Coherence modulation by deterministic rotating diffusers

Joonas Lehtolahti; Markku Kuittinen; Jari Turunen; Jani Tervo

doi:10.1364/OE.23.010453

1. Introduction

A rotating random diffuser is the most widely used, and the most straightforward method to reduce the spatial coherence of a light beam. Such coherence control is essential, for example, in speckle reduction [1], and the theoretical treatment is well understood [2, 3]. For example, if the incident Gaussian laser-beam is essentially spatially coherent, the coherence properties of the beam after the rotating diffuser obey closely the so-called Gaussian Schell-model [4].

The coherence of a light beam can be tailored, not only by random diffusers, but also by deterministic components whose transmission properties are altered fast compared to the integration time of the detector. Early realizations of such deterministic diffusers include the use of traveling ultrasonic waves, see [5] and the work cited therein. By deterministic modulation of the acoustic wave form [6], one can generate a wide class of coherence functions of either one-dimensional or separable form. For instance, transformation of coherent Gaussian beams to Gaussian Schell-model beams has been demonstrated [7].

Perhaps the most straightforward, and most flexible, way to implement custom-made coherence distribution is to make use of a spatial light modulator [8, 9] or a digital micromirror device [10]. In this way one is free from limitations such as separability of the coherence function. However, we show that interesting classes of coherence functions and partially coherent beams can be generated using more restricted schemes involving rotating deterministic diffusers. This scheme has recently been suggested for generation of partially coherent annular sources [11, 12]. Here we consider the case in which the diffuser is illuminated by an extended coherent beam (instead of an annular aperture) and evaluate coherence functions that can be achieved by rotating spiral-like diffusers. An experimental demonstration is provided to confirm the theoretical results.

2. Theory

We consider the following geometry (see Fig. 1): a spatially completely coherent paraxial quasi-monochromatic beam with complex amplitude U₀(ρ, ϕ) is incident on a diffuser with a complex-amplitude transmittance function t(ρ, ϕ). Here ρ and ϕ are the radial and azimuthal polar coordinates, respectively. The origin of the coordinate system is in the center of the diffuser, that is rotated around the longitudinal z axis.

Fig. 1 The geometry of illuminating a rotating diffuser with a coherent beam and the setup for measuring the spatial coherence properties of the resulting secondary source. L: Laser source, E: Rotating diffuser element, A: Aperture, M: Micromirror device, C: Camera.

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In principle, the velocity difference between the center and the edge of the diffuser leads to a Doppler shift, but such an effect can be ignored provided that the angular velocity is low enough. Hence, denoting the temporal rotation period of the diffuser by D, the transmitted field at time instant τ is given by

U (ρ, ϕ; τ) = t (ρ, ϕ - 2 π τ / D) U_{0} (ρ, ϕ) .

If the detector averages over a large number of rotation periods, the spatial coherence of the transmitted field, characterized by the zero-time mutual coherence function, can be calculated by time-averaging over a single period:

J (ρ_{1}, ϕ_{1}, ρ_{2}, ϕ_{2}) = \frac{1}{D} \int_{0}^{D} U^{*} (ρ_{1}, ϕ_{1}; τ) U (ρ_{2}; ϕ_{2}; τ) d τ .

Inserting the expression of the transmitted field from Eq. (1) to Eq. (2) and denoting the mutual intensity of the incident field by

J_{0} (ρ_{1}, ϕ_{1}, ρ_{2}, ϕ_{2}) = U_{0}^{*} (ρ_{1}, ϕ_{1}) U_{0} (ρ_{2}, ϕ_{2})

we have

J (ρ_{1}, ϕ_{1}, ρ_{2}, ϕ_{2}) = J_{0} (ρ_{1}, ϕ_{1}, ρ_{2}, ϕ_{2}) \frac{1}{D} \int_{0}^{D} t^{*} (ρ_{1}, ϕ_{1}, - 2 π τ / D) t (ρ_{2}, ϕ_{2} - 2 π τ / D) d τ .

To get further insight into the theory, we next express the complex-amplitude transmittance function in the form of an azimuthal Fourier series

t (ρ, ϕ) = \sum_{m = - \infty}^{\infty} G_{m} (ρ) exp (i m ϕ),

where

G_{m} (ρ) = \frac{1}{2 π} \int_{0}^{2 π} t (ρ, ϕ) exp (- i m ϕ) d ϕ

are the Fourier coefficients that depend on the radial coordinate. With the help of Eq. (5), we may now perform the time integration in Eq. (2), which gives

J (ρ_{1}, ϕ_{1}, ρ_{2}, ϕ_{2}) = J_{0} (ρ_{1}, ϕ_{1}, ρ_{2}, ϕ_{2}) \sum_{m = - \infty}^{\infty} G_{m}^{*} (ρ_{1}) G_{m} (ρ_{2}) exp (i m Δ ϕ)

with Δϕ = ϕ₂ − ϕ₁.

