28) can be written in terms of the following Fourier transform: Un ðkq=f ; n; zn Þ ¼ cn FfAðrÞT 00 ðr; nr Þgja¼kq=f ;
ð29aÞ
where cn ¼
kan exp ½ik ð f þ zn Þ: if
ð29bÞ
Finally, from Eq. (29a) it follows that T 00 ðr; nr Þ ¼ F1 fUn ðkq=f ; n; zn Þg=½cn AðrÞ:
ð30Þ
Eq. (30) confirms that, with a correction for the input aperture function AðrÞ, the transmittance T 00 ðr; nr Þ is indeed proportional to the inverse Fourier transform of the desired propagation-invariant distribution at the equidistant focal planes [see Eq. (20)]. Furthermore, analyzing Eqs. (28) and (29) along with Eq. (21), one can conclude that the intensity distributions of a generated beam at the equidistant focal planes well-separated along the propagation direction is really much close to a desired propagation-invariant pattern provided that jan j2 ’ const. In fact, the indicated equations describe in paraxial approximation the focal-plane complex amplitude distributions of the same generalized beam as that has been analyzed earlier in nonparaxial scalar treatment [see Eq. (8)]. It is easy to show that the complex aperture-form function for the nth nonuniform phase-distorted spherical wavelet of that beam is related, in paraxial approximation, to the beamforming-systemÕs characteristics and the complex amplitude distribution at the corresponding focal plane as: R R R f A ; nR ¼ an A rb T 00 rb ; nr ¼ F1 fUn ðkq=f ; n; zn Þg; ð31Þ Rn Rn Rn k where rb is the boundary aperture radius of the DMF lens and of the auxiliary filter with transmittance T 00 ðr; nr Þ. The last expression was obtained by comparing the components of Eq. (28) and the paraxial form of Eq. (8) with taking into account Eqs. (2), (21) and (29) choosing Dz ¼ f , and omitting unessential phase constants. Clearly, a pattern of intensity distributions at the transverse focal planes is strongly dependent on the Fourier coefficients bm ðrÞ of the function T 00 ðr; nr Þ. In general case, the pattern is azimuth-modulated and may have an arbitrary form. However, of special interest are single-mode beams characterized by a rotationally symmetric pattern with only one coefficient bm ðrÞ not being zero. The beams of zero-order mode with b0 ðrÞ being real and nonzero relate to the simplest form of periodically focused beams with multiple stigmatic ray convergence and have already been demonstrated (mainly in paraxial approximation) in [1].
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4. Multiple on-axis lens imaging Let us now discuss the issue of resemblance of generalized periodically focused wave fields and the wave fields inherent in Talbot self-imaging effect. Under collimated monochromatic illumination with light of the wavelength k, periodic objects with sufficiently large transverse dimensions are known to be self-imaged at regularly spaced transverse planes. For example, the pattern of a periodic grating is reconstructed (without transverse shift) at multiple transverse planes separated by the Talbot distance [8] ZT ¼ 2K2u =k;
ð32Þ
where Ku is the period of the grating. This effect is observed in the near field without the aid of lenses or other optical accessories and may be therefore referred to as lensless multiple Fresnel imaging. In using the beamforming system of Fig. 2, the minimum distance between the regularly spaced transverse planes, at which the pattern of an object can be reproduced (with N being odd), is defined as 2
Z ¼ f 2 =F ¼ 2ðf =Kr Þ k:
ð33Þ
Eqs. (32) and (33) disclose that the distances ZT and Z have quite different dependence on the wavelength k. Besides, the important feature of the latter system is the fact that the reconstructed pattern is directly associated with not the object itself but with its Fourier image. In Fig. 2, the Fourier image is represented by an objectÕs mask, which has the transmittance T 00 ðr; nr Þ and is placed at the plane of the DMF lens; the reproduction being realized by means of an auxiliary Fourier transform lens L. In this connection, such reproduction may be called lens multiple Fourier imaging. If the multiple imaging of an object itself is desirable, there have to be obvious modifications in the optical system. To associate the object with its intermediate Fourier image at the plane of the DMF lens, one more Fourier transform refractive lens may be introduced, for example. In this way, we arrive at a classical two-Fourier-lens imaging system (coherent optical processor) [2] considered yet in extended treatment as a system with plurality of output focal planes and employing the DMF lens as a spatialfrequency filter (see Fig. 3). The complex amplitude distribution at the plane of the object to be reproduced may now be related to the complex amplitude distribution at the plane situated just before the DMF lens through the suitable inverse Fourier transform. Thus, the transmittance of a new amplitude objectÕs mask t00 ðq0 ; nq0 Þ turns out to be also proportional to the inverse Fourier transform of the transmittance T 00 ðr; nr Þ of the auxiliary filter used in the previous system of Fig. 2:
Fig. 3. Multiple imaging system with a filter DMF lens having the phase transmittance T 0 ðrÞ. At the equidistant transverse planes z ¼ z1 ; z ¼ z2 ; . . . ; z ¼ z7 , the system provides the invariant projection imaging of an arbitrary object represented by a mask with the amplitude transmittance t00 ðq0 ; nq0 Þ.
