Diffraction of partially-coherent light beams by microlens arrays

Nikolai I. Petrov; Galina N. Petrova

doi:10.1364/OE.25.022545

1. Introduction

Micro-optical elements are widely used in modern optical systems, such as light homogenizers, light emitting diodes (LEDs), etc [1,2]. Another application is the 3D image systems [3–5]. In [6] a microlens array with different curvature unit lenses was studied. Such microlens arrays can be usefull for improving the performance of optical microsystems, such as real-time 3D imaging and parallel laser processing. In [7] a speckle reduction scheme using microlens arrays as screen material for application in laser-based projection systems was proposed. Different types of optical diffusers are used for beam shaping. Ground and opal glasses scatter light in all directions, but have limited light-control capabilities and also they often have low optical efficiency. Diffractive elements are the most powerful method for beam shaping, however, there are some issues such as the limitation to monochromatic illumination, limited divergence angles, and zero order, typically due to fabrication errors. Beam shapers such as periodic microlens arrays, have high transmission efficiency, controlled angular distribution and homogenized light. These structures are non-wavelength dependent and work equally well in white and monochromatic light. High transmission efficiency exceeding 90% can be achieved for wavelengths from 365nm to 2000nm.

The design and modeling of micro-optical systems are still challenging tasks because classical methods such as ray tracing do not take into account diffraction and other coherent effects which appear, for example, in the presence of micro-optical array systems. On the other side, there exist scalar or rigorous diffraction theories to model optical systems. But they are also limited in their applications because they either neglect non-paraxial effects or the calculation time is too high for a practical use.

Different methods for the simulation of micro-optical systems including geometrical optical and wave-optical methods have been applied in [8–12]. In spite of large quantity of publications devoted to the simulations of micro-optical systems, there are still a lot of problems to overcome. Usually publications are devoted to the analysis of microlens array (MLA) with low aspect ratio h/d << 1, where h is the height and d is the footprint width of the microlens. Such microlens arrays give small spreading angles of the scattered light beam. For h/d ~1 and h/d > 1 the methods used should be reconsidered. Existing methods do not take into account random fluctuations of surface profile, polarization effects as well as effects of light source coherence. Although the effects of light source coherence for microlens arrays were recently studied in the experiment [7], they were not previously investigated theoretically.

In this paper effects of spatial coherence of light source and randomization of MLA parameters are considered using analytical, semi-analytical and numerical methods. The influence of the parameters of radiation source (wavelength, wavefront curvature, beam radius, degree of coherence, etc.) and microlens array (periodic or random, sag, pitch size, refractive index, array surface shape and profile (convex, concave), etc.) on the output parameters (intensity distribution, radiation pattern, optical efficiency) of a diffracted beam is investigated. It is shown that the spreading angle and output efficiency of the diffracted beam is more sensitive to the aspect ratio of MLA. The bigger aspect ratio the higher spreading angle and lower output efficiency.

The synthesis method including wave-optics and ray-tracing is developed for the acceleration of simulation of micro-optical systems. The diffraction, coherence and non-paraxial effects should be taken into account for the modeling of microlens arrays. It is shown that the laser source beam, which is diffracted by the periodical microlens array, has non-uniform intensity distribution. Intensity profile and radiation pattern of a LED source beam diffracted by the periodical microlens array are highly uniform and insensitive to the variation of wavelength of incident beam.

It is established by the simulations that the spreading angle can be controlled by the change of sag value. For a given spreading angle the sag can be decreased substantially if the high-index material is used. High-index glasses coated with anti-reflection layer can be considered for manufacturing. The simulations with randomized lens parameters (lens radius, lens curvature radius, etc.) with uniform and Gaussian probability density functions are carried out. Lens array structures with different boundary shapes (square, hexagonal, etc.) are designed.

2. Problem formulation

Consider the beam created by coherent (Laser, Laser Diode (LD)) and partially coherent sources (LED, etc.) which is incident on a microlens array. Our purpose is to analyze the influence of the parameters of a radiation source (wavelength, wavefront curvature, beam radius, degree of coherence, etc.) and microlens array (periodic or random, sag, pitch size, refractive index, array surface shape (spherical, parabolic, aspheric) and profile (convex, concave), etc.) on the output parameters (intensity distribution, radiation pattern, optical efficiency) of a diffracted beam.

Schematic representation of the problem is shown in Fig. 1.

Fig. 1 Optical scheme of the problem.

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The periodic lens array consists of aspheric microlenses (Fig. 1) which have been traditionally defined with the surface profile (sag) given by [13]

s (x, y) = \frac{c_{x} {(x - x_{c})}^{2}}{1 + \sqrt{1 - (1 + κ_{x}) {(x - x_{c})}^{2} c_{x}^{2}}} + \frac{c_{y} {(y - y_{c})}^{2}}{1 + \sqrt{1 - (1 + κ_{y}) {(y - y_{c})}^{2} c_{y}^{2}}} + \sum_{p} A_{x p} {(x - x_{c})}^{p} + A_{y p} {(y - y_{c})}^{p},

where κ is the conic constant, κ = 0 – sphere; κ = −1 – parabola, c_x, c_y are curvatures along x and y axes, respectively, A_xp, A_yp are the aspheric coefficients.

3. Methods and algorithms

Wave propagation and ray-field tracing methods are described in this section. These models take into account the coherence properties of the light beam and geometric parameters of the MLAs.

3.1. Wave propagation

Coherent (lasers and laser diodes) and partially-coherent light sources (LEDs, lamps, etc.) are considered below.

3.1.1. Coherent source

The ground mode of a laser source is well defined with a fundamental Gaussian beam. The complex amplitude u(x, y, 0) of such a Gaussian beam is described by [14]

u (x, y, 0) = A_{0} \exp [- (\frac{x^{2}}{w_{x}^{2}} + \frac{y^{2}}{w_{y}^{2}})] \exp [i \frac{π}{λ} (\frac{x^{2}}{R_{x}} + \frac{y^{2}}{R_{y}})],

where w_x and w_y are the beam radii, R_x and R_y are the radii of wavefront curvatures, x and y are the transverse coordinates, λ is the wavelength.