The normalized mutual intensity, or complex degree of spatial coherence, of the transmitted field now takes on the form

\begin{array}{l} j (ρ_{1}, ϕ_{1}, ρ_{2}, ϕ_{2}) & = \frac{J (ρ_{1}, ϕ_{1}, ρ_{2}, ϕ_{2})}{\sqrt{J (ρ_{1}, ϕ_{1}, ρ_{1}, ϕ_{1}) J (ρ_{2}, ϕ_{2}, ρ_{2}, ϕ_{2})}} \\ = exp [i Φ (ρ_{1}, ϕ_{1}, ρ_{2}, ϕ_{2})] \frac{\sum_{m} G_{m}^{*} (ρ_{1}) G_{m} (ρ_{2}) exp (i m Δ ϕ)}{\sqrt{\sum_{m} {| G_{m} (ρ_{1}) |}^{2} \sum_{m} {| G_{m} (ρ_{2}) |}^{2}}}, \end{array}

where Φ(ρ₁, ϕ₁, ρ₂, ϕ₂) = arg[U₀(ρ₂, ϕ₂)] −arg[U₀(ρ₁, ϕ₁)] vanishes if the phase of the incident field is constant at the plane of the diffuser.

We can study the far-zone properties of the transmitted field by the angular correlation function [4] in cylindrical coordinates, which, for an arbitrary source with mutual intensity J(ρ₁, ϕ₁, ρ₂, ϕ₂), can be expressed as

\begin{array}{l} T (f_{1}, φ_{1}, f_{2}, φ_{2}) & = \frac{1}{{(2 π)}^{4}} \iint_{0}^{\infty} \iint_{0}^{2 π} J (ρ_{1}, ϕ_{1}, ρ_{2}, ϕ_{2}) exp [i f_{1} ρ_{1} cos (ϕ_{1} - φ_{1})] \\ \times exp [- i f_{2} ρ_{2} cos (ϕ_{2} - φ_{2})] ρ_{1} ρ_{2} d ϕ_{2} d ϕ_{1} d ρ_{2} d ρ_{1} . \end{array}

Let us denote the Cartesian components of the unit position vector ŝ = r/r by (ŝ_x, ŝ_y, ŝ_z) = (sin θ cos φ,sin θ sin φ, cos θ). Then the quantity T(k sinθ₁, φ₁, k sinθ₂, φ₂), where k is the wave number, is directly related to the far-zone mutual intensity (cf. [4], p. 273) and the radiant intensity is given by

J (θ, φ) = {(2 π k)}^{2} cos θ T (k sin θ, φ, k sin θ, φ) .

Furthermore, the complex degree of coherence in the far zone is

j (θ_{1}, φ_{1}, θ_{2}, φ_{2}) = \frac{T (k sin θ_{1}, φ_{1}, k sin θ_{2}, φ_{2})}{\sqrt{T (k sin θ_{1}, φ_{1}, k sin θ_{1}, φ_{1}) T (k sin θ_{2}, φ_{2}, k sin θ_{2}, φ_{2})}} .

3. Examples of diffuser functions

In the following, we study the properties of some fundamental types of deterministic diffusers. Firstly, if the complex-amplitude transmittance function is radially invariant with t(ρ, ϕ) = t(ϕ), we have

G_{m} (ρ) = G_{m} = \frac{1}{2 π} \int_{0}^{2 π} t (ϕ) exp (- i m ϕ) d ϕ

and Eq. (8) reduces to

j (ρ_{1}, ϕ_{1}, ρ_{2}, ϕ_{2}) = exp [i Φ (ρ_{1}, ϕ_{1}, ρ_{2}, ϕ_{2})] \frac{\sum_{m} {| G_{m} |}^{2} exp (i m Δ ϕ)}{\sum_{m} {| G_{m} |}^{2}} .

Therefore the (equal-time) degree of coherence |j(ρ₁, ϕ₁, ρ₂, ϕ₂)| is modulated only in the azimuthal direction.

If, on the other hand, the complex-amplitude transmittance function is invariant in the azimuthal direction, i.e., t(ρ, ϕ) = t(ρ), we find that G_m(ρ) = t(ρ)δ_m,0, where δ_m,n is the Kronecker delta symbol. Now

j (ρ_{1}, ϕ_{1}, ρ_{2}, ϕ_{2}) = exp [i Φ (ρ_{1}, ϕ_{1}, ρ_{2}, ϕ_{2})],

where

Φ (ρ_{1}, ϕ_{1}, ρ_{2}, ϕ_{2}) = Φ (ρ_{1}, ϕ_{1}, ρ_{2}, ϕ_{2}) + arg [G_{0} (ρ_{2})] - arg [G_{0} (ρ_{1})] .

