Open Access
Share with Facebook
Tweet This
Post on reddit
Share with LinkedIn
Add to CiteULike
Add to Mendeley
Add to BibSonomy
Abstract
Speckle patterns are used in a broad range of applications including microscopy, imaging, and light–matter interactions. Tailoring speckles’ statistics can dramatically enhance their performance in applications. We present an experimental technique for customizing the spatio-spectral speckled intensity statistics of optical pulses at the output of a complex medium (a disordered multimode fiber) by controlling the spatial profile of the input light. We demonstrate that it is possible to create ensembles of independent speckle patterns with arbitrary statistics at a single wavelength, simultaneously at multiple decorrelated wavelengths, and even tailored statistics across an entire pulse spectrum.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. INTRODUCTION
Speckle formation is ubiquitous, irrespective of wave medium or even wave nature. Whenever a coherent wave scatters from a large rough surface, the resulting random partial waves interfere in the far field, creating an irregular granular pattern known as a speckle. If the random partial waves are statistically independent, with uniformly distributed phases between zero and $2\pi$, the resulting speckle pattern is called “fully developed.” While the likelihood of finding two identical fully developed speckle patterns in nature is effectively zero, nearly all speckle patterns obey Rayleigh statistics. Typical Rayleigh speckles have a circular Gaussian field probability density function (PDF), a negative exponential intensity PDF, $P(I/\langle I\rangle) = {\exp}[- I/\langle I\rangle]/\langle I\rangle$, and lack any significant spatial coherence outside the diffraction limited speckle grains [1–3]. Due to both their near-universal statistics and their pervasiveness, speckles have been utilized in numerous practical applications. In optics, for example, speckle patterns have been used in imaging/wide-field microscopy applications [4–24], as well as in speckle-based sensing techniques [25–34]. Speckles have been adapted for use in optical trapping and micro-manipulation [35–38], as well as to control living cells in active media [39–41].
Even though Rayleigh speckles are the nearly universal family of speckles found in nature, over the years, attempts have been made to create non-Rayleigh speckles [42–48]. Early research demonstrated that non-Rayleigh speckles arise when the speckled field is either under-developed (not fully randomized or the sum of a small number of partial waves) or partially coherent (the sum of incoherent partial waves) [42–47]. After the advent of liquid-crystal-based wavefront shaping, a straightforward technique for creating fully developed non-Rayleigh speckle patterns in free space was developed [48], where the non-Rayleigh speckles could adhere to an intensity PDF with a tail decaying either faster (sub-Rayleigh) or slower (super-Rayleigh) than a negative-exponential function. Subsequent to this, a method for creating fully developed speckle patterns with arbitrary intensity PDFs was developed [49]. These, and other contemporary works [48–62], opened the door to many practical applications involving bespoke speckle patterns. Introducing tailored speckles into standard speckle imaging applications can improve the signal-to-noise, resolution, and/or the overall image quality [20,63–70]. For example, in the context of parallelized nonlinear pattern-illumination microscopy (stimulated emission depletion, reversible saturable optical fluorescence transitions, etc.), tailored speckles can produce a spatial resolution three times higher than the optical diffraction limit, before an equivalent Rayleigh speckle approach can become diffraction limited [59].
Fig. 1. (a) Experimental setup and (b)–(e) standard pulse properties. Light from a transform-limited pulsed laser source is split into two beams. One beam is wavefront shaped by a SLM and injected into a disordered MM fiber, while the other is used as an interferometric reference beam. The output of the disordered fiber is imaged by a camera, where the two beams are recombined and used to measure the pulsed field output from the fiber. A typical spatially averaged spectrum, measured using the interferometer, is shown in (b). Under random input conditions, (d), (e) the normalized speckled intensity patterns output from the fiber (c) adhere to Rayleigh intensity statistics. The measured intensity PDF (purple line) follows the theoretical prediction for Rayleigh speckles (black dashed line). The spectral resolution (0.8 nm) is significantly smaller than the spectral decorrelation length (4 nm) of the pulse output from the fiber. In (d), (e) the white circle marks the fiber core.
Generally, existing speckle customization techniques are intended for use in free space, with monochromatic or narrowband light (nanosecond pulses), i.e., light lacking spatio-spectral complexity. On the other hand, broadband spatio-spectrally complex light is used in a number of applications [71,72]. For example, in snapshot hyperspectral imaging [73–75], the spatial or spectral scanning required in conventional multispectral imaging is replaced by a single image acquisition. This is accomplished by leveraging speckled spatio-spectral complexity to perform measurements in parallel. Bespoke speckles have the potential to enhance broadband speckle imaging applications such as this, as they have done for monochromatic imaging applications. There is also a general interest in controlling short pulse behavior in complex media—such as multimode (MM) fiber—due to the potential applications in optical telecommunications [76], and MM fiber high-powered pulsed lasers [77–79]. Current speckle customization techniques cannot be applied to these research areas, however, because they typically involve broadband pulses that can be spatio-spectrally complex. Furthermore, it is not known if the speckle statistics in a broadband pulse can be controlled, and if this is possible, to what extent. Even in the context of monochromatic speckle customization, important questions remain unanswered. Whether it is possible to arbitrarily customize the intensity statistics of speckle patterns after a complex medium remains unknown. Along these lines, the influence of common effects such as incomplete control—due to either imperfect or insufficient light coupling into a medium, as well as index mismatch at the surfaces—on controlling speckles at the output of a complex medium remain unknown.
In this work, we experimentally demonstrate the ability to customize the intensity statistics of spatio-spectrally speckled pulses at the output of a disordered MM fiber. To accomplish this, we leverage the ability of a disordered MM fiber to convert spatial degrees of freedom—controlled using a phase-only spatial light modulator (SLM) modifying the light entering the fiber—into spatio-spectral degrees of freedom, which are in turn used to customize the speckle statistics across the pulse. Because the spectral decorrelation length of the fiber (4 nm) is smaller than the spectral width of the pulse (20 nm), we can generate independent speckle patterns with distinct statistics at different wavelengths within the pulse, using a single SLM pattern. We show that it is even possible to create non-Rayleigh speckles across the entire spectrum of the pulse. To create these distributions, we start with an ensemble of speckled pulses output from the fiber and tailor the spectrally dependent statistics of the speckles using sequential PDF transformations regularized by a spatio-spectral field-mapping matrix that characterizes the MM fiber. An interesting feature of all the results presented here is that the light output from the fiber has more degrees of freedom than the input light has degrees of control. Thus, this work also raises foundational questions for future studies, such as, “How many degrees of freedom are needed to generate speckle patterns with specified intensity statistics?”
