Multifunctional binary diffractive optical elements for structured light projectors

Omri Barlev; Michael A. Golub

doi:10.1364/OE.26.021092

1. Introduction

Three-dimensional (3D) sensing with reconstruction of depth data of a 3D scene is a rapidly expanding field with a variety of methods and growing number of applications. The most recent and popular applications are in entertainment [1], posture recognition [2], tracking of hand articulations [3], and facial recognition system [4,5]. Stereovision [6], a common method for 3D sensing, triangulates the 3D positions of the scene based on recording images of the scene from two or more points of view and finding correspondences between them. Today, in the emerging structured light (SL) technique, regular illuminators of a 3D scene are substituted by specialized structured light projectors that sequentially project a set of tailored SL light patterns onto the scene. Subsequent recording of respective set of images enables computational reconstruction of the depth data of the scene. A basic and common sets of SL patterns are “binary-encoded” patterns which consist of stripes [7,8], wherein the spatial density of stripes increases by a factor of two at every consecutive pattern, as shown in Fig. 1. Deviations of the stripes from their regular positions encodes depth data of the scene and supports its computational reconstruction. This way, the accuracy of acquiring the depth data by triangulation is improved, and becomes highly dependent on the sharpness, uniformity, and contrast of the projected SL patterns.

Fig. 1 Set of binary-encoded SL patterns for 3D sensing.

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The creation of high contrast SL patterns with high power, frame rate, and large fanout angle is quite challenging. Dynamic projectors based on spatial light modulators, digital micro mirror devices, or scanning mirrors may have some disadvantages as SL projectors. For example, in mobile application they overshoot for the creation of pre-determined patterns which comes on the expense of cost, complexity and bulkiness. Moreover, they have limited time frame rate and power limitation, as well as wavelength range restrictions, especially in IR. The latter may be crucial for outdoor, and mid-range scanning (5-10 meters) applications. In particular, the liquid-crystal have power threshold and absorb infrared light while the scanning mirror projectors may have laser safety problems. Relatively small number of pixels in SLMs and large few-micrometer pixel pitch restricts space-bandwidth product and fidelity of the SL patterns. In contrary, modern static DOEs enjoy sub-nanometer spatial accuracy, which enables diffraction limited fidelity of SL patterns. Further to advantages of static DOEs, the costs of SLMs and electronics far exceed costs of the binary DOEs in mass production. Static projectors based on computer generated “scalar“ diffractive optical element (DOE) are relatively simple, compact, and have low cost [9]. However, they have tough limits in fanout angle, spatial resolution of patterns, and light efficiency. For example, binary zone plate efficiency is capped at 40% for a relatively small NA of ~0.035 [10]. The on-axis Kinoform structure with continuous surface relief described at [11] experimentally demonstrated beam splitting of 1 to 9 with 0.6° fanout with high diffraction efficiency, albeit with even lower effective NA=0.005. Diffractive diffusers and beam splitters are [12–14] capable of large fanout angles. However, the diffusers feature a kind of speckle noise and random spots whereas, the beam splitters are essentially restricted to patterns composed of points. Further to that, the scalar DOEs, diffractive diffusers and beam splitters all require additional bulky optical components for laser light collimation and projection of the SL patterns, as in the recently debuted iPhone X dot projector [15].

The challenge in modern diffractive optics, in our opinion, is in combination of large fanout angles, high NA, separation of useful diffraction order from ghost ones, high diffraction efficiency, capability for creation of complex and high-resolution light patterns. Well-known scalar diffractive optics, based on Fourier optics and iterative Fourier transform algorithms, fails in matching the challenge, because of physical, computational and technological limitations. This paper meets the challenge by using off-axis design for non-paraxial diffractive lens with high NA of ~0.17, Gaussian-to-top-hat shaper and binary beam splitter. The fine structure of diffractive zones was designed with rigorous diffraction theory that is a must for large fanout angles and off-axis configuration with high diffraction efficiency. High diffraction efficiency of the DOEs compared to SLMs enables to use lower power light sources for achieving same output powers at the SL patterns.

For a sole DOE to shape structured light patterns in laser light, it must combine at least three functions: imaging of the point source (laser emitter or fiber edge) to the plane where patterns are expected to be sharp, converting the point source to a certain structure intensity distribution and finally, multiplexing that structure to compose SL pattern.

Nano-technologies have rapidly advanced in the last decade and are enabling new DOEs that overcome usual restriction of traditional ones. Such new DOEs include high contrast dense gratings [16–18], metalenses [19–21], grating waveguide structures [22] and resonance domain surface relief structures [23–26]. Specifically, resonance domain DOEs have a period comparable to the wavelength, feature Bragg effects, have high diffraction efficiency [23], fast focusing with high numerical aperture (NA) [27,28], tens of degrees separation angles between the diffraction orders, tailored aberrations and polarization control [29]. Their fabrication as binary structures on fused silica (FS) with well-established techniques of electron beam lithography (EBL) and reactive ion etching (RIE) is rather straight forward [30].

