Quasi-achromatic Fresnel zone lens with ring focus A. Vijayakumar* and Shanti Bhattacharya Centre for NEMS and Nanophotonics, Department of Electrical Engineering, Indian Institute of Technology Madras, Chennai 600036, India *Corresponding author:
[email protected] Received 18 November 2013; revised 8 February 2014; accepted 17 February 2014; posted 18 February 2014 (Doc. ID 200279); published 20 March 2014
The phase of a standard Fresnel zone lens (FZL) is periodically modulated in the radial direction using the phase of a binary fraxicon. The resulting element (rf -FZL) focuses light into a ring. The ring is found to be quasi-achromatic, in that the diameter is wavelength independent but its location is not. The binary rf -FZL is fabricated using electron beam direct writing. Experimental results confirm the generation of a wavelength-independent ring pattern at the focus of the rf -FZL. An efficiency of 24% was obtained. The variation in radius of ring pattern was reduced from 61 μm to less than 10 nm for a corresponding wavelength variation from 532 to 633 nm. © 2014 Optical Society of America OCIS codes: (050.1965) Diffractive lenses; (230.1950) Diffraction gratings; (220.4000) Microstructure fabrication. http://dx.doi.org/10.1364/AO.53.001970
1. Introduction
Fresnel zone lenses (FZLs) have long been used for focusing electromagnetic waves in numerous applications [1–4]. While a FZL is designed to have a point focus, axicons, on the other hand, are used for the generation of a Bessel intensity profile in the near field and a ring pattern in the far field [5–7]. The ring pattern directly obtained from an axicon is not focused and cannot be used for many applications. Therefore, axicons are normally used together with a lens [5,7]. In 1990, a laser delivery system utilizing the ring pattern generation property of an axicon for corneal surgery was proposed and demonstrated by Ren and Birngruber [8]. The optics configuration was bulky and involved many optical components. Ring pattern generation is also possible using different schemes [9–11]. However, in all these techniques, although only a single element is used for the generation, the ring pattern is not focused. To overcome this difficulty, we propose a composite optical element based on a FZL that will focus light into a ring. 1559-128X/14/091970-05$15.00/0 © 2014 Optical Society of America 1970
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Recently, composite optical elements using FZLs have been reported for applications in photomixing generation of THz radiation and optical trapping [12,13]. Fresnel axicons (fraxicons) [14,15] and diffractive axicons [16,17] for generation of Bessel beams and ring patterns have also been proposed and demonstrated. In this work, we report on the design of a ring focus FZL (rf -FZL) created by adding a binary fraxicon phase to the phase profile of a standard FZL. The modulation is introduced in the radial direction. The resulting element yields a ring focus, rather than a point, when illuminated. In all the above cases, the light was collimated using a lens before incidence on the axicon. In this case, the FZL is designed for nearfield illumination, which we call u − v configuration. Therefore, no additional collimating optics is required [18]. The light is focused into a ring at the image plane of the rf -FZL. 2. Ring Focus FZLs A. Design of a Ring Focus FZL
The optics configuration for generation of a ring pattern using a rf -FZL is shown in Fig. 1. A FZL is designed to be a phase-only element [1,19] in the u − v configuration [18] for the phase condition shown in
Fig. 1. Optics configuration for generation of a focused ring pattern at the image plane of a rf -FZL (u − v configuration).
kun vn − ku v 2nπ;
(1)
where u and v are the object and image distances from the FZL plane, un and vn are the optical path lengths of the nth ray, and k 2π∕λ, where λ is the wavelength of the source. The radii of the 2π period zones ρn of the FZL are given by 1∕2 C2ρ − 4u2 v2 ; ρn 4u2 v2 Cρ
(2)
where Cρ n2 λ2 2nλu 2nλv 2uv, and n is the 2π period zone number. In this case, the phase profile of a binary FZL (BFZL) can be described by ΦBFZL ρ
Φ1 0
ρn ≤ ρ ≤ ρn12 n 0; 1; 2; 3; …: (3) elsewhere
A binary fraxicon (BF) with a period of Λ can generate a ring pattern [6,9]. The phase profile of a BF is described by ( ΦBF ρ
Φ2 0
0 ≤ ρ ≤ Λ2 Λ 2
(4)
The value of the phase (Φ1 or Φ2 ) controls the efficiency of the BFZL [15] or the BF. The maximum efficiency of a binary structure in either of the first diffraction orders (1 or −1) is 40% when the phase is equal to π. However, in a BF, the first-order ring pattern consists of both positive and negative first orders, yielding an efficiency of 80%. The functions of a BFZL and a BF are combined by 2π-modulo phase addition of the phase profile of each of these elements. From Eqs. (3) and (4), the phase profile of the rf -BFZL can be described by Φrf -BFZL ΦBFZL ΦBF 2π .
