Super resolved optical system for objects with finite sizes using circular gratings

Asaf Ilovitsh; Vicente Mico; Zeev Zalevsky

doi:10.1364/OE.23.023667

1. Introduction

The resolution of an imaging system with perfect lenses and finite aperture is limited by diffraction. The diffraction limit can be defined as the minimum distance between two nearby points which still allow separation between them. It was pronounced by Abbe to be proportional to the optical wavelength and to the F number of the imaging system [1], and was later mathematically supported by Lord Rayleigh [2]. This limitation can be understood as a cutoff frequency truncation in the object’s spatial spectrum imposed by the finite aperture of the imaging system.

Super resolution (SR) techniques aim to overcome this limitation by generating synthetic aperture which expands the cutoff frequency and thus increase the resolution. The main concept in SR approaches is that high resolution (HR) spatial information can be reconstructed if some a priori information on the inspected object exists [3–9]. By stating that the number of degrees of freedom of an imaging system remains constant, Lukosz theorized that any parameter in the system can be extended above the classical limit if any other degree is proportionally reduced [10,11]. Using the a priori knowledge that the imaged object is independent to a given degree of freedom, this degree may be sacrificed in order to achieve more information in the spatial domain. The sacrificed axes, which may be used for multiplexing the additional high spatial resolution, can be for instance, time [11–14], wavelength [15,16], field of view [10,17], polarization [18–20], dynamic range [21,22], and spatial multiplexing [10,23,24]. In addition, it is possible to use SR techniques in order to achieve synthetic aperture in digital holography [25].

Time multiplexing is the most common SR technique [11–14,26–32], and methods like structured illumination microscopy (SIM) [33–35] are now commercially available [36]. However, as their name suggest, these methods require time for the SR process. Many scenarios require real time imaging, and thus time multiplexing is not suitable for them. Field of view (FOV) multiplexing offers a real time all optical SR system, without any moving parts. Instead of using the time domain, the resolution improvement is achieved by exploiting unused parts of the FOV. In FOV SR, diffractive gratings are used in order to optically encode (and later on decode) the HR spatial data that is diffraction limited by the aperture. The HR information is also encoded into other areas in the FOV, thus there is a trade-off between the possible resolution improvement and the size of the inspected object.

The original FOV SR setup evolved along the years. Some modifications consider the use of static gratings [37–39], or were implemented in digital holography [40–45]. In this paper we propose the use of circular gratings in order to improve the resolution of an optical system which has a circular aperture. The proposed method has several advantages over the previous ones, where 2D Dammann Cartesian gratings [46,47] were used for the SR process. Mathematically, the proposed circular gratings are 1D gratings as they are only radius dependent. As such, their design is simplified. In addition, since the gratings are rotating angle independent, the optical setup requires less calibration and alignment. Furthermore, the circular gratings generate synthetic circular duplications of the aperture. Thus, they seem to be the best choice for an optical system which has a circular aperture.

The paper is organized as follows: First, the theoretical background for spatially coherent followed by spatially incoherent, and white light illuminations are presented. Second, numerical simulations for spatially coherent and polychromatic spatially incoherent illuminations are presented. Third, the proposed method is validated experimentally.

2. Theoretical background

An illustration of the optical system for the proposed SR method is presented in Fig. 1. The setup is a 4F optical system, where F is the lenses’ focal length, D is the aperture diameter, and λ is the illumination wavelength. The 4F system has additional two identical circular gratings which are placed at the same distance z₀ before the object and image planes.

Fig. 1 An illustration of the 4F optical system for the proposed SR approach, where two additional gratings are placed before the object and the image planes.

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The analysis for the spatially coherent case is presented first. For mathematical simplicity, the analysis starts at the object plane, and then goes back to the first grating. The input field distribution of the object can be expressed as an inverse Fourier transform of the spectrum:

E_{o b j} (x, y, z = 0) = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} {\tilde{E}}_{0} (u, v) e^{2 π i (x u + y v)} d u d v

Free space propagation (FSP) a distance of −z₀ to the first grating may be expressed by the angular spectrum equation [48]:

E_{0} (x, y, - z_{0}^{-}) = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} {\tilde{E}}_{0} (u, v) e^{2 π i [x u + y v - \frac{z_{0}}{λ} \sqrt{1 - λ^{2} (u^{2} + v^{2})}]} d u d v

