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This study explores two-dimensional binary sub-wavelength diffractive lenses (BSDLs) for implementing long focal depth and high transverse resolution based on the rigorous electromagnetic theory and the finite-difference time-domain method. Focusing performances, such as the actual focal depth, the ratio between the focal depth of the designed BSDL and the focal depth of the conventional sub-wavelength lens and the spot size of the central lobe at the actual focal plane, for different f-numbers, have been studied in the case of TE incidence polarization wave. The rigorous numerical results indicate that the designed BSDLs indeed have long focal depth and high transverse resolution by modulating the binary sub-wavelength characteristic sizes. Because BSDLs have the ability for monolithic integration and can require only single step fabrication, the investigations may provide useful information for BSDLs’ application in micro-optical systems.
©2008 Optical Society of America
1. Introduction
Diffractive lenses whose diffracting feature sizes smaller than the wavelength of illumination have many distinctive features, such as infrared photo-detection, and spectral synthesizing that make them potentially useful for integrated and miniaturized optical systems[1–3]. Binary sub-wavelength diffractive lenses (BSDLs) whose profile consists of binary features that are less than a wavelength are easy to be fabricated by using modern micro photolithography and fabrication techniques, like electron beam writing and reactive ion etching[4]. BSDLs have many advantages, such as high fill factor, easy design for more general beam controlling function, and requiring only one step fabrication [5–7]. In recent years, various diffractive lenses with long focal depth and high transverse resolution have been extensively studied owing to their wide applications in high precision optical alignment systems; in profile measurement systems and in optical disk readout systems [8–12]. Among these, Davidson et al. [8] presented a holographic axilens with long focal depth and high transverse resolution based on an empirical formula, further, Sochacki et al. [9] obtained the axilens with the logarithmic phase retardation function by using a differential relation, and then, a non-paraxial design method of axilens with constant axial intensity according to the laws of geometrical optics was presented [10]. But when lenses’ scales move closer to that of illumination wavelength, it is necessary to use a rigorous electromagnetic theory of diffraction to design and analyze lenses’ performance [13–15]. Recently, researchers have designed and analyzed the micro-lenses with long focal depth in the non-paraxial case by presetting focal depth based on a rigorous electromagnetic theory [16–19]. We also designed two-dimensional and continuous profile’s diffractive lenses with different focal depth and high transverse resolution by using rigorous electromagnetic theory [20]. Compared with binary sub-wavelength profile, however, the continuous profile lenses are not very easy to be fabricated. So motivated by the wide applications and advantages of BSDLs with long focal depth and high transverse resolution, this paper presents the rigorous vector analysis and design of BSDLs that are finite in extent and have binary sub-wavelength structures by using a finite difference time domain (FDTD)method[21] and the angular spectrum propagation(AS)method[22–26]. The element analyzed in this paper indeed is an axicon, but of sub-wavelength structures and different parameters. Due to its small size and sub-wavelength profile, the geometrical-optics approach is not suitable, by using rigorous numerical simulations, the results have shown that the BSDLs’ focal depth can get modulating and get required focusing characteristics by setting the preset focal depth and adjusting the binary sub-wavelength structures. In this study, we use the focal depth over which the intensity is greater than 80% of maximum intensity along the light axis (i.e., the y-axis); use the diffraction-limited spot size to describe the transverse resolution, respectively.
This paper is organized as follows: In Section 2 we discuss the basic formulas used in our study and describe the binary profile of BSDLs. In Section 3 we provide the results of non-paraxial, rigorous designs and analysis of BSDLs with long focal depth and high transverse in detail. Finally, a brief conclusion and our contribution are given in Section 4.
2. Profile of BSDLs and theoretical formulas
In this study, aperiodic two-dimensional diffractive lenses with binary sub-wavelength features are designed and analyzed. Lenses are illuminated by a normally incident TE polarization plane wave.
