Optimal conditions for using the binary approximation of continuously self-imaging gratings

Martin Piponnier; Guillaume Druart; Nicolas Guérineau; Jean-Louis de Bougrenet; Jérôme Primot

doi:10.1364/OE.19.023054

1. Introduction

Diffractive Optical Elements (DOE) that generate a propagation-invariant intensity pattern are very appreciated for their optical properties. They have been studied theoretically in details [1–3]. The most common solution is the zero-order (J₀) Bessel beam. The axicon, introduced by McLeod [4], is a good approximation of a DOE producing such a J₀ Bessel beam and has been widely studied [4–6]. In this paper, we will focus on a class of DOE which generates propagation-invariant spot arrays [7]. The Continuous Self-Imaging Grating (CSIG), a DOE introduced by Guérineau [8,9], belongs to this class. It is commonly used in the field of optical metrology for wavefront sensing [10] or detector’s spatial response measurement [11]. The performances of CSIG’s, in term of imaging applications, have been explored in order to develop compact, robust and low-cost imaging systems. Imaging properties [12], zooming capability [13] and good angular tolerance [14] have already been demonstrated. From a technical point of view, it is still difficult to implement the continuous transmittance of a DOE, even if some progress has been achieved recently [15]. The quantization of the transmittance is a solution that has been widely studied [16,17] and that allows practical implementation. The two-level quantization gives the easiest implementation and it has already been studied for J₀ Bessel beams [18,19]. In this paper, we will analyse the consequence of the binarization on CSIG’s, and study different kinds of coding (amplitude, phase) in order to determine which coding gives a binary CSIG closer to the ideal CSIG.

In Section 2, we will examine two different ways to code the binary transmittance (in amplitude and in phase) of a CSIG and examine the consequence of the binarization on its spatial frequency spectrum. Then, we will define a criterion that will allow us to compare the different binary CSIG’s to the ideal one. This will lead us to verify that the phase coding yields the best approximation of an ideal CSIG. In section 3, we will analyse the influence of the binarization on the propagated transverse intensity pattern. We will demonstrate that the main consequence of the binarization is that the intensity pattern is no longer continuously propagation-invariant under monochromatic illumination. Theoretical analysis will be compared to experimental results obtained using phase and amplitude binary CSIG’s. We will see that under polychromatic illumination the propagation-invariant regime is recovered after a given distance, whatever the coding, allowing the generation of a theoretical non-diffracting beam even with a simplified binary implementation of CSIG’s.

2. Influence of the binarization on the transmittance of a CSIG

2.1 Ideal CSIG

Ideal CSIG’s are defined as a class of gratings that generate a propagation-invariant and biperiodic transverse intensity pattern. Montgomery [1] and Durnin [2] demonstrated that the scalar fields, which Fourier Transform (FT) is located on a ring, give a propagation-invariant transverse intensity pattern. The J₀ Bessel beam is the most common solution of such a scalar field. Moreover, if the transverse scalar field is a biperiodic function of period a₀, its FT is located on a Cartesian grid of pitch 1/a₀. So, ideal CSIG’s are defined, in the Fourier domain, by the intersection between a Montgomery ring of radius α and a two-dimensional Dirac comb of pitch 1/a₀ [8]. This intersection yields a finite set of N Dirac peaks. From the most general point of view, CSIG’s can also be defined as a sum of N Dirac peaks, where each peak is characterized by its complex weight c_k and its location in the Fourier domain given by a couple of integers (p_k, q_k) that must verify:

\begin{matrix} \forall k \in [1, N] : & \frac{p_{k}^{2}}{a_{0}^{2}} + \frac{q_{k}^{2}}{a_{0}^{2}} = \frac{η_{k}^{2}}{a_{0}^{2}} = α^{2}, \end{matrix}

where p_k, q_k and η² are integers, a₀ is the period of the grating, α the radius of the Montgomery ring and N the number of diffraction orders of the CSIG. N is implicitely defined by Eq. (1). It is the number of couples (p_k, q_k) solutions of Eq. (1) and it depends only on the parameter η². The transmittance of the CSIG, t_CSIG, and its Fourier Transform, T_CSIG, can also be written as following:

{\begin{matrix} t_{C S I G} (r) = \sum_{k = 1}^{N} c_{k} \exp (i 2 π \frac{p_{k} x + q_{k} y}{a_{0}}) \\ T_{C S I G} (σ) = \sum_{k = 1}^{N} c_{k} δ (σ_{x} - \frac{p_{k}}{a_{0}}, σ_{y} - \frac{q_{k}}{a_{0}}) \end{matrix} .

From the definition of the transmittance and its FT, we can define the Point Spread Function (PSF), as demonstrated in [9]. From the expression of the PSF, we can also define the Modulation Transfer Function (MTF) of a CSIG as following:

{\begin{array}{l} P S F (r) = {| t_{C S I G} (r) |}^{2} \\ M T F (σ) = | F T [P S F] (σ) | = | T_{C S I G} (σ) * T_{C S I G}^{*} (σ) | \end{array},

where * represents the convolution product.