Hence |j(ρ₁, ϕ₁, ρ₂, ϕ₂)| = 1 and the field remains completely coherent. In other words, radial modulation of the diffuser has no effect on the degree of coherence. This result follows from the fact that due to the azimuthal invariance of t(ρ) the field at two positions (ρ₁, ϕ₁) and (ρ₂, ϕ₂) behind the diffuser has a fixed phase relation for any rotation angle of the diffuser.

3.1. Cosinusoidal transmission

Let us proceed to investigate some more specific transmission functions. We first consider the complex-amplitude transmission function

t (ρ, ϕ) = cos (α ρ + q ϕ),

where α is real and q is an integer. This gives a radial (cosinusoidal) grating if q = 0, an azimuthal grating if α = 0, and a spiral grating if both α and q are non-zero. As noted earlier, the radial grating (q = 0) does not exhibit coherence modulation and hence such a case is not of interest in the present context. Assuming q ≠ 0, Eq. (6) gives

G_{m} (ρ) = \frac{1}{2} exp (i α ρ m / q)

when m = ±q and G_m = 0 otherwise. Assuming that the incident field has a constant phase at the element plane, Eq. (8) gives

j (ρ_{1}, ϕ_{1}, ρ_{2}, ϕ_{2}) = j (Δ ρ, Δ ϕ) = cos (α Δ ρ + q Δ ϕ),

where Δρ = ρ₂ − ρ₁. Remarkably j(Δρ, Δϕ) has the same functional form as t(ρ, ϕ). It depends on coordinate differences only, and we can adjust the balance between radial and azimuthal coherence scales by appropriate choices or α and q.

We next turn to examine the far-zone radiation assuming illumination by an on-axis Gaussian beam with its waist at the element plane, i.e.,

U_{0} (ρ, ϕ) = U_{0} exp (- \frac{ρ^{2}}{w^{2}}),

where w is the beam waist width. The beam immediately behind the element now has the mutual intensity

J (ρ_{1}, ϕ_{1}, ρ_{2}, ϕ_{2}) = \frac{1}{2} {| U_{0} |}^{2} exp (- \frac{ρ_{1}^{2} + ρ_{2}^{2}}{w^{2}}) cos (α Δ ρ + q Δ ϕ) .

Inserting this expression into Eq. (9) and evaluating the azimuthal integrals with the help of the Jacobi–Anger expansion [13] gives us the angular correlation function

\begin{array}{l} T (f_{1}, φ_{1}, f_{2}, φ_{2}) & = \frac{{| U_{0} |}^{2}}{8 π^{2}} \iint_{0}^{\infty} ρ_{1} ρ_{2} exp (- \frac{ρ_{1}^{2} + ρ_{2}^{2}}{w^{2}}) J_{q} (f_{1} ρ_{1}) J_{q} (f_{2} ρ_{2}) \\ \times cos (α Δ ρ + q Δ φ) d ρ_{1} d ρ_{2}, \end{array}

where J_q(f_iρ_i) are Bessel functions of the first kind. The radiant intensity is now independent of the azimuthal angle φ:

J (θ, φ) = \frac{1}{2} k^{2} {| U_{0} |}^{2} cos θ {| \int_{0}^{\infty} ρ exp (- \frac{ρ^{2}}{w^{2}}) J_{q} (k ρ sin θ) exp (i α ρ) d ρ |}^{2} .

3.2. Continuous phase-only transmission

The transmission function defined in Eq. (16) is not the most convenient one from a fabrication point of view since it has both amplitude and phase variations. Considering a phase-only function

t (ρ, ϕ) = exp [i (α ρ + q ϕ)],

we obtain

j (Δ ρ, Δ ϕ) = exp [i (α Δ ρ + q Δ ϕ)],

and therefore |j(Δρ, Δϕ)| ≡ 1. This result follows from the fact that now U(ρ, ϕ; τ) = exp(−iq2πτ/T)U(ρ, ϕ; 0) and hence the relation of field values at any two positions remains fixed.

Assuming again Gaussian illumination defined in Eq. (19), we find that the angular correlation function is

\begin{array}{l} T (f_{1}, φ_{1}, f_{2}, φ_{2}) = & \frac{{| U_{0} |}^{2}}{{(2 π)}^{2}} exp (i q Δ φ) \iint_{0}^{\infty} ρ_{1} ρ_{2} exp (- \frac{ρ_{1}^{2} + ρ_{2}^{2}}{w^{2}}) J_{q} (f_{1} ρ_{1}) J_{q} (f_{2} ρ_{2}) \\ \times exp (i α Δ ρ) d ρ_{1} d ρ_{2} . \end{array}

Since this can be factorized into form T(f₁, φ₁, f₂, φ₂) = V^*(f₁, φ₁)V(f₂, φ₂), the field is completely coherent in the far zone and the radiant intensity is

J (θ, ϕ) = k^{2} {| U_{0} |}^{2} cos θ {| \int_{0}^{\infty} ρ exp (- \frac{ρ^{2}}{w^{2}}) J_{q} (k ρ sin θ) exp (i α ρ) d ρ |}^{2} .