In this paper, we present DOEs that comprise all imaging, shaping and multiplexing functions in a single binary surface relief layer. In our knowledge, it is the first time that a DOE with complicated wide-angle fanout pattern and high diffraction efficiency is proposed, theoretically analyzed, designed, fabricated and optically tested with success. The DOEs project a sequence of binary-encoded light patterns at the visible wavelength. The patterns reach up to 64 tophat stripes at a distance of one meter with 20° fanout out angle. Sections 2 and 3 are dedicated to derivation of the phase function of the DOE as a mathematical superposition of an imaging diffractive lens, map-transformation Gaussian-to-top-hat beam shaper [31–33], and Dammann grating beam splitter [34]. In section 4 we use numerical calculations with rigorous conical diffraction to estimate the efficiency. The design was implemented in section 5 as a single layer of transmissive binary surface relief profile. Finally, section 6 presents the optical experiments. The results show fine resolution, high contrast and high efficiency of up 88%.

2. Gaussian-to-tophat imaging shaper

In this section we apply geometrical optics for design of the SL DOE with imaging and shaping functions, which we will call from now as an imaging shaper. The multiplexing by adding a phase function of a Dammann beam splitter will be considered in the next section. First, Eqs. (1)-(9) describe a model of a diverging polarized beam that is incident on the DOE. Then Eqs. (10)-(13) provide a phase function for imaging and shaping. Finally, Eq. (14) expresses the periods of a local diffraction grating.

The imaging shaper DOE converts an oblique diverging Gaussian light beam into a flat tophat (stripe) intensity pattern, as shown in Fig. 2. Figure 3 shows typical diffraction grooves of the imaging shaper, which can be considered as a continuous set of local diffraction gratings. In order to define the local diffraction gratings, we used geometrical optics to calculate the required smooth phase function.

Fig. 2 (a) Intensity distribution of the oblique incident Gaussian beam at the DOE plane. The DOE boundary corresponds to the intensity level exp(−8) ; (b) Tophat intensity at the output plane.

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Fig. 3 Imaging shaper as a set of adjacent local diffraction gratings.

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For the imaging shaper design in the resonance domain of diffraction, we used the off-axis optical scheme shown in Fig. 4. Figure 4(a) shows view from a top of an optical table, (b) shows a side view. We defined a Cartesian coordinate system $x, y$ with origin O at the DOE plane, axis $z$ serving as normal to the DOE and respective unit vectors $\hat{x}, \hat{y}, \hat{z}$ . For high diffraction efficiency, the chief ray of the incident beam subtends the Bragg angle of incidence

\sin θ_{B} = \frac{λ}{2 Λ_{0}},

to the DOE normal, where

Λ_{0}

is the base period of the diffraction grooves at the center of the imaging shaper and

λ

is the wavelength of the incident beam.

Fig. 4 Off-axis layout for the imaging shaper DOE design. (a) View from top; (b) side view.

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We assumed that the beam waist of the incident beam (e.g. cleaved end of a single mode fiber coupled to a laser) has the mode field semi-diameter $σ_{0}$ (where the intensity level is $e^{- 2}$ of that at the origin), and is placed in front of DOE at an oblique distance $l_{0}$ along the chief ray with angle $θ_{B}$ , as shown in Fig. 4. The unit polarization vector $e_{0}$ at the beam waist is defined with its orthogonal p and s polarization components $e_{0 p}, e_{0 s}$ ( $e_{0 p}^{2} + e_{0 s}^{2} = 1$ ) wherein s polarization corresponds to the axis y in Fig. 4. The propagation direction vector of the incident beam at the DOE point $x, y$ can be obviously expressed as

N_{i n c} (x, y) = \frac{(x + \sin θ_{B} l_{0}) \hat{x} + y \hat{y} + \cos θ_{B} l_{0} \hat{z}}{R (x, y, l_{0})},

where we defined a function

R (x, y, l) = \sqrt{l^{2} + x^{2} + y^{2} + 2 \sin θ_{B} l x},

such that

R (x, y, l_{0})

provides distance from the center of the incident beam waist to a point

x, y

at the DOE plane. To propagate the oblique incident beam towards the DOE plane, we used equations detailed in [35] for a vectorial Gaussian beam at the far-field zone, that should be applied to Cartesian coordinates

x^{'}, y^{'}, z^{'}

where

z^{'}

is aligned with the chief ray of the oblique incident beam and defined with same origin O at the DOE as coordinates

x, y

. Then, with the coordinate transformation

x^{'} = \cos θ_{B} x, y^{'} = y, z^{'} = \sin θ_{B} x

we obtained expressions for intensity

I_{i n c} (x, y)