2. EXPERIMENTAL SETUP
The experimental setup used in this work is illustrated in Fig. 1(a). An all-normal-dispersion fiber laser [80] and grating compressor generate a train of approximately transform-limited pulses [temporal duration 150 fs full-width at half-maximum (FWHM), repetition rate 35 MHz, and representative average spectrum shown in Fig. 1(b)]. The pulses are split into two separate beams using a $\lambda /2$ plate and a polarizing beam splitter (PBS). One beam illuminates the surface of a phase-only SLM (Meadowlark P1920-1064), while the second is used as a reference. The liquid crystal display on the SLM is fully illuminated, and a circular array of 156 macro-pixels controls the light reflected from the panel (circle radius is $\approx 7$ macro-pixels, and each macro-pixel consists of $64 \times 64$ SLM pixels). Within each macro-pixel, a stepped diffraction grating is displayed to control the amplitude and phase of the light in the first-order diffraction of each macro-pixel. Using a $f = 15\;{\rm{cm}}$ relay lens, we image the SLM macro-pixel array onto the back focal plane of an objective lens ($10 \times$, NA = 0.25)—via the first-order diffraction pattern—which then excites a graded index MM fiber (Thorlabs GIF 625) with the Fourier transform. The 1.4 m long fiber supports $\approx 325$ modes per polarization, and it is clamped at numerous locations along the fiber to introduce strong mode mixing. Specifically, the characteristic disorder length $\ell$ (often called the transport mean free path [81] or the equilibrium modal distribution length [82]) is approximately the length of the fiber. As a result, effectively any input spatial distribution is completely randomized when it reaches the end of the fiber. We image a single polarization of light output from the fiber onto a CMOS camera (Allied Vision Alvium U-511) using an objective lens ($20 \times$, NA = 0.40) with a $f = 15\;{\rm{cm}}$ tube lens. The second/reference beam is path-length-matched to the light imaged by the camera, and recombined with the light output from the fiber using a beam splitter before the camera. The combination of a variable delay stage along the beam path with the near plane-wave spatial profile on the camera enables off-axis digital holography to reconstruct the spatio-spectral field output from the fiber [83,84]. With this setup, we measure the spatio-spectral field-mapping tensor $({\hat x},{\hat y},{\hat \lambda})$, which can predict the full spatio-spectral field output from the disordered fiber for any input field modulation pattern created by the SLM macro-pixel array (see Supplement 1, Section 1). It is worth pointing out that this operator can be measured with a wavelength tunable CW laser instead of a pulsed laser. Nevertheless, with this operator, we study the properties of light traversing the disordered fiber. All speckle patterns presented in this work, and their associated statistics, are generated using the measured field-mapping tensor.
3. RAYLEIGH PULSE PROPERTIES
The spatio-spectral properties of light output from the disordered fiber under Rayleigh speckle illumination conditions (created by displaying uniformly distributed random phase patterns on the SLM array) are shown in Figs. 1(b)–1(e). The spatially averaged spectrum of the pulse output from the fiber [Fig. 1(b)] is unchanged relative to that of the input pulse. The intensity PDF of the normalized spectrally dependent spatially speckled fields is obtained after ensemble averaging 1000 independent realizations [purple line in Fig. 1(c)]. This PDF adheres to Rayleigh statistics (black line). To normalize the output intensity patterns, we use $\overline {I(x,y)/\langle I(x,y)\rangle} = 1$, where $\langle \cdots \rangle$ denotes ensemble averaging, and the overline indicates spatial averaging. This normalization is always used unless stated otherwise. For analysis of the statistics without normalization and effects on the method presented in this work, see Supplement 1, Section 2. Visual inspection of the output spectrally dependent spatial intensity patterns [Figs. 1(d) and 1(e)] indicates that the speckle patterns generated at the output of the disordered fiber are fully developed. This observation is rigorously confirmed in Supplement 1, Section 3.
One of the important properties of the speckled light output from the disordered fiber, relative to what can be created with a SLM and free-space optics, is the spatio-spectral complexity of the pulse. This property can be quantified by comparing the FWHM of the magnitude-squared spectral field-correlation function $|{C_E}(\Delta \lambda {)|^2}$ to the pulse bandwidth (20 nm). The spectral correlation function is defined as
4. METHOD
Our goal is to customize the intensity PDFs of the spectrally dependent speckles output from the fiber, and the technique for accomplishing this is an adaptive local intensity transformation algorithm based on spatio-spectral field mapping.