kun vn − ku v 2nπ − Φ2 :
(6)
Solving Eq. (6) for the radii ρ0n of zones of the BFZL, we obtain an equation similar to Eq. (1), replacing n by (n − 0.5): ρ0n
1∕2 2 2 C02 ρ − 4u v ; 4u2 v2 C0ρ
(7)
where C0ρ n − 0.52 λ2 2n − 0.5λu 2n − 0.5λv 2uv. The spherical aberration introduced by the glass substrate on which the device is fabricated is corrected by modifying the radius of BFZL according to [18]. The phase profile of the FZL is synthesized for u 5 mm, v 30 mm, and λ 633 nm, while the phase profile of the BF is synthesized for Λ 50 μm. For a BF, the radius of the ring pattern depends on the period Λ. An equivalent dependency of the radius of the ring pattern on the phase modulation period of the rf -BFZL can be obtained. The phase profile of the rf -BFZL swaps between positive and negative BFZL phase values within this period. The maximum efficiency of this binary pattern is only 32% (0.4 of BFZL × 0.8 of BF). B. Quasi-achromatic Ring Focus FZL
≤ρ≤Λ
ΦBF ρ ΦBF ρ lΛ where l 0; 1; 2…:
effect, the phase profile of the rf -BFZL can be generated by an exclusive OR (XOR) operation of the phase profiles of a BFZL and a BF. In all these four combinations, the resultant values are still binary (0, π). The calculation of the phase profile of rf -FZL is shown in Fig. 2. This addition is realized by modification of the radii of the zones wherever it encounters the additional phase of the BF. When an extra phase value of Φ2 is added to the standard BFZL zones, Eq. (1) changes to
The rf -BFZL, when designed for f configuration (u → ∞), exhibits a quasi-achromatic property. This property can be understood by comparing the radii of the ring patterns generated by a BF and by the combined element. In the former case, the radius can be calculated using the diffraction equation of a grating. In the above design, we are interested only
(5)
The 2π-modulo addition of these two phase profiles has four different combinations: f0 02π 0g, f0 π2π πg fπ 02π πg, and fπ π2π 0g. In
Fig. 2. Generation of the phase profile of a rf -FZL (u − vconfiguration) from the phase profiles of a BFZL and a BF. 20 March 2014 / Vol. 53, No. 9 / APPLIED OPTICS
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in the first-order ring pattern, as the higher orders’ intensity values are negligible compared to it. Hence for m 1, from the diffraction equation of a grating and trigonometry, the radius r1 of the ring pattern is calculated as vλ r1 p : Λ 2 − λ2
(8)
The focal length of a FZLs in f configuration (u → ∞) can be obtained from Eq. (1) as follows: f
ρ2n − n2 λ2 : 2nλ
(9)
From Eqs. (8) and (9), when u → ∞, we obtain the effective expression for the radius of the ring pattern r1
ρ2n − n2 λ2 p : 2n Λ2 − λ2
(10)
For the previous design values, Eq. (10) can be approximated to be r1 ≅
ρ2n : 2nΛ
(11)
From Eq. (9), it is obvious that different wavelengths come to focus at different points on the axis, resulting in chromatic aberration. To study this, two wavelengths, namely λ1 532 nm and λ2 633 nm, were considered for experimental verification. In order to compare the radius of the ring pattern for these two wavelengths, a rf -BFZL with a focal length of f 30 mm was considered. The focal length for n 1 in Eq. (9), as a function of wavelength, is shown in Fig. 3. The shift of the focal plane for the wavelengths 532 to 633 nm is 5.7 mm. The radius of the first-order diffraction ring in the case of only a
Fig. 3. Variation in the focal length of a rf -FZL (f -configuration) as a function of incident wavelength. 1972
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BF [Eq. (8)] is compared, in Fig. 4, with that of the rf -BFZL (f configuration) [Eq. (11)] for the wavelength variation from 400 to 800 nm. For the wavelengths 532 and 633 nm, the difference in radii is 61 μm for the BF but only 8 nm for the rf -BFZL element. The element appears to have a quasiachromatic behavior. This can be understood by the following explanation. When the wavelength of the source increases, the radius of the first-order diffraction ring [Eq. (8)] increases in the case of the BF. However, in the rf -BFZL, the increase in wavelength decreases the focal length (Fig. 3), resulting in formation of the ring pattern closer to the FZL plane, which in turn decreases the radius of the diffraction ring. These two effects cancel each other out, rendering a quasi-achromatic behavior. This wavelengthindependent behavior can be obtained even for higher-order diffraction rings by proper choice of design parameters. The undesirable chromatic aberration therefore greatly reduces the wavelengthdependent nature of the radius of the diffraction ring. This is a very useful result. It implies that a shift in focus, which does not affect the radius of the firstorder diffraction ring pattern, could be achieved simply by tuning the wavelength. The intensity in the first-order varies only by 10% when the wavelength changes from λ1 to λ2 . 3. Fabrication of Ring Focus FZL
The rf -BFZLs were designed with a diameter of 2 mm, such that the intensity throughput is greater than 98%. The outermost half-period zone width was 1.29 μm, with 390 half-period zones (after aberration correction) for the BFZL designed for u–v configuration for the previous design values. The BFZL designed for f configuration has an outermost zone width of 9.5 μm with 54 half-period zones. Polymethyl methacrylate (PMMA) 950 K with Anisole 8% (Microchem) was used as the electron beam resist. The thickness of the electron beam resist
Fig. 4. Variation of the radius of the first-order ring patterns of a rf -FZL (f -configuration) (blue color) and a BF (dashed green color) when the wavelength is varied between 400 and 800 nm. Colors are available in the online version.