, semi-diameter

σ (x, y, l_{0})

, phase

φ_{i n c} (x, y)

and unit polarization vector

e (x, y)

of the incident beam at the DOE plane as

I_{i n c} (x, y) = I_{0} \exp [- 2 \frac{\cos^{2} θ_{B} x^{2} + y^{2}}{σ^{2} (x, y, l_{0})}], σ (x, y, l_{0}) = \frac{λ}{π σ_{0}} R (x, y, l_{0}),

φ_{i n c} (x, y) = k [R (x, y, l_{0}) - l_{0}], k = \frac{2 π}{λ},

e (x, y) = \frac{(\cos θ_{B} l_{0} e_{0 p} - \sin θ_{B} y e_{0 s}) \hat{x} + (\sin θ_{B} x + l_{0}) e_{0 s} \hat{y} - [(x + \sin θ_{B} l_{0}) e_{0 p} + \cos θ_{B} y e_{0 s}] \hat{z}}{\sqrt{R^{2} (x, y, l_{0}) - {(- y e_{0 p} + \cos θ_{B} x e_{0 s})}^{2}}} .

Based on Eq. (5) we chose the boundary of the DOE as a contour line $\cos^{2} θ_{B} x^{2} + y^{2} = 4 σ^{2} (x, y, l_{0})$ where the Gaussian beam intensity level is $\exp (- 8)$ relative to its peak intensity. This contour line with an implicit equation

{(\frac{x - x_{0}}{A})}^{2} + {(\frac{y}{B})}^{2} = 1

describes an ellipse with major and minor half-axes

A

and

B

and a lateral x-shift

x_{0}

that are defined by

A = \frac{2 λ l_{0} \cos θ_{B} \sqrt{σ_{0}^{2} π^{2} - 4 λ^{2}}}{\cos^{2} θ_{B} σ_{0}^{2} π^{2} - 4 λ^{2}}, B = \frac{2 λ l_{0} \cos θ_{B}}{\sqrt{\cos^{2} θ_{B} σ_{0}^{2} π^{2}_{B} - 4 λ^{2}}}, x_{0} = \frac{4 λ^{2} \sin θ_{B} l_{0}}{\cos^{2} θ_{B} σ_{0}^{2} π^{2} - 4 λ^{2}} .

To implement functionality of imaging from the off-axis point source located at distance $l_{0}$ in front of the DOE to a distance $l$ in our actual optical arrangement of Fig. 4, we have added the off-axis lens [36] with off-axis angles $θ_{B}$ . Such lens images the center of the beam waist from $l_{0}$ in front of the DOE to $l$ after the DOE, i.e. converts phase $φ_{i n c} (x, y)$ in Eq. (3) of diverging incident beam to the phase $- k [R (x, y, l) - l]$ of a converging output beam, and accordingly has the phase function

φ_{l e n s} (x, y) = - k [R (x, y, l) - l + R (x, y, l_{0}) - l_{0}],

where the function

R (x, y, l)

was defined in Eq. (3).

To proceed with shaping of the tophat, we used one-dimensional on-axis paraxial solution for the far-field tophat shaper detailed in [37], with the phase function

φ (x; a, σ, C) = \frac{k a}{l e r f (\sqrt{2} C / σ)} {x e r f (\frac{\sqrt{2} x}{σ}) - \frac{σ}{\sqrt{2 π}} [1 - \exp (- \frac{2 x^{2}}{σ^{2}})]}

that converts the Gaussian intensity with parameter

σ

and aperture size

2 C

to flat tophat intensity distribution with size

2 a

at distance

l

. Respective two-dimensional (2D) paraxial solution for the on-axis tophat shaper is separable in coordinates

x, y

. Further to that, the angular tilt

θ_{B}

of the beam shaper to the incident beam in our off-axis optical scheme leads to modified equations wherein coordinates

x, y

exchange to

x \cos θ_{B}, y

per Eq. (4). Accordingly, the phase function of the 2D far-field tophat shaper that converts the Gaussian intensity with parameter

σ_{D O E}

and aperture size

2 A \cos θ_{B} \times 2 B

to flat top-hat intensity distribution with size

2 a \times 2 b

at distance

l

can be expressed as

φ_{t h, F a r} (x, y) = φ (x \cos θ_{B}; a, σ_{D O E}, A \cos θ_{B}) + φ (y; b, σ_{D O E}, B), σ_{D O E} = σ (0, 0, l_{0}) .