For laser source the field at the observation plane is given by the Fourier transform of the surface-relief structure or shape of the structured array s(ξ,η), given by the diffraction integral:

E (x, y, z) = \frac{E_{0}}{i λ z} \sum_{j} \iint_{D_{j}} \exp (- \frac{ξ^{2} + η^{2}}{a_{0}^{2}}) \exp [i φ] d ξ d η,

where

φ = k [n (λ) - 1] s_{j} (ξ, η) - \frac{π}{λ z} [{(x - ξ)}^{2} + {(y - η)}^{2}]

, k = 2π/λ, n(λ) is the refractive index, a₀ is the beam radius, the coordinates (ξ,η) define a point in the plane of the micro-lens array, s(ξ,η) is the surface profile, which is divided into N lenslets with a local surface shape s_j, j = 1, …, N and D_j is the surface of the single lenslet.

3.1.2. Partially-coherent sources

Partially coherent electromagnetic field is described by the coherence function [15, 16]

Γ (r, r^{'}, z) = 〈 E^{*} (r, z) E (r', z) 〉,

where E(r,z) is the electric field.

The radiation of partially-coherent light source can be described by Schell-Gauss model beam [15,16]

Γ ({\vec{r}}_{1}, {\vec{r}}_{2}) = I_{0} \exp {- \frac{{\vec{r}}_{1}^{2} + {\vec{r}}_{2}^{2}}{a_{0}^{2}} - \frac{{({\vec{r}}_{1} - {\vec{r}}_{2})}^{2}}{r_{0}^{2}} - \frac{i π}{λ R_{f}} ({\vec{r}}_{2}^{2} - {\vec{r}}_{1}^{2})},

where a₀ is the beam radius, r₀ is the coherence radius, R_f is the wavefront curvature radius,

{\vec{r}}_{1} \equiv (x_{1}, y_{1})

,

{\vec{r}}_{2} \equiv (x_{2}, y_{2})

.

When the correlation radius tends to infinity (r₀ → ∞), the expression (3) will describe the coherence function of completely coherent source.

Intensity distribution in the case of partially-coherent source is described by

\begin{array}{l} I (r, z) = 〈 {| E (r, z) |}^{2} 〉 = \frac{c ε_{0}}{2} {(\frac{k}{2 π z})}^{2} \iint Γ_{0} (r_{1}', r_{2}') \\ 〈 \exp i [Φ (r_{1}') - Φ (r_{2}')] 〉 G^{*} (r_{1}', r, z) G (r_{2}', r, z) d^{2} r_{1}' d^{2} r_{2}' \end{array}

where the angle brackets denote averaging over the field ensemble and over the ensemble of the random medium,

Γ_{0} (r_{1}', r_{2}')

is the coherence function of the light source at initial plane,

Φ (r)

is the phase change caused by the screen (

Φ (r) = k Δ n s (x, y), Δ n = n_{2} - n_{1}

, where n₁ is the refractive index of surrounding medium, n₂ is the refractive index of MLA material), s(x, y) is the MLA surface profile given by Eq. (1),

G (r_{1}, r, z)

is the Green’s function, r(x,y) is the coordinate in transverse plane, z is the longitudinal coordinate, c is the light velocity in free space, ε₀ is the dielectric constant.

The Green’s function is given by the expression

G (x', y', x, y, z) = \frac{1}{i λ} \frac{z \cdot \exp (i k R)}{R^{2}},

where

R = {[{(x - x')}^{2} + {(y - y')}^{2} + z^{2}]}^{\frac{1}{2}} .

Calculation of the intensity distribution using (6) and (7) allows the nonparaxial effects to be taken into account. However, the calculation time in this case is too high for a practical use. Recently, the spectrum of classes of point emitters has been introduced as a novel tool for the numerical modeling of nonparaxial wave fields in any state of spatial coherence [17]. This method might be useful for design of optical devices at the micro- and nanoscales.

In the case of microlenses with d >> λ the paraxial approximation (Fresnel diffraction) can be used. The Green’s function in the paraxial approximation is described by

G (x', y', x, y, z) = \frac{k}{2 π i z} \exp {i k z + \frac{i k}{2 z} [{(x - x')}^{2} + {(y - y')}^{2}]} .

Let’s consider the lens surface profile with cylindrical symmetry.

The intensity distribution of diffracted beam is described by the expression

I (x, y, z) = \frac{I_{0}}{{(λ z)}^{2}} \iint_{} d y_{1}^{'} d y_{2}^{'} F_{y} (y_{1}^{'}, y_{2}^{'}) \tilde{G} (y, y_{1}^{'}, y_{2}^{'}, z) \iint_{D_{m n}} d x_{1}^{'} d x_{2}^{'} F_{x} (x_{1}^{'}, x_{2}^{'}) \tilde{G} (x, x_{1}^{'}, x_{2}^{'}, z),

where

\begin{array}{l} F_{x} (x_{1}^{'}, x_{2}^{'}) = \exp {- \frac{x_{1}^{' 2} + x_{2}^{' 2}}{a_{0 x}^{2}} - \frac{{(x_{1}^{'} - x_{2}^{'})}^{2}}{r_{0 x}^{2}} - \frac{i k}{2 R_{f x}} (x_{1}^{' 2} - x_{2}^{' 2})} \exp {i k Δ n [s_{m} (x_{1}^{'}) - s_{n} (x_{2}^{'})]} \\ F_{y} (y_{1}^{'}, y_{2}^{'}) = \exp {- \frac{y_{1}^{' 2} + y_{2}^{' 2}}{a_{0 y}^{2}} - \frac{{(y_{1}^{'} - y_{2}^{'})}^{2}}{r_{0 y}^{2}} - \frac{i k}{2 R_{f y}} (y_{1}^{' 2} - y_{2}^{' 2})}, \\ \tilde{G} (x, x_{1}^{'}, x_{2}^{'}, z) = \exp {i k z + \frac{i k}{2 z} [{(x - x_{1}^{'})}^{2} - {(x - x_{2}^{'})}^{2}]} \end{array}

\tilde{G} (y, y_{1}^{'}, y_{2}^{'}, z) = \exp {i k z + \frac{i k}{2 z} [{(y - y_{1}^{'})}^{2} - {(y - y_{2}^{'})}^{2}]}

, a_0x and a_0y are the beam radii, r_0x and r_0y are the coherence radii, R_fx and R_fy are the wavefront curvature radii in the x and y coordinate directions, accordingly.