Figure 1 illustrates one elementary cell of the transmittance (a) and of the PSF (b), for the CSIG of parameter η² = 9425. T_CSIG is composed by N = 48 orders of equal real weight (c) and the MTF is composed by N’ = 1153 Dirac peaks (d). From Eq. (3), we can notice that the PSF is a biperiodic function of period a₀, as t_CSIG. We can also observe that, as the MTF is the autocorrelation of T_CSIG, it is also a finite sum of N’ Dirac peaks and can be written as following:

M T F (σ) = \sum_{k = 1}^{N'} D_{k} δ (σ_{x} - \frac{p'_{k}}{a_{0}}, σ_{y} - \frac{q'_{k}}{a_{0}}),

with

Fig. 1 Illustration of one elementary cell of size a₀  a₀ of t_CSIG (a) and of the PSF (b) for a CSIG of parameter η² = 9425. T_CSIG (c) and the MTF (d) are also represented. For a better visualization, T_CSIG is zoomed of a factor × 2 compared to the MTF.

Download Full Size | PDF

N' = 1 + N + (N - 2) \times \frac{N}{2} = \frac{N^{2}}{2} + 1.

The coefficient D_k represents the complex weight of each Dirac peak of the MTF. The couples of integers (p’_k, q’_k) are the coordinates of each peak in the frequency domain (see Fig. 1-d). The MTF is the autocorrelation of T_CSIG. So it is made up of one central peak corresponding to the intersection of two coincident circles, by N peaks corresponding to the intersection of two tangent circles and (N-2) × N/2 peaks corresponding to the intersection of two secant circles. This lead to the expression of N’ given by Eq. (5). The PSF given in Eq. (3) depends neither on z nor λ, it is therefore propagation-invariant and achromatic. An optical system with a long depth of focus and a sparse MTF can also be designed by using a CSIG as the only optical element.

2.2 Binary CSIG

In the rest of the article, we will consider an ideal CSIG with all c_k coefficients equal to 1/N. The transmittance t_CSIG is also a two-dimensional continuous function, with real values included in [-1, 1]. Figure 2 illustrates that the ideal CSIG can be decomposed in two components, one coding the sign of the transmittance and the other one coding its amplitude. From a practical point of view, it is difficult to implement the amplitude mask. We have also decided to use the simplified two-level function based on the sign function [9], because it is much easier to implement and because the pattern of the transmittance seems to be conserved.

Fig. 2 The transmittance t_CSIG (a) can be considered as the product of its two levels sign function (b) by its absolute value (c) function (in gray scale). Only a part of the elementary cell of t_CSIG is represented for a CSIG of parameter η² = 9425.

Download Full Size | PDF

This two-level function can be coded in amplitude (transmission value are also 0 or 1) or in phase (transmission values are also 1 or −1). Binary amplitude and phase transmittances are linked to ideal transmittance by the following expressions:

{\begin{array}{l} t_{a m p l i t u d e} (r) = \frac{s i g n [t_{C S I G} (r)] + 1}{2} \\ t_{p h a s e, λ} (r) = \exp (i π \frac{λ_{c o d i n g}}{λ} (\frac{1 - s i g n [t_{C S I G} (r)]}{2})) \end{array} .

We dispose of two CSIG’s of parameters η² = 9425 and a₀ = 7.5mm either coded in amplitude or in phase. The amplitude coding has been made by the deposit of a total reflecting layer on a transparent substrate. The metal is arbitrary laid where the sign function equals −1. The phase coding has been made by the deposit of a dielectric layer on a transparent substrate where the sign function equals −1. The thickness of this layer is calculated in order to introduce a phase-shift of π, at the particular wavelength, λ_coding = 532nm. If the wavelength of the incident light source, λ, is equal to λ_coding, the transmittance is exactly the sign function described Fig. 2. Otherwise, the phase-shift is no longer π and the transmittance is a complex function. As a consequence, the transmittance is achromatic over a large spectral range for an amplitude coding, but is wavelength dependent for a phase coding.

The binary transmittance is a discontinuous stepped function, so its FT will contain residual diffraction orders (cf Fig. 3 ) in addition to the N main orders of the ideal CSIG. The binary transmittance is still biperiodic of period a₀, so additional orders are located on the whole Cartesian grid of pitch 1/a₀ in the Fourier domain. As they are not located on a single Montgomery ring, the self-imaging property will be lost. All diffraction orders will interfere with each others leading, in the propagated intensity pattern, to the appearance of a parasitic Talbot effect in addition to the propagation-invariant intensity pattern. As a part of the diffracted energy is contained in the propagation-dependant intensity pattern, the propagation-invariant intensity pattern will have a reduced visibility factor compared to an ideal CSIG.

