Focusing of light by zone plates in Selfoc gradient-index lenses
José Manuel Rivas-Moscoso, Daniel Nieto, and Carlos Gómez-Reino
GRIN Optics Group, Departamento de Física Aplicada, Escola de Óptica e Optometría, Universidade de Santiago de Compostela,Campus Sur, E15782 Santiago de Compostela, Spain
Carlos R. Fernández-Pousa
Departamento de Ciencia y Tecnología de Materiales, Universidad Miguel Hernández de Elche,Avenida Ferrocarril s/n, E03202 Elche (Alicante), Spain
The zone plate (ZP) is a classic topic in optics andhas attracted widespread attention. There are vari-ous types of ZPs. Generally speaking, an amplitudeZP is a diffractive device consisting of a series of con-centric ring-shaped zones of radii alternately r2j21 �h1�2j 2 1�1�2 and r2j � ��2j 2 1�h1
2 1 e�1�2, where jassumes an integer value, that absorb or transmit ra-diation. The period of the ZP is p � 2h1
2, and e is thesquare width of the even zones, so, when e�p � 0.5, thesquare widths of the absorbing and the transmittingzones coincide and we can speak of Fresnel ZPs. Ananalogy with the conventional lens law can be made;thereby, for radiation of wavelength l0, the plate actsas a multifocus lens with foci fm � h1
2�ml0, wherem is an integer. For Fresnel ZPs m assumes an oddvalue.
The objective of this Letter is to report on anexperimental study of a simple hybrid diffractive–gradient-index (GRIN) element. Although purelydiffractive and hybrid diffractive–refractive elementsconstitute well-established technologies,1 – 3 hybriddiffractive–GRIN devices have been scarcely ex-plored.2 These devices, however, potentially couldsignificantly extend the range of applications of the in-tegration technology. In this context, Rivas-Moscoscoet al. previously derived the evolution of the ZPdiffraction orders as well as their multifocusing effectin a hybrid-structure ZP–GRIN medium.4,5 Here wepresent a simple experiment in which we validatesome of those results for a Selfoc GRIN rod lens.
The experimental conf iguration is shown schemati-cally in Fig. 1; basically, it is similar to a previoussetup used for exploration of Talbot images in a SelfocGRIN lens.6 A uniform beam of light departingfrom a He–Ne laser of wavelength l0 � 632.8 nm isincident upon a negative amplitude ZP with periodp � 0.503 6 0.013 mm2 and e�p � 0.45 6 0.02.The diffracted beam subsequently impinges upona 103 0.35-N.A. microscope objective �M1�, whichforms an image of the ZP inside a quarter-pitchSelfoc GRIN rod lens of diameter 2 mm and totallength LT � 5.168 mm fabricated by Nippon SheetGlass.7 The radial refractive-index distribution ofthe Selfoc lens follows this profile7,8:
where the index at the optical axis, n0 is 1.6073 andgradient function g is 0.304 mm21.
After the Selfoc lens, we mount a microscope with a3.53 0.10-N.A. objective �M2� upon an xyz stage withthe aim of probing the lens and inspecting the planesinside it. We define inside distance L as the distancebetween an arbitrary inside plane p0 and exit planepE , and outside distance s as the distance from ob-jective M2 to pE . The objective image is formed on aplane pI at a set distance s0 from objective M2. TheABCD matrix that relates planes p0 and pI is∑
A BC D
∏�
∑1 s0
0 1
∏ ∑1 0
21�f 1
∏
3
∑1 s0 1
∏ ∑1 00 n0
∏
3
∑cos�gL� sin�gL��g
2g sin�gL� cos�gL�
∏, (2)
where the last matrix is the well-known ray-transfermatrix in a Selfoc medium8 and f is the M2 micro-scope objective’s focal length. By imposing the imag-ing condition B � 0, we f ind the relationship betweendistances inside �L� and outside �s� the microlens to be
tan�gL� � n0gD , (3)
Fig. 1. Geometry of the input and readout systems: M1,focusing objective; p0, pE, and pI , object plane, exit planeof the microlens, and image plane, respectively; M2, obser-vation objective.
