Broadband photon sieves imaging with wavefront coding

Xiaonan Zhao; Feng Xu; Jingpei Hu; Chinhua Wang

doi:10.1364/OE.23.016812

1. Introduction

Diffractive optical elements (DOEs) have increasingly become a main type of optical components in modern optical systems, such as in high-resolution microscopy, spectroscopy, X-ray or EUV lithography that are difficult, or even impossible, with conventional glass-based refractive optics [1], because of their unique characteristics of compact size, light weight, and high degree of design flexibility. Photon sieves, which is evolved from the traditional Fresnel zone plate (FZP), was proposed by Kipp in 2001 [2] as a new class of diffractive focusing element. Unlike a FZP, the photon sieves has no connected regions, thus permitting the fabrication of a single surface without any supporting struts required. Cao et al [3,4] developed both paraxial and nonparaxial theoretical models for calculating and designing the photon sieves including the exact selection conditions of the pinhole size and the radial position as well as the validity range of the theory. Following those works, several theoretical and experimental studies on different types of the photon sieves were carried out. Andersen [5] demonstrated a large optical photon sieves in the visible regime with a diameter of 10cm and numerical aperture (NA) of 0.05 using electron-beam lithography. Menon et al [6] fabricated a high NA photon sieves with a small diameter of 72 $μ m$ using scanning-electron-beam lithography and applied the photon sieves in a scanning-optical-beam-lithography system. To increase the aperture size and hence the optical resolution of a photon sieves, and also to ease the fabrication difficulty, Chen et al [7] proposed an ultra-large multi-region photon sieves and experimentally demonstrated the fabrication feasibility using UV lithography. However, as a DOE, photon sieves suffers from large chromatic aberration, and therefore, all those mentioned photon sieves can work only at a designed single wavelength, which significantly limits the applications of the photon sieves, especially in optical imaging. Zhou et al [8] designed and fabricated a discrete three-wavelength photon sieves imaging system based on a random-area-divided approach, where the whole aperture of the multi-wavelength imaging photon sieves is divided into multiple discrete spaces corresponding to the number of the selected working wavelengths. Essentially, this device is a structural combination of three single-wavelength photon sieves. In 2006, Gimenez et al [9] showed that a fractal photon sieves (FPS) exhibits an extended depth of field and a reduced chromatic aberration. The FPS produces a sequence of subsidiary foci around each major focus. In all cases these subsidiary foci of the FPS provide an extended depth of focus for each wavelength that partially overlaps with the other ones, creating an overall extended depth of focus that is less sensitive to chromatic aberration [10]. However, the price paid to gain depth of field and reduce chromatic aberration is a reduced resolution, low energy efficiency, and strong background stray light. In 2007, Andersen et al [11] 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 that incorporates a second DOE designed to compensate for the dispersion of the photon sieves primary and two mirrors with an even larger aperture than the photon sieve primary for collimating and refocusing the highly divergent beam resulting from the DOE must be employed. In this paper, we proposed and experimentally demonstrated a novel method of extending the applications of photon sieves to broadband optical imaging in a relatively simple way using wavefront coding which was first proposed by Dowski and Cathey in 1995 [12] to extend the depth of focus (DOF) of an optical system and to correct axial chromatic aberrations [13]. The major function of the wavefront coding is to modify the point spread function (PSF) of the system in such a way that it becomes invariant over a range of distances around the image plane. The final image with diffraction limit quality is obtained by using digital filtering from the coded images. Both theoretical simulation and experimental results show that broadband photon sieves imaging can be obtained with wavefront coding of a simple cubic phase mask without complicated optical system to compensate the large chromatic aberration of the photon sieves, breaking the limitations of conventional single-wavelength photon sieves imaging. Using an UV-lithography fabricated photon sieves of a focal length 500mm and a diameter 50mm at designed wavelength 632.8nm and a diamond-turning fabricated cubic phase mask of a phase parameter $α = 20 π$ , an extension of imaging bandwidth of the proposed photon sieves system is at least 88 times that of the conventional one with almost the same optical resolution and much increased energy efficiency. It should be noted that the demonstrated method works generally for all types of DOEs (e.g., conventional FZP) because of the same physical mechanism. The proposed method opens a new direction of extending the applications of DOEs to work in a broadband wavelength range in a generalized way in contrast to conventionally either work at a single wavelength only or complicatedly designed system.