Fig. 3 3D representation of T_CSIG for an ideal CSIG of parameter η² = 9425 (a), for the associated binary amplitude CSIG (b), and for the binary phase CSIG coded at λ_coding = 532nm and illuminated at λ = 532nm (c) or λ = 600nm (d).

Download Full Size | PDF

The binary transmittance is biperiodic of period a₀, so it can be written from the most general point of view as following:

{\begin{matrix} t_{b i n a r y} (r) = \sum_{(k, l) \in ℤ^{2}} c_{k, l} \exp (i 2 π \frac{k x + l y}{a_{0}}) \\ T_{b i n a r y} (σ) = \sum_{(k, l) \in ℤ^{2}} c_{k, l} δ (σ_{x} - \frac{k}{a_{0}}, σ_{y} - \frac{l}{a_{0}}) \end{matrix},

where c_k,l coefficients depend on the coding. These coefficients can be deduced analytically in the case of axicons, where the binary transmittance is periodic with circular symmetry [19]. In the case of a CSIG, the pattern is more complicated and c_k,l coefficients cannot be easily deduced analytically from T_CSIG, but can be found by numerical simulation. Both for amplitude and phase coding, we observe on Fig. 3 the N main diffraction orders of the ideal CSIG (a), a zero order due to the mean value of the transmittance defined in Eq. (6) and additional small parasitic orders appear. For the amplitude coding, the zero order, equal to 0.5, is predominant. For the phase coding, the zero order depends on the wavelength of the incident light source by following relationship:

a (λ) = \iint_{p a t t e r n} t_{p h a s e, λ} (r) d r = \cos (π \frac{λ_{c o d i n g}}{2 λ}) \exp (i π \frac{λ_{c o d i n g}}{2 λ}) .

At λ = λ_coding, there is no zero order (c) but the more different from λ_coding the wavelength is, the more important the zero order is (d). From these observations we can transform Eq. (7) to emphasize the importance of the zero order and the N orders of the ideal CSIG in comparison with all other orders:

{\begin{array}{l} t_{b i n a r y} (r) = a + b t_{C S I G} (r) + \sum_{k, l} c_{k, l} \exp (i 2 π \frac{k x + l y}{a_{0}}) \\ T_{b i n a r y} (σ) = a δ (σ) + b T_{C S I G} (σ) + \sum_{k, l} c_{k, l} δ (σ_{x} - \frac{k}{a_{0}}, σ_{y} - \frac{l}{a_{0}}) \end{array},

where a, b and c_k,l are real or complex coefficients that depend on the coding (amplitude or phase), on the wavelength in the case of a phase coding (see Table. 1) and on the parameter η² of the CSIG. In Eq. (9), the sum is done over all integer couples (k,l) except couples corresponding to the zero order and the N main diffraction orders of T_CSIG.

To determine the coding that gives the binary transmittance closest to the ideal transmittance, we calculate the ratio R of the “energy” of the N main orders over the total “energy”, where the “energy” of each Dirac peak of T_binary is defined by the square value of its module.

R = \frac{N {| b |}^{2}}{{| a |}^{2} + N {| b |}^{2} + \sum_{k, l} {| c_{k, l} |}^{2}}

Table 1 gives the value of the ratio R for different coding. For the ideal CSIG, a and c_k,l coefficients are null, so the ratio R reaches 100%. For binary CSIG, the best coding is the one that gives a ratio R closest to 100%. The transmittance of a phase binary CSIG is chromatic, cf Eq. (6), so its ratio R depends on the wavelength, however we can notice, Table 1, that this ratio is better than the one of a binary amplitude CSIG, over the entire visible spectrum. Phase coding gives also the device closest to the ideal CSIG.

Table 1. Ratio R of Energy of N Main Peaks over Total Energy for CSIG of Parameter η² = 9425

View Table | View all tables in this article

3. Influence of the binarization on the intensity pattern generated by a CSIG

3.1 Propagation-dependent regime under incoherent monochromatic illumination

From features given Table 1, we can observe for the amplitude coding that the N main diffraction orders and the additional zero order represent about 80% of the energy. This means that these N + 1 diffraction orders mainly contribute to the transverse intensity pattern. A simplified model that neglects all residual diffraction orders can be first studied. In this model, only the interference between the zero order and the N main orders is taken into account. Let us consider a monochromatic plane wave at normal incidence, of wavelength λ and intensity I₀, illuminating a binary amplitude CSIG (a and b are real). The transmitted scalar field is

u_{t r a n s} (r) = \sqrt{I_{0}} \times t_{b i n a r y} (r) .