The complete far-zone coherence is, of course, a direct consequence of the complete source-plane coherence. Remarkably, the radiant intensity is of the same functional form as in the preceding case dealing with cosinusoidal transmission function (save for the factor 1/2 arising from the absorption in the former case), although the coherence properties are radically different.

3.3. Continuous amplitude-only transmission

Let us then consider an amplitude-only version of the function in Eq. (16) by simply adding a bias:

t (ρ, ϕ) = \frac{1}{2} [1 + cos (α ρ + q ϕ)],

which leads to

j (Δ ρ, Δ ϕ) = \frac{1}{3} [2 + cos (α Δ ρ + q Δ ϕ)]

for q ≠ 0. Now |j(Δρ, Δϕ)| is modulated, but only in the range 1/3 ≤ |j(Δρ, Δϕ)| ≤ 1.

With the same illumination as in the preceding two examples, we find that the angular-correlation function takes on the form

\begin{array}{l} T (f_{1}, φ_{1}, f_{2}, φ_{2}) & = \frac{{| U_{0} |}^{2}}{16 π^{2}} \iint_{0}^{\infty} ρ_{1} ρ_{2} exp (- \frac{ρ_{1}^{2} + ρ_{2}^{2}}{w^{2}}) [J_{0} (f_{1} ρ_{1}) J_{0} (f_{2} ρ_{2}) \\ + \frac{1}{2} J_{q} (f_{1} ρ_{1}) J_{q} (f_{2} ρ_{2}) cos (α Δ ρ + q Δ φ)] d ρ_{1} d ρ_{2} \\ = \frac{{| U_{0} |}^{2} w^{4}}{64 π^{2}} exp [- \frac{w^{2}}{4} (f_{1}^{2} + f_{2}^{2})] + \frac{{| U_{0} |}^{2}}{32 π^{2}} \iint_{0}^{\infty} ρ_{1} ρ_{2} exp (- \frac{ρ_{1}^{2} + ρ_{2}^{2}}{w^{2}}) \\ \times J_{q} (f_{1} ρ_{1}) J_{q} (f_{2} ρ_{2}) cos (α Δ ρ + q Δ φ) d ρ_{1} d ρ_{2} \end{array}

and the radiant intensity is

\begin{array}{l} J (θ, φ) = & k^{2} cos θ \frac{{| U_{0} |}^{2} w^{4}}{16} exp [- \frac{w^{2}}{2} k^{2} {sin}^{2} θ] \\ + k^{2} cos θ \frac{{| U_{0} |}^{2}}{8} {| \int_{0}^{\infty} ρ exp (- \frac{ρ^{2}}{w^{2}}) J_{q} (k ρ sin θ) exp (i α ρ) d ρ |}^{2} . \end{array}

3.4. Binary-phase transmission

The transmission functions in the all of the three examples discussed above are continuous. However, binary elements are much easier to fabricate with lithographic techniques. As shown above, continuous phase-only spiral element does not have any coherence-modulating feature, but the situation with corresponding binary phase-only element is different. Consider the phase-only transmission function

t (ρ, ϕ) = exp [i ϑ (ρ, ϕ)],

where

ϑ (ρ, ϕ) = arg [sin (α ρ + q ϕ)] \frac{Δ ϑ}{π}

assumes only two discrete values based on the sign of sin(αρ + qϕ). Obviously, Δϑ is the phase shift between the two binary levels. Thus, t(ρ, ϕ) = 1 when sin(αρ + qϕ) ≥ 0, and t(ρ, ϕ) = exp(iΔϑ) when sin(αρ + qϕ) < 0. We also demand q ≠ 0 to make coherence modulation possible.

Evaluating Eq. (6) for q ≠ 0 is not as straightforward as in the previous examples. In the following, we summarize the most essential results, whose detailed derivations are given in Appendix A. We first obtain

G_{m} (ρ) = {\begin{array}{l} \frac{1}{2} [1 + exp (i Δ ϑ)] & if m = 0, \\ \frac{i q}{m π} exp (i m α ρ / q) [1 - exp (i Δ ϑ)] & if m / q is an odd integer, \\ 0 & otherwise . \end{array}

Consequently, the complex degree of coherence is

j (Δ ρ, Δ ϕ) = \frac{1}{2} {1 + cos (Δ ϑ) + [1 - cos (Δ ϑ)] y (α Δ ρ + q Δ ϕ)},

where

y (ε) = \frac{8}{π^{2}} \sum_{p = 0}^{\infty} \frac{cos [(2 p + 1) ε]}{{(2 p + 1)}^{2}}

is the normalized symmetric triangle wave function, with −1 ≤ y(ɛ) ≤ 1. Now, as long as π/2 ≤ Δϑ ≤ 3π/2, this element provides the full range of coherence modulation, 0 ≤ |j(Δρ, Δϕ)| ≤ 1.