Calculating the integrals over dy₁’dy₂’, we obtain

I (x, y, z) = \frac{I_{0}}{{(λ z)}^{2}} A_{y} \sum_{m n} \iint_{D_{m n}} d x_{1}^{'} d x_{2}^{'} F_{x} (x_{1}^{'}, x_{2}^{'}) \exp {\frac{i k}{2 z} [{(x - x_{1}^{'})}^{2} - {(x - x_{2}^{'})}^{2}]},

where

\begin{array}{l} A_{y} = \frac{π}{\sqrt{a_{y} {\tilde{a}}_{y}}} \exp {- \frac{(a_{y r} - 1 / r_{0 y}^{2}) y^{2}}{2 a_{y} {\tilde{a}}_{y}} \cdot \frac{k^{2}}{z^{2}}}, \\ a_{y} = \frac{1}{a_{0 y}^{2}} + \frac{1}{r_{0 y}^{2}} - \frac{i k}{2 R_{f y}} + \frac{i k}{2 z}, {\tilde{a}}_{y} = \frac{1}{a_{0 y}^{2}} + \frac{1}{r_{0 y}^{2}} + \frac{i k}{2 R_{f y}} - \frac{i k}{2 z} - \frac{1}{r_{0 y}^{4} a_{y}}, a_{y r} = \frac{1}{a_{0 y}^{2}} + \frac{1}{r_{0 y}^{2}}, \\ a_{x} = \frac{1}{a_{0 x}^{2}} + \frac{1}{r_{0 x}^{2}} + \frac{i k}{2 R_{f x}} - \frac{i k}{2 z}, {\tilde{a}}_{x} = \frac{1}{a_{0 x}^{2}} + \frac{1}{r_{0 x}^{2}} - \frac{i k}{2 R_{f x}} + \frac{i k}{2 z} - \frac{1}{r_{0 y}^{4} a_{x}}, a_{x r} = \frac{1}{a_{0 x}^{2}} + \frac{1}{r_{0 x}^{2}} . \end{array}

Rewriting (10) in new coordinates

\vec{r}' = {\vec{r}}_{1}' - {\vec{r}}_{2}', \vec{R}' = \frac{1}{2} ({\vec{r}}_{1}' + {\vec{r}}_{2}')

, we obtain

I (x, y, z) = \frac{I_{0}}{{(λ z)}^{2}} A_{y} \sum_{m n} \iint_{D_{m n}} d r' d R' F_{x} (r', R') \exp {\frac{i k}{2 z} (2 R' r' - 2 x r')} .

a) semi-analytical algorithm

For certain conditions the calculation of the intensity (11) can be significantly simplified. For example, in the case of parabolic lens profiles the integration may be performed analytically. For a parabolic profile we have

F_{x} (r', R') = \exp {- \frac{2 R'^{2}}{a_{0 x}^{2}} - \frac{r'^{2}}{2 a_{0 x}^{2}} - \frac{r'^{2}}{r_{0 x}^{2}} - \frac{i k R' r'}{R_{f x}} + \frac{i k Δ n}{2 R_{s c}} [2 R' r' - 2 R' (x_{0 m} - x_{0 n}) - r' (x_{0 m} + x_{0 n}) + (x_{0 m}^{2} - x_{o n}^{2})]} .

For the sources with the correlation radius r₀ < R_L, there is no contribution to the integrals over dr’ at the bounds of lens cells. This indicates that the integration over dr’ may be extended to infinity, i.e. the integral over dr’ can be calculated analytically.

Thus, the integral (11) reduces to

I (x, y, z) = \frac{I_{0}}{{(λ z)}^{2}} A_{y} \sqrt{\frac{π}{a}} \sum_{m, n} \int_{a_{m}}^{b_{m}} d R' f (R'),

where

\begin{array}{l} f (R') = \exp {\frac{b^{2}}{4 a} - \frac{2 R'^{2}}{a_{0 x}^{2}}} \cos [\frac{k Δ n}{R_{s c}} (x_{0 m}^{2} - x_{0 n}^{2}) - \frac{k Δ n}{R_{s c}} R' (x_{0 m} - x_{0 n})], \\ a = \frac{1}{r_{0}^{2}} + \frac{1}{2 a_{0 x}^{2}}, b = \frac{i k}{R_{f x}} R' - \frac{i k Δ n}{R_{s c}} R' + \frac{i k}{z} (R' - x) + \frac{i k Δ n}{2 R_{s c}} (x_{0 m} + x_{0 n}) . \end{array}

b) numerical algorithm (arbitrary surface profile)

In the case of arbitrary surface profile the intensity distribution is described by

I (x, y, z) = \frac{I_{0}}{{(λ z)}^{2}} A_{y} \sum_{m n} \iint_{D_{m n}} d r' d R' F_{x} (r', R') \exp {\frac{i k}{2 z} (2 R' r' - 2 x r')} .