By using the approach of angular spectrum of plane waves [20] and by neglecting all c_k,l coefficients in the model of the binary transmittance given in Eq. (9), we get the simplified expression of the propagated scalar field at distance z,

u_{p r o p} (r, z) = \sqrt{I_{0}} [a \exp (i 2 π \frac{z}{λ}) + b \exp (i 2 π \frac{z}{λ} \sqrt{1 - λ^{2} \frac{η^{2}}{a_{0}^{2}}}) t_{C S I G} (r)],

and the expression of the transverse intensity pattern and its FT:

{\begin{array}{l} I_{p r o p} (r, z) = I_{0} [a^{2} + 2 a b \cos (2 π \frac{z}{Z_{T}}) t_{C S I G} (r) + b^{2} P S F (r)] \\ {\tilde{I}}_{p r o p} (σ, z) = I_{0} [a^{2} δ (σ) + 2 a b \cos (2 π \frac{z}{Z_{T}}) T_{C S I G} (σ) + b^{2} M T F (σ)] \end{array} .

The transverse intensity pattern generated by the ideal CSIG is the PSF. In this approximation, we observe that the binarization introduces in addition to the PSF, an offset term (a²) that is propagation-invariant and a cosine term that is propagation-dependent and chromatic. This term stresses the interference between the zero order and the N main diffraction orders leading to a residual Talbot effect [9] with following period:

Z_{T} = \frac{λ}{1 - \sqrt{1 - λ^{2} \frac{η^{2}}{a_{0}^{2}}}} \approx \frac{2 a_{0}^{2}}{λ η^{2}} .

The importance of this term depends on the coding (amplitude or phase). In particular, this term is null for a phase coding and with λ = λ_coding. In the Fourier domain, we can notice, Eq. (13), that the beating term is located at the N spatial frequencies of T_CSIG and the propagation-invariant term is located at the spatial frequencies of the MTF. The supports of T_CSIG and the MTF are distinct (see Fig. 4 ), therefore beating and propagation-invariant terms are uncorrelated in the Fourier domain, which is due to the lacunarity of the MTF of the CSIG and which is specific of the CSIG compared to non-lacunar DOE such as axicons. So, by extracting only the MTF frequencies, we could recover the propagation-invariant PSF.

Fig. 4 Illustration in the Fourier domain of the FT of the propagated intensity pattern (a). Small dots represent propagation-invariant Dirac peaks located at MTF’s spatial frequencies and big dots represent the propagation-dependant peaks located at the N spatial frequencies of T_CSIG. (b) represents the FT of an experimental image recorded at 20mm of the amplitude CSIG of parameters η² = 9425 and a₀ = 7.5mm.

Download Full Size | PDF

However the residual diffraction orders can affect the propagation-invariant frequencies of the MTF and a more complete description has to be considered in order to evaluate their effects. Let us consider the general expression of T_CSIG, given by Eq. (7), that takes into account all residual diffraction orders. We get also the following general expression for the propagated intensity pattern:

{\begin{array}{l} I_{p r o p} (r, z) = I_{0} \sum_{(k, l, k', l') \in ℤ^{4}} c_{k, l} c_{k', l'}^{*} \exp (i 2 π \frac{z}{Z_{k, l, k', l'}}) \times \exp (i 2 π \frac{(k - k') x + (l - l') y}{a_{0}}) \\ {\tilde{I}}_{p r o p} (σ, z) = I_{0} \sum_{(k, l, k', l') \in ℤ^{4}} c_{k, l} c_{k', l'}^{*} \exp (i 2 π \frac{z}{Z_{k, l, k', l'}}) \times δ (σ_{x} - \frac{k - k'}{a_{0}}, σ_{y} - \frac{l - l'}{a_{0}}) \end{array},

with

Z_{k, l, k', l'} = \frac{λ}{\sqrt{1 - λ^{2} \frac{k^{2} + l^{2}}{a_{0}^{2}}} - \sqrt{1 - λ^{2} \frac{k'^{2} + l'^{2}}{a_{0}^{2}}}} \approx \frac{2 a_{0}^{2}}{λ (k'^{2} + l'^{2} - k^{2} - l^{2})} .

The terms of the double sum, in Eq. (15), can be separated in two groups. The first group is composed by all terms which couples (k, l, k’, l’) verify k² + l² = k’² + l’². In this case, the phase term is rigorously null, so these terms contribute to the propagation-invariant intensity pattern. Second group is composed by all terms which couples (k, l, k’, l’) verify k² + l²≠k’² + l’². In this case, the phase term is non-null, so these terms contribute to the propagation-dependent pattern. The coefficients of this second sum can be gathered two by two to form a cosine function, as in the simplified model given in Eq. (13). The FT of the propagated intensity pattern, given Eq. (15), can also be written as following:

{\tilde{I}}_{p r o p} (σ) = I_{0} | \begin{array}{l} \sum_{k^{2} + l^{2} = k'^{2} + l'^{2}} c_{k, l} c_{k', l'}^{*} δ (σ_{x} - \frac{k - k'}{a_{0}}, σ_{y} - \frac{l - l'}{a_{0}}) + \\ 2 \sum_{k^{2} + l^{2} \neq k'^{2} + l'^{2}} ρ_{k, l, k', l'} \cos (2 π \frac{z}{Z_{k, l, k', l'}} + φ_{k, l, k', l'}) \times δ (σ_{x} - \frac{k - k'}{a_{0}}, σ_{y} - \frac{l - l'}{a_{0}}) \end{array},

with,

c_{k, l} c_{k', l'}^{*} = ρ_{k, l, k', l'} \exp (i φ_{k, l, k', l'}) .