A. Field-Mapping Matrix
The first step in this approach is to define an appropriate field-mapping matrix that relates the macro-pixel array on the SLM to the spatio-spectral fields we wish to control at the fiber output, and to construct a matrix representing the inverse relationship. While the array on the SLM is two dimensional, we represent it as a 1D vector by assigning each macro-pixel’s field modulation to an index, $n$, out of $N$ total ($N = 156$ for our system). Similarly, we represent the output at each spectrally dependent camera pixel of interest $(x,y,\lambda)$ using a single index, $m$, out of $M$ total. Here $M$ is the number of camera pixels measuring the light in the core of the fiber, ${M_{{\rm core}}}$, multiplied by the number of different spectrally dependent fields we are attempting to control, ${M_\lambda}$. It is worth noting that the unperturbed fiber mode basis may seem more appropriate than the spatial basis. In practice, there are two main drawbacks to using the fiber mode basis. First, with only 156 macro-pixels available on the SLM, we do not have complete control over the ${\sim}325$ single-polarization modes at the input of the fiber. Second, to calculate the intensity PDF, we need real-space representation of the field. On the other hand, there is one restriction on using a spatially indexed representation of the fields: the spatial sampling grid ($x,y$) must satisfy the Nyquist sampling theorem. Experimentally, we sample the fields at the output of the fiber at a rate of ${\sim}18$ points (along a single dimension) per average speckle grain width. We can therefore represent the multiwavelength field mapping from the SLM macro-pixels to the camera pixels with a single rectangular ($M \times N$) matrix using this basis and experimentally measure the field-mapping tensor to determine the matrix elements. Because it is rectangular, the field-mapping matrix does not have a well defined inverse. Nevertheless, we can still calculate its Moore–Penrose inverse matrix [85], often called the “pseudo-inverse” matrix. To a good approximation, applying the pseudo-inverse matrix to any field created by applying a SLM pattern to the field-mapping matrix results in the SLM pattern. When the pseudo inverse is applied to an arbitrary/artificial field, it gives a “best guess” for the SLM pattern that generates the field. However, the obtained pattern will not necessarily generate the arbitrary field. Therein lies the need for our method to be adaptive.
B. Adaptive PDF Transformation
Our next step is to define a formalism for obtaining a function that transforms a set of scalar intensity values, $I(x,y,\lambda)$, adhering to a given intensity PDF, $P(I)$, into a new set of scalar intensity values, $J(x,y,\lambda)$, adhering to the desired intensity PDF, $F(J)$. This type of local intensity transformation [49,58,62] is obtained by equating an infinitesimally narrow area element in each PDF, $P(\tilde I)d\tilde I = F(\tilde J)d\tilde J$, integrating
and solving the integral equation to write $J$ as a function of $I$. In addition to the standard PDF normalization condition, i.e., $\int_0^\infty P(I)dI = 1$, it is generally good practice to ensure that the PDFs have the same average intensity value $\langle I\rangle = 1$. Equation (2) can be numerically solved for any physically allowed $F(J)$ and $P(I)$, and the solution $J(I)$ is the desired local intensity transformation. Certain intensity PDFs are unphysical, however. For example, a zero in the PDF [$P(I) = 0$, when $0 \lt I \lt {I_{{\max}}}$] would necessitate a discontinuity in the field and is impossible for a classical light field. One of the advantages of using a local intensity transformation to alter the intensity PDF of a speckle pattern, or an ensemble of speckle patterns, is that it is a scalar function. As such, if the original ensemble of speckles consists of statistically stationary and ergodic random variables, which is the case for fully developed Rayleigh speckles, then the transformed variables are also stationary and ergodic [49].Fig. 2. Monochromatic speckle patterns with tailored intensity PDFs output from a disordered fiber. Output speckle patterns adhering to (a) linearly decreasing, (b) uniform, (c) linearly increasing, (d) unimodal, and (e) bimodal intensity PDFs are shown. Below each output pattern, the associated ensemble-averaged intensity PDF (purple line) is compared with the ideal PDF (gray dots), and a Rayleigh PDF (black line). The white circles are the boundary of the fiber core. One hundred independent speckle patterns are used to calculate each PDF. All speckle patterns are generated from the measured field-mapping tensor.
C. Iterative Algorithm
With the field-mapping operators defined and an adaptive PDF transformation technique identified, we can iteratively use the two numerical operators to arbitrarily customize ensembles of speckles. We start by numerically creating an ensemble of 100 independent speckle realizations, each generated by applying a different random SLM phase array to the field-mapping matrix. The intensity PDF, of the entire ensemble of fields predicted by the field-mapping matrix, is then calculated. The obtained intensity PDF is used in conjunction with the desired PDF to obtain a local intensity transformation function. The local intensity transformation is applied to the amplitude values of the field in each speckled realization, transforming the ensemble’s intensity PDF to the desired PDF. The phase value associated with each amplitude value is left unchanged. The result is an ensemble of “artificial” fields that are not necessarily possible to generate with the SLM, as discussed previously. Applying the pseudo-inverse matrix to each transformed field yields a best guess for a generating SLM field. By reapplying the field-mapping matrix to each best guess, we obtain an ensemble of non-artificial fields. The resulting fields do not exhibit the desired statistics; however, they also do not follow the original intensity PDF. In fact, they are incrementally closer to the desired statistics. By numerically iterating this process of generating a large ensemble of speckle realizations using independent SLM patterns, applying a local intensity transformation, applying the pseudo inverse, and repeating with the new SLM patterns, we eventually obtain a set of independent SLM patterns that will generate an ensemble of speckle realizations adhering to a desired intensity PDF at the output of the fiber. This numerical method can be modified to generate different PDFs (at different wavelengths, for example) by applying a different intensity transformation to different elements (wavelengths) of a given speckle realization.