A similar analysis for binary-amplitude element defined by

t (ρ, ϕ) = \frac{1}{2} {1 + sgn [sin (α ρ + q ϕ)]}

with q ≠ 0 gives

G_{0} (ρ) = \frac{1}{2}

G_{Nq} (ρ) = \frac{- i}{N π} exp (i N α ρ),

when N = 2p + 1, and G_m(ρ) = 0 for other values of m. Furthermore, the degree of coherence from Eq. (8) evaluates to

j (Δ ρ, Δ ϕ) = \frac{1}{2} [1 + y (α Δ ρ + q Δ ϕ)],

and thus 0 ≤ j(Δρ, Δϕ) ≤ 1. Interestingly the result in Eq. (39) is the same as for binary phase element with phase shift Δϑ = π/2.

Evaluation of the angular correlation function for the binary phase element considered earlier, again assuming on-axis Gaussian beam illumination, yields

T (f_{1}, φ_{1}, f_{2}, φ_{2}) = \frac{{| U_{0} |}^{2}}{2} {[1 + cos (Δ ϑ)] I_{1} (0) I_{2} (0) + [1 - cos (Δ ϑ)] Y (q Δ φ, I_{1} (N) I_{2} (N))},

where

Y (ε, R (N)) = \frac{8}{π^{2}} \sum_{p = 0}^{\infty} \frac{cos [(2 ρ + 1) ε]}{{(2 p + 1)}^{2}} ℜ {R (2 p + 1)}

and

I_{1} (N) = \int_{0}^{\infty} ρ exp (- \frac{ρ^{2}}{w^{2}} - i N α ρ) J_{Nq} (f_{1} ρ) d ρ

I_{2} (N) = \int_{0}^{\infty} ρ exp (- \frac{ρ^{2}}{w^{2}} + i N α ρ) J_{Nq} (f_{2} ρ) d ρ

are the integrals for f₁ and f₂ separated for brevity.

In the special case of f₁ = f₂, needed in the calculation of the degree of coherence by Eq. (11), we see that $I_{1} (N) = I_{2}^{*} (N)$ and as such I₁(N)I₂(N) = |I₁(N)|² = |I₂(N)|² =: |I(N)|². This leads to

T (f, Δ φ) = \frac{{| U_{0} |}^{2}}{2} {[1 + cos (Δ ϑ)] {| I (0) |}^{2} + [1 - cos (Δ ϑ)] Y (q Δ ϕ, {| I (N) |}^{2})} .

While the sum in Eq. (41) seems challenging, it can be numerically seen that with reasonably high values of q, the product I₁(N)I₂(N) is very close to zero for all N > 1, and thus only p = 0 needs to be computed. For instance, with the numbers from our experiment, f = 60969 m⁻¹,α = 16π/5 mm, w = 0.32 mm, and q = 16, we find that I₁(1)I₂(1)/I₁(3)I₂(3) > 5000.

With this discovery, inserting the three variations of Eq. (44) into Eq. (11), the far-zone spatial degree of coherence in the ideal case Δϑ = π is approximately modulated in azimuthal direction simply as

j (Δ φ) \approx cos (q Δ φ)

for all values of the radial component f.

4. Experimental verification

In the following we study, making use of a binary phase-modulating spiral diffuser, how well the theoretical predictions of the coherence patterns can be realized in practice. We first fabricated the diffuser on a fused silica substrate using electron beam lithography and reactive ion etching. For this test sample we chose parameters q = 16 and α = 16π/R, where R = 5 mm is the radius of the pattern. Designed for laser with wavelength λ = 633 nm, we dry etched the profile 693 nm deep in order to get a phase shift of π radians between the binary levels. Figure 2 illustrates the diffuser structure.

Fig. 2 The schematic structure of the diffuser. The dark and light gray areas mean phase delays of zero and π radians.

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After the fabrication, we constructed the optical measurement device. The center of the sample has to be positioned very precisely to match the rotation axis that, in turn, must coincide with the optical axis. These demands were met by including adjustment screws for tilt and position into the rotating shaft. In essence, imprecise positioning of the diffuser causes an overall irregular decrease in the spatial degree of coherence in output.

The diffraction patterns generated by the diffuser both while stationary and while rotating are illustrated in Fig. 3. Based on numerical simulations the center of the pattern should be completely dark, but evidently a central spot is present in our setup. This is caused by limitations in the fabrication process: at the very center of the pattern the spiral lines cannot go infinitely narrow but rather there exists a small area at the center with no depth modulation, allowing a small portion of the incident light to be transmitted without diffraction (in addition, at the very center, our complex-amplitude transmission function approach is not valid since the structural details are in the wavelength scale). We performed the measurement of the degree of coherence by dynamic Young’s double slit experiment with a digital micromirror device (DMD), described in [14]. According to Eqs. (40) and (41), the coherence modulation for spiral-patterned binary phase diffuser is separable in radial and azimuthal directions, and thus it is enough to only consider purely radial modulation and purely azimuthal modulation to get full information of the coherence properties in the spatial domain.

