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Abstract
Conventional diffractive optical elements suffer from large chromatic aberration due to its nature of severe dispersion so that they can only work at a single wavelength with near zero bandwidth. Here, we propose and experimentally demonstrate an achromatic imaging in the full-visible wavelength range with a single dual-pinhole-coded diffractive photon sieve (PS). The pinhole pattern (i.e., distribution of the position and size of each pinhole) is generated with dual wavelength-multiplexing coding (WMC) and wavefront coding (WFC), in which WMC makes multiple wavelengths that are optimally selected within the full visible range focus coherently on a common designed focal length while WFC expands the bandwidth of the diffracted imaging at each of the selected wavelengths. Numerical simulations show that when seven wavelengths (i.e., 484.8, 515.3, 547.8, 582.4, 619.1, 658.1 and 699.5 nm) within the visible range between 470 nm to 720 nm and a cubic wavefront coding parameter α = 30π are selected, a broadband achromatic imaging can be obtained within the full range of visible wavelength. Experimental fabrication of the proposed dual-pinhole-coded PS with a focal length of 500 mm and a diameter of 50 mm are performed using the mask-free UV-lithography. The experimental imaging results agree with the numerical results. The demonstrated work provides a novel and practical way for achieving achromatic imaging in the full visible range with features of thin, light and planar.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Diffractive optical elements (DOEs) have been used nowadays in numerous application areas, especially in some fields that conventional glass-based refractive optics are difficult, or even impossible, such as, X-ray microscopy [1], spectroscopy [2], virtual and augmented reality [3], and ultra-large space telescope primaries [4]. Unfortunately, chromatic aberration in diffractive imaging optics is more serious than in refractive imaging optics due to different dispersion mechanism. The most common method for correcting chromatic aberration in the refractive imaging optics is using different glass-based materials that exhibit complementary dispersion, as that in an achromatic doublet or triplet [5,6]. This technique is, however, not applicable in a fully diffractive imaging system due to the fact that dispersions of diffractive lenses are non-complementary. Many works have been done for correcting chromatic aberration in diffractive imaging optics, such as photon sieve imaging system [7,8,9]. In 2007, Andersen et al [10] proposed a broadband photon sieve telescope system consisting of an antihole photon sieve with very large holes centered on “dark” Fresnel zones. However, in order to obtain a useful bandwidth, a complex corrective system with an even larger aperture than the photon sieve primary must be employed. In 2008, Chung et al [11] and Zhou et al [12] fabricated and tested the multi-wavelength photon sieves by means of mixing the pinholes for several designed individual wavelengths in a random pattern. The drawback in this method is that each set of pinholes works only for one single wavelength while forms a strong background for other wavelengths. In 2018, Li et al [13] proposed and demonstrated a multispectral achromatic imaging with a single harmonically diffractive photon sieve, in which four separated spectral bands centered at 437.5, 500, 583.3, and 700 nm in the visible range can be focused simultaneously and achromatically onto the same focal plane. The working wavelengths are determined solely by the harmonic diffraction order which results in a lowered diffraction efficiency due to the increased space between adjacent pinhole-rings. Recently, Doskolovich et al further proposed a method for designing diffractive lenses having a fixed focal length at several discrete wavelengths, in which the operating wavelengths can be prescribed as needed [14]. Achromatic imaging in a continuous broadband wavelength range has been a hot and practically interested topic in recent years. In 2015, Zhao et al [15] proposed and demonstrated a broadband imaging photon sieve in which a separated cubic phase mask was placed in front of a conventional photon sieve. Results of the experiment show that the working bandwidth of the phase coded photon sieve system reaches ±14 nm at central wavelength of 632.8 nm, which is ∼88 times that of a conventional photon sieve. In 2016, Zhao et al further integrated the cubic phase directly into the distribution of pinholes of the photon sieve for achieving the broadband imaging with a single photon sieve, from which a bandwidth of 28 nm was achieved. It was further shown that a bandwidth of 300 nm can also be obtained when the phase coded photon sieve is integrated with a refractive element [16]. From 2016, Menon et al conducted a series of investigations on broadband achromatic imaging using phase-type planar diffractive lens in the visible [17,18], and near/long-wave infrared range [19,20], respectively. In their design, the on-axis focusing efficiency averaged over all the selected design wavelengths was used as the metric for optimization of the multi-level profile height (the width of rings is fixed) in which a grayscale of 100 was employed for a maximum height of ∼10 μm. Hundreds of height parameters at each ring of the lens must be searched and optimized in order to obtain an optimized focusing efficiency, which brings heavy computational workload.