5. RESULTS
A. Monochromatic Customization
Using the method described in the last section, we customize the monochromatic ($\lambda = 1020.4\,\,\rm nm$) intensity statistics across the entire fiber core. In Fig. 2(a), we present an example speckle pattern that adheres to a linearly decreasing intensity PDF, $P(I/\langle I\rangle) = (2/9)(3 - I)$, over a finite intensity range, $0 \le I/\langle I\rangle \le 3$. Unlike a Rayleigh PDF (black line) which monotonically decreases over an infinite range of intensity values, the custom intensity PDF (purple line) monotonically decreases over a finite range and then is zero for larger values of $I$. The monotonically decreasing nature of the custom PDF preserves the spatial structure/topology of a typical Rayleigh speckle pattern, an interconnected sea of low-intensity values surrounding isolated bright intensity islands, while the truncation of the custom PDF regularizes the peak intensity of the speckle grains. This can be seen by comparing Figs. 1(d) and 1(e) with Fig. 2(a). Changing the intensity PDF from decreasing to uniform, $P(I/\langle I\rangle) = (1/2)$, over finite intensity range, $0 \le I/\langle I\rangle \le 2$, changes the speckle topology but still regularizes the high-intensity grain values, as shown in Fig. 2(b). For this custom PDF, the speckle grains irregularly merge, breaking apart the interconnected low-intensity channel structure. By customizing the intensity PDF to linearly increase, $P(I/\langle I\rangle) = (8/9)(I)$, over a finite intensity range, $0 \le I/\langle I\rangle \le 1.5$, the spatial structure is inverted relative to Rayleigh speckles, as shown in Fig. 2(c). Here the speckle pattern consists of a sea of interconnected high-intensity grains surrounding isolated optical vortices. The interconnected structure in the speckle pattern need not be at the minimal or maximal intensity values, as shown in Fig. 2(d). Here the intensity PDF is unimodal, $P(I/\langle I\rangle) = \sin{(\pi I/2)^2}$, over finite intensity range $0 \le I/\langle I\rangle \le 2$. As a consequence, the average intensity values form a conjoining web while the vortices and the high-intensity regions remain isolated. Introducing a second maximum, as in the bimodal intensity PDF case shown in Fig. 2(e), creates a second intensity web structure. It is worth pointing out that Fig. 2(e) is an example of what happens when an unphysical intensity PDF, $P(I/\langle I\rangle) = \sin{(\pi I)^2}$ over $0 \le I/\langle I\rangle \le 2$, is used in our algorithm. The impossible zero in the intensity PDF is removed, and the exact PDF is not obtained when the iterative algorithm converges. The speckle patterns shown in Figs. 2(a)–2(e) are representative examples taken from ensembles of 100 statistically independent speckle patterns, all of which adhere to the purple intensity PDFs shown in the bottom row of Fig. 2. The speckles in each ensemble are fully developed, stationary, and ergodic (see Supplement 1, Section 3). Taken together, Figs. 2(a)–2(e) indicate that the intensity PDF of the speckles can be arbitrarily tailored. While this type of arbitrary customization has been shown before in free space [49], this is the first demonstration after a disordered medium, and it has been accomplished without dividing the output into “target” and “junkyard” regions; the full output intensity pattern is customized. Finally, while the customization shown was performed on spatially normalized intensity values $I(x,y)/\langle I(x,y)\rangle$, in Supplement 1, Section 2, we show that the same results can be obtained without spatially normalizing the output.
Fig. 3. Spectral detuning of customized speckles. The output from the fiber, associated with a customized speckle pattern (a) at $\lambda = 1020.4\,\,{\rm{nm}}$, is compared to speckle patterns at (b)–(e) spectrally detuned wavelengths generated by the same SLM pattern. Below each wavelength-dependent pattern, the associated ensemble-averaged intensity PDF (purple line) illustrates the speckles’ reversion back to Rayleigh statistics (black line)—effectively obtained after (e)—within one spectral decorrelation length, 4 nm. The white circles are the boundary of the fiber core. One hundred independent speckle patterns are used to calculate each PDF. All speckle patterns are generated from the measured field-mapping tensor.
B. Spectral Detuning
The tailored speckle PDFs of Fig. 2 are obtained at a single wavelength. The pulse at the output of the fiber has a relatively broad bandwidth (20 nm) when compared to the spectral decorrelation length (4 nm), so the customized statistics do not persist across the spectrum. In Fig. 3, the spectral dependence of a customized speckle pattern is presented. Specifically, the case of a customized speckle pattern with a uniform intensity PDF (purple line) is shown in Fig. 3(a). For a small wavelength detuning of 0.8 nm (b), many of the speckle grains retain the same topological shape as in Fig. 3(a), but the speckle grain maxima are more irregular in Fig. 3(b). The shape retention is a consequence of the low-intensity part of the PDF ($I/\langle I\rangle \le 1.5$) remaining almost flat, while the increased maxima irregularity is reflected in the loss of the sharp cutoff at high-intensity values ($I/\langle I\rangle \ge 1.5$) of the PDF. While the spectrally detuned pattern does not strictly adhere to the customized PDF, it is still non-Rayleigh. With greater spectral detuning, Figs. 3(c)–3(e), the speckles gradually revert and become Rayleigh when the spectral decorrelation length $\Delta \lambda = 4\,\,\rm nm$ is reached [Fig. 3(e)]. For further increases in the detuning, the outputs adhere to Rayleigh statistics.
An important conclusion to draw from Fig. 3 is that monochromatic speckle customization influences the statistics of other wavelengths within the spectral coherence length. Specifically, the functional form of the intensity PDF and the speckle topology are largely preserved within a quarter of the spectral decorrelation length, $\Delta \lambda = \pm 1\,\,\rm nm$. This has two implications for broadband speckle customization, which are explored in the next two sections. First, if different intensity PDFs are desired at different wavelengths, in principle, the PDFs can be arbitrarily chosen if their wavelengths are separated by more than the spectral decorrelation length. If the wavelength separation is less, however, the PDF choice will face constraints. For example, the intensity moments of the PDFs, $\langle {I^n}\rangle = \int_0^\infty P(\tilde I){\tilde I^n}{\rm d}\tilde I$, may need to have similar values depending on the exact wavelength separation. Second, when performing broadband speckle customization, targeting the same PDF across the pulse spectrum, the statistics of the speckles within a quarter of the spectral decorrelation length, $\Delta \lambda = \pm 1\,\,\rm nm$, do not need to be directly modified. This is because the intensity PDF is largely preserved between wavelength separations less than a quarter of the spectral coherence length.
Fig. 4. Independent customization of the intensity PDFs at different wavelengths, with the same SLM pattern. Output intensity patterns at (a) 1020.4 nm and (b) 1038.4 nm—(c) opposite sides of the spectrum—are tailored to have (d) unimodal and bimodal intensity PDFs, respectively. The white circle is the boundary of the fiber core. One hundred independent speckle patterns are used to calculate each PDF. All speckle patterns are generated from the measured field-mapping tensor.