A.G. Sedukhin / Optics Communications 229 (2004) 39–57
t00 ðq0 ; nq0 Þ / F1 fT 00 ðkr=f ; nr Þg;
49
ð34Þ
0
where q and nq0 are, respectively, the radial and azimuth coordinates at the plane of the object. The overall process of monochromatic multiple image replication, as applied to an imaging system of Fig. 3, can also conveniently be described within the framework of Fourier optics, if the following simplifications are met: (i) both refractive lenses do not introduce the essential confinement in the passage of useful spectral components and (ii) the image planes are well separated from one another to avoid overlapping of adjacent on-axis point spread functions. In this case, the complex amplitude distribution just behind the plane of DMF lens will be proportional to the product FfUðq; n; 3f Þgja¼kr=f ;n¼nr T 0 ðrÞ, with Uðq; n; 3f Þ Uðq0 ; nq0 Þ denoting the complex amplitude distribution just behind the objectÕs mask. Taking into consideration Eq. (29a), the complex amplitude distribution at the nth equidistant image plane z ¼ zn may be expressible as 9 #> kr n ðnÞ = Un ðq; n; zn Þ / F FfUðq; n; 3f Þgj a¼kr=f T 0 ðrÞ exp i > > 2F n¼nr ; : a¼kq=f ( ) kr2 n0 ðnÞ 0 ¼ F FfUðq; n; 3f Þgj a¼kr=f F T ðrÞ exp i 2F n¼nr a¼kq=f a¼kq=f Uðq; n; 3f Þ an dðqÞ ¼ an Uðq; n; 3f Þ; 8 > <
"
2 0
ð35Þ
where is the 2-D convolution operator, dðqÞ is the Dirac delta function, the dependence n0 ðnÞ is defined by Eq. (15), and the quadratic exponential term accounts for the axial focal shift of the nth image. When deriving Eq. (35), we assume that for the nth complex beamlet generating the nth image, FfT 0 ðrÞ exp ½ikr2 n0 ðnÞ=ð2F Þgja¼kq=f ¼ PSFn ðq; zn Þ an dðqÞ, with PSFn ðq; zn Þ denoting the amplitude point spread function of the imaging system for the nth beamlet. The change of sign at the argument q in Eq. (35) implies that the generated image will be inverted with respect to the objectÕs image. With double Fourier transform processing performed in the system by refractive lenses, the reproduction in view may be called lens multiple projection imaging. In common with conventional non-aberrated imaging systems with a single output focal plane, the quality of reproduction may be characterized in the last instance by the relationship between the maximum spatial frequency of an object and the boundary spatial frequency of the imaging system. An additional point to be emphasized is that the multiple projection imaging can be realized not only with two-Fourierlens systems but also with single-lens ones. For example, a single-lens system presented in [1, Fig. 4] is well suited for this purpose, if one places a reconstructed object at the plane of a point source S. Alternatively, a filter approximating the Fourier transform of the object can likewise be placed at the front focal plane of the output refractive lens L. An apparent advantage of the lens multiple imaging over the lensless Talbot multiple imaging is the fact that a pattern to be reproduced may have an arbitrary form and may also be imaged as both scaled up and down with respect to the objectÕs sizes. The beamforming systems based on DMF and refractive lenses and having the on-axis character of image replication are, in fact, complementary to the well-known in-plane multiple imaging systems based on Dammann gratings [9,10]. It is notable also that a closely similar system with on-axis multiple imaging has recently been proposed and investigated in [11–14]. A distinctive key component of that system is a Fabri– Pero interferometer acting as a comb-like spatial-frequency filter and originating behind the output refractive lens a collection of interfering zero-order Bessel beams. However, the narrow-band nonlinear character of the transmittance of the filter gives rise to the reduced light efficiency of the imaging system as
50
A.G. Sedukhin / Optics Communications 229 (2004) 39–57