Then, the phase function of the imaging shaper

φ_{t h}

is a superposition of the off-axis lens and the 2D far-field beam shaper

φ_{t h} (x, y) = φ_{t h, F a r} (x, y) + φ_{l e n s} (x, y) .

The spatial frequency vector

ν (x, y)

and period

Λ (x, y)

of the local diffraction grating of the imaging shaper vary with coordinates

x, y

and can be calculated through the 2D gradient of the smooth phase function of Eq. (13)

ν (x, y) = \frac{1}{2 π} \nabla_{⊥} φ_{t h} (x, y), Λ (x, y) = \frac{1}{| ν (x, y) |}, \nabla_{⊥} = (\frac{\partial}{\partial x}, \frac{\partial}{\partial y}),

such that angular orientation of the normal to the fringes of the local diffraction gratings is

ν (x, y) Λ (x, y)

, as shown in Fig. 3. The base period in these notations is simply

Λ_{0} = Λ (0, 0)

.

3. From single tophat to multi-stripe patterns

Off-axis imaging and shaping of the previous section provided single tophat stripe of the SL pattern. This section deals with multiplexing to convert the single tophat stripe to entire SL pattern with number of stripes of nearly equal power. In a set of subsequent mathematical equations, Eqs. (15) and (16) describe a model of the SL DOE, which generates entire pattern that comprises of $N$ tophat stripes. For that, Eqs. (17), (18) and Table 1 define binary phase of Dammann grating in a single grating period. Finally, Eq. (19) explains how to build the diffraction grooves of the SL DOE.

Table 1. Coordinates of alternating values 0 and π in Dammann gratings, following [34].

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The transition from the imaging shaper to the SL DOE that reconstructs $N$ tophat stripes is done by adding a phase function $φ_{s p l}^{(N)} (x)$ of the N-th order binary Dammann grating [34] to the phase $φ_{t h} (x, y)$ of the imaging shaper per Eq. (13). Accordingly, the phase function of the SL DOE is

φ_{s l}^{(N)} (x, y) = φ_{t h} (x, y) + φ_{s p l}^{(N)} (x) = φ_{t h, F a r} (x, y) + φ_{l e n s} (x, y) + φ_{s p l}^{(N)} (x) .

The phase function of the Dammann grating splits the light power into

N

equal intensity diffraction orders and yields multiplexed system of the tophat stripes instead of originally single one.

The phase of the even-order Dammann grating beam splitter [34], in our off-axis arrangement, is a periodic function of the coordinate $x \cos θ_{B}$ with period $Λ_{s p l}^{(N)}$ defined as

Λ_{s p l}^{(N)} = \frac{λ l}{0.5 Δ^{(N)}}

for an even number of top-hats, where

Δ^{(N)} = b / N

is a distance between centers of adjacent tophat stripes in the image plane. The phase of the Dammann beam splitter can be calculated through a function

φ_{1}^{(N)} (χ)

of a dimensionless variable

0 \leq χ \leq 1

reduced to a single period with length 1.0

φ_{s p l}^{(N)} (x) = φ_{1} (χ), χ = \frac{1}{Λ_{s p l}^{(N)}} \mod (x \cos θ_{B}, Λ_{s p l}^{(N)}),

where

\mod (\cdot, Λ_{s p l}^{(N)})

is a minimal non-negative residue of the first argument after subtraction of multiples of

Λ_{s p l}^{(N)}

. In case of even number of tophats,

φ_{1} (χ)

is calculated as

φ_{1} (χ) = {\begin{matrix} φ_{0.5} (χ), 0 \leq χ \leq 0.5 \\ φ_{0.5} (χ - 0.5) + π, 0.5 < χ \leq 1 \end{matrix} .

The piecewise constant function

φ_{0.5} (χ)

with alternating values 0 and π is defined on half of the interval

0 \leq χ \leq 0.5

. The normalized values of

N

coordinates where the phase

φ_{1} (χ)

or

φ_{0.5} (χ)

changes from 0 to π are summarized in Table 1 based on [34].

For design of diffraction grooves of the SL DOE, we first used Eq. (13) for the smooth phase function of the tophat beam shaper and solved numerically the isoline equation

φ_{t h} (x, y) = 2 π \cdot n,

where n is number of the diffraction groove in the imaging shaper. The width of each diffraction groove was chosen with duty cycle (DC) 0.5, i.e. as half of the local period per Eq. (16). Important to mention, that an optimal design for maximal efficiency should use a varying, rather than a fixed, DC and slant angle of diffractive grooves. However, these may cause substantial technological complications in DOE fabrication and is outside the scope of this paper. The addition of the Dammann beam splitter phase

φ_{s p l}^{(N)} (x)

with alternating values of 0, π was implemented by inverting the depth of binary diffraction grooves where the Dammann splitter value was π which is equal to shifting the diffraction grooves by half of the local period.

