1 Introduction

It is a common phenomenon that propagation of coherent light through a scattering medium will produce speckle patterns [1,2,3]. Laser speckle statistics were used to measure the surface roughness of materials by the speckle contrast measurements and the speckle pattern correlation [4, 5]. However, in some applications, such as laser-based display, the speckle is harmful [6]. Compared with completely coherent light, the partially coherent light is able to suppress speckle noise and improve the quality of the image [7]. In addition, the partially coherent beam can be generated by adding a random phase on the completely coherent beam. Three elements, including rotating ground glass (RGG), dielectric elastomeric actuator(DEA), and spatial light modulator (SLM), have been applied in generating partially coherent light [6,7,8,9,10,11,12]. It has shown in a previous study that the stochastic electromagnetic beams [10], partially coherent beams with controllable time-dependent coherence [11], and nonuniformly correlated partially coherent light beams can be generated using an SLM [12].

In 2007, Vellekoop and Mosk showed that the speckle pattern formed by coherent beam through scattering medium can be shaped into a bright focal spot [13]. The input beam is divided into N segments, and a spatial light modulator (SLM, PLUTO-VIS-016) is introduced to shape the wavefront of the segments. A focal spot with intensity enhancement larger than 1000 can be achieved by applying an appropriate phase modulation on the input coherent beam [14]. Further study showed that the focal spot can be smaller than that of diffraction limit [15]. The feedback wavefront shaping has important applications in imaging, optical manipulations et al. [16, 17]. Different algorithms, including stepwise sequential algorithm [18], genetic algorithm [19, 20], and particle swarm optimization [21, 22], have been developed and applied in feedback wavefront shaping. However, there are few researches on the focusing of the partially coherent light through the scattering medium now.

Since the speckle can be suppressed by laser beam with a lower coherence, we are interested in its influence on the focusing properties. In this paper, we investigate the effect of coherence on the focusing property of laser beam through scattering medium. The modulation of the coherence of beam is realized by introducing dynamic random phase on the completely coherent beam. The speckle property of partially coherent beam is investigated. The focal spot is obtained by introducing a SLM to shape the wavefront of the beam.

2 Generation of partially coherent light by employing SLM

In general, completely coherent light can be described by

$$U(r)=A(r)\exp [i\varphi (r)],$$
(1)

where A(r) is the amplitude and φ(r) is the phase of U(r). Different from completely coherent light, partially coherent light cannot be described Eq. (1), due to the phase or/and amplitude experiencing fast fluctuation. Partially coherent light should be expressed by statistical analysis, that is

$$W({r_1},{r_2},\omega )=\left\langle {{U^*}({r_1},\omega )U({r_2},\omega )} \right\rangle ,$$
(2)

< > stands for an ensemble average, asterisk “*” denotes complex conjugate; r1, r2 represent the two points vectors in the free space, respectively [22]. Upon substituting Eq. (1) into Eq. (2), we obtain

$$W({r_1},{r_2})=A(r1)A(r2)\left\langle {\exp [ - i\varphi ({r_1})]\exp [i\varphi )({r_2})]} \right\rangle .$$
(3)

Recalling the Zernike’s definition of the degree of the coherence µ is [23]

$$\begin{aligned} \mu ({r_1},{r_2}) & =\frac{{W({r_1},{r_2})}}{{\sqrt {W({r_1},{r_1})W({r_2},{r_2})} }} \\ & =\left\langle {\exp [ - i\varphi ({r_1})]\exp [i\varphi ({r_2})]} \right\rangle . \\ \end{aligned}$$
(4)

In Eq. (4), the degree of the coherence µ ranges from 0 to 1. In addition, the degree of coherence µ can be measured by the Young’s double-pinhole interferometer [24]. Each pinhole is a circular hole with 0.1 mm side length, and the separation distance between the two pinholes is 1.0 mm.