well as to the limitation of the full depth of a generated field and to the moderate invariance of relevant images. The imaging properties of such the system are confined due to the fact that its point spread functions possess marked transverse sidelobes (for systems with rotational symmetry these functions are close to the zero-order Bessel function), the main intensity maxima of these functions having approximately linear growth with propagation distance. The advantages of imaging systems with DMF lenses are thus the enhanced light efficiency, the large full depth of field, and the high quality of image replication due to high linearity of transmitting the spatial frequencies. 5. Numerical simulations Let us now consider examples of synthesizing some finite-extent periodically focused beams which possess a high degree of propagation invariance, but lack stigmatic on-axis ray convergence. For a rotationally symmetric system of Fig. 2, one can take a ring-like intensity distribution as being the simplest suitable pattern reproduced at the equidistant focal planes. A mathematically idealized desired distribution of the complex amplitude at the nth plane may then be represented in the form: Uðq; n; zn Þ ¼ dðq qring Þ exp½iWðnÞ; ð36Þ where WðnÞ is the arbitrary phase function which can be expanded in a Fourier series in circular harmonics. For simplicity of further derivations, consider the single-mode cases with WðnÞ ¼ mn. Additionally, suppose that the incident wave is the plane one and has a constant amplitude within input aperture, so that AðrÞ ¼ circðr=rb Þ. Using Eqs. (30) and (36) and omitting the amplitude and phase constants, we then get that the transmittance T 00 ðr; nr Þ must be consistent with the mth order Bessel function: T 00 ðr; nr Þ / Jm kqring r=f expðimnr Þ: ð37Þ When using the further asymptotic expansion of the Bessel function Jm ðkqring r=f Þ at large values of its argument and going to the complex representation with dropping the slowly varying amplitude factor, we find the following approximate expression for the transmittance function: T 00 ðr; nr Þ exp½iðkqring r=f mnr mp=2 p=4Þ þ exp½iðkqring r=f þ mnr mp=2 p=4Þ:
ð38Þ
It is easy to see that the first and second phase-only components of this function represent, respectively, the converging and diverging pure-conical (m ¼ 0) or helical-conical (m 6¼ 0) waves. Being passed through the output refractive lens, each wave is reshaped and provides, at the equidistant focal planes, the in-phase beam focusing to a ring of radius qring . Because of this, either or both components of Eq. (38) are suitable for the desired focusing. Besides, numerical simulations reveal that the shape of the focal-plane intensity distributions in the vicinity of radius qring is scarcely affected by the quantity m, if only qring a1 b or, differently, if qring Wring , where Wring is the full width of the ring at half intensity maximum. On this basis, one may set m ¼ 0, when synthesizing the beams with large-ring intensity patterns. Then, the simplest transmittance function generating, e.g., a converging pure-conical wave will be that of a refractive axicon: T100 ðrÞ ¼ b0 ðrÞ ¼ exp½iu001 ðrÞ ¼ exp ikqring r=f : ð39Þ In the back focal plane of a positive refractive lens the axicon is known indeed to form a ring-like intensity distribution with rather weak transverse diffraction sidelobes [15]. The plots of the amplitude components of the aperture functions and the front shapes of relevant wavelets, which have been demonstrated in Fig. 1 earlier, represent just the same asymptotic curves as those related to a periodically focused beam produced by the beamforming system with an auxiliary filter having the transmittance according to Eq. (39). Neglecting the beam vignetting by the positive refractive lens L and using Eqs. (39) and (28), the ring-like intensity distribution at the nth output focal plane and the full width of this distribution at half intensity maximum can be assessed, respectively, as
A.G. Sedukhin / Optics Communications 229 (2004) 39–57
51
2 Z rb k kqring r kqr In ðq; n; zn Þ ¼ an exp i r dr ; J0 f f f 0
ð40Þ
Wring ¼ 6:223=ab ;
ð41Þ