The circular grating can be expressed as a Fourier series with a basic frequency of υ₀:

g (r) = \sum_{m} A_{m} e^{2 π i r m υ_{0}} = \sum_{m} A_{m} e^{2 π i r [\cos^{2} (θ) + \sin^{2} (θ)] m υ_{0}} = \sum_{m} A_{m} e^{2 π i [x \cos (θ) + y \sin (θ)] m υ_{0}}

where x = r∙cos(θ), and y = r∙sin(θ). One should bear in mind that although the final expression in Eq. (3) appears to be x,y and θ dependent, it is actually only r dependent. The field after the first grating becomes:

E_{0} (x, y, - z_{0}^{+}) = \sum_{m} A_{m} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} {\tilde{E}}_{0} (u, v) e^{2 π i {x [u + \cos (θ) m υ_{0}] + y [v + \sin (θ) m υ_{0}] - \frac{z_{0}}{λ} \sqrt{1 - λ^{2} (u^{2} + v^{2})}}} d u d v

FSP a distance of + z₀ to the object plane, using angular spectrum yields:

E_{0} (x, y, z = 0) = \sum_{m} A_{m} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} {\tilde{E}}_{0} (u, v) e^{2 π i {x [u + \cos (θ) m υ_{0}] + y [v + \sin (θ) m υ_{0}] - \frac{z_{0}}{λ} ϕ_{1}}} d u d v

where

ϕ_{1}

is defined to be:

ϕ_{1} = \sqrt{1 - λ^{2} (u^{2} + v^{2})} - \sqrt{1 - λ^{2} {{[u + \cos (θ) m υ_{0}]}^{2} + {[v + \sin (θ) m υ_{0}]}^{2}}}

By defining µ = λF[u + cos(θ)mυ₀] and η = λF[v + sin(θ)mυ₀], FSP of Eq. (5) to the Fourier plane, and adding the circular aperture gives:

E_{0} (μ, η, z = 2 F^{+}) = \sum_{m} A_{m} {\tilde{E}}_{0} [\frac{μ}{λ f} - \cos (θ) m υ_{0}, \frac{η}{λ f} - \sin (θ) m υ_{0}] e^{- 2 π i (\frac{z_{0}}{λ} ϕ_{2})} c i r c (\frac{\sqrt{μ^{2} + η^{2}}}{Δ ρ / 2})

where 'circ' describes a round aperture with a diameter of Δρ in the Fourier plane, and

ϕ_{2}

(which is

ϕ_{1}

redefined using µ and η) is defined by:

\begin{array}{l} ϕ_{2} = \sqrt{1 - λ^{2} {{[\frac{μ}{λ F} - \cos (θ) m υ_{0}]}^{2} + {[\frac{η}{λ F} - \sin (θ) m υ_{0}]}^{2}}} - \sqrt{1 - λ^{2} [{(\frac{μ}{λ F})}^{2} + {(\frac{η}{λ F})}^{2}]} \\ \overset{}{} \approx \frac{λ [μ \cos (θ) m υ_{0} + η \sin (θ) m υ_{0}]}{F} - \frac{λ^{2} m^{2} υ_{0}^{2}}{2} \end{array}

where in order to simplify

ϕ_{2}

, the Taylor series approximation √(1−x²)≈1−x²/2 was taken. FSP to the image plane, without yet taking into account the effect of the second grating:

\begin{array}{l} E_{0} (x, y, z = 4 F) = \sum_{m} A_{m} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} {\tilde{E}}_{0} [\frac{μ}{λ F} - \cos (θ) m υ_{0}, \frac{η}{λ F} - \sin (θ) m υ_{0}] \\ \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \cdot e^{2 π i (\frac{x μ}{λ F} + \frac{y η}{λ F} - \frac{z_{0}}{λ} ϕ_{2})} c i r c (\frac{\sqrt{μ^{2} + η^{2}}}{Δ ρ / 2}) d μ d η \end{array}