The propagated intensity pattern is also made up of a propagation-invariant term (first sum) and a propagation-dependent and chromatic term (second sum), but both terms are located on the same Cartesian grid of pitch 1/a₀. In particular, there are Talbot terms, with different Talbot period Z_{k,l,k’,l’}, at spatial frequencies corresponding to the MTF: the MTF is no longer propagation-invariant and the property of infinite depth of field is lost under monochromatic illumination. Figure 5 represents the numerical simulation of the evolution with the propagation distance z of some Dirac peaks of the FT of I_prop described Eq. (17) for a CSIG of parameters η² = 9425 and a₀ = 7.5mm. The N’ = 1153 peaks located at spatial frequencies of the MTF plus the N = 48 peaks located at spatial frequencies of T_CSIG are represented. They are sorted according to their polar coordinates (ρ, θ) in the Fourier domain, by increasing ρ, and for a given ρ, by increasing θ. We can also associate a number k to each peak: 0 corresponds to the zero order, 369 to 416 correspond to the N peaks located at the frequencies of T_CSIG and 1200 to the last peak of the MTF.

Fig. 5 Evolution along the z axis of the peaks of the MTF for the binary CSIG of parameter η² = 9425 and a₀ = 7.5mm. On the vertical axis, z is ranging from 17 to 110mm. On the horizontal axis all 1201 peaks sorted, from left to right, by polar coordinates. Black lines localize peaks n°390 and n°665.

Download Full Size | PDF

To compare this simulation with the experience, we have recorded the transverse intensity pattern, generated by our phase CSIG, of parameters η² = 9425 and a₀ = 7.5mm, for different distances z ranging from a minimum distance of 17mm to a maximum distance of 110mm with a pitch of 1mm. The light source is an incoherent quasi monochromatic LED of central wavelength λ_LED = 635nm placed at a distance d = 4m in front of the CSIG. For each image, we have calculated the FT and extracted the value of the 1201 Dirac peaks (N’ MTF’s peaks + N main peaks). Figure 6 gives a scheme of the experimental set-up and theoretical and experimental data are compared in Fig. 7 . The curves of Fig. 7(a) represent the simulation of the evolution of a peak, the peak n°390 located at one of the spatial frequency of T_CSIG (cf Fig. 5), pointed out by our simplified model in Eq. (13). The cosine term is present for the phase coding, and this term is phase-shifted compared to the amplitude coding, because λ≠λ_coding. Figure 7(b) shows a very good agreement between simulation and experimental acquisition: the Talbot distance defined Eq. (14) is equal to 18.8mm under these experimental conditions. On the experimental curve, a period Z_T = 18.75mm can be measured. The curves of Fig. 7(c) represent the simulation of the evolution with z of one of MTF’s peaks: the peak n°665. This evolution is propagation-invariant for the ideal CSIG but propagation-dependant for binary CSIG’s. The evolution with z does not depend on the coding, however the mean value and the amplitude of this evolution are multiplied by 4 in the case of the phase CSIG compared to the amplitude one, leading to more important beatings but also to a better Signal/Noise Ration (SNR). This is a consequence of the definition of the coding, given in Eq. (6). Figure 7(d) shows once again a very good agreement between simulated and measured data, and point out the fact that under monochromatic illumination, the MTF is no longer propagation-invariant.

Fig. 6 Scheme of the experimental set-up containing a LED of wavelength λ_LED = 635nm located at a distance d = 4m of a CSIG of parameters η² = 9425 and a₀ = 7.5mm. A matrix detector is mounted on a translation stage.

Download Full Size | PDF

Fig. 7 (a) and (c) illustrate the evolution versus z of the peaks n° 390 and n° 665 of the MTF for the CSIG of parameter η² = 9425. This evolution is given for three different coding: ideal, binary amplitude and binary phase coded at 532 nm. A comparison between simulation and experimental data obtained with the phase CSIG is given for the peaks n° 390 (b) and n° 665 (d).

Download Full Size | PDF

So we have demonstrated that under monochromatic illumination, the transverse intensity pattern is no more propagation-invariant. As a consequence, the depth of focus of the imaging system composed by a CSIG is drastically reduced. In addition, we have demonstrated that the dependence in z of MTF’s peaks is the same whatever the binary coding but the phase coding presents higher MTF’s values, which gives a better SNR for the MTF measurement.