Table 2 details the set of parameters 2a, 2b, and $Λ_{s p l}^{(N)}$ for a set of the eight SL DOEs that generates SL pattern set with up to 64 stripes for the fanout angle of 20°. DOE#1 with N = 0 corresponds to a fully uniformly illuminated SL pattern within entire fanout angle. The DOE#2 with N = 1 corresponds to SL pattern with one stripe corresponding to half of the fanout angle. Both DOE#1 and 2 provide single tophat stripes and accordingly do not need a Dammann beam splitter phase.

Table 2. DOEs parameters for binary-encoded stripe patterns

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4. Efficiency in rigorous diffraction

In order to calculate the diffraction efficiency $η_{s l}^{(N)}$ of the SL DOE, we calculated separately the efficiency of the imaging shaper $η_{t h}^{(N)}$ and the Dammann beam splitter $η_{s p l}^{(N)}$ and multiplied them

η_{s l}^{(N)} = η_{t h}^{(N)} η_{s p l}^{(N)} .

For the calculation of the diffraction efficiency

η_{t h}^{(N)}

of the imaging shaper we approximated the imaging shaper DOE as a set of adjacent local resonance domain diffraction gratings with same groove depth

h

, and DC but slowly varying period, as shown in Fig. 3. Based on the propagation vector

N_{i n c} (x, y)

per Eq. (2) and polarization vector

e (x, y)

per Eq. (7) of the incident beam and the periods

Λ (x, y)

and angular orientation of the groove fringes

ν (x, y) \cdot Λ (x, y)

per Eq. (14), we calculated parameters of conical diffraction at distinct points

x, y

at the DOE. The parameters are as follows: the angle of incidence

0 \leq θ_{i n c} \leq π / 2

referenced to the normal z to the DOE, the azimuthal angle

0 \leq ϕ_{i n c ν} \leq 2 π

in the DOE plane referenced to the spatial frequency

ν (x, y)

of the local diffraction grating and the polarization angle

β_{i n c}

of the incident wave, which is defined in the plane that is perpendicular to

N_{i n c}

and referenced to the plane of incidence. With such input, we used the rigorous coupled-wave analysis (RCWA) method and Rsoft DiffractMOD software to calculate the local diffraction efficiencies

η_{1}^{} (x, y)

of the local gratings. Finally, the diffraction efficiency of the imaging shaper was calculated as

η_{t h}^{(N)} = \frac{\iint η_{1}^{} (x, y) I_{i n c} (x, y) d x d y}{\iint I_{i n c} (x, y) d x d y} .

An important parameter for optimal diffraction efficiency is the DOE base period $Λ_{0} = Λ (0, 0) = λ / (2 \sin θ_{B})$ at the DOE origin, which support off-axis arrangement of Fig. 4(a) with Bragg angle of incidence per Eq. (1). The base period should be chosen to get diffraction efficiency highest available at $θ_{B}$ and remaining such at slightly changed periods $Λ$ and angles $θ_{i n c}$ that occur at the DOE periphery. We consider here transverse electric (TE) polarized light at the beam waist of the incident beam, while generalization to the transverse magnetic (TM) polarized light can be done in a straightforward way. Figure 5 shows results for the first order diffraction efficiency of a resonance domain diffraction grating as a function of $θ_{i n c}$ , Λ for λ = 0.6328µm, groove depth h = 1.082µm, and DC 0.5, for TE polarization in the model of the classical rigorous diffraction, as applicable for the local gratings of our imaging shaper at $y = 0$ and arbitrary x. Also plotted in Fig. 5 are curves for angle of incidence $θ_{i n c}$ as a function of the local grating period $Λ (x, 0)$ of DOE #1 with $l_{0} = 7.6 mm$ , $l = 1 m$ , for given distinct base periods $Λ (0, 0)$ of 0.51µm, 0.61µm and 0.71µm. Results of Fig. 5 illustrates that the plot with optimal base period $Λ (0, 0) = 0.61 μm$ is confined to the region with the highest diffraction efficiency.

Fig. 5 TE first order diffraction efficiency of resonance domain diffraction gratings as a function of $θ_{i n c}$ , Λ, for λ = 0.6328µm, h = 1.082 µm, DC = 0.5 ; Plotted curves show $θ_{i n c}$ as a function $Λ (x, 0)$ for DOE#1 with $l_{0} = 7.6 mm$ , $l = 1 m$ , $σ_{0} = 2.06 μm$ for basic period $Λ_{0}$ of 0.51(dotted line), 0.61 (dashed line), and 0.71µm (dot dash).