Fig. 3 Diffraction pattern generated by the fabricated diffuser while stationary (left) and rotating (right). The vertical line shows the measurement range for radial modulation with the reference point marked with a dot; the circle shows the measurement range for azimuthal modulation.

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In the radial measurements we fixed f₁ and measured the coherence as a function of f₂. The resulting coherence modulation calculated from the fringe visibilities and comparison to the modulation numerically computed using Eq. (40) are given in Fig. 4. It can be seen that the measured degree of coherence deviates from the theoretical when the normalized light intensity drops below 0.3. This is most likely due to the increased relative noise. It should also be noted that the theoretical coherence maximum around −0.08° is unobservable due to near-zero intensity in the dark central region of the pattern. The coherence minimum around +0.08° is not reached in the experiment, most likely because of the phase shift generated by the element is not exactly π, but closer to 1.08π, caused by fabrication errors. Although not immediately evident from Eq. (40), coherence modulation in the radial direction is essentially a function of radial-coordinate-difference f₂ − f₁, and thus slightly changing the reference point f₁ only shifts the modulation curve accordingly.

Fig. 4 Left: The absolute value of spatial degree of coherence in radial direction as a function of divergence angle difference to the reference point. Right: Light intensity in the measured range of radial modulation. Measured data marked with dots, theoretical values as solid lines.

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In the azimuthal measurements, we fixed f₁ = f₂ and varied angles φ₁ and φ₂ such that φ₁ = −φ₂. This allows the two points to always share the same y coordinate, making the double-slit experiment practical to conduct. The measured coherence data together with the values computed using Eq. (45) are shown in Fig. 5. While the basic shape of the measured graph resembles that given by the theory, the maximum degree of coherence is seen to decrease as Δφ approaches 180°. One possible explanation for this type of behavior is a slight misalignment of the system: the measurement points are furthest away from each other at Δφ = 180°, and hence the fringe pattern is densest and most prone to measurement errors. Moreover, since the slits created by the micromirror device have lengths of tens of micrometers, the slit directions cause them to “see” a wider angular part of the field, which reduces the observed coherence.

Fig. 5 The absolute value of spatial degree of coherence in azimuthal direction as a function of difference in the azimuthal coordinate φ. Measured data marked with black, theoretical value with red.

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However, these are not the only explanations. As we performed the same measurement for the static diffraction pattern, which should give a uniformly coherent result for a coherent laser source, there was a similar dip in the degree of coherence for distant measurement points. Hence the most likely explanation is a limitation of the measurement setup itself, being unable to resolve the dense fringe pattern accurately enough. Also, as briefly mentioned before, the positioning of the diffuser along the rotational axis is very intolerant for errors. This can be seen by comparing Figs. 5 and 6 where in the latter the sample has been deliberately positioned 50 μm off axis.

Fig. 6 The measured response similar to that of Fig. 5 with the exception that the diffuser was offset by 50 μm from the rotational axis.

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5. Final remarks

In this work we analyzed only systems with central symmetry, i.e., the optical axis of the incident field coincided with the rotation axis of the diffuser. This allowed us to express the correlation function in rather simple forms in several cases of interest. It is, of course, also possible to intentionally misalign the axes of the incident beam and the diffuser, but in that case rather computationally intensive calculations are required. Also, only rather simple diffuser structures were investigated. More complicated diffusers would naturally allow a larger degree of freedom in coherence control.

Appendix A. Detailed mathematics

In this appendix we go through the mathematical steps required to reach the results given in subsection 3.4. This starts by deriving the Fourier series coefficients for the complex transmission function of the binary phase profile

t (ρ, ϕ) = exp [i ϑ (ρ, ϕ)],

where

ϑ (ρ, ϕ) = arg [sin (α ρ + q ϕ)] \frac{Δ ϑ}{π} .

The special case of m = 0 can be easily obtained from Eq. (6), considering the periodical discrete nature of ϑ(ρ, ϕ) and the physical interpretation of Riemann integral:

G_{0} (ρ) = \frac{1}{2 π} \int_{0}^{2 π} t (ρ, ϕ) d ϕ = \frac{1}{2 π} [π + π exp (i Δ ϑ)] = \frac{1}{2} [1 + exp (i Δ ϑ)] .

For other values of m the case is not as simple. Firstly, to get ρ outside the integral, we can do a coordinate shift

ϕ \to ϕ - \frac{α ρ}{q}

without changing the integration limits since the integration with respect of ϕ for 2π periodical function can be done on any contiguous range of length 2π. The modified integral is

G_{m} (ρ) = C_{m} (ρ) \int_{0}^{2 π} exp [i arg (sin (q ϕ)) \frac{Δ ϑ}{π} - i m ϕ] d ϕ,

where

C_{m} (ρ) = \frac{1}{2 π} exp (\frac{i m α ρ}{q}) .