Fig. 5. Independent control of three different statistics, at three different wavelengths, with a single SLM pattern. Output intensity patterns at (a) 1024.4 nm, (b) 1028.5 nm, and (c) 1032.6 nm are tailored to have (d) linearly increasing (green), flat (blue), and linearly decreasing (purple) intensity PDFs, respectively. The dashed lines represent ideal PDFs, while solid lines represent the obtained PDFs. In (e), the intensity PDFs of output speckle patterns with wavelengths between 1024.4 and 1032.6 nm are shown (black lines). In (a)–(c), the white circle is the boundary of the fiber core. One hundred independent speckle patterns are used to calculate each PDF. All speckle patterns are generated from the measured field-mapping tensor.
C. Multichromatic Customization
By incorporating a second wavelength into the field-mapping matrix used in the algorithm, we can arbitrarily customize the intensity statistics of speckle patterns at different wavelengths simultaneously. With a single SLM field modulation pattern, the field output from the fiber can have distinct intensity statistics at different wavelengths. As an example, a single speckled output designed to obey a unimodal intensity PDF at 1020.4 nm while simultaneously obeying a bimodal intensity PDF at 1038.4 nm is shown in Fig. 4. The spectral separation of these outputs, 18 nm, is approximately the pulse bandwidth [Fig. 4(c)]. Between the two customization wavelengths, the speckles follow Rayleigh statistics in accordance with the detuning behavior shown in Fig. 3. In Supplement 1, Section 4, we show that as long as the wavelength separation is equivalent to, or greater than, the spectral decorrelation length, the wavelengths can be arbitrarily chosen with no effect on the intensity PDFs. Furthermore, we have not observed any restrictions on the choice of PDFs when tailoring speckles at two wavelengths, relative to one wavelength, under the same condition.
Comparison of the ensemble average intensity PDFs in Fig. 4(d) with Figs. 2(d) and 2(e) does not indicate any major changes in our ability to control the intensity PDF when increasing to multiple wavelengths. The unimodal intensity PDFs presented in the two figures are almost indistinguishable, a trend generally seen with other intensity PDFs generated. The bimodal intensity PDF in Fig. 4(d) does exhibit slight differences from the PDF in Fig. 2(e). Specifically, in Fig. 2(e), the maxima of the peaks in the PDF are approximately twice the value of the probability density at $I/\langle I\rangle = 1$, while in Fig. 4(d), they are approximately 1.75 times as large as $P(I/\langle I\rangle = 1)$. Furthermore, in the multichromatic customization case, the bimodal intensity PDF has a more prominent high-intensity tail than the monochromatic case. Differences arise for the bimodal case, and not the others, because our algorithm cannot create a speckle pattern that adheres to the “impossible” intensity PDF given to the algorithm, as discussed previously. Factors such as the ratio of peak intensities and the high-intensity tail are good qualitative measures for changes in our ability to customize the intensity statistics of a speckle pattern. These differences indicate that customizing the speckles at two wavelengths incurs only a small reduction in the degree of control that is possible.
By further increasing the number of independent wavelengths in the field-mapping matrix, we can simultaneously tailor the intensity statistics at three different wavelengths output from the fiber, using a single spatial profile at the input. In Fig. 5, the spectrally resolved intensity patterns at 1024.4 nm (a), 1028.5 nm (b), and 1032.6 nm (c) are tailored to have linearly increasing (green line), flat (blue line), and linearly decreasing (purple line) intensity PDFs (d), respectively. Across the 8 nm wavelength band, shown in Fig. 5(e), the output speckle PDFs smoothly transition between the bespoke statistics. Outside the band, the statistics revert back to Rayleigh. There is nothing unique about the PDFs shown; the spectrally dependent statistical evolution shown in Fig. 5 can be created with different PDFs (see Supplement 1, Section 4).
Contrary to the monochromatic case (Fig. 2) or the two-wavelength case (Fig. 4), in the three-wavelength case (Fig. 5), limitations on the ability to dictate the precise PDF(s) of the output speckles arise. Comparison between the ideal (dashed lines) and obtained PDFs (solid lines) in Fig. 5(d) shows that while the speckles still predominantly adhere to the desired statistics, deviations begin to appear near the discontinuities in the PDFs. Specifically, this occurs at $I/\langle I\rangle = 2$ for the speckles with a flat PDF, and $I/\langle I\rangle = 1.5$ for the speckles with a linearly increasing PDF.
Fig. 6. Full spectrum PDF customization. For a single SLM pattern, every wavelength of the pulse has (a)–(d) an output with (e) an intensity PDF tailored to decrease linearly rather than exponentially. The white circle is the boundary of the fiber core. One hundred independent speckle patterns are used to calculate each PDF. All speckle patterns are generated from the measured field-mapping tensor.
D. Full-Spectrum Customization
The experimental limit of our ability to control spatio-spectral intensity statistics is illustrated in Fig. 6, where we tailor the speckles to obey non-Rayleigh statistics across the entire pulse spectrum (1019.6–1040.8 nm). Specifically, we tailor the speckles to have a linearly decreasing intensity PDF at every wavelength in the pulse output from the fiber, like the PDF shown in Fig. 2(a). In Figs. 6(a)–6(d), different custom monochromatic speckles output from the fiber are shown, generated by a single SLM pattern. The customized speckles have completely different spatial profiles, since they are separated by more than one spectral decorrelation length. However, the intensity PDFs [Fig. 6(e)] are the same across the entire spectrum. To accomplish this, we use every-other spectral component of the measured field-mapping tensor (approximately 1.6 nm separation per wavelength) in the optimization matrix. When plotting the PDFs in Fig. 6(e), we use a 0.8 nm spectral resolution. Because the PDFs in Fig. 6(e) are the same, we know that the intensity statistics apply to every wavelength within the range.