with ab krb =f . In the first numerical example, suppose that the transmittance of the auxiliary filter is given by Eq. (39) with a desired ring distribution, to be formed at the equidistant focal planes, having the maximum intensity radius qring ¼ 0:5 mm and the full width of the ring at half intensity maximum Wring ¼ 10 lm (when the condition qring Wring holds). From Eq. (41), we readily find the preliminary value of the required boundary spatial frequency: ab ¼ 622:3 mm1 . To characterize the beam, it is essential also to specify such its parameters as the design wavelength k, the number of useful foci N , the axial position of a nearest useful focus z1 , the maximum propagation distance zN , and the distance between adjacent useful foci Z. For the sake of convenience, we set these parameters to be consistent with those of a fundamental-mode periodically focused beam, which has been investigated in [1]. Namely, we assume that k ¼ 0:6328 lm, zN ¼ 1 m, N ¼ 9, Z ¼ 100 mm, and z1 ¼ 2Z ¼ 200 mm. Using a calculation procedure presented in the reference 2 mentioned, pffiffiffiffiffiffiffiffiffi one can immediately find that f ¼ ðzN þ z1 Þ=2 ¼ 0:6 m, F ¼ f =Z ¼ 3:6 m, and Kr ¼ 2kF ¼ 2:135 mm. Let the DMF lens be of continuous-relief profile and has the following phase function: " 2 # 4 X e Þþ2 e 0 Þ cos kr n0 ; e u0 ðrÞ ¼ arg e a 0 expði w a n0 expði w ð42Þ 0 n 2F n0 ¼1 e ;...; w e g ¼ f0:906; 2:966; 1:601; 1:015; where fe a0; . . . ; e a 4 g ¼ f1:002; 0:976; 0:969; 0:936; 1:030g and f w 0 4 0:488g. We derived the last function from the generating phase function of a 1-D fan-out grating which, in turn, was designed by the virtual-source approach [16] for the fan-out N ¼ 9 [17]. The theoretical light efficiency of the DMF lens is equal to 99.28%. In Figs. 4(a) and (b), the different views of the central portion of the phase function are shown. pIn final ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi stage, we find the values of other quantities: P ¼ Round½Za2b =ð4pkÞ ¼ 310, a0b ¼ 4pkP =Z ¼ 621:9 mm1 , rb ¼ f a0b =k ¼ 37:582 mm, and Rb ¼ zN a0b =k qring þ pBzN =ðf a0b ÞjB¼14 ¼ 62:3 mm. Using Eqs. (23) and (24), the calculation of a generated wave field was carried out. The entire longitudinal intensity profile of this field at maximum intensity radius qring and the superimposed entire transverse intensity profiles at equidistant focal planes z ¼ z1 ; z ¼ z2 ; . . . ; z ¼ z9 are shown, respectively, in Figs. 5(a) and (b). The superimposed local longitudinal intensity profiles I1 ¼ Iðqring ; z z1 Þ; . . . ; I9 ¼ Iðqring ; z z9 Þ and the superimposed local transverse intensity profiles I1 ¼ Iðq qring ; z1 Þ; . . . ; I9 ¼ Iðq qring ; z9 Þ are represented, respectively, in Figs. 5(c) and (d). The last plots characterize the degree of scattering of the useful intensity point spread functions of the beamforming system both in magnitude and also in longitudinal and transverse displacements. As another example, which might be useful, e.g., for precise alignment, we now consider the special case of generating a singular longitudinally periodic beam with strictly zero intensity on the optical axis and ring-like intensity distributions at the focal planes, with a rather small maximum intensity radius qring . In this case, the condition qring Wring may does not hold and the transmittance function T200 ðr; nr Þ, coming from either component of Eq. (38) at qring ! 0 and m 6¼ 0, has the form T200 ðr; nr Þ ¼ bm ðrÞm6¼0 ¼ exp½iu002 ðrÞ ¼ exp ðimnr Þ:
ð43Þ
As can be seen, the filter with such transmittance originates an mth order helical wavefront. From Eqs. (28) and (43) it follows that, in the idealized case of the total absence of beam vignetting by the positive refractive lens L, the intensity distribution at the nth output focal plane will take the form:
52
A.G. Sedukhin / Optics Communications 229 (2004) 39–57
Fig. 4. Halftone representation of the central portion of the phase function of a continuous-relief zero-order DMF lens (a) and its radial section (b). The first three Fresnel zones are shown only. The design parameters are as follows: N ¼ 9, Kr ¼ 2:135 mm, and P ¼ 310 (with rb ¼ 37:582 mm).