FSP a distance of −z₀ to the second grating plane:

\begin{array}{l} E_{0} (x, y, z = {[4 F - z 0]}^{-}) = \sum_{m} A_{m} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} {\tilde{E}}_{0} [\frac{μ}{λ F} - \cos (θ) m υ_{0}, \frac{η}{λ F} - \sin (θ) m υ_{0}] \\ \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \cdot e^{2 π i (\frac{x μ}{λ F} + \frac{y η}{λ F} - \frac{z_{0}}{λ} ϕ_{2} - \frac{z_{0}}{λ} ϕ_{3})} c i r c (\frac{\sqrt{μ^{2} + η^{2}}}{Δ ρ / 2}) d μ d η \end{array}

where

ϕ_{3}

is defined to be:

ϕ_{3} = \sqrt{1 - λ^{2} [{(\frac{μ}{λ F})}^{2} + {(\frac{η}{λ F})}^{2}]} \approx 1 - \frac{μ^{2} + η^{2}}{2 F^{2}}

Multiplying the field by the second grating, which is equal to the first grating, gives:

\begin{array}{l} E_{0} (x, y, z = {[4 F - z_{0}]}^{+}) = \sum_{m} \sum_{n} A_{m} A_{n} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} {\tilde{E}}_{0} [\frac{μ}{λ F} - \cos (θ) m υ_{0}, \frac{η}{λ F} - \sin (θ) m υ_{0}] \\ \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \cdot e^{2 π i {x [\frac{μ}{λ F} + \cos (θ) n υ_{0}] + y [\frac{η}{λ F} + \sin (θ) n υ_{0}] - \frac{z_{0}}{λ} ϕ_{2} - \frac{z_{0}}{λ} ϕ_{3}}} c i r c (\frac{\sqrt{μ^{2} + η^{2}}}{Δ ρ / 2}) d μ d η \end{array}

FSP a distance of + z₀ to the image plane:

\begin{array}{l} E_{0} (x, y, z = 4 F) = \sum_{m} \sum_{n} A_{m} A_{n} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} {\tilde{E}}_{0} [\frac{μ}{λ F} - \cos (θ) m υ_{0}, \frac{η}{λ F} - \sin (θ) m υ_{0}] \\ \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \cdot e^{2 π i {x [\frac{μ}{λ F} + \cos (θ) n υ_{0}] + y [\frac{η}{λ F} + \sin (θ) n υ_{0}] - \frac{z_{0}}{λ} ϕ_{2} - \frac{z_{1}}{λ} ϕ_{3} + \frac{z_{1}}{λ} ϕ_{4}}} c i r c (\frac{\sqrt{μ^{2} + η^{2}}}{Δ ρ / 2}) d μ d η \end{array}

where

ϕ_{4}

is defined to be:

\begin{array}{l} ϕ_{4} = \sqrt{1 - λ^{2} {{[\frac{μ}{λ F} + \cos (θ) n υ_{0}]}^{2} + {[\frac{η}{λ F} + \sin (θ) n υ_{0}]}^{2}}} \\ \overset{}{} \approx 1 - \frac{λ^{2}}{2} {{[\frac{μ}{λ F} + \cos (θ) n υ_{0}]}^{2} + {[\frac{η}{λ F} + \sin (θ) n υ_{0}]}^{2}} \end{array}

Changing the variables to $\hat{u}$ = µ/λF−cos(θ)mυ₀ and $\hat{v}$ = η/λF−sin(θ)mυ₀ results in:

\begin{array}{l} E_{0} (x, y, z = 4 F) = \sum_{m} \sum_{n} A_{m} A_{n} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} {\tilde{E}}_{0} (\hat{u}, \hat{v}) e^{2 π i {x [\hat{u} + \cos (θ) (m + n) υ_{0}] + y [\hat{v} + \sin (θ) (m + n) υ_{0}] - ϕ_{t o t}}} \\ \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \cdot c i r c {\frac{\sqrt{{[\hat{u} + \cos (θ) m υ_{0}]}^{2} + {[\hat{v} + \sin (θ) m υ_{0}]}^{2}}}{Δ ρ / 2 λ F}} d \hat{u} d \hat{v} \end{array}

where

ϕ_{tot}

is defined to be:

ϕ_{t o t} = \frac{z_{0}}{λ} ϕ_{2} + \frac{z_{1}}{λ} ϕ_{3} - \frac{z_{1}}{λ} ϕ_{4}