Let’s consider the lens base of the square symmetry (Fig. 2). The intensity distribution of a beam diffracted by micro-lens array of square symmetry is described by

I (x, y, z) = \frac{I_{0}}{{(λ z)}^{2}} \iint_{D_{p l}} d y_{1}^{'} d y_{2}^{'} F_{y} (y_{1}^{'}, y_{2}^{'}) \tilde{G} (y, y_{1}^{'}, y_{2}^{'}, z) \iint_{D_{m n}} d x_{1}^{'} d x_{2}^{'} F_{x} (x_{1}^{'}, x_{2}^{'}) \tilde{G} (x, x_{1}^{'}, x_{2}^{'}, z),

where

\begin{array}{l} F_{x} (x_{1}^{'}, x_{2}^{'}) = \exp {- \frac{x_{1}^{' 2} + x_{2}^{' 2}}{a_{0 x}^{2}} - \frac{{(x_{1}^{'} - x_{2}^{'})}^{2}}{r_{0 x}^{2}} - \frac{i k}{2 R_{f x}} (x_{1}^{' 2} - x_{2}^{' 2})} \exp {i k Δ n [s_{m} (x_{1}^{'}) - s_{n} (x_{2}^{'})]}, \\ F_{y} (y_{1}^{'}, y_{2}^{'}) = \exp {- \frac{y_{1}^{' 2} + y_{2}^{' 2}}{a_{0 y}^{2}} - \frac{{(y_{1}^{'} - y_{2}^{'})}^{2}}{r_{0 y}^{2}} - \frac{i k}{2 R_{f y}} (y_{1}^{' 2} - y_{2}^{' 2})} \exp {i k Δ n [s_{p} (y_{1}^{'}) - s_{l} (y_{2}^{'})]} . \end{array}

Rewriting the Eq. (15) in new coordinates

\vec{r}' = {\vec{r}}_{1}' - {\vec{r}}_{2}', \vec{R}' = \frac{1}{2} ({\vec{r}}_{1}' + {\vec{r}}_{2}')

, we have

I (x, y, z) = \frac{I_{0}}{{(λ z)}^{2}} \iint_{D_{p l}} d ρ' d S' F_{y} (ρ', S') \exp {\frac{i k}{2 z} (2 S' ρ' - 2 y ρ')} U,

where

\begin{array}{l} U = \iint_{D_{m n}} d r' d R' F_{x} (r', R') \exp {\frac{i k}{2 z} (2 R' r' - 2 x r')}, \\ F_{x} (r', R') = \exp {\begin{cases} - \frac{2 R'^{2}}{a_{0 x}^{2}} - \frac{r'^{2}}{2 a_{0 x}^{2}} - \frac{r'^{2}}{r_{0 x}^{2}} - \frac{i k R' r'}{R_{f x}} + \\ \frac{i k Δ n}{2 R_{s c}} [2 R' r' - 2 R' (x_{0 m} - x_{0 n}) - r' (x_{0 m} + x_{0 n}) + (x_{0 m}^{2} - x_{o n}^{2})] \end{cases}} \end{array}

and

F_{y} (ρ', S') = \exp {\begin{cases} - \frac{2 S'^{2}}{a_{0 y}^{2}} - \frac{ρ'^{2}}{2 a_{0 y}^{2}} - \frac{ρ'^{2}}{r_{0 y}^{2}} - \frac{i k S' ρ'}{R_{f y}} + \\ \frac{i k Δ n}{2 R_{s c}} [2 S' ρ' - 2 S' (y_{0 p} - y_{0 l}) - ρ' (y_{0 p} + y_{0 l}) + (y_{0 p}^{2} - y_{o l}^{2})] \end{cases}} .

In the case of low coherence source (r₀ << R_L) we can calculate integrals over dρ and dr’ analytically, so the integral (16) reduces to

I (x, y, z) = \frac{I_{0}}{{(λ z)}^{2}} \sum_{m n}^{} \iint_{D_{m n}} d S' d R' f (S', R', x, y, z),

where

\begin{array}{l} f (S', R', x, y, z) = \frac{π}{\sqrt{a \tilde{a}}} \exp {- \frac{b_{i}^{2}}{4 a} - \frac{2 R'^{2}}{a_{0 x}^{2}} - \frac{{\tilde{b}}_{i}^{2}}{4 \tilde{a}} - \frac{2 S'^{2}}{a_{0 y}^{2}}} \cos [\frac{k Δ n}{R_{s c}} \cdot Φ], \\ Φ = \frac{x_{0 m}^{2} - x_{0 n}^{2}}{2} - R' (x_{0 m} - x_{0 n}) + \frac{y_{0 m}^{2} - y_{0 n}^{2}}{2} - S' (y_{0 m} - y_{0 n}), \\ b_{i} = \frac{k}{R_{f x}} R' + \frac{k}{z} (R' - x) - \frac{k Δ n}{2} (\frac{2 R'}{R_{s c}} - \frac{x_{0 m} + x_{0 n}}{R_{s c}}), \\ {\tilde{b}}_{i} = \frac{k}{R_{f y}} S' + \frac{k}{z} (S' - y) - \frac{k Δ n}{2} (\frac{2 S'}{R_{s c}} - \frac{y_{0 m} + y_{0 n}}{R_{s c}}), a = \frac{1}{r_{0 x}^{2}} + \frac{1}{2 a_{0 x}^{2}}, \tilde{a} = \frac{1}{r_{0 y}^{2}} + \frac{1}{2 a_{0 y}^{2}} . \end{array}

Note that the above considered scalar wave approach does not take into account the polarization effects. Effects due to the polarization are considered below in the section 4.4.

Fig. 2 The region of integartion.

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3.2. Ray-field tracing

For consideration of large aperture micro-optical systems, conventional methods, such as physical optics, become overly time-consuming. Beam-mode representations do not provide right alternative, since the beam modes cannot be tracked simply through the arbitrary curved surfaces. Conventional ray tracing methods can be used to simulate micro-lens arrays as long as diffraction effects can be neglected and if the light is incoherent. Synthesis methods including wave optics and geometrical optics (ray tracing) are needed for the wave-optical simulation of micro-optical systems. Discrete phase-space methods have been proposed as an efficient alternative. These methods represent fields as discrete and finite superposition of elementary Gaussian beams that can be traced easily in a complicated environment. Geometrical ray tracing takes into account non-paraxial effects but it does not take into account diffraction effects. Below (section 3.2.2) we propose the coherent states method, which allows both the non-paraxial effects caused by the surface structure and diffraction to take into account. Moreover, this approach takes into account the Fresnel losses at boundaries depending on the polarization of an incident beam.