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Figure 6 shows first order local diffraction efficiencies $η_{1}^{} (x, y)$ of top-hat beam shapers #1,2,3,8 per Table 2, with the parameters $Λ_{0} = 0.61 μm$ , $l_{0} = 7.6 mm$ , $l = 1 m$ , $h = 1.082 μm$ , $σ_{0} = 2.06 μm$ and $λ = 0.6328 μm$ , for the case of TE polarization ( $e_{0 p} = 0, e_{0 s} = 1$ ) at the beam waist of the incident beam.

Fig. 6 (a)-(d) Local diffraction efficiency $η_{1}$ of imaging shapers of SL DOEs #1,2,3,8 with parameters; $Λ_{0} = 0.61 μm$ , $l_{0} = 7.6 mm$ , $l = 1 m$ , $D C = 0.5$ , $h = 1.083 μm$ , $λ = 0.6328 μm$ , $l = 1 m$ , $σ_{0} = 2.06 μm$ , and TE polarization.

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The diffraction efficiency $η_{s p l}^{(N)}$ of the Dammann beam splitter is defined as the ratio between powers of the N useful diffraction orders to the total power of all diffraction orders. Our numerical calculations of $η_{s p l}^{(N)}$ are specified in Table 3 and match those published in [34].

Table 3. Parameters and theoretical performance data for SL DOEs.

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Table 3 summarizes our theoretically estimated diffraction efficiencies for the set of the SL DOEs per Eqs. (20) and (21).

5. Fabrication

Eight DOEs #1 to #8 were fabricated with the following parameters; $λ = 0.6328 μm$ , $θ_{B} = 31^{o}$ $σ_{0} = 2.06 μm$ , $l_{0} = 7.6 mm$ , $Λ_{0} = 0.61 μm$ , $h = 1.082 μm$ , $D C = 0.5$ , $2 A = 3.6 mm$ , $2 B = 3 mm$ , $x_{0} = 217 μm$ , $l = 1 m$ , $b = 176 mm$ , $a^{(0)} = b$ , $a^{(N)} = b / (2 N)$ , $N = 1, 2, 4, 8, 16, 32, 64$ in accordance to parameters in Table 2. The ratios $b / l$ and $a^{(0)} / l$ correspond to $\pm 10^{0}$ fanout angle. Based on the DOEs parameters and our in-house software we created GDSII files that controlled direct e-beam writing of the spatial pattern in the EBL process. The fabrication process consisted of the following steps: (a) direct e-beam writing of the spatial pattern in e-beam resist layer (ZEP 520A) with EBL machine Raith EBPG5200, (b) transferring the recorded pattern to a fine Chrome mask by with ion beam milling with ATC-2020-IM machine from AJA, and (c) transferring the spatial pattern from the Chrome mask to the 0.5mm thick FS substrate by CHF3 plasma RIE with Nextral NE 860 machine from Unaxis. The etch-depth in FS was controlled by duration of the RIE process applied to calibration samples in advance. The etch-depth was measured with environmental scanning electron microscopy (ESEM) at the cross section after mechanical cutting. Figure 7 shows surface relief profile of the SL DOE#8 with N = 64 stripes, wherein (a) shows plot of theoretical phase $φ_{s p l}^{(64)}$ of the Dammann grating over half of a period $Λ_{s p l}^{(64)} / \cos θ_{B}$ , (b) shows top-view ESEM image, wherein inversions of the depth of the diffraction grooves are in match to the theoretical phase of the Dammann grating per (a), (c) provides zoom-in of a region of image (b), and (d) shows mechanically cut cross section of a sample SL DOE.

Fig. 7 ESEM images of SL DOE#8 that projects N = 64 stripes. (a) Plot of theoretical phase $φ_{s p l}^{(64)}$ of the Dammann grating over half of a period $Λ_{s p l}^{(64)} / \cos θ_{B}$ ; (b) Top-view image, wherein depth inversion of the diffraction grooves is linked to theoretical phase per (a); (c) zoom-in of a region of image (b); (d) mechanically cut cross section of a sample SL DOE.

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6. Experiments

This section is dedicated to experimental investigation of the performance of SL DOEs. In particular, we provide the main performance data: the patterns contrast, uniformity, and sharpness. For the optical characterization of SL DOEs we used a pigtailed laser diode with wavelength of 642nm (Thorlabs LP642-SF20) coupled to a polarization maintaining single mode fiber (SMF) as light source, a power sensor (Thorlabs S120C), a focusing lens with focal length 30mm and a square aperture with dimensions 25.4mm x 25.4mm.

