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Abstract
Static laser speckle suppression using multimode fibers has practical limitations as the technique requires an extremely long fiber to achieve an acceptable speckle contrast. An effective method based on liquid light guides was developed in this study to suppress laser speckle. In this study, a speckle simulation model of the liquid light guide was established for numerically calculating the speckle contrast without solving the analytical solution of the photon diffusion equation. The obtained simulation results were compared with the experimental results for the dependence of speckle contrast on the required length and numerical aperture with different liquid core types of liquid light guides. A speckle contrast of 12% and a speckle suppression efficiency of 5 was achieved at the end of a 2.4 m long liquid light guide. For the same fiber length, liquid light guides were found to suppress speckle more efficiently when compared to multimode fibers.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The commonly used light sources for projectors can be divided into three types: traditional light sources, light-emitting diodes (LEDs), and lasers. Traditional light sources have a relatively short lifetime of generally a few thousand hours, and the brightness decreases quickly after long-term use. Although LEDs have the advantages of high energy conversion efficiency and low cost, they have lower brightness when compared to lasers. Lasers have many advantages, such as high brightness, wide color gamut, low energy consumption, long lifetime, and small size. Therefore, lasers are considered as superior light sources for next-generation projectors.
However, the speckle caused by laser coherence, the mismatch in the lifetime of lasers with different wavelengths, and the lower brightness of green lasers are still issues restricting the wide usage of laser displays, in addition to the relatively high price of lasers. Laser speckle is considered as a key issue and has been studied widely in recent years [1–10] because it not only affects the image quality of projectors but also causes a series of adverse reactions such as dizziness and nausea in the viewers [11]. In this study, we aim to develop a complete static speckle suppression method without any moving components.
The speckle is usually assessed by the speckle contrast (C), which is defined [1] as the ratio of the light intensity standard deviation (σ) and the light intensity average $({\overline I } )$, that is, $c = {\sigma / {\overline I }}$. The speckle contrast of the superposition of N incoherent speckle patterns is defined as ${1 / {\sqrt N }}$ of the original value. The current technical solutions for suppressing laser speckle are as follows: introduction of polarization or angular diversity, reduction of the temporal and spatial coherence of lasers, and a combination of the above solutions. As there are only two incoherent polarization states, the speckle can only be at most reduced to ${1 / {\sqrt 2 }}$ of the original value. Hence, the polarization diversity cannot be used alone to suppress the speckle. The decrease of spatial coherence of lasers, which related to time average of speckle patterns, angular diversity and wavelength diversity, can provide more options to reduce speckle. Moving diffusers [2,3], light pipes [4], and specially designed diffractive optical elements (DOE) [5–9] can be used to remove the speckle. From the perspective of implementation, the afore-mentioned speckle suppression schemes can be divided into dynamic and static schemes. Dynamic schemes have been extensively studied [2,4,8,9]. For example, a vibrating screen was used to achieve good speckle suppression, but it required a high vibration speed [2]. A moving light pipe was proposed to achieve uniform light by suppressing speckle [4]. Our group previously proposed a binary sequence flexible DOE loop with tracked motion, which can effectively suppress speckle in the entire visible spectrum. The size and power consumption of the device can meet the requirements of portable projectors [8]. However, the jointed slit of the flexible DOE loop affects the sharpness of the image. The previous solutions for static speckle suppression were mainly based on electro-optic, magneto-optic, acousto-optic, and other modulation technologies. The static method of filling the scattering medium in the projection screen can effectively reduce the speckle contrast, but it causes blurring of the projection images and greatly increases the difficulty of screen manufacturing [12,13]. The other approach is the use of a multimode fiber to destroy the temporal coherence of lasers [10,14–16]. Although the speckle contrast of 1% was not achieved as Goodman’s theoretical prediction [14] in a practical projecting system, this method has achieved good results in speckle suppression and demonstrates advantages in terms of low weight, low loss, and low power consumption [15]. However, this method requires a long fiber length with a high light coupling efficiency. In order to increase the light coupling efficiency, a multimode fiber bundle was used to suppress the speckle instead of a single multimode fiber [15,16]. However, it suffers the light loss caused by the gaps between the monofilaments in the fiber bundle as well as the breakage of monofilaments. Moreover, some compound methods combining two or more speckle suppression approaches to achieve a better suppression effect have been proposed, such as a compound method which introduces time and space averaging simultaneously. For example, the use of a moving screen and rotating diffusers together introduced angular diversity and time averaging to achieve better overall speckle suppression [17]. However, compound methods usually require complicated optical systems, larger volume, and higher power consumption.
