1.

Introduction

Although the contrast of polychromatic speckle patterns has been used for numerous practical applications,^{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} there are still situations, such as Fresnel diffraction, where simple and practical analytical expressions have not yet been derived. To fill this gap we have developed a simple geometrical model utilizing a Fresnel-zone analogy for quantitative estimation of the contrast.

The speckle contrast is of great interest in various industrial^{1, 2} and biomedical applications.^{3, 4, 5, 6, 7, 8, 9, 10} By measuring its value as a function of either tissue thickness or the coherence length of the light source, the absorption and scattering coefficients of the diffusive media can be derived.⁴ Temporal averaging of contrasts for dynamic biospeckle has been used to monitor tissue conditions,⁵ visualize internal inhomogeneities,^{6, 7, 8, 9} and evaluate blood flow velocity.^{10, 11}

From a metrological point of view, the speckle contrast contains useful information when it differs from unity. It is well known that for all interference phenomena contrast reduction occurs when a coherent addition of light waves starts to compete with an incoherent one. This effect takes place when the mean optical-path-length difference is of order smaller than the coherence length of the light source.

Variations of the ratio of optical-path-length differences to coherence lengths can have a number of causes. Some of them, such as surface roughness and low coherence of the illuminating light, have been studied.¹² Most analytical solutions for these studies were obtained for the far-field diffraction zone.^{12, 13, 14} In another study a general formula for the intensity correlation function, based on the theory of linear systems, was considered, and analytical formulas were derived for the imaging geometry.¹⁵

In a previous paper¹⁶ we studied optical geometry as another factor that can significantly influence the contrast of polychromatic speckle within a Fresnel diffraction zone. In this article we extend this approach by developing a simple physical model of speckle formation that is an analogy of the Fresnel-zone technique.

This paper is organized as follows: after a short review (Sec. 2) on the background concept of forming polychromatic speckles as a sum of $N$ independent speckle patterns,^{12, 17, 18} we show how to construct zones geometrically in such a way that each zone creates an individual fully developed speckle (Sec. 3). The model is applied to contrast calculations, and some immediate practical applications are discussed. Next we prove that the model is consistent with our previous theoretical formulation¹⁶ (Sec. 4). Finally, experimental validation using different light sources—including a diode laser (DL) and a relatively new type of low-coherence light source, a superluminescence diode (SLD)—is presented (Sec. 5).

2.

Background

The contrast $C$ of a speckle pattern¹² is defined as

The simplest speckle, which is also known as fully developed or Gaussian speckle, is created by coherent monochromatic laser light. The contrast $C$ of this kind of speckle equals unity and does not depend on any physical parameters. In other words, it does not carry information about the scattering medium or optical geometry. However, analyzing speckle patterns where the contrast differs from unity allows us to get useful information about the sample under investigation.

Goodman¹² shows that the contrast $C$ of $N$ independent and fully developed speckle patterns each having equal mean intensity $⟨I_n⟩=I_0$ can be expressed as

The known sources of speckle contrast reduction are coherence (angular and wavelength diversity), polarization diversity, temporal and spatial averaging, and others. Some of these factors lead to the generation of independent speckle patterns. Examples of determining $N$ [Eq. 2] for contrast reduction due to increased surface roughness can be found in Refs. 13, 14 as side remarks on contrast calculation. In Refs. 17, 18 $N$ is calculated for polychromatic image-plane speckle.

Below, we have reformulated the results of Refs. 13, 14 to emphasize the concept of exploiting the same generic approach of polychromatic speckle formation despite the fact that the optical situations are completely different: Specifically, the preceding studies^{13, 14} investigated speckle in the Fraunhofer diffraction zone, whereas we are primarily interested in light properties outside of the Fraunhofer zone.

The general solution for $C$ in Refs. 13, 14 is reduced to

Figure 1 helps us interpret the physical meaning of Eq. 4. A light source illuminates a rough surface from the left side. Scattered light creates a speckle pattern in the Fraunhofer field. Consider that the rough surface can be split vertically into layers of thickness $L_c∕2$ . Different layers are depicted by different gray tones in the inset of Fig. 1. All elementary waves scattered from a single layer are considered mutually coherent. They create a fully developed speckle pattern with unit contrast. Similarly, the elementary waves reflected from the next layer are mutually coherent and generate the second speckle pattern. To simplify the analysis, we consider that the first and second speckle patterns are naturally incoherent and do not interfere with each other. Analogously, we can form the third and additional subsequent speckle patterns up to $N=\beta (\Delta ∕2)∕(L_c∕2)$ , where $\beta$ depends on the spectral profile and is of the order of unity. Overlaying these $N$ speckle patterns results in the final pattern with a reduced contrast.

Fig. 1

Formation of polychromatic speckle by light reflected from rough surface.

In summary, the appropriate choice of coherent layers allows us to model the formation of a speckle pattern in a simple manner and to calculate its contrast. The ratio $\Delta ∕L_c$ is a key element in the formulation, and we follow the same rationale for the following sections.

3.