2. Working principle of a broadband photon sieves imaging

Figure 1 shows the schematic of a wavefront coded broadband photon sieves imaging system, in which a cubic phase mask is placed in front of a photon sieves. A plane wave illumination is considered. The incident plane wave propagates through the cubic phase mask, the photon sieves and is then focused onto the image plane. The total diffracted field distribution of a photon sieves at the focal plane is a summation of those individual diffracted fields from different pinholes [2–4]. According to the individual far-field model [4], one can obtain the diffractive field at the image plane, Fig. 1:

U (X, Y) = \sum_{m = 1}^{M} U_{m} (X, Y) = \sum_{m = 1}^{M} \frac{k A_{m} a_{m}^{2} q}{H^{2}} \exp [j k (L_{m} + H)] J i n c (\frac{k a_{m}}{H} ρ)

where

U_{m} (X, Y)

is the individual diffracted field from the mth pinhole, M is the number of pinholes,

k = 2 π / λ

is the wave number,

λ

is the wavelength, A_m is real amplitude and L_m is the eikonal of the illuminative beam at the photon sieves, j is the imaginary unit, α_m is the radius of the pinhole, q is the distance between the photon sieves and the image plane,

H = {[q^{2} + {(X - x_{m})}^{2} + {(Y - y_{m})}^{2}]}^{\frac{1}{2}}

,

ρ = {[{(X - x_{m} - g_{m} H)}^{2} + {(Y - y_{m} - h_{m} H)}^{2}]}^{\frac{1}{2}}

, x_m and y_m is central location of the mth pinhole,

g_{m} = (\frac{\partial L}{\partial x}) |_{x_{m}, y_{m}}

,

h_{m} = (\frac{\partial L}{\partial y}) |_{x_{m}, y_{m}}

, and

J i n c (\cdot) = \frac{J_{1} (\cdot)}{(\cdot)}

is the Jinc function and

J_{n} (\cdot)

is the nth-order Bessel function of the first kind.

Fig. 1 Schematic of a wavefront coded broadband photon sieves imaging system.

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The transmission function $t (x, y)$ of a cubic phase mask can be expressed below:

\begin{array}{l} t (x, y) = \exp (j α \frac{x^{3} + y^{3}}{R^{3}}) \\ = \exp (j \frac{2 π}{λ} ξ (n - 1) \frac{x^{3} + y^{3}}{R^{3}}), x^{2} + y^{2} \leq R^{2} \end{array}

where

α

is a phase parameter of the cubic phase mask, R is the size of radius, n is the refractive index of the phase mask. In our experiment, the phase mask is made of an optical plastic polymethyl methacrylate (PMMA) with n = 1.49@632.8nm.

According to Eq. (2), one can get the complex amplitude distribution after the cubic phase mask if the complex amplitude of the incident plane wave is assumed to be $E_{0} (x, y)$ :

\begin{array}{l} E (x, y) = E_{0} (x, y) t (x, y) \\ = E_{0} (x, y) \exp (j \frac{2 π}{λ} ξ (n - 1) \frac{x^{3} + y^{3}}{R^{3}}) \end{array}

It is noted that $E (x, y)$ is the complex amplitude distribution of the illumination beam at the photon sieves plane. We can then obtain the eikonal at the photon sieves plane:

L_{m} (x, y) = ξ (n - 1) \frac{x^{3} + y^{3}}{R^{3}}

g_{m} = (\frac{\partial L}{\partial x}) |_{x_{m}, y_{m}} = \frac{3 ξ (n - 1)}{R^{3}} x_{m}^{3}

h_{m} = (\frac{\partial L}{\partial y}) |_{x_{m}, y_{m}} = \frac{3 ξ (n - 1)}{R^{3}} y_{m}^{3}

With the help of Eqs. (4)–(6), the total wavefront coded diffracting field at the image plane can be calculated from Eq. (1), which forms the theoretical basis for optical design and behavior analysis of the broadband imaging system. The PSF distribution of the system in the image plane can be calculated below:

I = {| U (X, Y) |}^{2}

3. Simulations

To give a more detailed illustration of the theory, a concrete design example with an aperture of 50mm, focal length of 500mm and working wavelength of 632.8nm is given. The total ring number of the photon sieves in conventional type is 987 and the minimum pinhole size is 6.3282 $μ m$ [Fig. 2(a) ]. Figure 2(b) shows the schematic of a cubic phase mask with a phase parameter $α = 20 π$ and an aperture size of 50mm.

Fig. 2 Schematics of (a): a photon sieves and (b): a cubic phase mask with a phase parameter $α = 20 π$ .