It is also noted that metasurface or metamaterial based metalenses have attracted lots of attentions for the purpose of planar imaging in recent years [21,22,23,24,25]. Compared with all those above-mentioned diffractive planar imaging based on scalar and polarization-insensitive theory, metalenses are different type of planar imaging that is based on nano- or subwavelength structure generated abrupt phase shift within a subwavelength interface, such as localized surface plasmonics [21–23] or circular polarization induced geometric phase [24,25]. By nature, these metasurface based elements are polarization sensitive, which is excellent when vector properties of light must be manipulated. Scalar diffractive optics is a better alternative for planar imaging in many cases when considering its advantages of scalar diffractive theoretical frame, polarization independent and far more simple and large-area fabrication [26].
In this paper, an achromatic imaging in the full-visible wavelength range with a single dual-pinhole-coded diffractive photon sieve (PS) is proposed and demonstrated, in which the pinhole pattern is generated with dual wavelength-multiplexing coding (WMC) and wavefront coding (WFC). The WMC makes multiple wavelengths that are optimally selected within the full visible range focus coherently on a common designed focal length while WFC expands the bandwidth of the diffracted imaging at each of the selected wavelengths so that diffraction-limited imaging at all continuous wavelengths within the designed wavelength range can be obtained. Compared with all those previous reported methods, the proposed method achieves broadband achromatic imaging with much less designing parameters than those with direct binary search method, which provides a much simplified and more efficient method for broadband achromatic imaging with diffractive lens. Experimental fabrication of the proposed dual-pinhole-coded PS with a focal length of 500 mm and a diameter of 50 mm are implemented using the mask-free UV-lithography. The experimental results agree excellently with the numerical results. The demonstrated work provides a simple and practical way for achieving achromatic imaging in the full visible range with features of thin, light, planar, large aperture and easy fabrication.
2. Principle of broadband imaging with dual WMC-WFC photon sieve
Physically, achromatic imaging means that different wavelengths will have a same phase delay at the focal plane after passing through an optical imaging element including diffractive optical elements (DOEs). This suggests that a phase pattern in a DOE that works simultaneously for multiple wavelengths can be optimally generated through wavelength-multiplexing coding (WMC). The phase distribution provided by the WMC DOE (under plane wave illumination at normal incidence) can be written as:
For a converging lens, the phase distribution for the lens at working wavelength λn can be written as:
Fig. 1. Illustration of the binary phases of (a) conventional PS; (b) WMC PS; and (c) D-WMC/WFC PS; and their corresponding pinhole patterns: (d) CPS; (e) WMC PS; and (f) D-WMC/WFC PS. The design focal length is 500 mm, and the aperture is 50 mm. Multiplexing wavelength at 484.8, 515.3, 547.8, 582.4, 619.1, 658.1 and 699.5 nm are assumed in WMC PS, and a cubic phase coding parameter α = 30π is assumed in D-WMC/WFC PS. For clarity, only the innermost rings are shown.