Zone Model

The coherence length is closely related to the mutual coherence function. Like wavelength, amplitude, phase, and polarization, the mutual coherence function is one of a wave’s basic characteristics, which carries information about the wave and about the objects or processes that the wave interacts with during its propagation. Recently a class of new measuring techniques utilizing mutual coherence functions has been established. These techniques cover areas such as low-coherence holography, low-coherence speckle, low-coherence tomography, and others.¹⁹ Like amplitude, the mutual coherence function obeys wave equations. This fact gives us an opportunity to develop an analogy to interference measurements, where the coherence length behaves similarly to the effective wavelength. Utilizing that analogy, we thus propose a multizone construction for polychromatic speckling.

The polychromatic speckle contrast degrades not only because of roughness but also because of geometry factors. When an illuminated area is large enough and the diffuser-source distance or the receiver-source distance is finite, the optical path length of a light ray passing through the peripheral zone can differ significantly from the optical path length through the central part of the surface. If the optical-path-length difference is greater than the coherence length, rays will not interfere with each other.

Let us calculate the contrast due to the mentioned geometrical consideration as a sum of $N_{\mathrm{geom}}$ fully developed patterns where each pair of patterns is mutually incoherent. To do that we introduce coherent zones: each coherent zone creates one speckle pattern. A geometrical construction is shown in Fig 2. Coherent zones can be applied to both transmission and reflection geometry; however, only the transmission geometry is shown in Fig. 2, to better illustrate the optical pathways. We suppose that a light ray emitted from the point $O$ (for example, at the end of a single-mode fiber) comes to the point $P$ through a thin diffuser located in the plane $S$ . We use a technique similar to the Fresnel zone construction.²⁰ First, we draw a sphere centered at the point $O$ with radius $z_0$ . Then we draw another sphere centered at $P$ in the opposite direction with radius $z$ . Note that $z_0+z$ is the optical path of the ray that passes through the central point $O^{\prime }$ on the surface $S$ . Let us choose a point $A_1$ that provides the ray route $O{A}_{1}P$ with the optical path length $z_0+{\Delta }_0+{\Delta }_P+z$ . We choose $A_1$ in such a way that the optical-path-length difference between $OP$ and $O{A}_{1}P$ is of the order of the light-source coherence length, i.e., ${\Delta }_0+{\Delta }_P=L_c$ . We define the coherence length²¹ as $L_c=c{\int }_{-\infty }^{\infty }{\mid \gamma \left(\tau \right)\mid }^2\mathrm{d}\tau$ , where $\gamma \left(\tau \right)$ is the coherence function and $c$ is the light speed. The new sphere with radius $P{A}_1$ crosses the diffuser $S$ , creating a circle with the radius $O^{\prime }{A}_1=R_1$ . Simplifying somewhat, we assume that the rays that pass through different points within this zone and reach the observer $P$ are coherent. The actual optical path of each ray includes a stochastic component related to the roughness of $S$ , but this will not influence our construction as long as the roughness height remains small compared to the coherence length. Interference of all these rays creates the first fully developed speckle pattern.

Fig. 2

Passage of light through the central and peripheral parts of an illuminated surface $S$ : (a) side view along the optical axis; (b) front view of the illuminated surface $S$ .

Continuing our construction, let us find another point $A_2$ that limits the second zone $(O^{\prime }{A}_2=R_2)$ , where ${\Delta }_{02}+{\Delta }_{P2}=2L_c$ . Because of the circular symmetry, any pair of rays passing through the second zone (which is a ring) will have a path-length difference that is less than the coherence length. All these rays create the second fully developed speckle pattern.

Other zones are constructed in the same fashion. Each zone creates an independent speckle pattern and contributes to the overall resultant pattern, wherein the contrast is reduced. Simple geometrical calculations show that the radius of zone $N$ is approximately

The total number of coherent zones due to the geometry influence, $N_{\mathrm{geom}}$ , is limited by the spot size $q$ of the illuminated area on the surface and can be calculated from the equation $R_N=q$ :

Equation 6 is valid for the reflectance as well. It is easy to show that the area $S_n$ of each flat ring zone is approximately the same: $S_1\approx S_2\approx \cdots \approx S_N$ (to prove that, we refer the reader to the Fresnel zone model²⁰). Therefore, if the illumination is uniform, each zone transmits (or reflects) equal amount of light, and the average intensity of each speckle pattern is the same. Our goal is to determine the contrast $C_{\mathrm{geom}}$ due to geometry-based parameters. Combining Eqs. 6, 2, and introducing $z_{\mathrm{eff}}=z{z}_0∕(z+z_0)$ , we obtain

Eq. 7 shows that the contrast is reduced when $q$ increases or when the coherence length and the effective distance $z_{\mathrm{eff}}=z{z}_0∕(z+z_0)$ decrease. The range of experimental parameters in Eq. 6 obeys the natural limitation $N_{\mathrm{geom}}>1$ (the number of summed patterns must be greater than one), which is the same as for Eq. 2.

Certain features concerning the generalization and application of this model can be observed:

Speckle contrast reduction can be easily observed when modern low-coherence light sources are applied with a large illumination spot and a small target distance. In practical terms this can be implemented with optical fiber illumination. Another obvious application of Eqs. 6, 7 is that the effective coherence length of a light source can be obtained without spectral devices, but rather through real-time speckle contrast measurements.