It is shown from Eq. (4) that by imposing dynamic random phase onto a completely coherent light, we can obtain partially coherent light. In our study, we introduce a dynamic random phase [11] loaded on the SLM, and the laser beam is modulated by the SLM to generate partially coherent light. The SLM that employed to modulate the phase of the incident beam is “PLUTO-VIS”, with 1920 × 1080 pixels, 8.0 µm pixel pitch, and the aperture of the SLM is 15.36 mm × 8.64 mm. It is shown that every dynamic random phase has different randomness degrees. The randomness degree is a parameter reflecting the controlled random phase which is expressed by a. The range of the randomness degree is [0, 1]. Figure 1a–c shows one of the frames of the dynamic random phase with different randomness degree a of 0, 0.5, and 1.0. In addition, the expression of random phase added onto the SLM can be expressed as

$$\left| {\varphi (r) - \pi } \right|<\pi a(r).$$
(5)

We impose the random phase onto the SLM, for modulating the phase of the completely coherent light for achieving partially coherent light. The degree of coherence of the generated partially coherent light can be obtained as follows [11]:

$$\mu ({r_1},{r_2})=\left\{ {\begin{array}{*{20}{c}} 1 \\ {\sin c\left[ {a({r_1})} \right]\sin c[a({r_2})]} \end{array}} \right..$$
(6)

when r1 = r2, the degree of coherence µ is equal to unity. Otherwise, the degree of the coherence µ is dependent on the randomness degree a(r). It is shown from Eq. (6) that the correlation function of a partially coherent light is Sinc type, which is presented in Fig. 1(d). When a = 0, the light is completely coherent light, and a = 1, the light is incoherent light. It is clear that the randomness of the phase increases with the increasing randomness degree. Imposing the dynamic random phase on a completely coherent beam can generate a partially coherent beam. In Fig. 1e, we give the experimental results of measuring the coherence degree as a function of randomness degree. The dot in the curve is the averaged results of 10 measurements, and the error bars on the dot indicates the errors of the measured results. It is found from Fig. 1e that the coherence of the partially coherent beam decreases with increasing randomness of the dynamic random phase.

Fig. 1
figure 1

a–c Examples of one frame of a resolution of 300 × 300 pixels with different gray levels. a Randomness degree is 0.0, whose phase is a constant π. b Randomness degree is 0.5, whose phase range is 0.5π–1.5π. c Randomness degree is 1.0, whose phase range is 0–2π. d Simulation of degree of coherence with randomness degree. e Measurement of degree of coherence with randomness degree

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3 Speckle pattern formed by partially coherent light through the scattering medium

The experimental setup can be divided into two measurement parts: one is to measure the partially coherent light passing through the scattering medium in Sect. 3, the other measurement part is a feedback system to generate a focal spot in Sect. 4. The schematic of the experimental setup is presented in Fig. 2. A beam expander comprised of two lenses L1 and L2 magnifies the incident laser beam so that the beam fills the whole aperture of SLM1. This SLM is used to modulate the coherence of incident laser beam. The light passes through scattering medium (DG05-220-MD, Thorlabs) produces a speckle pattern. To generate a bright focal spot, another SLM, which used to optimize the phase of the beam, should be introduced. The beam reflected by SLM1 is transmitted through a 4f system. Microscope objective O1 is introduced in front of the scattering sample to focus the incident laser beam. Light transmitted through the scattering sample is collected by a second microscope objective O2, and the light passing through the scattering medium is captured by a CCD, which is “PIKE F-421B”, with 2048 × 2048 pixels The pixel size of the CCD is 7.4 µm, the pixel number of the camera is 2048 × 2048, and the bit depth is 8 bit.

Fig. 2
figure 2

Schematic diagram of the experimental setup. Along the direction of propagation of laser, the devices of the setup, in turns, are, L1–L2, regular lenses with focal lengths of 100 and 150 mm, respectively; SLM1, SLM2; the two lenses in a 4f telescope configuration, L3–L4, with focal lengths of 300 mm; O1, 10X microscope objective lens with an NA = 0.25; S, a scattering medium; O2, a second 20X microscope objective lens with an NA = 0.40; CCD, a camera that records the speckle and focal spot

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The speckle patterns formed by light beam with different phase randomness are experimental measured, as shown in Fig. 3a–c. It is shown that the intensity of the speckle is more uniform with a lower degree of coherence. This indicates that the speckle can be suppressed by light beams with lower coherence.

Fig. 3
figure 3

Speckle patterns (a–c) formed by partially coherent beam generated with random phase in Fig. 2. a–c The scale bar is 500 µm

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The speckle contrast (C) is introduced to characterize the statistical property of the speckle pattern, which is expressed as follows [3]:

$$C=\frac{{\sqrt {\left\langle {{I^2}} \right\rangle - {{\left\langle I \right\rangle }^2}} }}{{\left\langle I \right\rangle }},$$
(7)

where the numerator is the standard deviation of the intensity fluctuation and the denominator is the mean intensity [6].