In this study, we focus on a static speckle suppression based on a liquid light guide, which provides shorter required length of light guide and higher light coupling efficiency when compared with a multimode fiber or a fiber bundle. We develop a liquid scattering model based on the log-normal distribution function, which describes the free path distribution of scattering. The temporal coherence of lasers is proposed to be destroyed by bulk scattering to achieve the effect of speckle suppression. On the basis of the developed liquid scattering model, speckle contrasts can be obtained by numerical calculation instead of solving photon diffusion equation for accurate analytical solution. The simulation results are then verified through experiments. Unlike multimode fiber bundles, the liquid core cross-section is made of a single material. Thus, the coupling loss caused by the gaps between the monofilaments in the multimode fiber bundle is avoided. In addition, the liquid light guide does not suffer from the problems of wire breakage and the decrease in light transmission efficiency due to the repeated bending of the fiber or fiber bundle. In this study, we thoroughly studied the influence of the parameters of the liquid light guide on the speckle-suppression efficiency. Experiments are conducted to investigate the dependence of the material, length, diameter, incident light angle and numerical aperture of the liquid light guide on the speckle contrast. We also compare the results with the effect of the multimode fiber for suppressing speckle.
2. Theoretical analysis
The liquid core of commercial liquid light guides is usually inorganic salt solution or oil. Considering the required index of liquid light guides, sucrose solution, glycerol, benzene are chosen as liquid core materials in our in-house fabricated liquid light guides. Although the scattering phenomenon in pure liquid [18] and the decoherence of laser beam by scattering medium [19] were studied, there is no speckle simulation model for analyzing the speckle contrast in a waveguide-like scattering systems, such as liquid light guide or microvascular in living organisms. We derive formulae for theoretical analysis and simulations of speckle contrast for liquid light guides in this study.
The scattering that produces speckles can be divided into surface and bulk scattering. Surface scattering mainly arises due to the random fluctuations in the height of the surface of an object, which introduces a path difference among the photons incident on the object surface. In bulk scattering, the photons undergo multiple scattering events when propagating inside an object. The random change of the path of the photons due to bulk scattering gives rise to independent speckle fields with different characteristics. As the random distribution caused by bulk scattering is much stronger than the surface scattering that occurs on a moderately rough surface [20,21], we will analyze the bulk scattering model in detail.
When the time delay introduced by bulk scattering is equal to or exceeds the coherence time of the laser source, the laser speckle is suppressed [1]. For a light source with limited coherence length, when the scattering approaches infinity, the speckle contrast tends to zero [3]. The beam emitted by lasers can be regarded as a stream containing a large number of photons that enter the liquid medium through the incident end of the liquid light guide and undergo random collisions and refraction in the liquid medium. It was reported that the particle model can be used to describe the statistical distribution of the partial coherent field for an obvious scattering medium, even if it cannot be used to directly describe the coherent field [4]. Therefore, the particle model was adopted in the present study to deal with the liquid light guide. The liquid light guide can be regarded as a sparse system, and thus the interaction between the particles is not considered because the liquid particles are far apart. At a certain time (t) and a certain point $({\overrightarrow r } )$ in the space, the light emissivity $L({\overrightarrow r ,\overrightarrow s ,t} )$ towards a certain direction $\overrightarrow s$ is isotropic, and the scattering is much stronger than the absorption. Therefore, the photon diffusion equation can be expressed as [21]:
In the Cartesian coordinate system, Eq. (1) can be rewritten as an equation that can be solved in a time series, so that the fluence rate of the next time step, ${\Phi _{n + 1}},$ can be solved according to ${\Phi _n}.$
Equation (3) shows the precise description of photon diffusion behavior, however, it is difficult to obtain an accurate analytical solution for practical applications such as for the liquid light guides, in which the situation is more complex with light leakage, bending of the light guide. Moreover, because only the statistical result of photon diffusion is concerned, we use the Monte Carlo algorithm to simulate the random walking process of a single photon in the liquid medium, and then solve the scattered photon distribution and the time distribution of movement. The random step length and random walking direction is determined using the corresponding probability model and random number generator.