4.

Theory

Our proposed model is consistent with a rigorous application of diffraction theory. If we illuminate a diffuser with a polychromatic light source characterized by half-width $\Delta \lambda$ , each wavelength produces a shifted speckle pattern and the final interference pattern becomes blurry. The intensity $I$ of the polychromatic speckles is the sum of the monochromatic speckle pattern intensities,

To calculate Eq. 9 we applied the Fresnel approach of diffraction theory for free-space geometry (Fig. 2).¹⁶ The correlation of intensity fluctuations in this case can be written as

Some important simplifications were made in order to obtain Eq. 11. We considered the case when the mean optical path difference is greater than the coherence length and the surface roughness. Also, to obtain an analytical solution we extended the integral limits to infinity, implying the condition $q^4∕{\left(z_{\mathrm{eff}}{L}_c\right)}^2\gg 1$ . This condition dictates the parameter ranges of Eq. 11 and leads to $C<1$ . Therefore, this formula describes an experimental situation with a wide illuminating beam, a low-coherence source, and a short source-detector distance.

The performed analysis shows that despite a number of simplifications, the zone model agrees well with the results of a rigorous application of theory.

5.

Experiment

An experimental validation of our model involves finding the contrast from the calculated number $N_{\mathrm{geom}}$ of coherent zones and comparing the experimental values of $C_{\mathrm{geom}}$ with the theoretical expectation as expressed in Eq. 2. The value of $N_{\mathrm{geom}}$ is determined by Eq. 6 according to variations in the geometric parameters $z_{\mathrm{eff}}$ , and $q$ .

The free-space geometry setup is shown in Fig. 3. Divergent waves illuminate a rough sample. Two low-coherence light sources were tested: a superluminescent diode (FSLD-3P-680, RPMC Inc., USA, $\lambda =683.6\mathrm{nm}$ with a half-width spectrum line $\lambda =9.5\mathrm{nm}$ and close to Gaussian spectral profile) with coherence length calculated as $L_c=33.2\mu \mathrm{m}$ ; and a laser diode (57PNL053/P4/SP, Melles Griot, USA, $\lambda =656\mathrm{nm}$ , multipeak spectral profile) with coherence length measured as $L_c=278\mathrm{mm}$ . The lens and iris were used to adjust the size of the illuminated spot $\left(q\right)$ and $z_{\mathrm{eff}}$ .

Fig. 3

The experimental setup.

A CCD camera without a lens recorded speckle patterns. We captured the spatial homogeneous portion of a speckle pattern. Therefore the ensemble averaging in Eq. 1 could be replaced by spatial averaging of all CCD camera pixels. The $8\text{-}\text{bit}$ CCD sensor had $512\times 486\text{pixels}$ , and the pixel size was made 5 times smaller than the average speckle size to avoid spatial averaging of the speckle contrast. The camera circuit allowed regulation of the dark current and gain, which were adjusted to supply a linear measurement mode.

The setup was designed mostly to remove contrast reduction factors except for the geometry parameters. In particular, the dark-current pedestal was subtracted from the speckle images. Any contrast reduction due to light depolarization was controlled by an output polarizer Pl. The angles of incidence $\left({\theta }_i\right)$ and reflection $\left({\theta }_r\right)$ were almost identical. We used metal roughness standards with profound roughnesses of 6 and $3.17\mu \mathrm{m}$ to reduce the specular component of the reflected light, which lowers the contrast. The roughness is still considerably less than the effective coherence lengths for both the DL and SLD light sources. Therefore, we can ignore the surface roughness effect in evaluating contrast changes.

The measured contrast versus the number of coherent zones [as calculated by Eq. 6] is plotted in Fig. 4. For both light sources the experimental points are consistent with the theoretical curve (solid line) $C=1∕\sqrt{N}$ , which validates Eqs. 6, 7. Note that the SLD contrast values display a slight downtrend. This degradation may be caused by the surface roughness due to the shorter $L_c$ of the SLD than the LD. More accurate theoretical calculation and modeling of compounding contrast, taking into account the combination of geometry, roughness, and other possible contrast reduction factors, is our next goal.

Fig. 4

The dependence of the speckle contrasts on the number of coherent zones. Triangles are the DL measurements, diamonds are the SLD data, and the solid line is the theoretical expectation.

6.

Conclusion

We have developed a zone model to explain quantitatively the polychromatic speckle contrast reduction when the geometry involves path-length differences large compared to the surface roughness of a diffuser. The model is analogous with the Fresnel-zone construction, with wavelength replaced by coherence length. The model is consistent with our theoretical results¹⁶ and has been validated experimentally.

Investigating the sources of contrast reduction is especially important for modern low-coherence light sources with short temporal coherence. The ideas and the formulas discussed may find immediate applications in measuring temporal coherence lengths without spectral devices, measuring geometrical parameters, and assisting experimental design to eliminate or suppress speckle. The proposed model could also be useful for separating different contrast reduction factors in situations involving complex contrast formation.