3.2.1. Gaussian beam propagation

The multi-Gaussian beam shape composed of a sum of Gaussian function components can be considered as a model for laser beam profiles. The general formula for the complex electric field distribution at an input plane is given by [18]

E (x) = E_{0} \frac{\sum_{m = - N}^{N} \exp [- {(\frac{x - m w}{w})}^{2}]}{\sum_{m = - N}^{N} \exp (- m^{2})},

where w is the radius of Gaussian field distribution.

Propagation of multi-Gaussian beams in free-space and optical systems can be analyzed analytically using ABCD matrices [19].

However, displaced Gaussian beams do not form the full set of the functions. Below the coherent states, representing elementary Gaussian beams with axis displacement and tilt are proposed for the calculations.

3.2.2. Coherent states approach

Consider the Gaussian wave packets, which are incident at the MLA surface. Note, that in a sense such Gaussian beams are similar to the complex rays used for simulation of reflection and transmission of beams at a curved interface in [20]. Coherent states (CS) form a full set of functions $π^{- 1} \int d^{2} α | α 〉〈 α | = 1$ , so the arbitrary incident field E(x,0) can be expanded into a set of CS [21]:

E (x, 0) = π^{- 1} \int d^{2} α 〈 x | α 〉 f (α)

where

〈 x | α 〉

is given by the expression (3),

d^{2} α = d (Re α) d (Im α)

is the elementary phase-space volume,

f (α) = \int d x 〈 α | x 〉 E (x, 0)

are the amplitudes of the expansion.Coherent state can be determined as the eigenfunction of the operator

\overset{⌢}{a}

:

\overset{⌢}{a} | α 〉 = α | α 〉,

where

\overset{⌢}{a} = \frac{\hat{x}}{w_{0}} + i \frac{k w_{0}}{2} {\hat{p}}_{x}, {\overset{⌢}{p}}_{x} = - \frac{i}{k} \frac{\partial}{\partial x} .

An explicit form of CS is given by a Gaussian beam function

| α 〉 = Ψ_{α} (x) = {(\frac{2}{π w_{0}^{2}})}^{\frac{1}{4}} \exp (- \frac{x^{2}}{w_{0}^{2}} + \frac{2 x}{w_{0}} α - \frac{α^{2}}{2} - \frac{{| α |}^{2}}{2})

where the complex eigenvalues

α = \frac{x_{0}}{w_{0}} + i \frac{k w_{0}}{2} p_{x 0} = | α | \exp (i ϑ)

determine the initial coordinates x₀ of the center of the elementary beam and the angle of its inclination p_x0 = n₀sinθ₀ to the z-axis, n₀ is the refractive index of medium, w₀ is the elementary beam width, k = 2π/λ is the wavenumber.

The calculation procedure consists of several steps. First, we apply the expansion of incident field into CS (Gaussian rays). Second, the CS rays are reflected and transmitted at the surface according the Fresnel laws. Third, total reflected and transmitted powers are determined by summation of all elementary beams (rays).

Note, in contrast to Fourier-expansion, there is no requirement for orthogonality of functions, owing to CS form overfull function system. The square of the modules ${| f (α) |}^{2} = {| 〈 α | E 〉 |}^{2}$ determine the incident beam power distribution between the elementary beams (CS). For the incident Gaussian beam $E (x, 0) = {(2 / π a_{0}^{2})}^{1 / 4} \exp (- x^{2} / a_{0}^{2})$ the amplitudes of expansion have the form

{| f (α) |}^{2} = \frac{2 w_{0} / a_{0}}{1 + w_{0}^{2} / a_{0}^{2}} \exp {- {| α |}^{2} + {| α |}^{2} \frac{1 - w_{0}^{2} / a_{0}^{2}}{1 + w_{0}^{2} / a_{0}^{2}} \cos 2 ϑ} .

For partially-coherent light beam we have

Γ (x_{1}, x_{2}) = 〈 E (x_{1}) E (x_{2}) 〉 = π^{- 2} \iint d^{2} α d^{2} β Ψ_{α}^{*} (x_{1}) Ψ_{β} (x_{2}) 〈 f (α^{*}) f (β) 〉,

where

〈 f (α *) f (β) 〉 = \iint d x_{1} d x_{2} Ψ_{α}^{*} (x_{1}) Ψ_{β} (x_{2}) Γ (x_{1}, x_{2}) \exp (\frac{1}{2} {| α |}^{2} + \frac{1}{2} {| β |}^{2}) .

Evolution of the coherence function with distance is given by

Γ (x_{1}, x_{2}, z) = 〈 E (x_{1}, z) E (x_{2}, z) 〉 = π^{- 2} \iint d^{2} α d^{2} β Ψ_{α}^{*} (x_{1}, z) Ψ_{β} (x_{2}, z) 〈 f (α^{*}) f (β) 〉,

where

Ψ_{α} (x, z) = \int d x^{'} G (x, x^{'}, z) Ψ_{α} (x^{'}, 0),

and

G (x, x^{'}, z) = \frac{1}{\sqrt{2 π i z / k}} \exp [i k z + i \frac{k (x^{2} + {x^{'}}^{2} - 2 x x^{'})}{2 z}]

is the Green’s function in free space, and the function

Ψ_{α} (x^{'}, 0)

is given by the Eq. (21).

Note that in the paraxial approximation the integral (24) can be calculated analytically. This allows the calculation time to be greatly reduced.

Note that for the consideration of propagation of partially coherent light beams in inhomogeneous medium the density-matrix formalism can be used [22].