In contrast to the single/multi-wavelength optimization shown in previous sections (Figs. 2, 4, and 5), in the full-spectrum optimization case, there are limits on what PDFs can be created. While we can generate unimodal intensity statistics across the spectrum (see Supplement 1, Section 5), we cannot create bimodal intensity statistics across the spectrum. To understand why this is the case, consider the relevant degrees of freedom. The SLM has 156 macro-pixels that can modulate the field incident on the fiber input. For a single polarization, the disordered fiber has about 325 spatial modes, and ${\sim}5$ spectrally distinct outputs (pulse bandwidth divided by spectral decorrelation length). Thus, the field we are attempting to control has approximately 1625 degrees of freedom, while the SLM has 156 degrees of control. Given the order-of-magnitude difference, it is not surprising that the full-spectrum customization cannot be done arbitrarily. In fact, it is remarkable that we can still manipulate properties of the PDF, such as the general shape (linear decay or unimodal). Furthermore, because we still possess some control over the shape of the PDF, we retain the ability to control properties such as the speckle contrast, $C = \sqrt {\langle {I^2}\rangle /{{\langle I\rangle}^2} - 1}$, similar to what was shown in [48] for monochromatic light, which is a sufficient degree of control for many exciting applications of tailored speckles.
6. CONCLUSION
In this work, important questions related to the generation of customized random light have been answered. First, we demonstrated that it is possible to arbitrarily tailor the statistics of monochromatic random light at the output of a complex medium (a disordered fiber). Not only was this done with fewer degrees of control over the input (156 SLM macro-pixels) than degrees of freedom in the output field (325 modes for one wavelength and polarization), but this arbitrary control was exerted over the entire output of the fiber core. Previous techniques for monochromatic speckle customization were either restrictive in terms of the obtainable statistics [48,50–56,60,61] or could generate arbitrary statistics in free space but only over a target region consisting of about 1/4 of the total far field controlled by the SLM [49,57–59,62]. Second, we demonstrated that with a single SLM pattern, multiple spectrally decorrelated outputs from the fiber can be generated with different arbitrary intensity statistics. Given that the generally accepted mechanism behind the creation of tailored speckle patterns is the presence of higher-order (fourth, sixth, eighth, etc.) correlations in the random partial waves that interfere to create the pattern [48,49,58,62], it was not obvious that this could even be done for the same statistics at decorrelated wavelengths—much less different statistics—since the wavelength-dependent field mappings are completely different. Nevertheless, it may be possible to connect the spatio-spectral ($x,y,\lambda$) statistical control demonstrated in our work to the volumetric ($x,y,z$) statistical control demonstrated in [62] via the chromato-axial memory effect [86]. Finally, we have demonstrated that it is possible to manipulate the speckle statistics across the entire pulse spectrum at the output of the fiber simultaneously.
Funding
Office of Naval Research (N00014-20-1-2789); National Science Foundation (ECCS-1912742).
Acknowledgment
N.B. thanks SeungYun Han and Hui Cao for insightful discussions on speckle statistics.
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.
Supplemental document
See Supplement 1 for supporting content.
REFERENCES
1. J. C. Dainty, Laser Speckle and Related Phenomena, Topics in Applied Physics (Springer Berlin Heidelberg, 2013).
2. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts and Company, 2007).
3. I. Freund, “’1001’ correlations in random wave fields,” Waves Random Media 8, 119–158 (1998). [CrossRef]
4. C. Ventalon and J. Mertz, “Quasi-confocal fluorescence sectioning with dynamic speckle illumination,” Opt. Lett. 30, 3350–3352 (2005). [CrossRef]
5. C. Ventalon and J. Mertz, “Dynamic speckle illumination microscopy with translated versus randomized speckle patterns,” Opt. Express 14, 7198–7209 (2006). [CrossRef]
6. J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802 (2008). [CrossRef]
7. D. Lim, K. K. Chu, and J. Mertz, “Wide-field fluorescence sectioning with hybrid speckle and uniform-illumination microscopy,” Opt. Lett. 33, 1819–1821 (2008). [CrossRef]
8. Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009). [CrossRef]
9. O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95, 131110 (2009). [CrossRef]
10. D. Lim, T. N. Ford, K. K. Chu, et al., “Optically sectioned in vivo imaging with speckle illumination hilo microscopy,” J. Biomed. Opt. 16, 016014 (2011). [CrossRef]
11. S. Bernet, W. Harm, A. Jesacher, et al., “Lensless digital holography with diffuse illumination through a pseudo-random phase mask,” Opt. Express 19, 25113–25124 (2011). [CrossRef]
12. E. Mudry, K. Belkebir, J. Girard, et al., “Structured illumination microscopy using unknown speckle patterns,” Nat. Photonics 6, 312–315 (2012). [CrossRef]
13. J. Min, J. Jang, D. Keum, et al., “Fluorescent microscopy beyond diffraction limits using speckle illumination and joint support recovery,” Sci. Rep. 3, 1–6 (2013). [CrossRef]
14. H. Yilmaz, E. G. van Putten, J. Bertolotti, et al., “Speckle correlation resolution enhancement of wide-field fluorescence imaging,” Optica 2, 424–429 (2015). [CrossRef]