2 Z rb k kqr Iðq; n; zn Þ an Jm r dr : f f 0
ð44Þ
Of particular interest is a maximum-compressed ring distribution corresponding to the first-order modes with m ¼ 1. In accordance with Eq. (44), the full width of the ring zone of this distribution at half intensity maximum amounts to Wring ¼ 2:623=ab :
ð45Þ
If need be, the other sizes can likewise be determined. For instance, the radius of the central dark spot at half intensity maximum and the maximum-intensity radius are found to be qdark spot ¼ 1:205=ab and qring ¼ 2:452=ab , respectively. Using the binary representation for the structure of the DMF lens introduced in [1], it makes sense to combine the elements with transmittances T 0 ðrÞ and T200 ðr; nr Þ into a single element with transmittance T ðr; nr Þ ¼ T 0 ðrÞT200 ðr; nr Þ ¼ exp½iuðr; nr Þ;
ð46aÞ
where uðr; nr Þ ¼
2 Q P Y pY kr sgn þ mnr 2pðuq þ p 1Þ 2 p¼1 q¼1 2F
ð46bÞ
is the binary phase function of the element, P is the number of Fresnel zones within the aperture of the element relative to its principal focus the distance F apart, Q is the number of transition points within one
A.G. Sedukhin / Optics Communications 229 (2004) 39–57
53
Fig. 5. Intensity patterns of the output field for a beamforming system corresponding to Fig. 2, designed for illumination with a uniform collimated input beam, and consisting of a DMF lens with the phase function of Fig. 4, a refractive axicon with the phase function u001 ðrÞ ¼ kqring r=F ¼ 0:00524r=k, and a refractive lens with f ¼ 600 mm and Rb ¼ 62:3 mm [the values of other design parameters are as follows: k ¼ 0:6328 lm, Z ¼ 100 mm, z1 ¼ 200 mm, zN ¼ 1 m, F ¼ 3:6 m, Wcirc ¼ 10 lm, and a0b ¼ 621:9 mm1 ]: the entire longitudinal intensity profile at the maximum intensity radius qring (a), the superimposed entire transverse intensity profiles at the equidistant focal planes z ¼ z1 ; z ¼ z2 ; . . . ; z ¼ z9 (b), the superimposed local longitudinal intensity profiles I1 ¼ Iðqring ; z z1 Þ; . . . ; I9 ¼ Iðqring ; z z9 Þ (c), and the superimposed local transverse intensity profiles I1 ¼ Iðq qring ; z1 Þ; . . . ; I9 ¼ Iðq qring ; z9 Þ (d).
period of the generating phase function of a 1-D fan-out grating at which the phase function undergoes abrupt changes, and 0 < uq 6 1 is the normalized position of the qth transition point within one grating period. We assume here that the arguments of the function uðr; nr Þ are varied in the ranges 0 6 r < rb and 0 6 nr < 2p only. The integrated element with transmittance of Eq. (46) may be referred to as an mth-order singular DMF lens. In the numerical simulation, the full width of the ring zone at half intensity maximum is chosen to be equal Wring ¼ 20 lm at m ¼ 1. The values of the main parameters of the beam to be synthesized and of the focal lengths derivable from them – k, z1 , zN , Z, N , f , and F – are assumed to be the same as in the previous case, when synthesizing the beam with large-ring focal patterns. The preliminary value of the required boundary spatial frequency can immediately be determined from Eq. (45) as ab ¼ 2:623=Wring ¼ 131:2 mm1 . From [18], we adopt the following normalized ordered set of transition points at which the generating phase function of a 1-D binary fan-out diffractive grating undergoes abrupt changes within a period: fu1 ; . . . ; u6 g ¼ f0:06668; 0:12871; 0:28589; 0:45666; 0:59090; 1:00000g (the theoretical light efficiency of this structure is equal to 72.49%). The values of other parameters are finally determined as: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P ¼ Round½Za2b =ð4pkÞ ¼ 14, a0b ¼ 4pkP =Z ¼ 132:17 mm1 , rb ¼ f a0b =k ¼ 7:987 mm, and Rb ¼ zN a0b =k þ pBzN =ðf a0b ÞjB¼14 ¼ 13:9 mm. Fig. 6 shows the central portion of the phase function of the calculated DMF element. Fig. 7(a) displays the map of intensity contours of the generated first-order singular beam in the meridional plane. The longitudinal intensity profile at radius qring ¼ 18:5 lm, where the main intensity maxima are observed, and the superimposed transverse intensity profiles at the equidistant focal planes z ¼ z1 ; z ¼ z2 ; . . . ; z ¼ z9 are shown, respectively, in Figs. 7(b) and (c).