6.1 Diffraction efficiency

The optical scheme for measurement of diffraction efficiency of SL DOEs is shown in Fig. 8. The DOE was mounted on a rotating stage, and the fiber polarization orientation was set in the direction of y-axis that is perpendicular to the optical Table 1. TE polarization. The square aperture was centered around the chief ray of the first diffraction order at a distance about 72mm from the DOE in order to block the ghost orders of the Dammann beam splitter. The focusing lens was placed adjacent to the square aperture to collect the light power $P_{1}$ onto the power sensor. As a reference, we measured the light power emitted from the SMF $P_{S M F}^{}$ , with the same lens and the power sensor placed right opposite the unobscured SMF incident beam. The diffraction efficiency of the SL DOE was calculated as

η_{s l}^{(N)} (θ_{B}) = \frac{P_{1}^{}}{η_{F r e s n e l}^{} P_{S M F}^{}},

where Fresnel transmission coefficient

η_{F r e s n e l}^{}

was measured according to the procedure detailed in [28] to compensate for the TE Fresnel losses incurred at the smooth surface of the FS DOE substrate.

Fig. 8 The optical scheme for measurement of the diffraction efficiency of the SL DOEs.

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Table 4 summarizes the results of the diffraction efficiency measurements. Some difference from the theoretical efficiencies can be explained by fabrication defects, which may be present in some parts of the DOE aperture.

Table 4. Theoretical and experimental diffraction efficiencies

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6.2 Pattern quality

Crucial parameters for determining the accuracy of 3D sensing using structured light patterns, are the contrast, uniformity, and sharpness of the projected patterns. In order to evaluate the SL DOEs patterns we used the optical scheme of Fig. 4 wherein a white screen was placed perpendicular to the chief ray of the diffracted beam at a distance l = 1m, in accordance to the design of previous section. Actual optical configuration is shown in Fig. 9. The SL pattern that was diffracted from the white screen was captured with a metal–oxide–semiconductor (CMOS) camera (Cannon EOS 30D) which was mounted on the same optical axis as the diffracted chief ray. Another CMOS sensor (Aptina MT9J003) was placed directly in the image plane, for accurate measurements of local intensity distribution in the pattern.

Fig. 9 Photo of the optical setup for experimental evaluation of the SL patterns.

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Figures 10 and 11 show the experimental structured light patterns that were reconstructed from DOE#1-8 with different numbers of the tophat stripes and common dimensions 352.7 mm x 352.7mm for the fanout angle of 20° at distance l = 1m. Some geometrical distortions that make the boundaries of the tophat stripes curved rather than straight lines can be explained by the use of the paraxial tophat shaper phase function $φ_{t h, F a r}$ in off-axis layout and can be corrected as was demonstrated in [38,39]. Few local defects in the patterns, most notable in Figs. 10(a) and 10(b) stem from some stitching errors and nonuniformity in EBL fabrication process of the DOEs.

Fig. 10 Experimental structured light patterns with dimensions 352 mm x 352 mm shot from a white screen at distance l = 1m, from DOEs with following numbers N of tophat stripes: (a) #1, N = 0; (b) #2, N = 1; (c) #3, N = 2; (d) #4, N = 4; (e) #5, N = 8; (f) #6, N = 16;

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Fig. 11 Experimental structured light patterns reconstructed from DOE#7 and 8 with 32 and 64 tophat stripes, at distance l = 1m with dimensions 352 mm x 352 mm. (a),(b) are photos that were shot from a white screen; (c),(g) local images of the patterns direct sensed by the CMOS camera, and (d),(h) their 1-D cross section plots; (e),(f),(i)-(k) magnified parts of (d),(h). The dashed lines show theoretical results whereas the thick line is 200 points moving average.

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For the SL pattern of Fig. 11 reconstructed from DOEs #7 and 8, the tophat widths 2a were small enough (5.5mm and 2.8 mm) to be directly captured locally with the CMOS sensor whose active sensor area is 6.4mm X 3.6mm, as shown in fragment images of the patterns in Figs. 11(c) and 11(g). Plots for intensity in the local cross sections of the pattern in Fig. 11(e), 11(f), 11(i), 11(j) and 11(k) show that lowest intensity is virtually zero and the noise level is quite low. Therefore, the resonance domain DOEs provide high 96% contrast in structured light patterns, even for dense 64 stripes. The widths of the rise zones at the boundaries at the tophat stripes, at levels 0.1 to 0.9 in normalized intensity, were calculated from the plots and summarized in Table 5. The difference between the theoretical and experimental widths of the rise zones can be attributed the DOEs chromatic dispersion, whose impact is estimated as 1.8 mm lateral shift of the chief ray for 1nm spectral width of the laser diode [40].