Now, since t(ρ, ϕ) only gets two values, 1 and exp(iΔϑ), and we know the ranges where the values are, we can write Eq. (50) as

\begin{array}{r} G_{m} (ρ) = C_{m} (ρ) [\int_{0}^{π / q} exp (- i m ϕ) d ϕ + exp (i Δ ϑ) \int_{π / q}^{2 π / q} exp (- i m ϕ) d ϕ \\ + \int_{2 π / q}^{3 π / q} exp (- i m ϕ) d ϕ + \dots + exp (i Δ ϑ) \int_{2 π - π / q}^{2 π} exp (- i m ϕ) d ϕ] . \end{array}

This is easily evaluated in the form

G_{m} (ρ) = C_{m} (ρ) \frac{i}{m} [1 - exp (i Δ ϑ)] {\sum_{n = 0}^{q - 1} exp [- (2 n + 1) i m π / q] - \sum_{n = 0}^{q - 1} exp [- 2 n i m π / q]},

which can be rewritten as

G_{m} (ρ) = C_{m} (ρ) \frac{i}{m} [1 - exp (i Δ ϑ)] [exp (- i m π / q) - 1] \sum_{n = 0}^{q - 1} exp [- 2 n i m π / q] .

Now if m = Nq, N ≠ 0, Eq. (54) becomes

G_{Nq} (ρ) = C_{Nq} (ρ) \frac{i}{N} [1 - exp (i Δ ϑ)] [exp (- i N π) - 1] .

It can be seen at once that G_Nq = 0 for all even values of N, and for odd values of N Eq. (55) gives

G_{Nq} (ρ) = \frac{i}{N π} [1 - exp (i Δ ϑ)] exp (i N α ρ) .

For all other values of m that are not whole multiples of q, the geometric sum in Eq. (54) is

\sum_{n = 0}^{q - 1} exp {(- 2 i m π / q)}^{n} = \frac{1 - exp {(- 2 i m π / q)}^{q}}{1 - exp (- 2 i m π / q)},

and it equals zero for all values of m. It should be noted that the case m = Nq was investigated separately since otherwise Eq. (57) would contain division by zero and is thus not valid for those orders.

To evaluate the degree of coherence for this element, we need to consider the summations in Eq. (8). For m = 0 we have

G_{0}^{*} (ρ_{1}) G_{0} (ρ_{2}) = \frac{1}{2} [1 + cos (Δ ϑ)] .

For the cases m = Nq and N = (2p + 1) we have

G_{Nq}^{*} (ρ_{1}) G_{Nq} (ρ_{2}) = \frac{2}{N^{2} π^{2}} [1 - cos (Δ ϑ)] exp (i N α Δ ρ) .

Let us write

S_{N} (ρ_{1}, ρ_{2}, Δ ϕ) = G_{Nq}^{*} (ρ_{1}) G_{Nq} (ρ_{2}) exp (i Nq Δ ϕ) .

While we need to consider both negative and positive values of N, it can be seen from Eq. (59) that

S_{- N} = S_{N}^{*}

and as such the degree of coherence can be written in form given by Eq. (34):

j (Δ ρ, Δ ϕ) = \frac{1 + cos (Δ ϑ) + [1 - cos (Δ ϑ)] y (α Δ ρ + q Δ ϕ)}{2} .

Assuming a Gaussian input beam centered on the rotational axis of the diffuser, Eq. (7) gives

J (ρ_{1}, ϕ_{1}, ρ_{2}, ϕ_{2}) = {| U_{0} |}^{2} exp (- \frac{ρ_{1}^{2}}{w^{2}}) exp (- \frac{ρ_{2}^{2}}{w^{2}}) \sum_{m = - \infty}^{\infty} G_{m}^{*} (ρ_{1}) G_{m} (ρ_{2}) exp (i m Δ ϕ),

which inserted into Eq. (9) gives

\begin{array}{l} T (f_{1}, φ_{1}, f_{2}, φ_{2}) = & \frac{{| U_{0} |}^{2}}{{(2 π)}^{2}} \iint_{0}^{\infty} ρ_{1} exp (- \frac{ρ_{1}^{2}}{w^{2}}) ρ_{2} exp (- \frac{ρ_{2}^{2}}{w^{2}}) \sum_{m = - \infty}^{\infty} G_{m}^{*} (ρ_{1}) G_{m} (ρ_{2}) \\ \times F_{m} (ρ_{1}, ρ_{2}, f_{1}, φ_{1}, f_{2}, φ_{2}) d ρ_{1} d ρ_{2}, \end{array}

where

\begin{array}{l} F_{m} (ρ_{1}, ρ_{2}, f_{1}, φ_{1}, f_{2}, φ_{2}) = & \iint_{0}^{2 π} exp (i m Δ φ) exp [i f_{1} ρ_{1} cos (ϕ_{1} - φ_{1})] \\ \times exp [- i f_{2} ρ_{2} cos (ϕ_{2} - φ_{2})] d ϕ_{1} d ϕ_{2} . \end{array}