3. Numerical simulation
To show the performance of the proposed broadband imaging with a D-WMC/WFC PS, numerical simulations of a D-WMC/WFC PS with a focal length of 500 mm and an aperture of 50 mm are conducted, along with a CPS and a WMC PS with the same focal length and aperture parameters for comparison. For the purpose of the broadband imaging in the visible wavelength range from 470 nm to 720 nm, seven multiplexing wavelengths, i.e., 484.8, 515.3, 547.8, 582.4, 619.1, 658.1 and 699.5 nm are selected, i.e., λn (n = 1, 2, …, 7) in Eq. (3) or Eq. (5). The designed working wavelength of the comparison CPS with the same focal length and aperture is assumed at 595 nm. The total ring number of the designed WMC PS is 1110, and the minimum pinhole size is 2.6 µm at the 1106th ring. The performance of the WMC PS was simulated using far-field theoretical model in that light field contributions from individual pinholes are coherently summed [27]. Figure 2 shows the intensity distribution at the focal point at Z=500 mm within the wavelength range from 470 nm to 720 nm when the WMC PS is illuminated by a normal and collimated incident light beam. The inset (upper picture) in Fig. 2(a) shows the focusing spot in the focal plane at different multiplexing wavelengths. It is seen that the focusing spots can only be obtained at the designed multiplexing wavelengths, the focal spot is rapidly diffused when the wavelength deviates from each designed multiplexing wavelengths. Figure 2(b) further shows the intensity distribution of the WMC PS along the optical axis (Z) near focal plane. As expected, the intensity peaks at the 7 multiplexing wavelengths appear coincidently at the designed focal length, i.e., Z=500 mm. The radius of the focal spot at each multiplexing wavelength is shown in Fig. 2(c) and is compared with that of the CPS. It is seen that the sizes of the focal spot at each multiplexing wavelength are all very close to the respective diffraction limited values, which is ∼14.52 μm with small differences owing to its wavelength dependent nature. The superior feature of the diffraction-limited focusing at the selected multiplexing wavelengths arises from the WMC phase distribution (i.e., $\Phi ({x,y} )$ in Eq. (1)) which is obtained from the summed optical fields at different selected wavelengths with different phase components φn. The wavelength-multiplexed phase ($\Phi ({x,y} )$) at each selected wavelength (i.e., λn) is actually dominated by the field generated by φn at wavelength λn, because the contributions of the optical fields from other wavelengths are all negligible when the separation between the selected wavelengths is not too small. As a result, the diffracted field of the WMC PS at the focal plane at each multiplexing wavelength should all be a diffraction-limited focusing spot, as witnessed in Fig. 2. This is a unique behavior which is totally different from that with geometrically averaged or optimized phase over multiple wavelengths.
Fig. 2. Numerical simulation of the focusing charactopticaleristics of a WMC PS. (a) The simulated intensity distribution with wavelength from 470 nm to 720 nm at the designed focal plane Z=500 mm. Upper: intensity distribution in the focal plane at 7 multiplexing wavelengths; lower: intensity spectrum at the focal point. (b) Intensity distribution of the 7 multiplexing wavelengths around the focal point Z=500 mm along Z axis. (c) The normalized focal spots at seven designed wavelengths in the focal plane.
Continuous broadband imaging can be implemented with the proposed D-WMC/WFC PS although the WMC PS can focus the lights at the selected discrete multiplexing wavelengths onto a common focal plane simultaneously with diffraction limited quality. In the D-WMC/MFC PS, additional cubic phase is added onto the phase distribution, as shown in Eq. (5) and Fig. 1(f), when compared with the WMC PS. Here, a typical cubic phase coding parameter is set as α=30π (different α could be used corresponding to different selected multiplexing wavelengths). The aperture size, the focal length and the multiplexing wavelengths are the same as those in the WMC PS used in Fig. 2, and the pinhole configuration of the D-WMC/WFC PS is shown in Fig. 1(f). Because the added cubic phase only slightly changes the position of the pinholes in the D-WMC/WFC PS while the size and the number of the pinholes are the same as those of the WMC PS, in which the pinhole pattern becomes symmetric about y = x in contrast to that circularly symmetric in WMC PS.