Thompson et al. [21] analyzed the parameters of several sets of scattering media and expressed the distribution of the optical path through a scattering medium in terms of the linear absorption coefficient ${\mu _a}$ and the reduced scattering coefficient $\mu _s^{\prime}.$ It was found that the distribution of the optical path showed a normal distribution trend, especially when the scattering coefficient was much larger than the absorption coefficient. However, there was a long “tail” on the right side of the optical path distribution function. Under this condition, a larger number of photons have a particularly long optical path. For this type of material, the optical path distribution of the photons was more dispersed under the influence of the scattering effect, and the time delay was expected to have a stronger effect.
The normal distribution function is symmetrical. Hence, when the mean path length is small, the normal probability distribution cannot be used to describe the “tail” on the right side of the function. The log-normal distribution exactly conforms to the shape of the function we need. Hence, we define the parameters of the log-normal distribution function based on the material parameters to describe the length probability distribution. With reference to the results obtained and considering the dependence of the scattering effect on the liquid material in [22–25], we use the mean ${l_{mfp}} = \frac{1}{{\mu _s^{\prime}}}$ and the log-normal distribution of standard deviation $\left( {{l_{std}} = \ln \frac{{\mu_s^{\prime}}}{{{\mu_a}}}} \right)$ to describe the free path distribution function, ${p_{fp}}(l )$.
In the scattering process, we define the angle between the moving direction of the photon before and after scattering as the scattering angle. The cosine distribution $({\cos \theta } )$ of the scattering angle θ is expressed as:
Fig. 1. Scattering model of n-th scattering with the scattering length ${l_n},$ scattering angle ${\theta _n},$ and azimuth angle ${\varphi _n}.$
Thus, we can obtain the scattering model, as shown in Fig. 1. The scattering path of the photon is equivalent to the movement of a generatrix on a cone surface. The radius of the bottom surface of the cone is $r = l\sin \theta ,$ and the walking length of the n-th scattering along the optical axis is ${z_{n}} = l\cos \left( {\sum {_{n = 1}^i\theta } } \right).$ When $\sum {{z_{n}}} = L,$ it is considered that the photons exit from the scattering medium with length L, and $\sum {{l_{n}}}$ represents the light path traveled by each photon. According to the numerical distribution of $\sum {{l_{n}}}$, the optical path probability distribution function $P({path} )$ of the scattering medium can be obtained according to the numerical distribution of ln. $P({path} )$ describes the superposition of n optical paths with different θn and φn, witch at the same time satisfy the free path distribution function, ${p_{fp}}(l )$.
The essence of speckle suppression by a scattering medium is to reduce the temporal coherence of lasers by introducing delays in the optical path. In the bulk scattering model, the optical path delay can approach many times the length of the scattering medium. The main factor controlling the frequency sensitivity of the speckle is the characteristic function ${M_l}({\Delta {q_z}} )$ of the optical path probability distribution $P({path} ).$
where $\Delta {q_z} = \frac{{2\pi ({n - 1} )\Delta \omega }}{c}$ represents the transmitted light path. For a Gaussian light source with a limited coherence length, the unilateral normalized power spectrum $G(\omega )$ can be expressed as:The autocorrelation function of the normalized power spectrum of the light source can be calculated as:
The initial speckle contrast, C0 is depend on the input laser spectral bandwidth. Once a certain laser is chosen, C0 is determined. In order to estimate the static speckle suppression using liquid light guide, speckle suppression efficiency, ${k_C} = {{{C_0}} / C}$ is defined to describe the speckle suppression effect in the subsequent experiments, where C is speckle contrast after the application of liquid light guides. Therefore, the impact of laser spectral bandwidth and uneven light emission can be avoided.
3. Simulations
The propagation of photons in the liquid light guide involves scattering as well as reflection, transmission, and absorption. For transmission, the critical angle of the total internal reflection can be calculated simply based on the refractive index of the core and cladding. When the angle of incidence of the photon on the cladding interface is smaller than the critical angle, it is considered that the photon will leave the liquid light guide and terminate its motion. For reflection under ideal conditions, an equivalent optical path is set, and its impact can be ignored in our analysis of the optical path probability function. For the absorption of photons, we can perform simulations by reducing the weight of light intensity after each scattering and the reflection of the photon. The transmittance of the liquid light guide was set to 85% per meter. To simplify the calculations, we extracted 85% of the photons with the shortest total distance for the simulations.