54
A.G. Sedukhin / Optics Communications 229 (2004) 39–57
ϕBF (rad)
(a)
π _ 2
0
_ –π 2 –3
(b)
–2
–1
0
1
2
3
Radius r (mm)
Fig. 6. Graphic representation of the central portion of the phase function of a binary-phase first-order DMF lens (a) and its radial section (b). The first three Fresnel zones are shown only. The design parameters are as follows: N ¼ 9, Kr ¼ 2:135 mm, and P ¼ 14 (with rb ¼ 7:987 mm).
6. Conclusions We have shown that, apart from sharply peaked monochromatic periodically focused beams with multiple stigmatic on-axis ray convergence, there can exist periodically focused beams having an arbitrary, but the same, form of a desired propagation-invariant pattern reproduced at the equidistant focal planes. In the general case such the beams are represented as a discrete set of nonuniform phase-distorted spherical wavelets which possess the property of mutual similarity of their complex aperture functions. The last property means that the spherical projections of the real-valued amplitude and phase components of the complex aperture functions of the wavelets onto unit spheres, centered relative to the respective regularly spaced coaxial focal points, are pairwise the same for all the wavelets. In alternative treatment, the angular spectrum of plane waves for each useful wavelet may be conceived as a nonuniform phase-distorted spherical wave diverging from the respective on-axis focal point and having a changeless complex amplitude distribution. At the output aperture plane of a beamforming system the field of any wavelet can be decomposed into two multiplicative constituents: the complex amplitude at the cut of a strictly spherical wavelet, converging to a respective on-axis focal point, and a complex aperture-form function proportional, in paraxial approximation, to the inverse Fourier transform of a desired pattern. The high-order angular constituents of the aperture-form function give rise, in turn, to phase singularities appearing at the equidistant focal planes as dark coaxial spots with bright rims. The simplest optical arrangement for generation of a finite-extent generalized periodically focused beam can be implemented in the form of a positive refractive lens and the combination of two spatial-frequency phase filters placed in the front focal plane of the lens. The transmittance of the first filter corresponds to that of a diffractive multifocal lens and is associated with the fields of the above strictly spherical wavelets. The transmittance of the second filter is associated with the above complex aperture-form function and is defined by the inverse Fourier transform of the desired pattern. Both filters can be integrated in a single diffractive element.
A.G. Sedukhin / Optics Communications 229 (2004) 39–57
55
300 Radius ρ (µm)
200 100 0 100 200 300 0.1
(a)
0.2
< 0.005
0.3
0.4
0.005 0.01
0.5
0.6 0.7 0.8 Propagation distance z (m)
0.01 0.02
0.02 0.05
ρring = 0.7 18.55 µm 0.8 0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.1
0.2
(b)
0.05 0.1 Intensities I1, ... , I9 (arb. units)
Intensity I (arb. units)
1 0.9
0.3
0.4
0.5
0.6
0.7
Propagation distance
0.8
z (m)
0.9
1
1.1
0.9
1.2
1.0
1.1
0.1 0.2
1.2
1.3
0.2 0.5
1.4
0.5 1.0
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
120 100 80
(c)
60
40
20
0
20
40
60
80 100 120
Radius ρ (µm)
Fig. 7. Intensity patterns of the output field for a beamforming system corresponding to Fig. 2, designed for illumination with a uniform collimated input beam, and consisting of a DMF lens with the phase function of Fig. 6 and a refractive lens with f ¼ 600 mm and Rb ¼ 13:87 mm [the values of other design parameters are as follows: k ¼ 0:6328 lm, Z ¼ 100 mm, z1 ¼ 200 mm, zN ¼ 1 m, F ¼ 3:6 m, Wcirc ¼ 20 lm, and a0b ¼ 132:2 mm1 ]: the map of the intensity contours of the generated field in the meridional plane (a), the longitudinal intensity profile at the maximum-intensity radius qring (b), and the superimposed transverse intensity profiles at the equidistant focal planes z ¼ z1 ; z ¼ z2 ; . . . ; z ¼ z9 (c).