Figure 3 shows the simulated focusing and imaging behaviors of the D-WMC/WFC PS. Figures 3(1a)-3(3a) show the simulated point spread functions (PSFs) of the D-WMC/WFC PS at the selected multiplexing wavelengths and the wavelengths deviating 10 nm from each multiplexing wavelengths. It is seen that all PSFs remain almost unchanged, although the PSFs are no longer the shape of a traditional circular airy spot. Figures 3(1b)-3(3b) show the blurred intermediate images, which are obtained by a convolution calculation between a target object (i.e., a patterned test plate) and the PSFs shown in Figs. 3(1a)-3(3a). The intermediate images are almost identical at the aforementioned wavelengths due to the consistency of the PSFs of the D-WMC/WFC PS. Figures 3(1c)-3(3c) show the restored images of the D-WMC/WFC PS. The image restoration is performed by a deconvolution operation (Wiener filtering) between the blurred intermediate images and the PSF (i.e., filtering function). It is seen that all the intermediate blurred images can be well restored by Wiener filtering.
Fig. 3. Simulated focusing and imaging behaviors of the D-WMC/WFC PS at different wavelengths. (1a, 2a, 3a): PSFs; (1b, 2b, 3b): Intermediate blurred images; (1c, 2c, 3c): Restored images.
To evaluate the imaging quality, MTF curves of the D-WMC/WFC PS by taking a Fourier transform of the PSFs. Figure 4(a) shows the MTFs achieved from the D-WMC/WFC PS imaging system, solid lines are the MTF curves at the designed wavelengths, dashed lines are the MTF curves at the wavelengths which deviate 10 nm from the designed ones, for clarity, only three representative wavelengths 474.8, 572.4, 709.5 nm are plotted. It is easy to find the consistency of the MTFs at different wavelengths in the D-WMC/WFC PS imaging system, although dashed lines have slight deviations at high frequencies. There is no zero points appeared in the MTFs from high to low frequency, so the blur images at different wavelengths can be restored by using an appropriately designed digital filter. Figure 4(b) shows the MTF curves of the comparison between WMC PS (solid line) and the D-WMC/WFC PS (dashed line) at different wavelengths deviated from the designed ones. As expected, the MTF curves of the WMC PS at these deviated wavelengths drop very fast, and have many zero points, which means these images will be very blurring and unrecoverable.
Fig. 4. (a) Simulated MTFs of the designed D-WMC/WFC PS at the designed wavelengths (solid lines) and the deviated wavelengths (dashed lines). (b) Comparison of simulated MTFs of the WMC PS (solid line) and the D-WMC/WFC PS (dashed line) at wavelengths deviated from the designed ones.
It is also noted from Fig. 4 that MTFs of D-WMC/WFC PS with a fixed deviation (i.e., 10 nm) from the designed wavelength at different wavelength band show wavelength dependent behavior (dashed lines in Fig. 4), i.e., the longer the design wavelength, the higher the cutoff frequency of the MTF. To keep the consistency of the MTF at different wavelength bands, the designed seven multiplexing wavelengths covering the full visible wavelength range from 470 nm to 720 nm are thus selected unequally at 484.8, 515.3, 547.8, 582.4, 619.1, 658.1 and 699.5 nm, respectively, as shown in Figs. 1 and 2. The detailed design process is as follows: the first design wavelength is selected at 484.8 nm because the bandwidth can be extended to ±14.8 nm (i.e., working wavelength range from 470 nm to 499.6 nm centered at 484.8 nm can be obtained) with a cubic phase coding parameter α=30π. With the fixed parameter α=30π, other design wavelengths and the extension of bandwidth at those design wavelengths can be calculated. The defocus aberration coefficient W20 =$\frac{{\mathrm{\Delta }\mathrm{\lambda }{\ast }{D^2}}}{{8{\ast }\mathrm{\lambda }{\ast f}}}$ [16] of a diffractive optical system is a constant at different wavelengths, so we have $\frac{{\mathrm{\Delta }{\mathrm{\lambda }_1}}}{{{\mathrm{\lambda }_1}}} = \frac{{\mathrm{\Delta }{\mathrm{\lambda }_2}}}{{{\mathrm{\lambda }_2}}} = \ldots = \frac{{\mathrm{\Delta }{\mathrm{\lambda }_N}}}{{{\mathrm{\lambda }_N}}}$. Therefore, the next design wavelength and its working bandwidth can be calculated by $\frac{{14.8}}{{484.8}} = \frac{{\mathrm{\Delta }{\mathrm{\lambda }_2}}}{{484.8\, + \,14.8 + \mathrm{\Delta }{\mathrm{\lambda }_2}}}$, which is λ2=515.3 nm and Δλ2=15.7 nm in order to form a continuous working wavelength connecting to that of centered at 484.8 nm of the first design wavelength. The third and also the other design wavelengths can be calculated with this recursion relationship. Figure 5 shows the MTFs of D-WMC/WFC PS at unequally selected multiplexing wavelengths with unequal bandwidths (dashed lines) and its comparison with those at the selected wavelengths. It is seen that the wavelength and bandwidth dependent MTFs of D-WMC/WFC PS can be manipulated to the same level by selecting unequally distributed multiplexing wavelengths and unequal bandwidths.
