Interfering optical coherence lattices by use of a wavefront-folding interferometer

Mengwen Guo; Daomu Zhao

doi:10.1364/OE.25.014351

1. Introduction

Optical lattices are spatially periodic interference patterns in two or three dimensions originating from the interference of a finite set of plane waves [1,2]. They have attracted considerable attention for diverse applications including trapping and cooling of atoms [3], lithographic fabrication of photonic crystal [4,5], microfluidic sorting [6], and lattice light-sheet microscopy [7]. A variety of techniques for generating optical lattice have been proposed and realized, such as the interference of multiple plane-waves [8–10], Fourier-filtering operation of phase patterns [11,12], launching a spatially modulated incoherent light into a self-focusing photorefractive crystal [13], and constructing the angular spectrum of the kernel-generated function [14]. Recently, there has been an increasing interest in engineering and generating optical lattices of partially coherent beams [15–20]. In this regard, a new kind of partially coherent beam with periodic spatial coherence, optical coherence lattice (OCL), was theoretically introduced [15] and its propagation properties in free space as well as in turbulent atmosphere have been studied [16,17]. It is well known that for a planar quasi-homogeneous source, the far-zone radiant intensity primarily depends on the spatial Fourier transform of the spectral degree of coherence of light across the source [21]. Therefore, the lattice-like intensity patterns of the far field can be realized by a random source with periodic [15] or alternating series-like [19] structures of the spectral degree of coherence.

The peculiar specularity feature of the correlation function was first found in the field produced by a space-time modulated source [22], which is then applied to the optical processing [23]. Soon the concept of specularity was extended to the cross-spectral density function (CSDF) that describes the coherence properties of the field in the space-frequency domain [24]. In particular, a method for generating specular fields is proposed, i.e., launching any partially coherent field into a wavefront-folding interferometer (WFI). Accordingly, some common source models like, e.g., Schell-model and quasi-homogeneous sources, which are not specular in general, are examined. Since then little attention has been paid to the phenomenon of specularity, with an important exception of the research on specular and antispecular partially coherent solitons [25]. Lately, the specular and anti-specular light fields from a Gaussian Schell-model beam were demonstrated experimentally [26], followed by the study on the particles trapping using this class of beams [27]. The influence of the turbulent media on the specularity or anti-specularity of these beams has also been investigated [28]. However, to our best knowledge, the specular and anti-specular fields from an OCL have not been investigated in the literature.

In this paper, we study the propagation characteristics of an interfering OCL produced by launching an OCL into a WFI. We find that the far-zone interferogram consists of two lattices symmetrical about the $z$ axis. In particular, we demonstrate a variety of intensity patterns by varying the path length in one arm of the interferometer. Furthermore, we show the intensity profiles with different weight distribution parameters in the non-uniform case.

2. Theory

The wavefront-folding interferometer [29] is mainly considered for the measurement of the spatial correlation of partially coherent fields, shown in Fig. 1. This device is basically a Michelson interferometer with two crossed right-angle prisms in the arms. The beam splitter divides the incident beam into two equal parts, which are folded by the right-angle prisms ${PR}_{x}$ and ${PR}_{y}$ . The waves from each arm are recombined at the beam splitter and observed from outside the interferometer. When these prisms are slightly tilted with respect to the optical axis, interference fringes can be observed. However, we assume this instrument perfectly aligned.

Fig. 1 Wavefront-folding interferometer. S is the source, BS is a non-polarizing beam splitter, PR_x and PR_y are right-angle prisms.

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Suppose a linearly polarized light with the electric field $E_{0} (x, y)$ incident on the interferometer. The fields coming back to the beam splitter are thus of the form $E_{0} (- x, y)$ (from ${PR}_{x}$ ) and $E_{0} (x, - y)$ (from ${PR}_{y}$ ). On recombination, the electric field $E (x, y)$ after the beam splitter corresponds to the superposition of the two fields at the output of the system [26]