3.1 Material parameter analysis
According to the above analysis, it can be concluded that the normalized scattering coefficient $\mu _s^{\prime}$ is the main factor affecting the mean free path, and it affects the standard deviation ${l_{std}}$ of the free path distribution function along with the absorption coefficient. Therefore, we can analyze the influence of a liquid material on the path length probability distribution and then calculate the speckle contrast by changing the ratio $({{L / {{l_{mfp}}}}} )$ of the length of the optical axis to the liquid mean free path and the standard deviation of the free path distribution $\left( {\ln \frac{{\mu_s^{\prime}}}{{{\mu_a}}}} \right).$
Figure 2 shows the simulation results of the path length probability distribution $P({path} )$ and the speckle contrast C by MATLAB programming when the diameter of the liquid light guide D = 5 mm, the incident numerical aperture of light guide NA = 0.5, and the length of the liquid core is in the range of 100 to 1500 times the scattering mean free path $({{l_{mfp}}} ).$ The results of $P({path} )$ are shown in Fig. 2(a), and the results of the speckle contrast C are shown in Fig. 2(b). It can be seen from Fig. 2(a) that the maximum probability of $P({path} )$ distribution continues to decrease with an increase in the length of the liquid core. According to Fig. 2(a) and $\int_{ - \infty }^\infty {P({path} )} = 1,$ it can be concluded that $P({path} )$ expands as ${L / {{l_{mfp}}}}$ increases, and the speed of expansion gradually slows down. At the same time, it can be seen from Fig. 2(b) that along with the decrease in C at the exit end of the liquid light guide, the speed gradually decreases. When ${L / {{l_{mfp}}}} \sim 1000,$ the speckle contrast C no longer changes.
Fig. 2. Influence of L/lmfp on (a) the path length probability distribution P(path) and (b) the speckle contrast C
For a liquid light guide with a fixed length of $L = 250{l_{mfp}},$ a diameter of D = 8 mm, NA = 0.5, and ${l_{std}} = \ln (2 )-{-}\ln ({40} ),$ the simulated results of $P({path} )$ and C are shown in Fig. 3. As shown in Fig. 3(a), the logarithmic standard deviation, ${l_{std}},$ of the free path distribution function has a relatively small influence on $P({path} ).$ However, $P({path} )$ has a stable trend of gradually moving to the left as ${l_{std}}$ increases, and its distribution curve becomes lower and wider. At the same time, the speckle contrast, C gradually decreases, and C reaches the saturation value around 23% when ${{\mu _s^{\prime}} / {{\mu _a}}} = 15$ (Fig. 3(b)).
Fig. 3. Influence of ${{\mu _s^{\prime}} / {{\mu _a}}}$ on (a) the path length probability distribution P(path) and (b) the speckle contrast C.
3.2 Geometric parameter analysis
In Section 3.2, we discussed the influence of the liquid material based on ${L / {{l_{mfp}}}}.$ Hereafter, we will investigate the influence of structured parameters, such as the diameter D, incident angle ${\theta _{in}},$ and numerical aperture, on the speckle contrast C. As the incident numerical aperture of the light guide can be expressed as a function of ${\theta _{in}},$ we choose ${\theta _{in}}$ as the influencing parameter for the simulations of the influence of the incident angle and the numerical aperture on the speckle contrast.
Figure 4 shows the numerical simulation of the dependence of the liquid light guide diameter D on the path length probability distribution and speckle contrast. When the diameter of the liquid light guide increases from 1 mm to 10 mm with a liquid length of $L = 250{l_{mfp}},$ $P({path} )$ gradually shifts to the right, but the height increases slightly. As shown in Fig. 4(a), the increase in the diameter of the liquid core does not widen $P({path} ),$ hence it has no effect on C, which remains at ∼28%. Figure 4(b) also verifies this result. Therefore, we can conclude that the speckle contrast is not influenced by the diameter of the liquid core with a liquid length of $250{l_{mfp}}.$
Fig. 4. Influence of the liquid core diameter D on (a) path length probability distribution P(path) and (b) the speckle contrast C.