Fig. 5. MTFs of D-WMC/WFC PS at unequally distributed multiplexing wavelengths with unequal bandwidths.
4. Experiments
According to above design, the corresponding WMC and D-WMC/WFC PS are fabricated on metallic chromium thin film on glass substrate using UV lithography. The designed parameters are the same as those shown in Fig. 1. Figure 6 shows the experimental setup of imaging system of WMC and D-WMC/WFC PS. The inset of Fig. 6(a) shows the fabricated D-WMC/WFC PS. A super-continuum laser (Fianium SC450) and a monochromator is employed as the wavelength adjustable illumination source. The spectral linewidth of the monochromator is less than 2.5 nm within the wavelength range from 400 nm to 1000 nm. A rotating diffuser is placed in the optical path to remove the laser speckle, and a convex lens is used to focus the monochromatic light onto the resolution target (USF1951). To demonstrate the achromatic imaging with polychromatic light, the super-continuum laser system (shown in dashed block) will be replaced by a tungsten halogen lamp (Thorlab SLS301), a 470 nm high performance longpass filter (Edmund 84743) and a 720 nm high performance shortpass filter (Edmund 86103), which is shown in Fig. 6(b) and (c). A light collimator with a focal length 550 mm and a diameter 55 mm is used for imaging. The fabricated WMC or D-WMC/WFC PS is placed next to the collimator, and a CCD with pixel size 4.54 μm (AVT Prosilica GX2750C) is placed at a distance of 500mm from the fabricated photon sieve.
Fig. 6. Experimental setup of imaging system of WMC and D-WMC/WFC PS at different wavelengths. Insets: Fabricated photon sieve and magnified pinhole rings. (a) Setup with a wavelength adjustable super-continuum laser source; (b) polychromatic light source of a tungsten halogen lamp; (c) Spectral transmissions of a longpass and a shortpass filter.
Figure 7 show the imaging of WMC PS at the seven discrete designed wavelengths, i.e., 484.8, 515.3, 547.8, 582.4, 619.1, 658.1 and 699.5 nm, and wavelengths deviated 10 nm from each designed wavelengths. The discrete monochromatic wavelength is obtained from the monochromator with a linewidth of less than 2.5 nm. It is seen that at the seven designed wavelengths, clear images that is close to diffraction-limited performance can be obtained (middle row in Fig. 7), which agrees well with the designed results in terms of both the designed wavelength and the image resolution (experimental resolution of 50.8 lp/mm corresponding to 16 μm focal spot versus the diffraction limited spot size of 14.52 μm at 595 nm). When working wavelength is deviated from each designed wavelengths by 10 nm, the images become very blurring (upper and lower row in Fig. 7), which means WMC PS can work only at designed wavelengths with narrow bandwidth (bandwidth can be evaluated by Δλ≈±2λ2f / D2.