E (x, y) = \frac{1}{\sqrt{2}} [E_{0} (x, - y) + E_{0} (- x, y) \exp (i ϕ)],

where

ϕ

is the phase difference introduced in the interferometer. As a result, the CSDF at the input and output plane of the interferometer can be associated by the following relation

\begin{matrix} W (x_{1}, y_{1}, x_{2}, y_{2}) = \frac{1}{2} [W_{0} (x_{1}, - y_{1}, x_{2}, - y_{2}) + W_{0} (- x_{1}, y_{1}, - x_{2}, y_{2})] \\ + \frac{1}{2} [W_{0} (x_{1}, - y_{1}, - x_{2}, y_{2}) \exp (i ϕ) + W_{0} (- x_{1}, y_{1}, x_{2}, - y_{2}) \exp (- i ϕ)], \end{matrix}

where

W_{0} (x_{1}, y_{1}, x_{2}, y_{2})

and

W (x_{1}, y_{1}, x_{2}, y_{2})

are the CSDF at the input and output, respectively. As can be seen from Eq. (2), whatever the partially coherent beam of the primary source is, the CSDF of the field at the secondary source is specular

W (- x_{1}, - y_{1}, x_{2}, y_{2}) = W (x_{1}, y_{1}, x_{2}, y_{2})

when

ϕ = 2 l π

and anti-specular

W (- x_{1}, - y_{1}, x_{2}, y_{2}) = - W (x_{1}, y_{1}, x_{2}, y_{2})

when

ϕ = (2 l + 1) π

. Here

l

is an integer.

We consider an OCL incident on the interferometer, whose CSDF is of the form [15]

W_{0} (X_{1}, Y_{1}, X_{2}, Y_{2}) = \prod_{s = X, Y} \sum_{n_{s}} \frac{v_{n_{s}}}{\sqrt{π}} \exp [- \frac{s_{1}^{2} + s_{2}^{2}}{2} - \frac{i π n_{s} (s_{1} - s_{2})}{a}],

where

X = x / σ_{0}

and

Y = y / σ_{0}

are dimensionless scaled Cartesian coordinates with

σ_{0}

being the rms width of the intensity profile at the source,

n_{s}

are non-negative integers,

v_{n_{s}}

specify the energy distribution among the pseudo-modes composing the lattice,

a

denotes a dimensionless lattice constant. Substitute Eq. (3) into Eq. (2), we can obtain the CSDF of an interfering OCL at the output plane of the interferometer

\begin{matrix} W (X_{1}, Y_{1}, X_{2}, Y_{2}) = \exp (- \frac{X_{1}^{2} + Y_{1}^{2} + X_{2}^{2} + Y_{2}^{2}}{2}) \sum_{n_{X}} \sum_{n_{Y}} \frac{v_{n_{X}} v_{n_{Y}}}{π} \\ \times {\cos [\frac{π n_{X}}{a} (X_{1} - X_{2}) - \frac{π n_{Y}}{a} (Y_{1} - Y_{2})] \\ + \cos [\frac{π n_{X}}{a} (X_{1} + X_{2}) - \frac{π n_{Y}}{a} (Y_{1} + Y_{2}) - ϕ]} . \end{matrix}

The spectral density of the interfering OCL is expressed as

I (X, Y) = \exp [- (X^{2} + Y^{2})] \sum_{n_{X}} \sum_{n_{Y}} \frac{v_{n_{X}} v_{n_{Y}}}{π} {1 + \cos [\frac{2 π}{a} (n_{X} X - n_{Y} Y) - ϕ]} .

By applying the Fresnel diffraction integral for partially coherent fields [21], we obtain the CSDF of this field in any transverse plane $z > 0$

\begin{matrix} W (P_{1}, P_{2}, Z) = \frac{1}{π σ^{2} (Z)} \exp [- \frac{i σ_{0}}{2 R (Z)} (P_{1}^{2} - P_{2}^{2}) - \frac{1}{2 σ^{2} (Z)} (P_{1}^{2} + P_{2}^{2})] \sum_{n_{X}} \sum_{n_{Y}} v_{n_{X}} v_{n_{Y}} \\ \times \exp [- \frac{π^{2} Z}{a^{2} R (Z)} (n_{X}^{2} + n_{Y}^{2})] {\cos h [H_{1} (P_{1}, P_{2}, Z)] + \cos h [H_{2} (P_{1}, P_{2}, Z) + i ϕ]}, \end{matrix}

where

σ (Z) = \sqrt{1 + Z^{2}}

,

R (Z) = σ^{2} (Z) / Z

,

H_{1} (P_{1}, P_{2}, Z) = \frac{π}{a R (Z)} [n_{X} (X_{1} + X_{2}) - \frac{i n_{X}}{Z} (X_{1} - X_{2}) - n_{Y} (Y_{1} + Y_{2}) + \frac{i n_{Y}}{Z} (Y_{1} - Y_{2})],

H_{2} (P_{1}, P_{2}, Z) = \frac{π}{a R (Z)} [n_{X} (X_{1} - X_{2}) - \frac{i n_{X}}{Z} (X_{1} + X_{2}) - n_{Y} (Y_{1} - Y_{2}) + \frac{i n_{Y}}{Z} (Y_{1} + Y_{2})] .