For m = −n, the total phase is $ϕ_{tot}$ = 0, and thus Eq. (15) becomes:

\begin{array}{l} E_{0} (x, y, z = 4 F) = \\ \sum_{n = - m} A_{m} A_{n} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} {\tilde{E}}_{0} (\hat{u}, \hat{v}) c i r c {\frac{\sqrt{{[\hat{u} + \cos (θ) m υ_{0}]}^{2} + {[\hat{v} + \sin (θ) m υ_{0}]}^{2}}}{Δ ρ / 2 λ F}} e^{2 π i (x \hat{u} + y \hat{v})} d \hat{u} d \hat{v} \end{array}

This expression is valid for all θ angles simultaneously. Therefore, by choosing the gratings basic frequency to be υ₀ = Δρ/λF a wider circular synthetic aperture, through which the spectral information of the object may pass, is achieved. For example, circular gratings with only three Fourier coefficients A_{0, ± 1} = 1 will generate:

\begin{array}{l} E_{0} (x, y, z = 4 F) = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} {\tilde{E}}_{0} (\hat{u}, \hat{v}) c i r c (\frac{\sqrt{{\hat{u}}^{2} + {\hat{v}}^{2}}}{Δ ρ / 2 λ F}) e^{2 π i (x \hat{u} + y \hat{v})} d \hat{u} d \hat{v} \\ \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} + \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} {\tilde{E}}_{0} (\hat{u}, \hat{v}) (c i r c {\frac{\sqrt{{\hat{u}}^{2} + {\hat{v}}^{2}}}{3 Δ ρ / 2 λ F}} - c i r c {\frac{\sqrt{{\hat{u}}^{2} + {\hat{v}}^{2}}}{Δ ρ / 2 λ F}}) e^{2 π i (x \hat{u} + y \hat{v})} d \hat{u} d \hat{v} \\ \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} \overset{}{} = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} {\tilde{E}}_{0} (\hat{u}, \hat{v}) c i r c (\frac{\sqrt{{\hat{u}}^{2} + {\hat{v}}^{2}}}{3 Δ ρ / 2 λ F}) e^{2 π i (x \hat{u} + y \hat{v})} d \hat{u} d \hat{v} \end{array}

An illustration of the widening example is presented in Fig. 2. Figure 2(a) represents the coherent transfer function (CTF) of the original 4F system with aperture diameter D, which has a circular cutoff frequency of D/2λF. Figure 2(b) represents the synthetic aperture generated simultaneously for all angles by the circular gratings with a basic frequency of twice the cutoff frequency. Figure 2(c) represents the final synthetic aperture that has a circular cutoff frequency of 3D/2λF, which is three times higher than the original cutoff frequency. One may notice that by using circular gratings, the duplication of the CTF fits perfectly around the original CTF. In order to avoid distortions of the synthetic aperture, the gratings Fourier coefficients should be A_m = 1. This effect may be achieved by using circular Dammann gratings [49].

Fig. 2 Circular synthetic aperture example for spatially coherent illumination. (a) CTF of the original 4F system, with circular cutoff frequency of D/2λF. (b) Synthetic aperture generated simultaneously for all angles by the circular gratings with a basic frequency of twice the cutoff frequency. (c) Final synthetic aperture, with circular cutoff frequency of 3D/2λF.

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In order to achieve the SR image, it was assumed in the summation that m = −n. However, there are terms for which m≠−n, which generates undesirable features in the image plane. These features are referred to as “ghost images”. The undesirable summation terms will be shifted to a location of:

r_{m, n} = λ z_{0} υ_{0} (m + n)

The closest undesirable term will appear where m + n = 1. Therefore, in order to avoid overlapping of the replications onto the desired object, the original object must have a restricted diameter size of:

d_{\max} = λ z_{0} υ_{0}

Since the basic frequency of the gratings needs to be twice the coherent cut-off frequency, and the wavelength is determined by the application, the gratings position determines the maximum size of the object.

An interesting debate exists regarding the inclusion of coherent SIM in the category of SR methods [50]. The point of contention is whether coherent SIM equals to oblique illumination. Here, the SR process is not based on time multiplexing as is SIM, but rather on FOV multiplexing. Thus, the comparison to oblique illumination is not valid, and the proposed method is therefore indeed a SR technique.