It was reported that an increase in the incident numerical aperture of the fiber reduces the speckle contrast at the exit end of the multimode fiber [15]. The increase in the optical path length in a multimode fiber mainly depends on the total internal reflection between the fiber core and the cladding, which is related to the numerical aperture. However, increasing the incident numerical aperture of the liquid light guide does not widen its path length probability distribution, as shown in Fig. 5(a), and the speckle contrast is also almost unchanged, as shown in Fig. 5(b). This can be explained by the fact that the scattering angle is random and restricted by the scattering process. In a medium with a strong scattering effect, it is scattering rather than reflection, which affects the widening of the path length probability distribution function. Therefore, changing the angle of the incident light, ${\theta _{in}},$ can be equivalently regarded as limiting the scattering angle θ. This effect is small in liquid light guides with a length exceeding 100 times the mean free path (that is, more than 100 scattering events occurs on average).
Fig. 5. Influence of the incident angle θin on (a) the path length probability distribution P(path) and (b) the speckle contrast C with D = 8 mm, L = 250lmfp, and lstd = ln(50).
From the above discussions, we can infer that the effect of the incident angle on the speckle contrast decreases with an increase in the light guide length. The simulation results with fiber lengths 10 to 80 times larger than the mean free path confirmed this conjecture. Figure 6 shows that the variation in the speckle contrast at the exit end of a liquid light guide is a function of the incident angle. It can be seen that as the length of the liquid core increases, the rate at which the speckle contrast decreases with ${\theta _{in}}$ gradually reduces.
Fig. 6. Influence of the incident angle ${\theta _{in}}$ on the speckle contrast C for different lengths of liquid light guide.
4. Speckle suppression experiment with liquid light guides
4.1 Experiments with the in-house fabricated liquid light guides
To test the influence of the parameters of the liquid light guide on the speckle contrast and to verify the simulation results, we fabricated a series of liquid light guides for experiments. The liquid light guide can reflect all the incoming light. The liquid light guide is similar to the single-core silica fiber, but it has a much larger diameter. Although the fiber bundles can have larger diameters than a single multimode fiber, the gaps between the monofilaments within a fiber bundle are the “dead corners” that cannot be used. The structure of the liquid light guide can be divided into three parts: cladding, liquid core, and optical window (i.e., the ends of the liquid light guide), as shown in Fig. 7. The cladding of the liquid light guide is generally a flexible polymer material, and the liquid core is composed of a high-refractive-index liquid. Therefore, it has better bending characteristics than a multimode fiber or fiber bundle.
We chose a transparent Teflon tube with diameter in the range of 5.5 mm to 10 mm as the cladding of the liquid light guide, which has a refractive index of 1.37. It is insoluble in any solvent and has excellent chemical stability, corrosion resistance, airtightness, and good anti-aging resistance. The light transmittance and refractive index are the main factors considered when choosing the liquid core material [18,19,26,27]. The higher the refractive index, the larger is the numerical aperture of the liquid light guide. For the experiments, we chose the following liquids: 50% sucrose solution with a refractive index of 1.42, analytical grade pure glycerol with a refractive index of 1.473, and analytical grade pure benzene with a refractive index of 1.5012. The numerical aperture (NAlcf) of the various liquid light guides can be calculated using the following formula:
where ${n_1}$ and ${n_2}$ are the refractive indices of the liquid core and cladding, respectively. The numerical apertures of the liquid light guide made of 50% sucrose solution, glycerol, and benzene were calculated to be 0.37, 0.54, and 0.61, respectively. We chose a heat-resistant transparent acrylic cylinder as the material for the light window.A Thorlabs semiconductor laser with a center wavelength of λ = 638 nm was used in the experiments. The light spot of the laser and its one-dimensional intensity distribution are shown in Fig. 8. The initial speckle contrast was measured as C0 = 60%, and the initial spot of the laser is not uniform at this time. In addition to the effect of speckle, the intensity fluctuations of initial spot is also caused by uneven light emission of the laser itself and insufficient collimation accuracy. However, when using the speckle suppression efficiency, ${k_C} = {{{C_0}} / C}$ to describe the speckle suppression effect, the uneven light intensity of the above-mentioned laser does not affect our conclusion.
Fig. 8. (a) The light spot of the laser, (b) one-dimensional intensity distribution of the laser light spot along the red dash line in (a).