Figure 8 shows the experimental PSFs and monochromatic imaging behaviors of D-WMC/WFC PS at different wavelengths. In order to compare with the WMC PS, both images at the designed wavelengths and the wavelengths deviating 10 nm from the designed ones are taken. Because of the characteristic of the cubic phase coding, the bandwidth at each designed wavelength is extended. It is, therefore, seen that the PSFs of D-WMC/WFC PS remain almost unchanged not only at the seven designed wavelengths but also at wavelengths nearby these seven designed wavelengths, as shown in Figs. 8(1a)-8(3a). The corresponding measured resolution target images also have the consistency as shown in Figs. 8(1b)-8(3b), which is very different from the images of WMC PS in Fig. 7. However, the images obtained directly by CCD are blurring, it is necessary to carry out restoration processing. The restoration is performed by a deconvolution operation between the blurred images (Figs. 8(1b)-8(3b)) and the PSF (Figs. 8(1a)-8(3a)). Figure 8(1c)-8(3c) show the restored images, from which the optical resolution is about 50.8 lp/mm (corresponding to 16 μm focal spot) at typical wavelength of 595 nm., which is close to the diffraction limited spot size of 14.52 μm.
Fig. 8. Experimental focusing and imaging behaviors of D-WMC/WFC PS at different wavelengths. (1a, 2a, 3a): PSFs; (1b, 2b, 3b): Intermediate blurred images; (1c, 2c, 3c): Restored images.
The experimental MTF curves (Fourier Transform of the measured PSFs) of the designed D-WMC/WFC PS at the different wavelengths (both designed and deviated wavelengths) are shown in Fig. 9. The consistency of all the curves and no zero points in these curves validates the designed D-WMC/WFC PS.
Imaging experiments of the fabricated D-WMC/WFC PS with polychromatic incident light ranging from 470 nm to 720 nm is also performed. Figure 10 gives the broadband images and the MTF of the measured images. The measured blurred image and the PSF under illumination of 250 nm bandwidth incident light in the visible are shown in Figs. 10(a) and 10(b), respectively. The restored image is shown in Fig. 10(c), and the corresponding MTF curves of the images before and after image restoration is shown in Fig. 10(d), which can be compared with that of a fabricated CPS at designed wavelength 595 nm.
Fig. 10. (a) Intermediate blurred image of D-WMC/WFC PS under illumination of 250 nm bandwidth. (b) PSF of D-WMC/WFC PS under 250 nm bandwidth. (c) Restored image under illumination of 250 nm bandwidth. (d) Measured MTFs of the D-WMC/WFC PS before (solid line) and after (dashed line) restoration and its comparison with that of a CPS.
Finally, arbitrary objects and color imaging are also conducted with the fabricated D-WMC/WFC PS and its comparison with that of a CPS under broadband illumination, shown in Fig. 11. Column (a) is the original object. For comparison, column (b) is the image of CPS with designed wavelength at 595 nm under a broadband illumination from 470 to 720 nm. Column (c) is the intermediate blurred image of the fabricated D-WMC/WFC PS, and column (d) is the restored color images. All the images in column (d) show the excellent capability of the proposed D-WMC/WFC PS for broadband achromatic imaging.
5. Conclusions
In summary, we have proposed and experimentally demonstrated an achromatic imaging in the full-visible wavelength range with a single dual-pinhole-coded diffractive PS. The distribution of the position and size of each pinhole is patterned with dual wavelength-multiplexing coding and wavefront coding, in which WMC makes multiple wavelengths that are optimally separated within the full visible range focus coherently on a common designed focal length while WFC expands the bandwidth at each of the selected wavelengths. Numerical simulations show that when seven wavelengths and a cubic wavefront coding parameter α = 30π are employed, a broadband achromatic imaging can be obtained within the full range of visible wavelength between 470 nm to 720 nm. Experimental fabrication and imaging of the proposed dual-pinhole-coded PS with a focal length of 500 mm and a diameter of 50 mm are performed and the experimental results agree excellently with the numerical results. The proposed method also works for the phase type of diffractive elements and other wavelength range, and the bandwidth can be manipulated with the number of designing wavelength and the wavefront coding parameter. In particular, the proposed method avoids complicated search algorithm for hundreds of structural parameters as those previously reported. The demonstrated work provides a novel and practical way for achieving achromatic imaging in different wavelength ranges with features of thin, light and planar.
Funding
National Natural Science Foundation of China (61775154); Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).
Disclosures
The authors declare no conflicts of interest.
Data availability
The data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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