For spatially incoherent illumination, and assuming the object is now a single point source located at the center, Eq. (17) provides the output amplitude:

\begin{array}{l} P (x, y, z = 4 F) = \\ \sum_{n = - m} A_{m} A_{n} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} c i r c {\frac{\sqrt{{[\hat{u} + \cos (θ) m υ_{0}]}^{2} + {[\hat{v} + \sin (θ) m υ_{0}]}^{2}}}{Δ ρ / 2 λ F}} e^{2 π i (x \hat{u} + y \hat{v})} d \hat{u} d \hat{v} \end{array}

And its intensity is given by:

S (x, y, z = 4 F) = {| P (x, y, z = 4 F) |}^{2}

Assuming the point source is located at x₀,y₀, within the maximum allowed object size (i.e. √(x₀² + y₀²)<r_max), and ignoring the ghost images which are outside the allowed region, the new “point spread function” S assumes a form which is convolution suitable:

S_{x_{0}, y_{0}} = S (x, y) * δ (x - x_{0}, y - y_{0})

Since an object which is illuminated by spatially incoherent illumination can be regarded as an ensemble of mutually incoherent point sources, the intensity of the object can be expressed as:

I_{o b j} (x, y, z = 0) = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} I_{o b j} (x_{0}, y_{0}, z = 0) δ (x - x_{0}, y - y_{0}) d x_{0} d y_{0}

Therefore, the output intensity for this object is:

I_{i m g} (x, y, z = 4 F) = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} I_{o b j} (x_{0}, y_{0}, z = 0) S (x - x_{0}, y - y_{0}, z = 4 F) d x_{0} d y_{0}

Meaning that the output intensity is a convolution of the input intensity and the new “point spread function” S. S is the SR optical transfer function (OTF) which is the normalized autocorrelation function of the SR amplitude transfer function [48]:

O T F (u, v) = \tilde{S} (u, v) = \frac{\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \tilde{P} (μ + \frac{u}{2}, η + \frac{v}{2}) {\tilde{P}}^{*} (μ - \frac{u}{2}, η - \frac{v}{2}) d μ d η}{\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} {| \tilde{P} (μ, η) |}^{2} d μ d η}

where P̃ is the CTF (a Fourier transform of P).

This suggests that the proposed method is valid for spatially incoherent illumination. An illustration of the aperture generation for the incoherent case is presented in Fig. 3. Figure 3(a) represents the OTF of the original 4F system with aperture diameter D, which has a circular cutoff frequency of D/λF. Figure 3(b) represents the synthetic aperture generated by the circular gratings with a basic frequency υ₀ which equals to the cutoff frequency, and Fig. 3(c) represents the final synthetic aperture, which has a circular cutoff frequency of 2D/λF. Although the final synthetic aperture suffers from some distortions (since it is not uniform), it still results in a wider aperture generation.

Fig. 3 Circular synthetic aperture example for spatially incoherent illumination. (a) Incoherent transfer function of the original 4F system, with circular cutoff frequency of D/λF. (b) Synthetic aperture generated simultaneously for all angles by the circular gratings with a basic frequency υ₀ which equals to the cutoff frequency. (c) Final synthetic aperture, with circular cutoff frequency of 2D/λF.

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White light illumination (or expanded polychromatic illumination) can be considered as an expansion of the monochromatic spatially incoherent case with additional wavelengths. Let us consider that the object is illuminated by three spatially incoherent light sources; red, green, and blue (denoted by R,G,B), and the basic frequency of the circular gratings equals to the cutoff frequency of the red wavelength. An illustration of this case is presented in Fig. 4. Figure 4(a) represents the OTF of the original 4F system with aperture diameter D for the three different wavelengths. The OTF for each wavelength is equal to the one depicted in Fig. 3(a) but with a circular cutoff frequency which is wavelength dependent: D/λ_R,G,BF. Figure 4(b) represents the synthetic aperture generated by the circular gratings with a basic frequency υ₀ which equals to the cutoff frequency of the red wavelength. The duplications are generated with respect to υ₀. This frequency was selected since the following simulations and experiments were performed with R wavelength. However, any other choice will also be applicable, yielding in a slightly different synthetic aperture result. Figure 4(c) represents the summation for each wavelength individually, and Fig. 4(d) represents the final synthetic aperture, which has a circular cutoff frequency of D(1/λ_R + 1/λ_B)/F. Similarly to the spatially incoherent case, the result has a wider synthetic aperture, with some distortions.