15. D. Phillips, R. He, Q. Chen, et al., “Non-diffractive computational ghost imaging,” Opt. Express 24, 14172–14182 (2016). [CrossRef]
16. T. Chaigne, J. Gateau, M. Allain, et al., “Super-resolution photoacoustic fluctuation imaging with multiple speckle illumination,” Optica 3, 54–57 (2016). [CrossRef]
17. A. K. Singh, D. N. Naik, G. Pedrini, et al., “Exploiting scattering media for exploring 3D objects,” Light Sci. Appl. 6, e16219 (2017). [CrossRef]
18. A. Vigoren and J. M. Zavislan, “Optical sectioning enhancement using higher-order moment signals in random speckle-structured illumination microscopy,” J. Opt. Soc. Am. A 35, 474–479 (2018). [CrossRef]
19. L.-H. Yeh, S. Chowdhury, N. A. Repina, et al., “Speckle-structured illumination for 3D phase and fluorescence computational microscopy,” Biomed. Opt. Express 10, 3635–3653 (2019). [CrossRef]
20. M. Pascucci, S. Ganesan, A. Tripathi, et al., “Compressive three-dimensional super-resolution microscopy with speckle-saturated fluorescence excitation,” Nat. Commun. 10, 1327 (2019). [CrossRef]
21. H. Zhang, S. Jiang, J. Liao, et al., “Near-field fourier ptychography: super-resolution phase retrieval via speckle illumination,” Opt. Express 27, 7498–7512 (2019). [CrossRef]
22. Y. Choi, M. Kim, C. Park, et al., “Wide-field super-resolution optical fluctuation imaging through dynamic near-field speckle illumination,” Nano Lett. 22, 2194–2201 (2022). [CrossRef]
23. K. Affannoukoué, G. Maire, T. Mangeat, et al., “Super-resolved imaging under total internal reflexion using random illumination microscopy (rim),” Proc. SPIE PC12144, PC121440J (2022). [CrossRef]
24. L. Zhu, F. Soldevila, C. Moretti, et al., “Large field-of-view non-invasive imaging through scattering layers using fluctuating random illumination,” Nat. Commun. 13, 1447 (2022). [CrossRef]
25. A. Fercher and J. D. Briers, “Flow visualization by means of single-exposure speckle photography,” Opt. Commun. 37, 326–330 (1981). [CrossRef]
26. K. Pan, C.-M. Uang, F. Cheng, et al., “Multimode fiber sensing by using mean-absolute speckle-intensity variation,” Appl. Opt. 33, 2095–2098 (1994). [CrossRef]
27. J. D. Briers and S. Webster, “Laser speckle contrast analysis (LASCA): a nonscanning, full-field technique for monitoring capillary blood flow,” J. Biomed. Opt. 1, 174–179 (1996). [CrossRef]
28. J. D. Briers, “Laser speckle contrast imaging for measuring blood flow,” Opt. Appl. 37, 139–152 (2007). [CrossRef]
29. K. Kim, H. Yu, J. Koh, et al., “Remote sensing of pressure inside deformable microchannels using light scattering in scotch tape,” Opt. Lett. 41, 1837–1840 (2016). [CrossRef]
30. P. Berto, H. Rigneault, and M. Guillon, “Wavefront sensing with a thin diffuser,” Opt. Lett. 42, 5117–5120 (2017). [CrossRef]
31. M. J. Murray and A. Davis, “Speckle-based strain sensing in multimode fiber,” Opt. Express 27, 28494–28506 (2019). [CrossRef]
32. Y. Baek, K. Lee, J. Oh, et al., “Speckle-correlation scattering matrix approaches for imaging and sensing through turbidity,” Sensors 20, 3147 (2020). [CrossRef]
33. S. Liu, Z. Liu, C. Hu, et al., “Spectral ghost imaging camera with super-Rayleigh modulator,” Opt. Commun. 472, 126017 (2020). [CrossRef]
34. Q. Luo, J. A. Patel, and K. J. Webb, “Super-resolution sensing with a randomly scattering analyzer,” Phys. Rev. Res. 3, L042045 (2021). [CrossRef]
35. V. G. Shvedov, A. V. Rode, Y. V. Izdebskaya, et al., “Selective trapping of multiple particles by volume speckle field,” Opt. Express 18, 3137–3142 (2010). [CrossRef]
36. F. Jendrzejewski, A. Bernard, K. Mueller, et al., “Three-dimensional localization of ultracold atoms in an optical disordered potential,” Nat. Phys. 8, 398–403 (2012). [CrossRef]
37. G. Volpe, L. Kurz, A. Callegari, et al., “Speckle optical tweezers: micromanipulation with random light fields,” Opt. Express 22, 18159–18167 (2014). [CrossRef]
38. A. Kotnala, P. S. Kollipara, and Y. Zheng, “Opto-thermoelectric speckle tweezers,” Nanophotonics 9, 927–933 (2020). [CrossRef]
39. K. M. Douglass, S. Sukhov, and A. Dogariu, “Superdiffusion in optically controlled active media,” Nat. Photonics 6, 834–837 (2012). [CrossRef]
40. C. Bechinger, R. Di Leonardo, H. Löwen, et al., “Active particles in complex and crowded environments,” Rev. Mod. Phys. 88, 045006 (2016). [CrossRef]
41. E. Pinçe, S. K. Velu, A. Callegari, et al., “Disorder-mediated crowd control in an active matter system,” Nat. Commun. 7, 10907 (2016). [CrossRef]
42. H. Fujii and T. Asakura, “Effect of surface roughness on the statistical distribution of image speckle intensity,” Opt. Commun. 11, 35–38 (1974). [CrossRef]
43. J. W. Goodman, “Dependence of image speckle contrast on surface roughness,” Opt. Commun. 14, 324–327 (1975). [CrossRef]
44. H. Fujii and T. Asakura, “Statistical properties of image speckle patterns in partially coherent light,” Nouv. Rev. Opt. 6, 5 (1975). [CrossRef]
45. E. Jakeman and J. G. McWhirter, “Non-Gaussian scattering by a random phase screen,” Appl. Phys. B 26, 125–131 (1981). [CrossRef]
46. B. Levine and J. Dainty, “Non-Gaussian image plane speckle: measurements from diffusers of known statistics,” Opt. Commun. 45, 252–257 (1983). [CrossRef]
47. E. Jakeman, “Speckle statistics with a small number of scatterers,” Opt. Eng. 23, 234453 (1984). [CrossRef]
48. Y. Bromberg and H. Cao, “Generating non-Rayleigh speckles with tailored intensity statistics,” Phys. Rev. Lett. 112, 213904 (2014). [CrossRef]
49. N. Bender, H. Ylmaz, Y. Bromberg, et al., “Customizing speckle intensity statistics,” Optica 5, 595–600 (2018). [CrossRef]