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Figures 3(a) and 3(b) show the PSF of a photon sieves imaging system with and without wavefront coding at different incident wavelengths ranging from $λ$ = 625.8 to 639.8nm. Figures 3(a) and 3(b) are calculated based on Eq. (7) using MATLAB computing program with $α$ = 0 or $α = 20 π$ for systems with and without wavefront coding, respectively. No surprising, the PSF of the conventional photon sieves is diffused with the increased wavelength deviation from the designed wavelength of 632.8nm [Fig. 3(a)]. In contrast, the PSF distribution of the wavefront coded photon sieves system remains almost unchanged within a wide range of the wavelengths from $λ$ = 627.8 to 637.8nm with slight deviation when $λ$ <627.8nm or >637.8nm. This wavelength range represents the boundary of the consistency of the PSFs at different wavelengths, which can thus be considered as the operation bandwidth. The phase parameter used in the cubic phase mask is $α = 20 π$ .

Fig. 3 The PSF of a photon sieves imaging system with and without wavefront coding at different incident wavelengths from $λ$ = 625.8 to 639.8nm. (a) conventional photon sieves imaging (without wavefront coding); (b) wavefront coded photon sieves imaging with a cubic phase parameter $α = 20 π$ .

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Figure 4 shows respectively the MTF curves (a Fourier transform of the PSF) of the conventional photon sieves imaging system and a wavefront coded broadband photon sieves system at different incident wavelengths. The MTF curves were obtained by taking a Fourier transform of the PSFs shown in Fig. 3. Same as those observed in Fig. 3, MTFs drop significantly and zeros appear in the MTFs of the conventional photon sieves when the incident wavelengths deviate from the designed one, resulting in the loss of spatial frequencies in the image. In contrast, MTFs are nearly identical in the case of wavefront coded broadband photon sieves system within the wavelength range from $λ$ = 627.8nm to 637.8nm with slight deviation at high frequencies when $λ$ <627.8nm or >637.8nm. Because of the consistency of the MTFs at different wavelengths in the wavefront coded system, and there is no zero points appeared in the MTFs from high to low frequency, the wavefront coded blur image at different wavelengths can be restored by using an appropriately designed digital filter. Hence the cubic phase mask can greatly reduce the sensitivity of photon sieves to incident wavelength, resulting in extension of the working bandwidth of a photon sieves imaging system.

Fig. 4 MTF of a conventional photon sieves imaging system and a wavefront coded broadband photon sieves imaging system with a cubic phase mask of phase parameter $α = 20 π$ at different incident wavelengths from $λ$ = 625.8 to 639.8nm.

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Figure 5 shows the simulated imaging behavior of a conventional photon sieves at different wavelengths ( $λ = 625.8 ~ 639.8 n m$ ). The simulated image of the photon sieves is obtained by a convolution calculation between a target object (i.e., a patterned test plate) and the PSF that is calculated using Eq. (7) and shown in Fig. 3. All the simulation is performed using MATLAB computing program. The images of the conventional photon sieves become increasingly fuzzy when the incident wavelength deviates from the designed wavelength increasingly. The bandwidth of the conventional photon sieves can be quantified by $Δ λ \approx \frac{2 λ^{2} f}{D^{2}} = 0.16 n m$ in our system [11].

Fig. 5 The simulated imaging behavior of a conventional photon sieves at different incident wavelengths from $λ$ = 625.8 to 639.8nm.

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Figure 6 shows the intermediate wavefront coded blur images of the broadband photon sieves system. As expected, all the images (fuzzy images) are almost identical at different wavelengths, which suggests a clear image restoration using a unified digital image filter can be obtained.

Fig. 6 The intermediate wavefront coded blur images of the broadband photon sieves image system at different wavelengths from $λ$ = 625.8 to 639.8nm.

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Figure 7 shows the restored images of the broadband photon sieves system. The image restoration is implemented with Wiener filtering technique [14] in which the blurred intermediate images shown in Fig. 6 are deconvoluted with the filtering function of an averaged PSF from $λ$ = 625.8nm to 639.8nm shown in Fig. 3(b). It is seen that all the blurred images can be well restored with a fixed filtering function for all the wavelengths and all the restored images have the almost same resolution as that of the conventional photon sieves in the designed wavelength (i.e., 632.8nm in our case), with very small deviation at $λ$ = 625.8nm or 639.8nm due to the small inconsistency of the MTF at large deviation from the designed wavelength. In our case with a wavefront coded photon sieves system of D = 50mm, f = 500mm, and $α = 20 π$ , the bandwidth can be as large as 14nm, which implies that the extension of the bandwidth of the wavefront coded broadband photon sieves imaging system is about 88 times that of the conventional one.