3.2 Propagation-invariant regime reached under panchromatic illumination

From Eq. (17), the propagated intensity pattern can be deduced by inverse Fourier transform:

I_{p r o p, λ} (r, z) = I_{λ} | \begin{array}{l} \sum_{k^{2} + l^{2} = k'^{2} + l'^{2}} c_{k, l} c_{k', l'}^{*} \times e^{i 2 π \frac{(k - k') x + (l - l') y}{a_{0}}} + \\ 2 \sum_{k^{2} + l^{2} \neq k'^{2} + l'^{2}} ρ_{k, l, k', l'} \cos (2 π \frac{z}{Z_{k, l, k', l'}} + φ_{k, l, k', l'}) \times e^{i 2 π \frac{(k - k') x + (l - l') y}{a_{0}}} \end{array} .

In this expression, Z_{k,l,k’,l’} depends on the wavelength as described Eq. (16) and Eq. (6) shows that c_k,l coefficients can also depend on the wavelength, according to the considered coding. This expression, valid for a monochromatic light source, must also be summed over the wavelength for a panchromatic light source of spectrum s(λ) and of total intensity I₀:

I_{p a n c h r o} (r, z) = \int_{ℝ} s (λ) I_{p r o p, λ} (r, z) d λ .

This lead to the following expression of the panchromatic propagated intensity pattern that contains an achromatic and propagation-invariant term, the first sum, and a chromatic and propagation-dependent term, the second sum.

I_{p a n c h r o} (r, z) = | \begin{array}{l} I_{0} \sum_{k^{2} + l^{2} = k'^{2} + l'^{2}} \int_{ℝ} s (λ) c_{k, l} (λ) c_{k', l'}^{*} (λ) d λ \times e^{i 2 π \frac{(k - k') x + (l - l') y}{a_{0}}} + \\ 2 \sum_{k^{2} + l^{2} \neq k'^{2} + l'^{2}} \int_{ℝ} s (λ) ρ_{k, l, k', l'} (λ) \cos (2 π \frac{z}{Z_{k, l, k', l'}} + φ_{k, l, k', l'} (λ)) d λ \times e^{i 2 π \frac{(k - k') x + (l - l') y}{a_{0}}} \end{array}

To illustrate the evolution of the chromatic term under panchromatic illumination, we will consider the particular case of a light source with a Gaussian spectrum and a binary coding for which c_k,l coefficient are achromatic. The spectrum, s(λ), and its FT, S(λ), are given by:

{\begin{array}{l} s (λ) = I_{0} \frac{1}{σ \sqrt{2 π}} \exp (- \frac{{(λ - λ_{0})}^{2}}{2 σ^{2}}) \\ S (Λ) = I_{0} \exp (- i 2 π λ_{0} Λ) \exp (- 2 π^{2} σ^{2} Λ^{2}) \end{array},

where λ₀ is the central wavelength and σ is the standard deviation of the Gaussian. σ is linked to the Full Width at Half Maximum (FWHM) of the Gaussian by the relation:

F W H M = 2 \sqrt{2 \ln (2) σ} .

We define the expression of the chromatic coefficient summed over λ, in Eq. (21), by:

f (z) = 2 ρ_{k, l, k', l'} \int_{ℝ} s (λ) \cos (2 π \frac{(k'^{2} + l'^{2} - k^{2} - l^{2}) z}{2 a_{0}^{2}} λ + φ_{k, l, k', l'}) d λ .

With the expression of the spectrum given in Eq. (22), this sum has an analytical expression and can be written as following:

f (z) = I_{0} 2 ρ_{k, l, k', l'} \exp (- 2 π^{2} \frac{z^{2}}{Z'^{2}}) \cos (2 π \frac{z}{Z_{T}} + φ_{k, l, k', l'}),

where the panchromatic distance Z’, originally introduced by Guérineau [9], and the Talbot distance Z_T are defined by:

{\begin{array}{l} Z' = \frac{2 a_{0}^{2}}{σ (k'^{2} + l'^{2} - k^{2} - l^{2})} \\ Z_{T} = \frac{2 a_{0}^{2}}{λ_{0} (k'^{2} + l'^{2} - k^{2} - l^{2})} \end{array} .

The width of source’s spectrum introduces a visibility factor, Eq. (25), that makes the function f decrease with the propagation distance z. In the expression of the intensity pattern given in Eq. (21), this leads to the extinction of the chromatic term beyond the characteristic distance Z’ given in Eq. (26). Z’ is inversely proportional to spectrum’s width σ and to the radius of the spatial frequency (in polar coordinates in the Fourier domain). For a given peak of the MTF, the wider the spectrum and the higher the spatial frequency are, the faster the theoretical achromatic and propagation-invariant regime is reached. This conclusion can be generalized to other spectrum’s shapes (top hat, blackbody). The transverse intensity pattern is also propagation-invariant after a certain propagation distance and we get again the very long depth of field that is desired for imaging applications:

{\tilde{I}}_{p a n c h r o} (σ, z) = | I_{0} \sum_{k^{2} + l^{2} = k'^{2} + l'^{2}} \int_{ℝ} s (λ) c_{k, l} (λ) c_{k', l'}^{*} (λ) d λ \times δ (σ_{x} - \frac{k - k'}{a_{0}}, σ_{y} - \frac{l - l'}{a_{0}}) .