Fig. 7. (a) Image and (b) intensity profile of the ring pattern generated by the rf -FZL (u − v-configuration). Fig. 5. Optical microscope image of the outermost part of the rf -FZL (u − v-configuration) fabricated by electron beam direct writing.
for maximum efficiency in a binary pattern was calculated to be λ∕2nr − 1 ∼ 630 nm, where nr (∼1.5) is the refractive index of the electron beam resist. PMMA was spin coated onto indium tin oxide (ITO) borosilicate glass substrates (transmittivity ∼86% in the visible region) with a hexamethyldisilazane prime layer to improve adhesion to the glass substrate. The rf -BFZL patterns were generated as bitmap images using Eqs. (2) and (7) after applying suitable aberration correction [18]. The bitmap images were converted into GDSII format with LinkCAD software (Version 7.1.8). PMMA was patterned using a RAITH 150 TWO system with 10 kV acceleration voltage, 15 mm working distance (magnification 50), and 30 μm aperture size. The pixel size was 30 nm. The writing time of the rf -BFZL element was 1 h and 50 min for an area dose of 40 μC∕cm2. The patterns were developed using MIBK (methyl isobutyl ketone):isopropyl alcohol (IPA) (1∶3). The optical mi-
Fig. 6. Optical microscope image of the rf -FZL (f -configuration) fabricated using electron beam direct writing.
croscope image of the outermost part of the rf -BFZL (u–v configuration) is shown in Fig. 5. The optical microscope image of the rf -BFZL (f configuration) is shown in Fig. 6. A confocal measurement showed a resist height of 640 nm with an error of 1%. The surface roughness was measured as 43 nm. 4. Evaluation of Ring Focus FZLs
The fabricated rf -BFZL (u–v configuration) was evaluated using a fiber-coupled laser source (635 nm) in an optics configuration similar to Fig. 1. The image of the ring pattern was recorded using a charge coupled device (CCD), and its intensity profile is shown in Fig. 7. The zero-order patterns are not completely cancelled due to the error in resist height. The experimental value of the radius of the ring pattern was found to be ∼390 μm, which was close to the estimated value of 379 μm for the specific u–v settings used. Typically, the 1∕e2 full width of the ring pattern
Fig. 8. (a) and (c) Image of the ring pattern generated by rf -FZL (f -configuration) for the wavelengths 635 and 532 nm, respectively, (b) and (d) give the normalized intensity profiles for these wavelengths. The ring pattern occurs at a different focal plane for each wavelength. This was captured by moving the position of a CCD through a distance of 5.9 mm. A video of the same can be seen in Media 1, online. 20 March 2014 / Vol. 53, No. 9 / APPLIED OPTICS
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is 1.65 times the diffraction-limited value [5]. However, in the u–v configuration, the width has to be calculated taking into account the magnification of the system. All of these parameters are taken care of in the simulation and the resulting width was 131 μm. The experimental value was found to be ∼153 μm [13]. The difference in these values may be due to fabrication errors and any remaining spherical aberration [18]. The wavelength-independent behavior of the element was tested using the rf -BFZL (f configuration) using the two reference wavelengths. The images of the ring pattern and their respective intensity profiles for the two wavelengths are shown in Fig. 8. In both cases, the radius of the ring pattern was found to be 378 μm. The measurement was carried out using a CCD with a pixel size of 4.6 μm. The 1∕e2 widths of the ring were found to be ∼19 μm and ∼23 μm, which is very close to the calculated values 17 and 20 μm (i.e., 1.65 times the diffraction-limited spot size). The experimental value of the focal shift was found to be 5.9 mm. The efficiency of the fabricated elements was found to be >24%. The discrepancy in the efficiency value is partly due to the transmittivity of the ITO glass substrate and partly due to fabrication errors. 5. Conclusion
A modified FZL that can focus light into a ring when used in the u–v configuration was designed and fabricated. Alternately, the element could be used in the f configuration, yielding a wavelength-independent ring in the focal plane. The maximum theoretical efficiency is 32%, not including the loss introduced by the ITO layer. For the wavelengths used, the ITO layer would introduce about 4% extra loss. Therefore, the maximum expected efficiency would be 28%. The evaluation results were promising, with a measured efficiency of 24%. The efficiency could be improved by using a gradient phase profile instead of a binary one [17]. The variation in radius of the ring pattern was reduced from 61 μm to 8 nm for a corresponding wavelength variation from 532 to 633 nm. This quasi-achromatic behavior can be extended to higher-order diffraction patterns as well, by proper choice of the design parameters. We believe that the rf -FZL could be used in various applications to replace a number of optical components, thus reducing the overall size and weight of a system. The rf -FZL also has potential for applications in lithography and optical trapping experiments [20], when designed to generate a ring pattern with a smaller radius.
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