The optical scheme used in the experiments is illustrated in Fig. 9. The collimated laser beam is first constrained by aperture stop 1 with diaphragm D1 to control the incident numerical aperture of the liquid light guide, and then coupled into the liquid light guide through the coupling lens with aperture D2 = 25.4 mm. The numerical aperture of the coupling lens should be equal to or slightly larger than that of the liquid light guide to make full use of the light-collecting ability of the liquid light guide. The laser beam emitted from the liquid light guide is then homogenized through the fly eye lens with aperture D3 = 50 mm and imaged on to the aperture stop 2 with diaphragm D4. The intermediate image is received by aperture stop 2 and then projected onto the screen through the objective lens with aperture D5 = 50.8 mm. D1 and D4 are adjustable. A CMOS (complementary metal-oxide-semiconductor) camera (Thorlabs DCC3260M) is used to capture images on the screen. There are two conjugate planes in the optical scheme: one is the exit end of the liquid light guide and the aperture stop 2, and the other is the aperture stop 2 and the screen. In addition, in Fig. 9, D6 is the aperture of the camera lens, S1 is the distance from the coupling lens to the front end of the liquid light guide and is equal to its focal length F1, S2 is the distance from the exit end of the liquid light guide to the fly eye lens, S3 is the distance from the fly eye lens to the aperture stop 2, S4 is the distance from the aperture stop 2 to the objective lens, S5 is the distance from the objective lens to the screen, S6 is the distance from the screen to the camera lens, and S7 is the distance from the camera lens to the detector array. And the focal length of objective lens is 60 mm.
Fig. 9. Optical scheme of speckle suppression experiments with a liquid light guide, where S1 = F1 = 45 mm, S2 = S3 = 80 mm, S4 = 62 mm, S5 = 1810mm, S6 = 1580 mm, S7 = 25 mm, and D6 = 1 mm.
Owing to the low spot uniformity presented by the in-house fabricated liquid light guides, a fly eye lens was adopted in the optical scheme to make the spot uniform. Our experiments prove that the fly eye lens alone does not have a speckle-suppression effect. A fly eye lens with a hexagonal arrangement of sub-eyes was used, which was made of PMMA (polymethyl methacrylate) material, with a diameter of 50 mm, a focal length of 40 mm, and a grain pitch of 0.3 mm.
Unlike ordinary multimode fibers, even under the incidence of parallel light, a liquid light guide made of 50% sucrose solution or glycerin or benzene still has a certain speckle suppression effect. The experimental results are shown in Fig. 10. The light spots of the liquid light guides with different liquid materials and with a diameter of 7 mm and a length of 0.3 m are shown in Fig. 10(a) to Fig. 10(c). For comparison, a polymer step multimode fiber with numerical aperture of 0.37 and fiber diameter of 1 mm was used in the experiments. The light spots of the multimode fibers with a length of 0.3 m and a length of 2 m are shown in Fig. 10(d) and 10(e). Figure 10(a) shows that the liquid light guide with 50% sucrose solution has a speckle contrast of 25.2% and a speckle suppression efficiency of 2.3. Figure 10(b) shows that the glycerol liquid light guide has a speckle contrast of 24.3% and a speckle suppression efficiency of 2.5. Figure 10(c) shows that the benzene liquid light guide has a speckle contrast of 21.2% and speckle suppression efficiency of 2.8. In contrast, when using multimode optical fibers with lengths of 0.3 m and 2 m, there is no obvious speckle suppression effect, as shown in Fig. 10(d) and Fig. 10(e).
Fig. 10. Images of light spots at the exit end of the liquid light guides or multimode fibers (top) and the one-dimensional light intensity distribution along the red dash lines on each spot image (bottom) for (a) liquid light guide with 50% sucrose solution and L = 0.3 m, (b) glycerol liquid light guide with L = 0.3 m, (c) benzene liquid light guide with L = 0.3 m, (d) multimode fiber with L = 0.3 m, and (e) multimode fiber with L = 2 m.
It can be seen from the experimental results that the speckle suppression efficiency increases with an increase in the numerical aperture of the liquid light guides. The spot uniformity of the glycerin and benzene liquid light guides are better than that of the 50% sucrose liquid light guide. For multimode fibers, the transmission length has a significant influence on the characteristics of the light spot. For the multimode fiber with a short length (e. g., 0.3 m), as there is no mode mixing caused by bending or vibration, the LP0n mode is excited first [28]. The dominant mode mixing is coherent superposition. Therefore, the outgoing spot appears as regular ring stripes (see Fig. 10(d)). When the optical fiber reaches a certain length, the difference in the transmission constant between the modes become sufficiently large for incoherent superposition between the modes. This results in a reduction in the coherence of light, and hence suppression of the speckle. Figure 10(e) shows the spot image and light intensity distribution at the exit end of a multimode fiber with a length of 2 m.