Figure 4a shows maximum and minimum of the intensity with partially coherent beam with different degree of coherence. It is shown from Fig. 4a that the maximum intensity decreases and the minimum intensity increases with decreasing coherence. To have a better understanding on the speckle reduction, we calculate the speckle contrast, as shown in Fig. 4b. The contrast of the speckle pattern decreases gradually with the decreasing coherence. The result indicates that the speckle can be reduced by decreasing the degree of coherence.

Fig. 4
figure 4

a Maximum and minimum of the speckle intensity (gray value) with different degree of coherence. b Evolution of speckle contrast with degree of coherence

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4 Focusing of partially coherent light through the scattering medium

In general, he incident completely coherent light passing through the scattering medium gives rise to speckle pattern as the result of the collective interference of scattered light. In Fig. 5a, we show the speckle pattern, obtained by passing completely coherent light passing through the scattering medium. The experimental setup is presented in Fig. 2. By modulating the wavefront of the incident completely coherent light using SLM2, we can reshape speckle pattern into a bright focal spot, as shown in Fig. 5b. To achieve a bright focal spot, the wavefront of the incident light on SLM2 is divided into N2 segments, and the optimized wavefront is done by the phase optimization process. Figure 5c is the generated optimized phase distribution for realizing bright focal spot.

Fig. 5
figure 5

a Speckle pattern formed by a laser beam through scattering medium. b Focal spot formed by a laser beam through scattering medium after phase optimization. c Phase distribution used to generate (b). The scale bar is 500 µm

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The optical field through the scattering medium in the target, Em, is a linear combination of the fields that originate from the N2 = 10 × 10 different segments of SLM2:

$${E_m}=\sum\limits_{{n=1}}^{N} {{t_{mn}}{A_n}\exp \left( {i{\varphi _n}} \right),}$$
(8)

where An and φn are the amplitude and phase of the light before the scattering medium, respectively. tmn represents the transmission matrix of light passing through the scattering medium [13]. In Eq. (10), if all the segments of the SLM2 interfere constructively in the target, then a bright focal spot with high intensity enhancement can be generated. In addition, the intensity enhancement is defined as the ratio of the maximum intensity of the target after the wavefront optimization to the averaged intensity of the speckle before optimization,

$$\eta =\frac{{{I_{\hbox{max} }}}}{{\left\langle {{I_0}} \right\rangle }}.$$
(9)

Note that when dynamic random phase (a = 0) is loaded on SLM1, the light from SLM1 is completely coherent light. In this case, we can produce the focal spot using a stepwise sequential algorithm to optimize the wavefront (see Fig. 6a). It is shown that the intensity enhancement is η = 24.8. Then we load the dynamic random phases with different a onto SLM1. The intensity distributions of the focal spot with a randomness degree of a = 0.5 and a = 1 are presented in Fig. 6b, c. It is shown that the intensity distribution of the focal spot decreases with the increasing of a. Compared with the speckle contrast, Fig. 6d shows the evolution of the intensity enhancement with the degree of coherence. It is shown that the intensity enhancement of the focal spot decreases with the decreasing degree of coherence of the beam. This can be understood by that phase matching deteriorate as the phase randomness increases.

Fig. 6
figure 6

Intensity of the focus under the illumination of beams with modulation degree of a 0.0; b 0.5; and c 1.0, respectively. d Evolution of intensity enhancement with degree of coherence. The scale bar is 500 µm

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Moreover, the focal spot has an obvious change in size from the inset of Fig. 6a–c. It is found that the focal spot size decreases with the decreasing of the degree of coherence of the light, as shown in Fig. 7. This can be understood that due to that light beam with lower spatial coherence results in larger numerical aperture of this focusing system. A substantially large numerical aperture is known to produce a considerably small focal spot. This result can explain why the focal spot decreases with a decrease in beam coherence. Therefore, the degree of coherence of laser has an impact on intensity enhancement and the focal size.

Fig. 7
figure 7

Focal spot size versus the degree of coherence

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5 Conclusions

We have studied the influence of the light coherence on the speckle pattern. The result showed that the intensity distribution of the speckle pattern is more uniform and the speckle contrast decreases with decreasing of the degree of coherence. This indicated that the speckle can be reduced by means of decreasing the degree of coherence. We also found that the partially coherent light beam can be focused through the scattering media. It was shown that the intensity enhancement of the focal spot decreases with decreasing of the degree of coherence. In particular, the size of the focal spot decreases with the decreasing of the degree of coherence. Our results could potentially be useful in the partially coherent light in optical phase optimization and optical imaging.