Fig. 4 Circular synthetic aperture example for white light illumination. (a) OTF of the original 4F system for RGB wavelengths, with circular cutoff frequency of D/λ_R,G,BF. (b) Synthetic aperture generated simultaneously for all angles by the circular gratings with a basic frequency υ₀ which equals to the cutoff frequency of the red wavelength. (c) Summation for each wavelength individually. (d) Final synthetic aperture, with circular cutoff frequency of D(1/λ_R + 1/λ_B)/F.

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The presented method includes two fixed gratings, where the first grating is placed outside the 4F system. It is possible to include all the gratings inside the 4F system, by using three fixed grating (similarly to [39]). This change could make the optical setup more applicable for other optical scenarios, like fluorescence microscopy.

3. Numerical simulations

The method was numerically simulated using Matlab (MathWorks, Natick, MA, USA). The optical simulation contained a 635nm light source illuminating a 4F optical system with 100mm lenses. A circular aperture with 1.4mm diameter was placed in the Fourier domain. The coherent cutoff frequency for these parameters was D/2λF = 11lp/mm. Two circular grating were placed 72mm before the object plane and 72mm before the image plane. In order to simplify the simulation (and later on the experiment) the gratings were rectangular circular amplitude gratings with a basic frequency which was twice the cutoff frequency of the 4F system (22lp/mm). The object maximum possible diameter (or maximum allowed FOV) for this configuration was d_max = λz₀υ₀ = 1mm. The object in the simulation was a USAF target with a maximum diameter of 0.65mm.

The simulation results for spatially coherent monochromatic illumination are presented in Fig. 5. Figure 5(a) is the HR reference object, Fig. 5(b) is the low resolution (LR) image, and Fig. 5(c) is the SR image, achieved using the proposed approach. This image is presented in Log scale in order to emphasize that the circular duplications (ghost images) limits the original FOV. Figure 5(d)-5(f) are x4 digital zooms of Fig. 5(a)-5(c) respectively, presented in normal intensity scale.

Fig. 5 SR numerical simulation results for spatially coherent monochromatic illumination. (a) HR reference image. (b) LR image. (c) SR image, presented in Log scale. (d-f) x4 digital zooms of (a-c) respectively, presented in normal intensity scale.

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For simulating white light spatially incoherent illumination, the object was illuminated by three wavelengths: 635nm, 535nm, and 435nm. In order to simulate the incoherency of the light sources, 1000 iterations of a different random phase was added for each wavelength. The obtained SR image was the intensity summation of all the iterations. The simulation results for the white light illumination are presented in Fig. 6. Figure 6(a) is the HR reference object, Fig. 6(b) is the LR image, and Fig. 6(c) is the SR image, presented in Log intensity scale. Since the duplications position are wavelength dependent [Eq. (20)], the shorter wavelength limits the allowed FOV. For these simulations parameters d_max = λ_Bz₀υ₀ = 0.69mm. Figure 6(d)-6(f) are x4 digital zooms of Fig. 6(a)-6(c) respectively, presented in normal intensity scale.

Fig. 6 SR numerical simulation results for white light illumination. (a) HR reference image. (b) LR image. (c) SR image, presented in Log scale. (d-f) x4 digital zoom of (a-c) respectively, presented in normal intensity scale.

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4. Experimental results

The proposed method has been tested using the experimental setup which is presented in Fig. 7(a). For the spatially coherent monochromatic case, the setup consisted of a red laser beam at a wavelength of 635nm (Thorlabs LDM635), and a × 10 beam expander (Thorlabs BE10M-A). The gratings were specifically designed and manufactured for this experiment by Suron A.C.A LTD. The first circular grating was a 22lp/mm rectangular grating, which was placed 72mm before the object plane. An image of the central part of the printed circular grating is presented in Fig. 7(b). The object was a USAF target (Thorlabs R3L3S1N). The 4F optical system contained two 100mm lenses (Newport KPX594), and a 1.4mm round aperture (Thorlabs SM1D12C). The second grating (equals to the first one) was placed 72mm before the image plane. The images were captured using a USB camera (Thorlabs DCC1545M). As in the numerical simulation, the coherent cutoff frequency of the 4F system was 11lp/mm, and the maximum allowed FOV was d_max = 1mm. In order to limit the size of the object, most of the target was blocked, and only ~0.7mm diameter was illuminated. The illuminated part contained elements 2-4 of group number 4 from the USAF target, which correspond to 17.95lp/mm, 20.16lp/mm, and 22.62lp/mm, respectively. For the white light spatially incoherent experiment, the laser source was replaced by a white light emitting diode (LED). The incoherent cutoff frequency of the 4F system for the red was D/λ_RF = 22lp/mm, and the rest of the setup remained the same.