Here

P = (X, Y)

denotes the transverse position vector,

Z = z / k σ_{0}^{2}

is a dimensionless propagation distance scaled to the Rayleigh range

z_{R} = k σ_{0}^{2}

of a fully coherent source. It is obvious from Eq. (6) that the specularity and anti-specularity of the interfering OCL always maintain during propagation.

Equation (6) can be used in the investigation of the spectral density and the spectral degree of coherence, which are defined as [30]

I (P, Z) = W (P, P, Z),

μ (P_{1}, P_{2}, Z) = \frac{W (P_{1}, P_{2}, Z)}{\sqrt{I (P_{1}, Z)} \sqrt{I (P_{2}, Z)}} .

3. Numerical results

In this section, we will demonstrate the evolution of the intensity profile and the spectral degree of coherence of an interfering OCL during propagation in free space. The parameters are chosen as follows: $a = 1$ , $σ_{0} = 1 mm$ . In the following simulations, we use the dimensionless scaled coordinates $X$ and $Y$ except in Fig. 4 which employs the coordinates $X {}_{1}$ and $X_{2}$ .

We first consider the interferometer incident by a uniformly distributed OCL which is composed of a finite number $N$ of equally weighted complex Gaussian pseudo-modes such that $v_{n_{X}} = v_{n_{Y}} = v_{0} = c o n s t$ with $0 \leq n_{X, Y} \leq N - 1$ . We set $v_{0} = 1$ and the number of lattice lobes $N = 4$ .

Figure 2 demonstrates the intensity profiles of the input and output beams of the WFI when propagating in free space. In rows 2, 3 and 4 of Fig. 2, we show the intensity profiles of a uniformly distributed interfering OCL (output) with three different phase differences $ϕ = 0, π / 2, π$ , respectively. In row 1 of Fig. 2, we illustrate the propagation of a uniformly distributed OCL (input) for comparison with the patterns in rows 2-4. It can clearly be seen that upon propagation of an OCL, a $4 \times 4$ lattice (in this case) in quadrant one is formed over the Rayleigh range [see Fig. 2(c1)] due to the lattice-like spectral degree of coherence at the source. However, in the case of an interfering OCL, two symmetrical $4 \times 4$ lattices distributed in quadrants two and four are formed over the Rayleigh range [see Figs. 2(c2), 2(c3) and 2(c4)]. This phenomenon can be explained by the folding effect of the right-angle prisms; one beam is inverted by the right-angle prism ${PR}_{x}$ in the $x$ direction and the other by the right-angle prism ${PR}_{y}$ in the $y$ direction. Besides, one can find that in the cases $ϕ = 0$ and $ϕ = π / 2$ the field has a bright lobe in the center while in the case $ϕ = π$ this lobe disappears completely. In this connection, it can be seen that after propagating a certain distance, the center lobe intensity of the interfering OCL decreases as $ϕ$ increases in the range of $0 \leq ϕ \leq π$ while that of the side lobes keeps invariant. It is because that the interference area of the two lattices only appears in the center. Thus, one can conclude that the intensity pattern of the interfering OCL in the far zone can be modulated by changing the path length in one arm of the interferometer.

Fig. 2 Intensity distributions of the input and output beams of the WFI in the uniform case. Row 1, evolution of intensity profile of a uniformly distributed OCL (input). The rows are from 2 to 4 the intensity distributions of the corresponding interfering OCL (output) with the phase difference $ϕ = 0, π / 2, π$ , respectively. The first images are calculated immediately behind the interferometer.

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Figure 3 demonstrates the modulus of the spectral degree of coherence of an interfering OCL between two symmetrical points in contrast with that of an OCL. Since the field is completely coherent between the symmetrical points in the specular and anti-specular cases, we choose a conventional case of $ϕ = π / 2$ . It can be seen that although the coherence distributions of the interfering OCL are greatly different from that of the OCL, they present a similar evolution, i.e., the lattice structure of the coherence at the source becomes aperiodic with the increase of the propagation distance. In Fig. 4, we show the modulus of the spectral degree of coherence of an interfering OCL in the specular case ( $ϕ = 0$ ) and anti-specular case ( $ϕ = π$ ). The coordinates are $X {}_{1}$ and $X_{2}$ to exhibit the specularity or antispecularity property of the field. The pictures show that during the transformation from the periodic structure to the aperiodic structure, the cross-like shape in the coherence distributions always maintains.