50. A. Dogariu and R. Carminati, “Electromagnetic field correlations in three-dimensional speckles,” Phys. Rep. 559, 1–29 (2015). [CrossRef]
51. R. Fischer, I. Vidal, D. Gilboa, et al., “Light with tunable non-Markovian phase imprint,” Phys. Rev. Lett. 115, 073901 (2015). [CrossRef]
52. J. P. Amaral and E. J. Fonseca, “Tailoring speckles with Weibull intensity statistics,” Phys. Rev. A 92, 063851 (2015). [CrossRef]
53. H. E. Kondakci, A. Szameit, A. F. Abouraddy, et al., “Sub-thermal to super-thermal light statistics from a disordered lattice via deterministic control of excitation symmetry,” Optica 3, 477–482 (2016). [CrossRef]
54. D. Di Battista, D. Ancora, M. Leonetti, et al., “Tailoring non-diffractive beams from amorphous light speckles,” Appl. Phys. Lett. 109, 121110 (2016). [CrossRef]
55. M. Guillon, B. C. Forget, A. J. Foust, et al., “Vortex-free phase profiles for uniform patterning with computer-generated holography,” Opt. Express 25, 12640–12652 (2017). [CrossRef]
56. D. Di Battista, D. Ancora, G. Zacharakis, et al., “Hyperuniformity in amorphous speckle patterns,” Opt. Express 26, 15594–15608 (2018). [CrossRef]
57. N. Bender, H. Ylmaz, Y. Bromberg, et al., “Introducing non-local correlations into laser speckles,” Opt. Express 27, 6057–6067 (2019). [CrossRef]
58. N. Bender, H. Ylmaz, Y. Bromberg, et al., “Creating and controlling complex light,” APL Photonics 4, 110806 (2019). [CrossRef]
59. N. Bender, M. Sun, H. Ylmaz, et al., “Circumventing the optical diffraction limit with customized speckles,” Optica 8, 122–129 (2021). [CrossRef]
60. L. Devaud, B. Rauer, J. Melchard, et al., “Speckle engineering through singular value decomposition of the transmission matrix,” Phys. Rev. Lett. 127, 093903 (2021). [CrossRef]
61. R. Liu, B. Qing, S. Zhao, et al., “Generation of non-Rayleigh nondiffracting speckles,” Phys. Rev. Lett. 127, 180601 (2021). [CrossRef]
62. S. Han, N. Bender, and H. Cao, “Tailoring 3D speckle statistics,” Phys. Rev. Lett. 130, 093802 (2023). [CrossRef]
63. J.-E. Oh, Y.-W. Cho, G. Scarcelli, et al., “Sub-Rayleigh imaging via speckle illumination,” Opt. Lett. 38, 682–684 (2013). [CrossRef]
64. S. Zhang, W. Wang, R. Yu, et al., “High-order correlation of non-Rayleigh speckle fields and its application in super-resolution imaging,” Laser Phys. 26, 055007 (2016). [CrossRef]
65. Z. Li, X. Nie, F. Yang, et al., “Sub-Rayleigh second-order correlation imaging using spatially distributive colored noise speckle patterns,” Opt. Express 29, 19621–19630 (2021). [CrossRef]
66. P. Liu, “Label-free storm principle realized by super-Rayleigh speckle in photoacoustic imaging,” Opt. Lett. 44, 4642–4645 (2019). [CrossRef]
67. E.-F. Zhang, W.-T. Liu, and P.-X. Chen, “Ghost imaging with non-negative exponential speckle patterns,” J. Opt. 17, 085602 (2015). [CrossRef]
68. K. Kuplicki and K. W. C. Chan, “High-order ghost imaging using non-Rayleigh speckle sources,” Opt. Express 24, 26766–26776 (2016). [CrossRef]
69. X. Nie, F. Yang, X. Liu, et al., “Noise-robust computational ghost imaging with pink noise speckle patterns,” Phys. Rev. A 104, 013513 (2021). [CrossRef]
70. J. Guilbert, “Super resolution imaging in coherent Raman scattering for biological tissues,” Ph.D. thesis (PSL Research University, 2022).
71. H. Cao and Y. Eliezer, “Harnessing disorder for photonic device applications,” Appl. Phys. Rev. 9, 011309 (2022). [CrossRef]
72. A. P. Mosk, A. Lagendijk, G. Lerosey, et al., “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6, 283–292 (2012). [CrossRef]
73. R. French, S. Gigan, and O. L. Muskens, “Speckle-based hyperspectral imaging combining multiple scattering and compressive sensing in nanowire mats,” Opt. Lett. 42, 1820–1823 (2017). [CrossRef]
74. S. K. Sahoo, D. Tang, and C. Dang, “Single-shot multispectral imaging with a monochromatic camera,” Optica 4, 1209–1213 (2017). [CrossRef]
75. K. Monakhova, K. Yanny, N. Aggarwal, et al., “Spectral diffusercam: lensless snapshot hyperspectral imaging with a spectral filter array,” Optica 7, 1298–1307 (2020). [CrossRef]
76. D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7, 354–362 (2013). [CrossRef]
77. L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9, 306–310 (2015). [CrossRef]
78. L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal mode-locking in multimode fiber lasers,” Science 358, 94–97 (2017). [CrossRef]
79. H. Haig, P. Sidorenko, A. Dhar, et al., “Multimode Mamyshev oscillator,” Opt. Lett. 47, 46–49 (2022). [CrossRef]
80. A. Chong, J. Buckley, W. Renninger, et al., “All-normal-dispersion femtosecond fiber laser,” Opt. Express 14, 10095–10100 (2006). [CrossRef]
81. W. Xiong, P. Ambichl, Y. Bromberg, et al., “Principal modes in multimode fibers: exploring the crossover from weak to strong mode coupling,” Opt. Express 25, 2709–2724 (2017). [CrossRef]
82. A. Garito, J. Wang, and R. Gao, “Effects of random perturbations in plastic optical fibers,” Science 281, 962–967 (1998). [CrossRef]
83. E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. 39, 4070–4075 (2000). [CrossRef]
84. M. Mounaix, H. Defienne, and S. Gigan, “Deterministic light focusing in space and time through multiple scattering media with a time-resolved transmission matrix approach,” Phys. Rev. A 94, 041802 (2016). [CrossRef]
85. R. Penrose, “A generalized inverse for matrices,” Math. Proc. Camb. Philos. Soc. 51, 406–413 (1955). [CrossRef]
86. L. Devaud, M. Guillon, I. Gusachenko, et al., “Chromato-axial memory effect in step-index multimode fibers,” APL Photonics 6, 126105 (2021). [CrossRef]