When using projection optical systems with diffractive multifocal lenses, the wave field of a generated finite-extent periodically focused beam is defined by structure of a planar object, for which the patterns reproduced at the equidistant focal planes of the beam are its own images. The effect of multiple projection imaging realized by these systems might be useful in numerous applications. As in the case of a conventional monochromatic imaging system with no spatial-frequency filters, the process of image formation at each equidistant focal plane is shown to be representable through the squared convolution of the complex amplitude distribution of an image to be reproduced and the point spread function of a multiple-imaging system, with only difference that the last function should be taken for the respective diffraction order.
Acknowledgement The author thanks the reviewer for its constructive comments.
Appendix A Consider the derivation of the direct and inverse Fourier transforms, when one is using the cylindrical coordinate system and the mathematical forms adopted in the present paper. By definition, the direct
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A.G. Sedukhin / Optics Communications 229 (2004) 39–57
Fourier transform of a function gðu; vÞ in the Cartesian system ðu; vÞ and the inverse Fourier transform of a function Gðxu ; xv Þ in the Cartesian system ðxu ; xv Þ can be represented as (see e.g. [19]): Z Z 1 1 gðu; vÞ exp ð ixu u ixv vÞ du dv; ðA:1Þ Ffgðu; vÞg ¼ Gðxu ; xv Þ ¼ 2p 1 Z Z 1 1 F1 fGðxu ; xv Þg ¼ gðu; vÞ ¼ Gðxu ; xv Þ exp ðixu u þ ixv vÞ dxu dxv ; ðA:2Þ 2p 1 with xu and xv being the angular frequencies. pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Going from the Cartesian system (u; v) to the polar coordinate system (r; nr ), such that r ¼ u2 þ v2 and nr ¼ arctanðv=uÞ, and going, in the frequency domain, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi from the Cartesian system (xu ; xv ) to the polar system (a; n), such that a ¼ x2u þ x2v and n ¼ arctanðxv =xu Þ, we rewrite Eqs. (A.1) and (A.2) as Z 2p Z 1 1 gðr; nr Þ exp ½ iar cosðnr nÞr dr dnr ; ðA:3Þ Gða; nÞ ¼ 2p 0 0 Z 2p Z 1 1 gðr; nr Þ ¼ Gða; nÞ exp ½iar cosðn nr Þa da dn: ðA:4Þ 2p 0 0 Using the identity Z 2p exp ½imf ix cosðf vÞ df ¼ ðiÞm 2p expðimvÞJm ðxÞ
ðA:5Þ
0
and expanding the functions Gða; nÞ and gðr; nr Þ in Fourier series 1 X Gða; nÞ ¼ Gm ðaÞ expðimnÞ
ðA:6Þ
m¼1
and
1 X
gðr; nr Þ ¼
gm ðrÞ expðimnr Þ;
ðA:7Þ
m¼1
with the following substitution into Eqs. (A.3) and (A.4) and changing the order of integration and summation, we get the following expressions for the direct and inverse Fourier transforms in the polar systems: Z 1 1 X m Ffgðr; nr Þg ¼ Gða; nÞ ¼ ðiÞ expðimnÞ gm ðrÞJm ðarÞr dr; ðA:8Þ 0
m¼1
F1 fGða; nÞg ¼ gðr; nr Þ ¼
im expðimnr Þ
m¼1
where gm ðrÞ ¼
1 X
1 2p
Gm ðaÞ ¼
1 2p
Z
Z
1
Gm ðaÞJm ðarÞa da;
ðA:9Þ
0
2p
gðr; nr Þ expðimnr Þ dnr ;
ðA:10Þ
0
Z
2p
Gða; nÞ expðimnÞ dn:
ðA:11Þ
0
References [1] A.G. Sedukhin, Periodically focused propagation-invariant beams with sharp central peak, Opt. Commun. 228 (2003) 231. [2] J.W. Goodman, Introduction to Fourier Optics, first ed., McGraw-Hill, New York, 1968 (Chapter 3).
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