In the non-paraxial approximation, the continuous profile function of diffractive lens can be written as [27, 28]:
where n 1 and n 2 are the refractive indices of lens’ medium and air, respectively; n 1>n 2; λ=λ0/n2 and λ0 is the free-space wavelength; m is the number of zones; D and f are the diameter and the focal length of lenses, respectively. Usually, the focal length is a constant for the conventional diffractive lenses, but in order to design diffractive lenses with long focal depth, we set the focal length as a continuous function [8]:
where f0 and df are the beginning focal length and the preset focal depth. Although this function is empirical, and the focal region of the lens is at least twice as large as it can be seen from the Ref. [9] which is based on the geometrical-optics approach, the designed BSDLs can indeed achieve a long focal depth based on rigorous electromagnetic theory, as will be shown in Section 3. Substituting Eq. (2) into Eq. (1), we can get the profile distribution of the diffractive lens as:
As we see, Eq. (3) will reduce to the conventional diffractive lenses’ profile distribution when the preset focal depth df is equal to zero. We can modulate the focal depth by adjusting the preset focal depth df. In order to obtain binary sub-wavelength structures for designing BSDLs, we approximate this continuous profile to a piecewise-linear profile, and then encode the individual linear segments as binary sub-wavelength structures by using the approach presented by Farn[29] as shown in Fig. 1, so when setting different df, the BSDLs’ structures will be modified also. The interaction between a BSDL and a normally incident TE mode plane wave is determined by the FDTD method. Once the electromagnetic fields in the FDTD region have reached steady state, the near field (at the output plane) will be transformed forward to the far field (at the observation plane) by using the angular spectrum propagation method, as shown in Fig. 2. In free space, each angular spectrum component will be propagated to the observation plane, and at the same time, a phase delay is caused by the transform function as Eq. (4):
where fx and d cos(α) are the angular spectrum’s spatial frequency and the direction cosine.
In the next section, we will design and analyze BSDLs with long focal depth and high transverse resolution by modulating the preset focal depth df and the width of a binary sub-wavelength feature.
Fig. 1. A binary sub-wavelength diffractive lens (BSDL) based on Farn method[29] and a continuous profile diffractive lens
3. Numerical results
In order to evaluate the focal depth and the transverse resolution, we calculate the actual depth , the ratio (here is the focal depth of a conventional binary sub-wavelength lens), and the spot size of the focused beam in the actual focal plane. The lens is assumed to fabricate into a fused silica substrate (n1=1.5) and the outside medium is air (n2=1.0), and the wavelength λ in air is 10µm. The f-number of a BSDL is defined as the ratio of its beginning focal length 0 f to its diameter D (f/#=f0/D). We will design three families of BSDLs with f-number f/0.6, f/1.0 and f/1.5, all having a 100µm diameter (D=100µm), so the beginning focal length f0 of BSDLs are 60µm, 100µm and 150µm, respectively.
Fig. 3. Binary sub-wavelength profile distribution of BSDLs for different preset focal depth (df=0, 9µm, and 18µm)
Firstly, to design BSDLs with long focal depth and compare with each other, we calculate the field intensity distribution of the BSDLs for f/0.6 with preset focal depth df as 0, 0.15f0 (9µm), and 0.3f0 (18µm) by using the FDTD method and the AS method. The BSDLs’ profiles are shown in Fig. 3 with different preset focal length and have different structures (although all these BSDLs have two zones, binary structures in each zone are different with different preset focal depth). The axial intensity distribution of the conventional BSDL (df=0µm) is calculated also and shown by curve c in Fig. 4. Curves a and b correspond to df=18µm and df=9µm, respectively. It is clear that the shape of curve a is wider than that of curve b which is wider than that of curve c. This means that when setting a positive df, we can design the BSDL with a longer focal depth. The focal depth of the curve a, curve b and curve c are 30µm, 26µm and 21µm, respectively. Values of Ra for two lenses are 1.43 and 1.24. The actual focal planes that are the position corresponding to the maximal intensity peak for curves a, b, and c in Fig.4 are located at y=58µm, 49µm and 47µm, respectively. These three BSDLs’ transverse intensity distributions at the focal planes are shown in Fig. 5. The sizes of central lobes are 7.9µm, 7.5µm and 7.3µm, which are similar to the diffractive-limited spot size λf0/D=6µm. The plots clearly demonstrate that the designed diffractive lenses represent good focusing characteristics, which illustrates that the designed BSDLs have high transverse resolutions. In order to get a regional view of both the axial resolution and the transverse resolution, we display the propagation plots of electric field intensity over a region surrounded with focal points in Fig. 6(a) for df=18µm, (b) for df=9µm and (c) for df=0µm, respectively. The bright regions correspond to high field values, and the dark regions correspond to low field values. As shown in Fig.6, we believe that our designed binary sub-wavelength diffractive lenses have special functions of long focal depth and high transverse resolution.