With the help of Jacobi–Anger expansion, Eq. (64) becomes

\begin{array}{l} F_{m} (ρ_{1}, ρ_{2}, f_{1}, φ_{1}, f_{2}, φ_{2}) = & \sum_{n, r} i^{n} {(- i)}^{r} J_{n} (f_{1} ρ_{1}) J_{r} (f_{2} ρ_{2}) exp (- i n_{1} φ_{1}) exp (i r φ_{2}) \\ \times \int_{0}^{2 π} exp [i (n - m) ϕ_{1}] d ϕ_{1} \int_{0}^{2 π} exp [i (m - r) ϕ_{2}] d ϕ_{2} . \end{array}

The two integrals in Eq. (65) are 0 for all values of n and r except for n = m and r = m when the integrals both give 2π. Thus the summation over all n and r leaves only

F_{m} (ρ_{1}, ρ_{2}, f_{1}, φ_{1}, f_{2}, φ_{2}) = {(2 π)}^{2} J_{m} (f_{1} ρ_{1}) J_{m} (f_{2} ρ_{2}) exp (i m Δ φ) .

Inserting Eq. (66) into Eq. (63) and moving the summation outside the integrals yields

T (f_{1}, φ_{1}, f_{2}, φ_{2}) = {| U_{0} |}^{2} \sum_{m = - \infty}^{\infty} exp (i m Δ φ) \iint_{0}^{\infty} ρ_{1} exp (- \frac{ρ_{1}^{2}}{w^{2}}) ρ_{2} exp (- \frac{ρ_{2}^{2}}{w^{2}})

\times G_{m}^{*} (ρ_{1}) G_{m} (ρ_{2}) J_{m} (f_{1} ρ_{1}) J_{m} (f_{2} ρ_{2}) d ρ_{1} d ρ_{2},

Since G₀ differs from other G_m significantly, it makes sense to split the summation over m to handle them separately. Based on Eqs. (58) and (59) we can write

T (f_{1}, φ_{1}, f_{2}, φ_{2}) = {| U_{0} |}^{2} T_{c} (f_{1}, f_{2}) + {| U_{0} |}^{2} \sum_{p = - \infty}^{\infty} T_{p} (f_{1}, f_{2}, Δ φ)

where

T_{c} (f_{1}, f_{2}) = \frac{1}{2} [1 + cos (Δ ϑ)] \int_{0}^{\infty} H (ρ_{1}) J_{0} (f_{1} ρ_{1}) d ρ_{1} \int_{0}^{\infty} H (ρ_{2}) J_{0} (f_{2} ρ_{2}) d ρ_{2},

and

\begin{array}{l} T_{p} (f_{1}, f_{2}, Δ φ) = & [1 - cos (Δ ϑ)] \frac{2 exp [i (2 p + 1) q Δ φ]}{{(2 p + 1)}^{2} π^{2}} \\ \times \int_{0}^{\infty} H (ρ_{1}) J_{(2 p + 1) q} (f_{1} ρ_{1}) exp [- i (2 p + 1) α ρ_{1}] d ρ_{1} \\ \times \int_{0}^{\infty} H (ρ_{2}) J_{(2 p + 1) q} (f_{2} ρ_{2}) exp [i (2 p + 1) α ρ_{2}] d ρ_{2}, \end{array}

with H(ρ) = ρ exp(− ρ²/w²). With little investigation it can be seen that

T_{p - 1} = T_{- p}^{*}

and so

\sum_{p = - \infty}^{\infty} T_{p} (f_{1}, f_{2}, Δ φ) = 2 \sum_{p = 0}^{\infty} ℜ {T_{p} (f_{1}, f_{2}, Δ φ)},

which brings the result as given by Eq. (40).

For the binary amplitude-modulating element the derivation follows exactly the same steps except that all occurrences of exp(iΔϑ) are replaced with 0 to signify not giving a phase modulation but instead blocking light. As the end results have Δϑappearing only as cos(Δϑ), the coherence modulation of amplitude spiral is the same as with phase-shift structures that generate phase shift of π/2 + nπ.

Acknowledgments

We are grateful to H. Partanen for help in the experiments. The work of JL was partially funded by the Academy of Finland through the Graduate School on Modern Optics and Photonics.

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Coherence modulation by deterministic rotating diffusers

Abstract

1. Introduction

2. Theory

3. Examples of diffuser functions

3.1. Cosinusoidal transmission

3.2. Continuous phase-only transmission

3.3. Continuous amplitude-only transmission

3.4. Binary-phase transmission

4. Experimental verification

5. Final remarks

Appendix A. Detailed mathematics

Acknowledgments

References and links

Cited By

Figures (6)

Equations (72)

Optics Express