Fig. 7 (a) The experimental setup: A red laser, expanded × 10, provides illumination in the 4F system. Two circular gratings are placed into the setup, the 1st grating is between the beam expander and the object, and the 2nd grating is between the final lens and the camera. (b) The central part of the printed 22lp/mm circular grating.

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The SR results, obtained using the proposed technique for the spatially coherent case, are presented in Fig. 8. Figure 8(a) is the HR reference image of the restricted sized object, captured with an open aperture. Figure 8(b) is the LR image, captured with a 1.4mm round aperture. Since the basic frequencies of these USAF elements are above the system’s cutoff frequency, they are all blurred. Figure 8(c) is the SR image, achieved using the proposed method. This image is presented in Log scale in order to emphasize the circular duplications. Figure 8(d)-8(f) are x4 digital zooms of Fig. 8(a)-8(c) respectively, presented in normal intensity scale.

Fig. 8 SR experimental results for spatially coherent monochromatic illumination. (a) HR reference image. (b) LR image. (c) SR image, presented in Log scale. (d-f) x4 digital zooms of (a-c) respectively, presented in normal intensity scale.

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According to the theory [Eq. (18) and Fig. 2(c)], the resolution improvement factor for the spatially coherent illumination case is 3. Therefore, the theoretical cutoff frequency of the synthetic aperture will be 33lp/mm. This value is high enough to resolve the 3 USAF elements inside the allowed FOV since their highest frequency is 22.62lp/mm.

The SR results for the white light case are presented in Fig. 9. Figure 9(a) is the HR reference image. Figure 9(b) is the LR image, captured with a 1.4mm round aperture. Since the incoherent cutoff frequency was 22lp/mm, element 2 of the group 4 (upper bars) is visible, element 3 (middle bars) are barely visible since it is the closest to the resolution limit, and element 4 (lower bars) are not resolved. Figure 9(c) is the SR image in Log intensity scale. Figure 9(d)-9(f) are x4 digital zooms of Fig. 9(a)-9(c) respectively, presented in normal intensity scale.

Fig. 9 SR experimental results for white light illumination. (a) HR reference image. (b) LR image. (c) SR image, presented in Log scale. (d-f) x4 digital zooms of (a-c) respectively, presented in normal intensity scale.

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In the white light spatially incoherent illumination experiment, the cutoff frequency of the imaging system was 22lp/mm. According to the theory [Eq. (26) and Fig. 4(d)], the resolution improvement factor in this case is 2, which means that the synthetic cutoff frequency is 44lp/mm. This value is high enough to resolve the elements included at Fig. 9.

5. Conclusions

A method based on the use of circular gratings for the FOV SR method was presented. The circular gratings are 1D gratings (only radial dependent) which generate 2D SR, by creating multiple circular duplications of the aperture. These circular duplications seem to be the best choice for an optical system with a circular aperture, rather than the Cartesian gratings that were previously reported in the bibliography. Similarly to other FOV SR methods, there is a trade-off between the resolution improvement and the object size. The mathematical background is presented in virtue of gratings that are rotating angle independent, thus providing a fresh look on the theory involving SR imaging with gratings. The proposed method was analytically presented, numerically simulated, and laboratory experimented for both spatially coherent monochromatic and white light spatially incoherent illuminations. The method can be further developed using circular Dammann gratings which have constant Fourier coefficients. These types of gratings will generate a uniform synthetic aperture without distortion of the high spatial frequency content, thus improving the finally retrieved SR image. In addition, by using three fixed grating instead of two, the optical setup can be applicable for other optical scenarios, like fluorescence microscopy.

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Super resolved optical system for objects with finite sizes using circular gratings

Abstract

1. Introduction

2. Theoretical background

3. Numerical simulations

4. Experimental results

5. Conclusions

References and links

Cited By

Figures (9)

Equations (26)

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