Fig. 3 The modulus of the spectral degree of coherence between two symmetrical points $P_{1} = (- X / 2, - Y / 2)$ and $P_{2} = (X / 2, Y / 2)$ . Rows 1 and 2 correspond to the coherence distributions of a uniformly distributed OCL and a uniformly distributed interfering OCL (with $ϕ = π / 2$ ), respectively.

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Fig. 4 The modulus of the spectral degree of coherence between two axial points $P_{1} = (X_{1}, 0)$ and $P_{2} = (X_{2}, 0)$ . Rows 1 and 2 correspond to the coherence distributions of a uniformly distributed interfering OCL with the phase differences $ϕ = 0$ and $ϕ = π$ , respectively.

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Let us now consider the non-uniform case. Suppose that the field consists of a finite number $N$ of complex Gaussian pseudo-modes which are weighted according to $v_{n_{s}} = γ^{n_{s}} / n_{s}!$ with $0 \leq n_{s} \leq N - 1$ . We set the number of lattice lobes $N = 4$ .

In Fig. 5, we display the intensity profiles of a non-uniformly distributed OCL with $γ = 1$ and that of the corresponding interfering OCL. It reveals the same key trends of the intensity profiles, as do the uniform case in Fig. 2. That is, for an interfering OCL, two lattices symmetric about the $z$ axis are formed over the Rayleigh range and the intensity profile of the center lobe can be modulated by changing the phase difference. In contrast with the uniform case, the only difference is that the intensity distributed in each side lobe is exactly the same in the uniform case while that in the non-uniform case is different due to the distribution of the modes.

Fig. 5 Intensity distributions of the input and output beams of the WFI in the non-uniform case. Row 1, evolution of intensity profile of a non-uniformly distributed OCL. The rows are from 2 to 4 the intensity distributions of the corresponding interfering OCL with the phase difference $ϕ = 0, π / 2, π$ , respectively. The weight distribution parameter $γ = 1$ .

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In Fig. 6, we show the intensity distributions of the non-uniform case with two different weight distribution parameters as the fields propagate into the far field ( $Z = 5$ ). In comparison with the figures in column 4 of Fig. 5, one can find that the intensity distributions exhibit different patterns when $γ$ is changed. Take the OCL case as an example [see Figs. 5(d1), 6(a1) and 6(a2)]. The intensity focuses on the left bottom lobes, middle lobes, and top right lobes when $γ =1, 2$ , and 3, respectively. The intensity distributions in the interfering OCL case show the similar trends as do the OCL case apart from that in the center lobe on account of the interference between the two lattices after the OCL beam is divided into two parts. Thus, it can be concluded that the intensity pattern can also be modulated by changing the weight distribution parameter $γ$ .

Fig. 6 Intensity profiles of non-uniformly distributed OCLs (column 1) and the corresponding interfering OCLs (columns 2 to 4) in the far zone ( $Z = 5$ ). Rows 1 and 2 correspond to $γ =2$ and $γ =3$ , respectively.

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4. Conclusions

In conclusion, we have investigated the evolution of the spectral density and the spectral degree of coherence of an interfering OCL by use of a wave-front interferometer. We have found that two lattices symmetrical about the $z$ axis are formed in the far zone, differing from the OCL case with only one lattice in quadrant 1. We have also shown that the intensity pattern of the interfering OCL in the far zone can be modulated by varying the phase difference between the two optical paths of the interferometer. In addition, we have revealed that for the non-uniform case, the intensity distribution can also be changed by varying the weight distribution parameter. The results obtained in this paper may find particular applications in image transmission and information encoding in free-space optical communications. Apart from the periodicity of the coherence at the source, information can also be encoded in the phase difference of the interferometer or the weight distribution parameter of the OCL source, which can then be decoded by directly observing the far-zone intensity pattern at the receptor. In addition, our results may also be applied in periodic trapping of micro-particles.

Funding

National Natural Science Foundation of China (NSFC) (11474253 and 11274273); Fundamental Research Funds for the Central Universities (2017FZA3005).

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Interfering optical coherence lattices by use of a wavefront-folding interferometer

Abstract

1. Introduction

2. Theory

3. Numerical results

4. Conclusions

Funding

References and links

Cited By

Figures (6)

Equations (10)

Optics Express