Figure 8 illustrates this for a binary phase CSIG illuminated by different sources: a quasi monochromatic LED of FWHM = 23nm and a polychromatic halogen lamp of FWHM = 150nm. If we compare the evolution with z of peaks n°390 and n°665 under monochromatic or panchromatic illumination, we can clearly notice that the propagation-dependent term weakens with a large spectrum and for the MTF peak n°665 only the propagation-invariant term remains. This experimental propagation-invariant MTF can be compared to the ideal CSIG’s MTF.

Fig. 8 (a) and (c) illustrate the evolution versus z of the peaks n° 390 and n° 665 of the MTF for the CSIG of parameter η² = 9425. This evolution is given for a monochromatic and a broadband source. A comparison between simulation and experimental data obtained with the phase CSIG is given for the peaks n° 390 (b) and n° 665 (d).

Download Full Size | PDF

Let’s consider the MTF peaks from the point of view of an optical imaging system. The central peak gives information on the mean value of the image. All other peaks contain information on the spatial frequencies, i.e on the details of the image. For an imaging application, it is interesting to measure the ratio R’ of the energy contained in the useful peaks (all peaks of the MTF except the central peak) over the energy contained in all peaks of the MTF. For an ideal CSIG, this ratio has an analytical expression:

R' = \frac{(\frac{N^{2}}{2} - N) \times 2^{2} + N \times 1^{2}}{1 \times N^{2} + (\frac{N^{2}}{2} - N) \times 2^{2} + N \times 1^{2}} = \frac{2 - \frac{3}{N}}{3 - \frac{3}{N}} .

This ratio reaches highest value, R’_max = 66.6%, for a number of main diffraction orders N close to infinity. For a CSIG of parameter η² = 9425 that diffracts N = 48 orders, this ratio R’ = 65.96% is very close to R’_max. In Table 2 , we verify once again that the phase coding is closer to the ideal CSIG than the amplitude coding. With phase coding, about 40% of the energy is contained in the “useful” peaks of the MTF, which is much better than the 16% of amplitude coding. The phase coding is also better than the amplitude coding, for imaging applications, because it maximizes the energy present at the useful spatial frequencies of the MTF.

Table 2. Ratio R’ of Energy of Useful Peaks over Total Energy of MTF for CSIG of Parameter η² = 9425.

View Table | View all tables in this article

4. Conclusion

In this paper, we have considered two ways to code a binary CSIG. Despite the fact that the amplitude coding gives an achromatic transmittance, in contrast to the phase coding, this study shows that the phase coding is the best, even if the incident light source has a wavelength different from the optimal wavelength λ_coding. The influence of the binarization on the intensity pattern generated by a binary CSIG has been studied. We have demonstrated that under monochromatic illumination the intensity pattern is no more propagation-invariant and the property of continuous self-imaging is lost. The propagation-invariant regime can however been recovered under panchromatic illumination and after a characteristic distance Z’ of the CSIG. In this configuration a simplified binary CSIG is nearly equivalent to an ideal CSIG. Thus, we noticed that under panchromatic illumination, the energy contained in MTF’s peaks different from the zero order, is higher for the phase coding than for the amplitude one. Practically this will be traduced by a better signal/noise ratio in the measurement of the Dirac peaks of the MTF. This binary CSIG is easy to manufacture and has same properties as ideal CSIG if used in the area of panchromatic regime. Its use is also clearly justified for metrology applications (wavefront sensing or detector’s spatial response measurement) and for imaging applications (detection or position sensor) where sparse MTF is sufficient for recovering the wanted information.

Acknowledgments

The authors are grateful to Emilie Bialic for having shared her knowledge in the practical realization of binary axicons and CSIG’s.

References and links

1. W. D. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. A 57(6), 772–778 (1967). [CrossRef]

2. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4(4), 651–654 (1987). [CrossRef]

3. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. 29(1), 44–46 (2004). [CrossRef] [PubMed]

4. J. H. McLeod, “The axicon: a new type of optical element,” J. Opt. Soc. Am. 44(8), 592–597 (1954). [CrossRef]

5. A. T. Friberg, “Stationary-phase analysis of generalized axicons,” J. Opt. Soc. Am. A 13(4), 743–750 (1996). [CrossRef]

6. Z. Jaroszewicz and J. Morales, “Lens axicons: systems composed of a diverging aberrated lens and a perfect converging lens,” J. Opt. Soc. Am. A 15(9), 2383–2390 (1998). [CrossRef]