Table 5. The contrast and rise length for the SL patterns reconstructed from DOEs #7, #8

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7. Conclusion

We designed, fabricated, and characterized a set of eight transmissive resonance-domain surface relief DOEs for binary-encoded structured light patterns. The patterns with 20° fanout angles consist up to 64 stripes with angular pitch of 0.32°. Each of the DOEs demonstrated combined optical functions of imaging, Gaussian-to-tophat beam shaping, and beam splitting, all implemented in a single layer of surface relief with binary profile, with experimental diffraction efficiency of up to 88%. The DOEs are implemented in fused silica material that is suitable for high power lasers. Combination of large fanout angles, separation of useful diffraction order from ghost ones, high diffraction efficiency, high resolution and contrast reveals a breakthrough of the described DOEs compared to wide-spread scalar diffractive optics.

The average efficiency presented in this paper of the eight DOEs is 74%. Several measures could further improve the efficiency, such as advancing to a continuous phase for beam splitting design, varying DC and slant angle of the grooves in binary design. However, varying DC would require more computational resources as numerical results scan an optimal DC for each local grating and, more importantly, substantial complicate the fabrication process. The fabrication of varying slant would be possible after substantial modification of RIE fabrication. Higher fanout angles are possible as well, but, on the expense of efficiency. Some geometrical distortions from straight lines that are seen in the stripes of the SL patterns could be optimized and tailored to the off-axis layout, in further research efforts.

Compactness of the SL projectors may be crucial for placing 3D sensor in mobile platforms, including smartphones with 3D camera for face recognition and virtual reality applications. High power could enable 3D sensors to perform outdoor and increase their working distance. The developed technology could also find applications in such fields as laser micromachining, beam shaping, optical trapping of atoms and particles, pattern recognition, and optical computing.

Funding

Kamin program of the Office of the Chief Scientist, Ministry of Economy.

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DOE#	N	Normalized coordinates for 0 to π phase changes
3	2	0.5.
4	4	0.22, 0.44, 0.5.
5	8	0.061, 0.176, 0.208, 0.317, 0.5.
6	16	0.140, 0.175, 0.221, 0.267, 0.356, 0.395, 0.439, 0.452, 0.5.
7	32	0.055, 0.089, 0.110, 0.133, 0.173, 0.195, 0.210, 0.230, 0.248, 0.330, 0.348, 0.401, 0.433, 0.441, 0.465, 0.484, 0.5.
8	64	0.010, 0.017, 0.029, 0.040, 0.047, 0.06, 0.07, 0.08, 0.09, 0.10, 0.12, 0.14, 0.212, 0.246, 0.262, 0.28, 0.288, 0.296, 0.317, 0.324, 0.337, 0.361, 0.382, 0.396, 0.412, 0.423, 0.433, 0.438, 0.448, 0.45944, 0.466, 0.474, 0.483, 0.489 0.5.

DOE#	N	$Λ_{s p l}^{(N)}$ , µm	$Δ^{(N)}, mm$	2a, mm	2b, mm
1	0	∞	-	352.7	352.7
2	1	∞	-	176.3
3	2	3.63	176.4	88.2
4	4	7.25	88.2	44.1
5	8	14.50	44.0	22.0
6	16	29.01	22.0	11.0
7	32	58.01	11.0	5.5
8	64	116.02	5.6	2.8

DOE#.	N	$η_{s p l}^{(N)}$	$η_{t h}^{(N)}$	$η_{s l}^{(N)}$
1	0	1.00	0.91	0.91
2	1	1.00	0.95	0.95
3	2	0.81	0.96	0.78
4	4	0.70		0.67
5	8	0.76		0.73
6	16	0.81		0.78
7	32	0.83		0.80
8	64	0.80		0.77

DOE#.	N	$η_{s l}^{(N)}$
DOE#.	N	Theoretical	Experimental
1	1	0.91	0.88
2	1	0.95	0.81
3	2	0.78	0.68
4	4	0.67	0.67
5	8	0.73	0.72
6	16	0.78	0.76
7	32	0.80	0.72
8	64	0.77	0.73

DOE No.	N	Experimental Contrast	Rise length (10% to 90%), mm
DOE No.	N	Experimental Contrast	Theoretical	Experimental
7	32	0.98	0.527		1.85
8	64	0.98	0.475		0.936

Multifunctional binary diffractive optical elements for structured light projectors

Abstract

1. Introduction

2. Gaussian-to-tophat imaging shaper

3. From single tophat to multi-stripe patterns

4. Efficiency in rigorous diffraction

5. Fabrication

6. Experiments

6.1 Diffraction efficiency

6.2 Pattern quality

7. Conclusion

Funding

References and links

Cited By

Figures (11)

Tables (5)

Equations (22)

Optics Express