Fig. 7 The restored images of the wavefront coded broadband photon sieves system at different wavelengths from $λ$ = 625.8 to 639.8nm.

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4. Experimental imaging demonstration

The wavefront coded broadband photon sieves imaging is demonstrated and compared with that of a conventional photon sieve of the same numerical aperture. A photon sieves with a focal length of 500mm and a diameter of 50mm at working wavelength 632.8nm is fabricated using UV lithography and a cubic phase mask of a phase parameter $α = 20 π$ is fabricated using diamond turning technique to validate the proposed method.

We first evaluate the performance of the conventional photon sieves at the designed single wavelength 632.8nm. The experimental setup is shown in Fig. 8 . The incident laser beam (632.8nm) is firstly focused by an objective and then passes through a pinhole and a rotating diffuser (to remove speckles). A collimator with a focal length 550mm and a diameter 55mm and a CCD with pixel size 4.54 $μ m$ (AVT Prosilica GX2750C) are used. The inset of Fig. 8 shows the magnified image of the central region of the fabricated photo sieves.

Fig. 8 Experimental setup of a photon sieves imaging system at designed wavelength 632.8nm.

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Figure 9 shows the PSF measurement and the imaging result of the conventional photon sieves with a diameter of 50mm at the designed single wavelength. The target used in the imaging experiment is USAF 1951. The target image produced by the photon sieves at 632.8nm is shown in Fig. 9(b). Figure 9(c) shows the magnified target image from the central region of Fig. 9(b). Detailed examinations show that the optical resolution of this photon sieves is about 50.8lp/mm (corresponding to 16 $μ m$ focal spot), which is very close to the diffraction limited spot size of 15.44 $μ m$ .

Fig. 9 (a) PSF of the photon sieves at 632.8nm; (b) Image produced by the conventional photon sieves; (c) Magnified image from the central region of (b). Other parameters are: focal length = 500mm, diameter = 50mm, and working wavelength 632.8nm.

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The wavefront coded broadband photon sieves (with cubic phase mask) imaging system is shown in Fig. 10(a) . Compared with the single wavelength imaging in Fig. 8, a broadband source is generated with a tungsten lamp filtered by a bandpass filter of central wavelength 632.8nm and a full width at half magnitude (FWHM) of 10nm (THORLABS, FL632.8-10). The transmission curve of the bandpass filter is shown in Fig. 10(b). A cubic phase mask is placed in the front of the photon sieves. The insets of Fig. 10 show the fabricated cubic phase mask and the bandpass filter.

Fig. 10 (a) Experimental setup of a wavefront coded broadband photon sieves imaging system; and (b) Transmission curve of a bandpass filter (central wavelength 632.8nm, FWHM 10nm).

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To show the effect of the broadband incident light on the image of a photon sieves system and compare with that is obtained at the designed single wavelength shown in Fig. 9, Fig. 11 gives the PSF measurement result and the imaging result of the photon sieves system without the cubic phase mask in the imaging optical path using the broadband light source. No surprising, as a diffractive optical element, the photon sieves suffers from large chromatic aberration. The broadband incident light centered at 632.8nm with a FWHM bandwidth of 10nm generates a blurred PSF spot [Fig. 11(a)] and also a blurred image [Figs. 11(b) and 11(c)] on the focal plane, which is in a significant contrast with that shown in Fig. 9.

Fig. 11 PSF and image measurement results of the photon sieves imaging system without wavefront coding under a broadband incident illumination. (a) PSF measurement; (b) imaging result; and (c) magnified image from the central region of (b). Incident illumination is centered at 632.8nm with a FWHM bandwidth of 10nm.