4. Simulations

The procedure of simulations includes the specifying the initial data (the parameters of radiation source: $λ$ - wavelength, wavefront curvature, a₀ - beam radius, coherence radius r₀, etc. and micro-lens array: periodic or random, sag, pitch size, refractive index, array surface shape – parabolic, spherical, aspheric and profile (convex, concave), etc.). The output parameters (intensity distribution, radiation pattern, optical efficiency) are obtained by calculations of integrals (Eqs. (13), (14), and (17)).

4.1. Lens surface shapes

Lenses with cylindrical, square, hexagonal and crossed-cylindrical base shapes are considered. Note that both rotationally symmetric and non-symmetric lens surfaces are considered. In Fig. 3 the array with convex surface profile is presented. In Fig. 4 the integration area for hexagonal lens base is shown. The parabolic surface profile is described by the relationship $s (x) = s_{0} - \frac{{(x - x_{0 m})}^{2}}{2 R_{s c}}$ , where R_L is the lenslet radius, R_sc is the lenslet surface curvature radius.

Fig. 3 Array with convex profile of microlenses: x, y, and z in µm.

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Fig. 4 Integration area for hexagonal lens base.

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4.2. Simulation results

In Fig. 5 the intensity distributions for different values of coherence radius r₀ are presented at the distance of z = 5 mm from the cylindrical MLA with parabolic profile: R_L = 10 μm, R_sc = 4μm. The aspect ratio is equal to h/d = 0.625.

Fig. 5 Intensity distributions at the distance z = 5 mm for different coherent radii r₀: (a) r₀ = 3 μm; (b) r₀ = 10 μm; (c) r₀ = 30 μm; (d) r₀ = 300 μm.

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It is seen from the Fig. 5 that the structured diffraction patterns appear in the intensity distributions with the increase of the coherence radius r₀ of the incident light beam. It is followed from the simulations that noticeable diffraction patterns appear when the coherence radius r₀ becomes of the order of the microlens footprint radius R_L. Similar result was obtained from the measurements in [7]. In the case of fully coherent incident field similar diffraction patterns were obtained in [10, 11].

In Fig. 6 the formation of independent beams during the propagation of light behind the MLA screen is shown. It is followed from the simulations that each of the beams propagate independently. The angular separation of the beams depends on the aspect ratio of MLA. For the considered values of R_L = 10 μm and R_sc = 3 μm the angular separation between spots is equal to 16.23 mrad.

Fig. 6 Intensity distributions at different distances z for R_L = 10 μm and R_sc = 3 μm: (a) z = 1 mm; (b) z = 5 mm.

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In Fig. 7 the intensity distributions and radiation patterns for coherent (a, c) and low-coherent (b, d) sources are presented.

Fig. 7 Intensity distributions (a, b) and radiation patterns (c, d) for LD (a, c) and LED (b, d) sources: z = 5mm.

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Note that the spatially separated spots can be observed at the far-field region for r₀ >> R_L. The angular separation of the spots depends on the parameters of the microlenses and increases with the increase of the sag value.

Thus, the structured diffraction pattern appears behind the MLA screen if the coherence radius r₀ is larger than the microlens radius R_L. Moreover, multiple beams are formed at far-field region. The appearance of the structured diffraction pattern when the radiation coherence radius r₀ becomes of the size of the same order or bigger than the microlens footprint width, was also shown in the experiment in [7].

4.3. Random MLA

Different types of randomization methods for the surface profile are considered: uniform and Gaussian probability distribution functions, etc. Randomization is necessary in order to increase the uniformity of diffracted light intensity distribution.

Random values for microlens radius R_L, curvature radius R_sc with a given rms values σ with uniform and Gaussian distributions were generated (Fig. 8). In Figs. 8(a) and 8(b) the intensity distributions for the coherent source r₀ >> R_L are shown for regular MLA and randomized MLA, accordingly. In Figs. 8(c) and 8(d) similar intensity distributions are presented for low-coherence source, when r₀ < R_L.

Fig. 8 Intensity distributions of diffracted light for coherent (a, b) and low-coherent (c, d) sources. (a, c) – regular MLA, (b, d) – random MLA.

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It is followed from the simulations that the small-scale intensity oscillations appear at the randomization of the microlens parameters for both coherent and incoherent sources. The contrast of the structured diffraction pattern which is caused by the regular structure of the MLA can be reduced if the irregular MLA with the variation in the size and shape of the different lenses is considered. Randomization of the parameters of MLA leads to the irregular intensity distribution, i.e. the diffraction pattern attains irregular speckle structure. Although the randomization of the parameters destroys the structured diffraction pattern, it does not allow us to obtain the intensity distribution with high uniformity. Tolerance analysis of the MLA parameters is conducted. The tolerances for lenslet parameters variation can be translated into the microlens array fabrication process. Note that various fabrication technologies allow lenslet parameters variation within a few percent.

To obtain the uniform intensity distribution the surface relief diffusers, such as diffraction diffusers and holographic diffusers can be used. Diffractive diffusers also known as computer generated holograms, phase plates and kinoforms. However diffraction diffusers can be used with monochromatic light only. Also it is difficult to eliminate the “zero-order” diffraction order, even at the design wavelength. Speckle noise reduction by adding random phases generated by a spatial light modulator into the illuminating beam is also known. In [23] speckle noise reduction method in digital holography by using digital image processing was proposed. Note that holographic diffusers [24] and RPC engineered diffusers provide the controlled light distributions and achromatic performance. Typically, holographic diffusers have high optical transmission efficiency on the order of 90%.

4.4. Optical efficiency

Optical efficiency of the MLA depends on the parameters of MLA and light source. Below we show that the CS method is an effective approach to calculate the reflection and transmission coefficients of light at MLA boundaries.

Consider the curved boundary between two different dielectric media (Fig. 9). The absence of media losses is assumed below. For simplicity, the two-dimensional periodically corrugated interface (y–independent) with the period d >> λ is considered, but the extension to the arbitrary profile of 3D case is straightforward.

Fig. 9 Geometrical configuration and coordinate system for a boundary.