Although the speckle contrast of 1% was predicted in Goodman’s theory [14], the reported experimental results are much higher than it [15,16,29,30]. In our previous researches [15,16], the speckle contrast C for multimode fibers with the length of 4 m, 7 m, 10 m and 15 m was measured. For the fiber length of 15 m, C = 15.6% and C = 13.9% were obtained for a blue semiconductor laser (PL 450B) and green semiconductor laser (PL 520B1) from OSRAM, respectively [15]. It has shown that even for large fiber lengths (L = 15 m), the speckle contrast was large, and it appears as if one should not expect a large gain in speckle suppression effect with further increase in the fiber length due to the saturation. In addition, as the fiber length increases, the optical attenuation and radiation loss increase and the coupling efficiency decreases. Therefore, a very long fiber is not technically preferable for a practical projector, especially for a pico-projector.
In order to increase the light coupling efficiency and decrease the length of fiber, a multimode fiber bundle was also investigated for a static speckle suppression [15]. For the fiber bundle with length of 0.4 m, we obtained values of 20% for speckle contrast C and 2.3 for speckle suppression efficiency kc for the same green laser, and corresponding values of 30% and 1.86 for the same blue laser. Although the fiber bundles can have larger diameters than a single multimode fiber, the gaps between the monofilaments within a fiber bundle cannot be used. Therefore, the speckle suppression effect were not satisfied.
From the preliminary results shown in Fig. 10(a) to Fig. 10(c), we think the speckle suppression using liquid light guide could be a better solution than a multimode fiber or fiber bundle. Unlike multimode fiber bundles, the liquid core cross-section of liquid light guide is made of a single material. Thus, the coupling loss caused by the gaps between the monofilaments in the multimode fiber bundle is avoided. In addition, the liquid light guide does not suffer from the problems of wire breakage and the decrease in light transmission efficiency due to the repeated bending of the multimode fiber or fiber bundle.
Figure 11 shows the speckle suppression results of a glycerol liquid light guide with a length of 0.3 m and diameter in the range of 5.5 mm–10 mm. It can be seen that as the diameter of the glycerol liquid light guide increases, the speckle contrast tends to decrease but does not change significantly. This result is different from the conclusion that the speckle contrast is not related to the diameter in the above simulation results. We think that this non-conformity is caused by the simplification of the reflection model mentioned in section 2, in which we ignored the light energy loss related to the reflection. As the liquid light guide with a small diameter imposes a stronger restriction on the range of photon motion, the speckle contrast is slightly higher. For the glycerin liquid light guide, the speckle contrast reaches the lowest value of $C = 21.94\%$ when L = 0.3 m.
Fig. 11. Speckle suppression by the glycerin liquid light guides with different diameters: (a) speckle contrast C and (b) speckle suppression efficiency kc.
In addition to the red laser as described above, a green semiconductor laser (PL 520B1) from OSRAM with a center wavelength of λ = 520 nm was also used in the experiments for the in-house fabricated liquid light guides with the liquid cores of glycerin and benzene, respectively. By using the optical scheme shown in Fig. 9 and with the fly eye lens, similar results with slightly lower speckle contrasts were obtained with 0.3 m length of the liquid light guide. According to the experimental results with the red and green lasers, a similar result can be expected for a blue semiconductor laser.
4.2 Experiments on the LUMATEC liquid light guide
Although the in-house fabricated liquid light guides provide more flexibility for our experimental research, the optical transmittance and the speckle suppression effect are limited by the production quality. Therefore, we purchased a series of commercial liquid light guides produced by LUMATEC with NAlcf = 0.6; diameters of 3 mm, 5 mm, 8 mm; and lengths of 1.2 m, 1.5 m, 1.8 m, 2 m, 2.4 m for follow-up experiments. All those purchased LUMATEC liquid light guides have transmittance of 85% per meter in the visible light range. The experimental optical scheme is similar to that shown in Fig. 9. The only difference is that no fly eye lens is used, considering the better uniformity of the light spot emitted by the commercial liquid light guide. The exit end of the liquid light guide was directly imaged on the screen through an objective lens. When the fly eye lens is removed, only the position of objective lens needs to be changed. The objective lens was moved forward so that the exit end of the liquid light guide keeps conjugate with the screen.