where D � sE 2 s and sE � s0f��s0 2 f �. Likewise,the transverse magnification of the observation sys-tem, MtS � A, can be written as
MtS�D� � M0�1 1 �n0gD�2�1�2, (4)
where M0 is the magnification of the objective. Periodp of the projected ZP inside the lens at p0 � pZP canbe obtained from the measured apparent period pap�D�on image plane pI from p�D� � pap�D��MtS
2�D�, wherethe square is accounted for by the quadratic period ofa ZP.
Experimentally, we obtain a sequence of diffractionpatterns consisting of concentric circles, some of whichshow an inner spot that corresponds to a focus alongthe optical axis (Fig. 2). Owing to the f inite lengthof the lens, the lower orders fall outside the exit face ofthe lens. To discover which diffractive order matchesevery focus, let us consider the equivalent systemshown in Fig. 3. In it, a point source a distance laway from the input face of the lens illuminates ahybrid-structure ZP–Selfoc lens of length LZP , whichforms the image of the object point source at variousdistances dm from the exit face outside the lens. TheABCD matrix for this system is∑A BC D
∏�
∑1 dm
0 1
∏ ∑1 00 n0
∏
3
∑cos�gLZP � sin�gLZP ��g
2g sin�gLZP � cos�gLZP �
∏
3
∑1 00 1�n0
∏ ∑1 0
22ml0�p 1
∏ ∑1 l0 1
∏, (5)
where the second-to-last matrix represents the ZP.The imaging condition, B � 0, provides the followingexpression for the position of the foci outside the lensas measured from the exit face:
dm �21
Um�LZP��
�p 2 2mll�DZP 1 lpn0
2g2lpDZP 2 �p 2 2ml0l�, (6)
where DZP � tan�gLZP ���n0g� and Um is the complexcurvature of the mth diffractive order.4 It can be eas-ily proved that order 0 (when it is outside the lens) isalways placed where the Fourier transform of the ZPtakes place. Once this position �d0� is found, solvingEq. (6) for l with m � 0 provides
l �sin�gLZP � 1 n0gd0 cos�gLZP �
n02g2d0 sin�gLZP � 2 n0g cos�gLZP �
. (7)
The Fourier-transform plane position was measuredto be d0 � 21.20 6 0.02 mm. To obtain the value ofl we must convert the distances s that can be mea-sured directly with the microscope into indirect, realdistances L inside the lens. The ZP position is thenLZP � 4.7183 6 0.0012 mm, so DZP � 14.90 6 0.04 mmand the illumination distance is calculated to be l �1.092 6 0.004 mm. The total magnif ication MtS�DZP �is 25.73 6 0.06, so pap�DZP� � 4.90 6 0.07 mm 2 andthe resultant period of the image of the ZP inside themicrolens is p�DZP � � �7.32 6 0.11� 3 1023 mm2. Now
we are in a position to label every focus and find thatthe lowest order that falls inside the microlens is m � 6.
Based on the equivalent system shown in Fig. 3, wecan also calculate the theoretical focal positions insidethe Selfoc lens,4,5 which are
zm � �1�g�tan21�n0glp��2ml0l 2 p�� . (8)
Figure 4 shows the dependence of the focal positions(from the ZP plane) inside the microlens on the diffrac-tive order. The results show excellent agreement be-tween the experimentally measured values and thosepredicted by Eq. (8).
Let us now measure the diffraction eff iciency of theZP inside the Selfoc lens. We remove the eyepiecefrom the microscope, replace it with a lens, and focusplane pI into a CCD camera. Both objective and CCDare mounted upon the same xyz stage as the micro-scope objective, so the distance from the objective tothe CCD camera is f ixed. The ZP limits the amountof light that passes through the focus. Consequently,the light converges to a spot whose size is that specified
Fig. 2. Photographs of (a) the ZP, (b) the plane wherediffractive order m � 7 is focused, and (c) a plane halfwaybetween the planes, where orders m � 6 and m � 7 focusinside a Selfoc GRIN lens.