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The calculation procedure of the reflected and transmitted powers consists of the following steps. At first, the incident beam field is expanded into coherent states, representing elementary Gaussian beams with axis displacement and tilt.

The Gaussian elementary beams (CS) pass through the interface as determined by the corresponding Fresnel coefficients that vary according to the angle of incidence. Usually the Fresnel formulae are known from the plane–wave limit. However, these formulae can be used also for localized wave beams with the beam waists w > λ [25, 26].

Finally, total reflected and transmitted powers are determined by a summation of powers of all elementary beams. The reflectance and transmittance are defined as the ratios of reflected and transmitted powers to the incident power, accordingly:

r = \frac{P^{r}}{P^{i}} = \frac{\int d^{2} α {| f (α) |}^{2} R (θ_{1}^{i})}{\int d^{2} α {| f (α) |}^{2}}, t = \frac{P^{t}}{P^{i}} = \frac{\int d^{2} α {| f (α) |}^{2} T (θ_{1}^{i})}{\int d^{2} α {| f (α) |}^{2}},

where

R (θ_{1}^{i})

and

T (θ_{1}^{i})

are the reflection and transmission coefficients,

θ_{1}^{i}

is the incident angle.

The reflection and transmission coefficients for TE (transverse electric) and TM (transverse magnetic) linearly polarized incident beams, accordingly, are given by the expressions [27]:

\begin{array}{l} R_{E} = \frac{{(n_{1} \cos θ_{1}^{i} - \sqrt{n_{2}^{2} - n_{1}^{2} \sin^{2} θ_{1}^{i}})}^{2}}{{(n_{1} \cos θ_{1}^{i} + \sqrt{n_{2}^{2} - n_{1}^{2} \sin^{2} θ_{1}^{i}})}^{2}}, T_{E} = \frac{4 n_{1} \cos θ_{1}^{i} \sqrt{n_{2}^{2} - n_{1}^{2} \sin^{2} θ_{1}^{i}}}{{(n_{1} \cos θ_{1}^{i} + \sqrt{n_{2}^{2} - n_{1}^{2} \sin^{2} θ_{1}^{i}})}^{2}}, \\ R_{H} = \frac{{(n_{2} \cos θ_{1}^{i} - \frac{n_{1}}{n_{2}} \sqrt{n_{2}^{2} - n_{1}^{2} \sin^{2} θ_{1}^{i}})}^{2}}{{(n_{1} \cos θ_{1}^{i} + \frac{n_{1}}{n_{2}} \sqrt{n_{2}^{2} - n_{1}^{2} \sin^{2} θ_{1}^{i}})}^{2}}, \\ T_{H} = \frac{4 n_{1} \cos θ_{1}^{i} \sqrt{n_{2}^{2} - n_{1}^{2} \sin^{2} θ_{1}^{i}}}{{(n_{2} \cos θ_{1}^{i} + \frac{n_{1}}{n_{2}} \sqrt{n_{2}^{2} - n_{1}^{2} \sin^{2} θ_{1}^{i}})}^{2}}, \end{array}

where n₁ and n₂ are the refractive indexes of media 1 and 2, accordingly.

For the surface-relief profile z = s(x), the ray (CS) with initial coordinates (x₀,θ₀) strikes the interface at the point (x₁, z₁), where the corresponding incidence angle $θ_{1}^{i}$ is uniquely determined. For example, the incident angle can be expressed as $θ_{1}^{i} = ϕ - θ_{0}$ , where $ϕ = \arctan [s^{'} (x)]$ , $s^{'} (x) = \frac{d s (x)}{d x}$ is the derivative of the surface profile function with respect to the x coordinate.

Results of simulation are presented for different surface profiles, wavefront curvature radiuses and polarizations of incident beam. The parameters for incident beam, surface-relief, and elementary beam are in the ratio a₀ > d >> w₀ > λ, where d is the diameter of the single element in corrugated surface (Fig. 9). In Fig. 10 the reflectance and transmittance as function of sag h of the surface relief $s (x) = h_{0} \cos^{2} (π x / d)$ with the period d = 50μm for TE and TM polarized beams with different wavefront curvature radiuses are presented. Analogical dependences are obtained for the parabolic surface profile $s (x) = s_{0} - \frac{{(x - x_{0 m})}^{2}}{2 R_{s c}}$ , where $s_{0} = \frac{R_{L}^{2}}{2 R_{s c}}$ , R_L = d/2 and R_sc are the radius and curvature radius of the single element, x_0m is the center coordinate of the single element. Lesser sensitivity of the reflection and transmission to the sag h and wavefront curvature radius R_f changes is observed for parabolic surface profile. Reflectance increases and transmittance decreases with the increase of sag h.

Fig. 10 Reflectance r (red curves) and transmittance t (blue curves) versus depth h at lower/higher index and higher/lower index interfaces for different values of incident wavefront curvature radiuses: left - TE polarized beam (a, c), right – TM polarized beam (b, d); curves 1, R_f = 1500 μm; curves 2, plane wavefront.

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The reflectance and transmittance are sensitive to the wavefront curvature radius and polarization of the incident beam. For lower-higher index interface the reflectance is lower and transmittance is higher for TM polarized beam if h < d. For higher-lower index interface there is no evident difference between TE and TM polarizations. It follows from the simulations that the decrease of the reflectance and increase of the transmittance take place with the increase of the sag h for TM polarization owing to Brewster angle effect.

Efficiency η is determined as a ratio of transmitted power P_t to total incident power P_in: η = P_t/P_in, where $P_{i n} = \iint d x d y {| E (x, y, 0) |}^{2}$ .

In Fig. 11 the optical efficiency and spreading angle as function of the aspect ratio h/d are presented for microlenses with a parabolic surface profile.

Fig. 11 Optical efficiency and spreading angle as function of aspect ratio.