Fig. 4. Optical field axial intensity distribution of BSDLs(f/0.6) with different preset focal depth df.
Fig. 5. Optical field transverse intensity distribution of BSDLs(f/0.6) with different preset focal depth df at the actual focal plane
Fig. 6. Propagation plots of the intensity of the electric field of BSDLs (f/0.6) with different preset focal depths: (a) df=18µm, (b) df=9µm, (c) df=0µm.
Secondly, the f-number is set as f/1.0 in the design, and the important feature of BSDLs remains unchanged. Fig. 7 shows the axial direction intensity of the BSDL for f/1.0 with different value of df: curve c is for the conventional BSDL (df=0µm), curves a and b are for df=30µm and 15µm, respectively. The actual focal depth of the curve a, curve b and curve c are 60µm, 35µm and 27µm, respectively, and values of Ra for curves a and b are 2.22 and 1.30. Compared with the focusing characteristic of the conventional BSDL, it is evident that the designed BSDLs can obtain longer focal depth according to the values of the preset focal depth df. The actual focal planes for curves a, b and c appear at y=95.9, 76.8, and 64.9µm, respectively. The transverse intensity distributions at actual focal planes of three BSDLs are shown in Fig. 8 and the spot sizes of central lobes are 12.1µm, 9.7µm and 8.9µm, similar to the diffraction-limited spot size of 10µm. As the first example, the plots evidently illustrate that the designed BSDLs can obtain long focal depth with high transverse resolution through setting a positive preset focal depth.
Fig. 7. Optical field axial intensity distribution of BSDLs (f/1.0) with different preset focal depth df.
Fig. 8. Optical field transverse intensity distribution of BSDLs(f/1.0) with different preset focal depth df at the actual focal plane
Finally, in order to compare focusing characteristics of different designed BSDLs with different f-number (f/#=0.6, 1.0, and 1.5) and with different preset focal depth, we summarized the numerical results in Table1. It is evident that the actual focal depth of the BSDL is increased by setting different preset focal depth and different binary sub-wavelength structures.
Table 1. Focusing characteristics of BSDLs with different preset focal depths for several f-numbers*
4. Conclusion
By using a rigorous electromagnetic computational model, we have presented non-paraxial designs and analyses of BSDL with long focal depth while keeping high transverse resolution. Binary sub-wavelength diffractive lenses with f-numbers=f/0.6, f/1.0, and f/1.5 have been designed through modulating different preset focal depths and binary sub-wavelength structures. The results of rigorous numerical simulations have illustrated that designed BSDLs can get required focusing characteristics. The combination of numerical and graphical results has successfully shown that the designed BSDLs have long focal depth and high transverse resolution. On the basis of the design and analyse results, out method may prove to be a very useful technique for rigorous design of BSDLs with long focal depth and high transverse resolution, and these BSDLs may be very useful for practical applications, such as in optical disk readout systems, and in high precision optical alignment systems etc.
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