Fig. 3. Equivalent system under uniform illumination:l, distance from the point source to the image plane of theZP inside the microlens; LZP , lens length from the imageof the ZP to the exit face; dm, image of the point sourcecorresponding to the mth diffractive order.
Fig. 4. Focal positions z inside a Selfoc GRIN lens as afunction of diffractive order m. Measured (experimental)positions and positions given by Eq. (8) are shown.
Fig. 5. ZP diffraction eff iciency ratios Rm versus diffrac-tive order m in a Selfoc GRIN lens for foci inside and outsidethe lens. Measured ratios and ratios given by Eq. (10) areshown. The shaded area represents the 95% conf idenceinterval on parameter estimate e�p. Inset, an enlarge-ment for the higher orders.
by the point-spread function of the ZP, which is simi-lar to the point-spread function of a lens with the sameaperture diameter.3 The Airy radius for the planesthat contain foci at a circular zone plate can be com-puted by standard techniques,8,9 yielding
r � 0.610l0
∑sin�gL�n0ga
1cos�gL�dm
a
∏, (9)
where a is the outer ZP radius and dm � 0 for insidefoci. When the foci are recorded and analyzed, themagnification of the Airy spot is due to the systemitself, up to plane pI [Eq. (4)], and to the CCD imaginglens. We took into account both magnif ications whenwe measured the energy delivered to each spot. The
theoretical ratio for the intensity delivered to order mrelative to order 0 in a generalized amplitude ZP, validfor either inside or outside foci, can be expressed as4
Rm � Im�I0 � sinc2�me�p� , (10)
so we can compare them with the measured intensityratios. In Fig. 5 we have plotted the ratio Rm in termsof the order m for foci inside and outside the GRINlens. A nonlinear curve f it produces the outcome thate�p � 0.41 6 0.01, where the shaded area displays the95% confidence interval on the nonlinear least-squaresparameter estimate e�p. Error bars represent the 2s
uncertainties in experimental Rm. Despite some dis-similarities that we attribute to the inf luence of thebackground energy that is due to adjacent orders; tothe small number of zones of the ZP; or to imperfec-tions of the ZP, agreement is reasonably good withinthe errors.
In conclusion, we have experimentally demonstratedthe multifocusing property of zone plates in Selfocgradient-index lenses. The experiment generatedresults, to our knowledge previously unreported, thatare in close accordance with those obtained throughprevious theoretical research on such properties asfocal positions and diffraction efficiency.
This study was supported by the Ministerio deCiencia y Tecnología, Spain, under contract TIC-2003-03041. We acknowledge helpful conversations withJ. Ares and the technical assistance of M. T. Flores-Arias, L. Chantada, and A. Castelo. The authors’e-mail addresses, in order, are [email protected], [email protected]. [email protected], and [email protected].
References
1. H. P. Herzig, ed., Micro-Optics: Elements, Systems andApplications (Taylor & Francis, London, 1998).
2. J. Turunen and F. Wyrowski, eds., Diffractive Op-tics for Industrial and Commercial Applications(Akademie-Verlag, Berlin, 1997), Chap. 1.
3. J. Ojeda-Castañeda and C. Gómez-Reino, eds., SelectedPapers on Zone Plates, Vol. MS 128 of SPIE MilestoneSeries (SPIE Press, Bellingham, Wash., 1996), and ref-erences therein.
4. J. M. Rivas-Moscoso, C. Gómez-Reino, C. Bao, and M. V.Pérez, J. Mod. Opt. 47, 1549 (2000).
5. J. M. Rivas-Moscoso, C. Gómez-Reino, M. V. Pérez, andC. Bao, J. Mod. Opt. 48, 915 (2001).
6. M. T. Flores-Arias, C. Bao, M. V. Pérez, and C. R.Fernández-Pousa, Opt. Lett. 27, 2064 (2002).