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Thus, the optical efficiency of MLA decreases with the increase of the aspect ratio. Note that the optical efficiency depends on the direction from which the light beam is incident. The efficiency is higher if the beam is incident from the lens surface profile side. This difference is stronger for bigger aspect ratios.

4.5. Interface development

User Friendly Layout for graphical and data outputs is developed (see, for example, Fig. 12). Interface includes window for introducing initial parameters of light source (wavelength, radius of beam, radius of wavefront curvature, coherence radius), window for introducing parameters of micro-lens array (lens radius R_L, lens surface curvature radius R_sc, periodic or random, concave or convex surfaces, frontal and outflow face illumination, refractive index of lens material), window for introducing the distance between the lens array and observation plane, and window for graphical and data outputs of simulated results (Intensity Profile, Radiation Pattern, Output Efficiency and Surface Sag Profile.

Fig. 12 User friendly interface: input data (a) and graphical outputs (b).

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4.6. Measurements

In Fig. 13 the measured intensity distributions of the diffracted radiation by microlens array are presented for the LD and LED sources. The microlens array with the diameter of the lens of d = 127 μm (200 lpi) is used in the measurements. As seen from the Fig. 13 the light beams from LD and LED sources are diffracted differently. In the case of periodic structure at $r_{c o h} > > d$ the non-uniformity in the intensity distribution of the diffracted beam is clearly observed. The periodic structure splits the incident coherent beam (LD) into multiple separate beams (spots), while uniform intensity distribution is observed for a low coherence source (LED) (Fig. 13(d)). The angular separation of spots is equal to $θ$ = 5.17 mrad. The spots are not resolved at z = 45 cm because the separation distances between the centers of spots are smaller than the spot radiuses.

Fig. 13 Measured intensity distributions of the diffracted radiation by microlens array at different distances with LD source: (a) z = 45cm; (b) z = 145cm; (c) z = 290cm; (d) LED source.

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For the microlens array with the footprint width d = 254 μm (100 lpi) the angular separation of spots is equal to $θ$ = 2.62 mrad. The distance between the spots is bigger for the arrays with a large number of microlenses per inch or with a smaller diameter d of the raster. These experimental observations are in good agreement with the calculation results. The structured diffraction patterns are not desirable in laser-based image projection systems [7]. However, this effect can be used for the splitting of laser beams into the multiple beams.

5. Summary

Wave-optics and ray-field tracing methods for the simulation of micro-lens arrays taking into account the coherence and polarization effects of light source, randomization of microlens array parameters are implemented. The new ray-field approach based on the coherent states representation is developed for calculation of the optical efficiency of the microlens arrays. Such wave beams can be tracked simply through the arbitrary curved surfaces. Results of simulations of intensity distribution and spreading angle of a diffracted beam are in good agreement with the measurements.

The influence of the parameters of radiation source (wavelength, wavefront curvature, beam radius, coherence radius, etc.) and micro-lens array (periodic or random, aspect ratio, pitch size, refractive index, array surface shape and profile (convex, concave), etc.) on the output parameters (intensity distribution, radiation pattern, optical efficiency) of a diffracted beam is investigated. It is shown that the spreading angle and the output efficiency of the diffracted beam is more sensitive to the aspect ratio of MLA. The bigger is the aspect ratio, the higher is the spreading angle and lower output efficiency.

The simulations clearly indicate that the intensity distributions and radiation patterns of the beams diffracted by the MLA are sensitive to the degree of coherence of light source. The laser source beam diffracted by the periodical microlens array has strongly non-uniform intensity distribution, so LED sources are more appropriate for display lighting systems. The intensity profile and radiation pattern of a LED source beam diffracted by the periodical microlens array are highly uniform and insensitive to the variation of wavelength of incident beam.

The randomization of lens parameters (lens radius and lens curvature radius) changes the intensity profile and radiation pattern. The uniformity of the intensity distribution mainly depends on the coherence of light source: lower coherence source results in higher degree of uniformity. The tolerance analysis of the MLA parameters is conducted. It is shown that the small variations (1-3%) of lens radiuses and lens curvature radiuses relative to mean values do not give visible changes in the intensity profile and radiation pattern. Micro-lens arrays with optimal parameters are designed using a developed simulation tool. There are different fabrication technologies of MLAs [28–33]. Existing technologies allow fabrication of MLAs with high aspect ratios. MLA diffuser was replicated from the nickel master mold and a radiation pattern with a scattering angle of 150° was showed in [30, 31].

It is established from the simulations that the spreading angle can be controlled by the change of sag value. For a given spreading angle the sag can be decreased substantially if the high-index material is used. High-index glasses (sapphire; LaSFN9; LaSFO15, OHARA, etc), LiNbO₃ and GaP coated with anti-reflection layer can be considered for manufacturing.

The simulation results are in good agreement with the measurements using existing MLAs. The splitting of the incident beam into the multiple spots takes place if the coherent source is used. The structured diffraction pattern appears behind the MLA screen if the coherence radius r₀ of the source is larger than the microlens radius R_L.

User Friendly Interface is developed for introducing initial parameters of light source and MLA array and for graphical and data outputs of simulated results.

The results obtained may be useful for various illumination systems, LED backlighting systems, for speckle reduction in laser-based projection systems, 3D displays, etc.

Funding

Russian Science Foundation (RSF) (17-19-01461).

References and links

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Diffraction of partially-coherent light beams by microlens arrays

Abstract

1. Introduction

2. Problem formulation

3. Methods and algorithms

3.1. Wave propagation

3.1.1. Coherent source

3.1.2. Partially-coherent sources

a) semi-analytical algorithm

b) numerical algorithm (arbitrary surface profile)

3.2. Ray-field tracing

3.2.1. Gaussian beam propagation

3.2.2. Coherent states approach

4. Simulations

4.1. Lens surface shapes

4.2. Simulation results

4.3. Random MLA

4.4. Optical efficiency

4.5. Interface development

4.6. Measurements

5. Summary

Funding

References and links

Cited By

Figures (13)

Equations (34)

Optics Express