7. V. Kettunen and J. Turunen, “Propagation-invariant spot arrays,” Opt. Lett. 23(16), 1247–1249 (1998). [CrossRef] [PubMed]

8. N. Guérineau and J. Primot, “Nondiffracting array generation using an N-wave interferometer,” J. Opt. Soc. Am. A 16(2), 293–298 (1999). [CrossRef]

9. N. Guérineau, B. Harchaoui, J. Primot, and K. Heggarty, “Generation of achromatic and propagation-invariant spot arrays by use of continuously self-imaging gratings,” Opt. Lett. 26(7), 411–413 (2001). [CrossRef] [PubMed]

10. J. Primot and N. Guérineau, “Extended hartmann test based on the pseudoguiding property of a hartmann mask completed by a phase chessboard,” Appl. Opt. 39(31), 5715–5720 (2000). [CrossRef] [PubMed]

11. N. Guérineau, S. Rommeluere, E. Di Mambro, I. Ribet, and J. Primot, “New techniques of characterization,” C. R. Phys. 4(10), 1175–1185 (2003).

12. G. Druart, N. Guérineau, R. Haïdar, J. Primot, A. Kattnig, and J. Taboury, “Image formation by use of continuously self-imaging gratings and diffractive axicons,” Proc. SPIE 6712, 1–11 (2007).

13. G. Druart, J. Taboury, N. Guérineau, R. Haïdar, H. Sauer, A. Kattnig, and J. Primot, “Demonstration of image-zooming capability for diffractive axicons,” Opt. Lett. 33(4), 366–368 (2008). [CrossRef] [PubMed]

14. G. Druart, N. Guérineau, R. Haïdar, J. Primot, P. Chavel, and J. Taboury, “Non-paraxial analysis of continuous self-imaging gratings in oblique illumination,” J. Opt. Soc. Am. A 24(10), 3379–3387 (2007). [CrossRef]

15. S. López-Aguayo, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Nondiffracting light on-demand,” Opt. Photonics News 21(12), 43 (2010). [CrossRef]

16. W. J. Dallas, “Phase quantization–a compact derivation,” Appl. Opt. 10(3), 673–674 (1971). [CrossRef] [PubMed]

17. F. Wyrowski, “Diffractive optical elements: iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. A 7(6), 961–969 (1990). [CrossRef]

18. J. A. Davis, E. Carcole, and D. M. Cottrell, “Intensity and phase measurements of nondiffracting beams generated with a magneto-optic spatial light modulator,” Appl. Opt. 35(4), 593–598 (1996). [CrossRef] [PubMed]

19. L. Niggl, T. Lanzl, and M. Maier, “Properties of Bessel beams generated by periodic gratings of circular symmetry,” J. Opt. Soc. Am. A 14(1), 27–33 (1997). [CrossRef]

20. J. W. Goodman, “Foundations of Scalar Diffraction Theory,” in Introduction to Fourier Optics, 3^rd ed. (Roberts & Company, 2005), pp. 31 - 62.

Coding	Ideal	Amplitude	Phase (λ_coding = 532nm)
Wavelength (nm)			400	532	800
\|a\|	0	0.500	0.495	0	0.502
\|b\|	1/N = 0.021	0.058	0.100	0.116	0.100
Max(\|c_k,l\|)	0	0.005	0.009	0.010	0.009
R (in %)	100.00	32.10	48.44	64.20	48.00

Coding	Ideal	Amplitude	Phase (λ_coding = 532nm)
Wavelength (nm)			532	625
Central peak	1/N	0.5	1	1
Middle peak	2/N²	0.0070	0.0271	0.0270
External peak	1/N²	0.0035	0.0135	0.0132
R’ (in %)	65.96	16.97	43.79	41.89

Coding	Ideal	Amplitude	Phase (λ_coding = 532nm)
Wavelength (nm)			400	532	800
\|a\|	0	0.500	0.495	0	0.502
\|b\|	1/N = 0.021	0.058	0.100	0.116	0.100
Max(\|c_k,l\|)	0	0.005	0.009	0.010	0.009
R (in %)	100.00	32.10	48.44	64.20	48.00

Coding	Ideal	Amplitude	Phase (λ_coding = 532nm)
Wavelength (nm)			532	625
Central peak	1/N	0.5	1	1
Middle peak	2/N²	0.0070	0.0271	0.0270
External peak	1/N²	0.0035	0.0135	0.0132
R’ (in %)	65.96	16.97	43.79	41.89

Optimal conditions for using the binary approximation of continuously self-imaging gratings

Abstract

1. Introduction

2. Influence of the binarization on the transmittance of a CSIG

2.1 Ideal CSIG

2.2 Binary CSIG

3. Influence of the binarization on the intensity pattern generated by a CSIG

3.1 Propagation-dependent regime under incoherent monochromatic illumination

3.2 Propagation-invariant regime reached under panchromatic illumination

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (8)

Tables (2)

Equations (28)

Optics Express