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Figure 12 shows the PSF measurement and the imaging result of the wavefront coded photon sieves imaging system under the broadband incident illumination with central wavelength at 632.8nm and a FWHM of 10nm. The phase parameter of the cubic phase mask employed in the system is $α = 20 π$ . Figure 12(a) shows the experimentally measured PSF of the wavefront coded system, which takes “L-shape”, same as those predicted in the simulation shown in Fig. 3(b). Figure 12(b) shows the intermediate blurred image (i.e., before digital image filtering) produced by the wavefront coded broadband photon sieves system. The restoration of the wavefront coded image is shown in Fig. 12(c). The restoration is performed using Wiener filtering technique in which the blurred intermediate image shown in Fig. 12(b) is deconvoluted with the filtering function of PSF shown in Fig. 12(a). Figure 12(d) shows the magnified restored image from the central region of Fig. 12(c). It is seen that the intermediate blurred image can be well restored and the restored image of broadband wavelength illumination shows the almost same resolution as that of the conventional photon sieves imaging at the designed single wavelength 632.8nm, shown in Fig. 9. It is estimated, from Fig. 12(c), that the optical resolution of the broadband photon sieves system is about 50.8lp/mm under broadband illumination of a FWHM bandwidth of 10nm. Detailed comparison of the image quality between the conventional photon sieves working at “single” wavelength and wavefront coded photon sieves system working at a “broadband” wavelength range can be performed with the MTFs from the experimental results shown in Figs. 9, 11, and 12 [15,16]. The MTFs are calculated from the Fourier transform of the experimental images of Figs. 9, 11, and 12. Figure 13 shows respectively the MTF curves of the conventional photon sieves imaging at the designed single wavelength 632.8nm (black line) and a broad bandwidth of 10nm (FWHM) centered at 632.8nm (red line), as well as the MTF curve of the image with wavefront coding (blue line) under broadband illumination. As expected, MTF drops rapidly and zeros appear in the MTF (red line) when the conventional photon sieves is illuminated with a broadband source, which is consistent with the blurred image seen visually in Fig. 11. In contrast, the MTF (blue line) of the wavefront coded image under broadband illumination is almost the same as that of the conventional photon sieves at single wavelength illumination (black line), which provides a quantitative evidence for the observed experimental images. It should be noted that the working bandwidth could be further extended when the wavefront coding parameter $α$ is further optimized. It is known that the larger the phase parameter $α$ , the more insensitive to defocus and wavelengths the PSF/MTF, and consequently, the lower the MTF, which results in the noise enhancement after deconvolution [17]. The effect of the phase parameter $α$ on the depth of field and working bandwidth and the image signal-to-noise ratio (SNR) can be optimized with the help of the calculation power of modern image processing and also the criteria of image SNR, to achieve a trade-off between maximal bandwidth extension and acceptable image SNR.

Fig. 12 (a) Measured PSF of the wavefront coded photon sieves system; (b) Intermediate blurred image produced by the broadband photon sieves system; (c) Restored image of the broadband photon sieves system; (d) Magnified image from the central region of (c). The incident FWHM bandwidth is 10nm centered at 632.8nm.

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Fig. 13 MTF curves of the conventional photon sieves imaging at designed single wavelength (black line) and broadband source (red line), as well as the wavefront coding imaging (blue line) with broadband source.

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5. Conclusions

We have proposed and experimentally demonstrated a novel method for extending the conventional single-wavelength working mode only to broadband wavelength working mode. Both theoretical simulation and experimental results show that broadband photon sieves imaging can be obtained with wavefront coding of a simple cubic phase mask without complicated optical system to compensate the large chromatic aberration of the photon sieves, which opens a new way for much more application possibilities of using photon sieves technology, breaking the limitations of single-wavelength operation only. This proposed method is simple and generic without the need of complicated design for each specific system. Using an UV-lithography fabricated photon sieves of focal length 500mm and diameter 50mm at wavelength 632.8nm and a diamond-turning fabricated cubic phase mask of a phase parameter $α = 20 π$ , the working bandwidth of the wavefront coded system reaches 14nm compared with 0.16nm in conventional system, which represents that an extension of imaging bandwidth of the proposed system is at least 88 times that of the conventional one with almost the same optical resolution and much increased energy efficiency. It should be noted that the working bandwidth could be further extended when the wavefront coding parameter is further optimized by taking into account the trade-off between the bandwidth and the acceptable SNR. Although the cubic phase mask is employed in this paper to demonstrate the working principle, other types of phase masks such as logarithmic masks, exponential masks, high-order polynomial masks, and quartic masks, etc., can also be used with appropriate design. It is thus concluded that the proposed method opens a new direction of extending the applications of DOEs to work in a broadband wavelength range in a generalized way in contrast to conventionally either work at a single wavelength only or complicatedly designed system.

Acknowledgments

The work was supported by the National Natural Science Foundation of China (No. 61378057), the National High Technology Research and Development Program of China (No. 2013AA031901), and the project of the Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institutions.

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Broadband photon sieves imaging with wavefront coding

Abstract

1. Introduction

2. Working principle of a broadband photon sieves imaging

3. Simulations

4. Experimental imaging demonstration

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (13)

Equations (7)

Optics Express