Figure 12 shows the simulated results (dashed lines) of the speckle contrast at the exit end of the liquid light guides under collimated incident conditions, as well as the experimental results (indicated by symbol *). It can be seen that for a liquid light guide with a diameter of 3 mm or 5 mm and a length of 1 m, the speckle contrast is ∼15.3%. When the length of liquid light guide is 1.8 m, the speckle contrast drops to 14%, and when the length is 2.4 m, the speckle contrast mildly drops to 12%. It was observed that our simulation model is consistent with the experimental results. It can be seen from Fig. 12 that the speckle contrast at the exit end of the liquid light guide slightly decreases with an increase in the length regardless of the change in liquid core diameter. It can be inferred that as the length of liquid light guides increases, the speckle contrast gradually approaches a constant value. When the length of the liquid light guide is larger than 1 m, the speckle contrast is almost unaffected by the diameter of the liquid light guide.
Fig. 12. The relationship between the speckle contrast C and the diameter and the length of the liquid light guides. Speckle contrast of a liquid light guide with diameter D = (a) 3 mm, (b) 5 mm, and (c) 8 mm. The numerical aperture of the liquid light guide is NAlcf = 0.6.
Figure 13 shows the influence of the incident numerical aperture on the speckle contrast at the exit ends of a multimode fiber and a liquid light guide having the same length of 2 m. The speckle contrast at the exit end of the multimode fiber decreases from 24% to 19% as the numerical aperture increases. Unlike the multimode fiber, the speckle contrast at the exit end of the liquid light guide is stable at about 12% regardless of the numerical aperture. The experiments prove that for the same length, the speckle suppression efficiency of liquid light guides is higher than that of multimode fibers.
Fig. 13. Comparison of speckle suppression performance of the liquid light guide and the multimode fiber. Both of them have the same length (L = 2 m).
For the liquid light guides, according to the obtained theoretical and experimental results, we predict the speckle contrast can be slightly lower than 12% as the improvement of their production quality. For the multimode fibers, speckle contrast of 13.9% and 15.6% were obtained in our previous work with the fiber length of 15 m for a blue laser (PL 450B) and a green laser (PL 520B1) from OSRAM, respectively [15]. And it has appeared that the speckle suppression effect tends to saturate with the fiber length around 15 m. For further improving the speckle suppression effect, a combination of multimode fiber or fiber bundle with additional optical elements such as DOE could be a technical solution [15]. However, even for the combination with other optical elements, the liquid light guides still have advantages in terms of their bigger aperture, higher transmittance and shorter required length.
5. Conclusions
In this study, we established a liquid scattering model to simulate and analyze the speckle contrast under a variety of conditions. It was found that our model greatly reduces the time complexity of the calculation because it obviates the need for solving analytical solution of complex diffusion equation. The theoretical results obtained were confirmed by the experimental results. We believe that the developed model is useful not only for laser projection but also for imaging applications based on scattering effects, such as bio-imaging and medical imaging.
The experimental studies on the in-house fabricated liquid light guides and commercial LUMATEC liquid light guides demonstrates that liquid light guides are highly effective for laser speckle suppression. By using the combination of a fly-eye lens with the in-house fabricated liquid light guide of 0.3 m length and 7 mm diameter, the speckle contrast of $C = 21.2\%$ and a speckle suppression efficiency of ${k_c} = 2.8$ could be achieved at the exit end of the benzene liquid light guide. At the exit end of the 2.4 m long commercial LUMATEC liquid light guide, a speckle contrast of 12% and a suppression efficiency of ${k_C} = 5$ could be achieved. The results also show that for liquid light guides longer than 1 m, the speckle suppression effect is not affected by its incident numerical aperture.
In summary, the use of a liquid light guide is a feasible technical solution for suppressing speckle. Compared with multimode fibers of the same length, the liquid light guide has a higher speckle-suppression efficiency. The transmittance of the liquid light guide is very high in the visible light range, and the speckle suppression effect is not sensitive to the light wavelength. It is worth noting that the speckle suppression method based on the liquid light guide has a few issues. The large diameter of the liquid light guide imposes a burden on the volume and weight of the projection equipment. In addition, the leakage, temperature, and pressure stability of the fibers need to be further improved.
Funding
National Key Research and Development Program of China (2018YFB0504600, 2018YFB0504603); National Natural Science Foundation of China (61975183).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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