Coherence of Light and Generation of Speckle Patterns in Photobiology and Photomedicine

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1 Coherence of Light and Generation of Speckle Patterns in Photobiology and Photomedicine Zeev Zalevsky 1* and Michael Belkin 1 Faculty of Engineering, Bar-Ilan University, Ramat-Gan 5900, Israel, Goldshleger Eye Research Institute, Tel-Aviv University, Tel-Hashomer, Israel ABSTRACT The use of diodes instead of lasers was recently suggested for phototherapeutic applications. This trend is due to economical and practical reasons and is based on the argument that lasers have no preference over diodes as light sources as the former lose their coherency upon penetrating biological tissues. This module supports this claim while providing a brief eplanation to non professionals on the meaning of coherence of light as well as the physics behind the generation of speckle patterns, and the relation of these physical entities to photomedicine. Keywords: Speckle; Coherence of light; Photomedicine; Photobiology. 1. INTRODUCTION Using light to biostimulate a cell is common approach in Low Level Laser Therapy (LLLT) [1]. The growing acceptance of incoherent light sources such as light emitting diodes (LEDs) in phototherapy continues to debate on the value of coherence in achieving beneficial results with light. Smith argues that the spatial coherence of lasers is not useful in LLLT []. Hode claims that coherence of laser light is not lost when the light enters tissues [3]. The purpose of this module is to clear up those issues and to eplain, to non professionals, the meaning of speckle and coherence light.. COHERENCE OF LIGHT Coherence of light occurs when all the light waves are "in phase" with one another along time, i.e., the crests of one wave are aligned with the crests of all the other waves, and similarly for the troughs of the waves. Figure 1 tends to illustrate this point by presenting, on one hand, waves having their direction of propagation changed with time (the incoherent case) and on the other hand waves preserving their in phase property versus time (coherent case). Thus, coherence of light is basically related to how randomly or how often the electric field of light is changing within the time of observation. Therefore, coherence is how correlative (mathematical measure that characteries similarity) or how synchronied two time varying distributions (of the electric field), that are measured at different spatial positions (coordinates) or temporal sites. In other words, coherence relates to the etent of similarity between time varying distributions of the electric field that can be measured at different spatial or temporal locations. Mechanisms for Low-Light Therapy VII, edited by Michael R. Hamblin, Juanita Anders, James D. Carroll, Proc. of SPIE Vol. 811, 8110J 01 SPIE CCC code: /1/$18 doi: / Proc. of SPIE Vol J-1

2 Spatially Incoherent t=t 0 t=t 1 t=t Time Spatially Coherent t=t 0 t=t 1 t=t Time Fig. 1: Schematic illustration of temporal change of electric field in spatially incoherent (upper plot) and spatially coherent (lower plot) beam of light. Therefore, when observing infinitely short pulse, every source is basically coherent since it does not have enough time to change the value of its electric field. However, over longer observation time, sources do vary from each other according to the rate that the field of light is changing with time and the way this change is spatially distributed within the given beam of light. Therefore, it is clear that the loss of coherence (temporal and spatial) of a given initially coherent source (e.g. laser) will strongly depend on the rate of the temporal variation of the medium through which the light is propagated. Following the above mentioned eplanation, one should distinguish between spatial and temporal coherence. Spatial coherence is related to how two spatial coordinates (i.e. locations) are temporally correlated to each other, i.e., how the changes of the field of light (usually due to changes of its phase) in these two spatial positions correlate to each other [4]. Temporal coherence is related to eamining the temporal rate at which the optical field is changing in a given spatial location (usually the change is in the phase of the optical field). One over the maimal rate of temporal change is called the coherence time. This coherence time can be translated into coherence length, simply by multiplying it with the speed of light. Mathematically, coherence is being defined by the mutual coherence function Γ [4] as follows: * Γ 1( τ) = upt ( 1, + τ) u( P, t) (1) where u(p,t) is an input comple field distribution (as every comple number it may be represented using polar representation by an amplitude and a phase), P represents the two dimensional spatial coordinate, t is the time ais, and τ is the time difference between two temporal points of measurement. < > describes ensemble averaging or averaging over time (the two types of averaging are equivalent for the discussed kind of physical process). For spatially incoherent field Γ 1 (τ)=0 for all τ 0, i.e. the two different spatial positions are not correlated to each other. The temporal coherence related function is obtained when we eamine the mutual coherence function Γ of Eq. (1) for P 1 =P, i.e. when we deal with the self coherence function (compare the correlation of a given spatial coordinate P 1 with itself versus time difference of τ). The width of the self coherence function Γ 11 (τ) along the τ coordinate is the temporal coherence length of the given beam. Therefore, since the temporal coherence is inversely proportional to the rate of temporal changes of the field, it is directly related to precise etent of the monochromaticity of the beam. The more polychromatic the light is, the larger is its rate of temporal change. Thus, a monochromatic beam is temporally coherent, and it has an infinitely long coherence length, since the change of its field (usually due to the change of its phase) over time is fully correlated and anticipated. The more polychromatic the illumination is the shorter is its temporal coherence length. Proc. of SPIE Vol J-

3 Therefore, one may, for instance, have spatially coherent light which is temporally incoherent. In this case, the light is polychromatic with large temporal rate of changing (in the value of its optical field) and in every spatial position the field of light is rapidly oscillating with time. But, in all the spatial positions along the beam of light, the rapid oscillations are the same, and thus the changes in various spatial locations are correlated to each other. On the other hand, one cannot have an optical field that is temporally fully coherent and spatially fully incoherent, since if it is temporally coherent, the field or its phase are not changing with time (the light is monochromatic) and thus one cannot get two spatial positions that are being decorrelated enough from each other (i.e., changing in different ways with time). For eample a laser has all of its spatial waves of light synchronied (in phase), and thus it is a spatially coherent source. Usually lasers are also temporally coherent (i.e., monochromatic), however some lasers have several spectral lasing lines (several output wavelengths), and thus their temporal coherence length is short, although they remain spatially very coherent. As an another clarification eample, we may say that coherent light that is being propagated through a biological tissue may eventually lose its spatial coherence since, as previously stated, there are temporal changes in the value of the optical field that are being involved with the tissue medium. Those temporal changes that eventually break the spatial coherence are related to flow of fluids through the biological tissue. In cases there are no flows, the spatial coherence will not be lost. In case there is a flow, the rate at which the spatial coherence is lost is directly related to the volumetric flow rate of the fluid [5]. This is also the operation principle of devices measuring movement based upon laser Doppler shift (a slight change in the wavelength of the laser due to the flow through the tissue). For instance, consider a spot of light having a diameter of mm shining perpendicularly on a blood vessel having a diameter of 3 mm and a blood flow of 5 litres/minute. This yields that a volume of a cylinder of 3 mm in diameter and a height of mm is moving perpendicularly to the illuminating laser beam. This cylinder changes every 170 μsec (volume of blood of π(3 mm)/4 mm=14.14 μlitre at flow of 5 litres/minute means that it is changed every μ/5 =.88 μminute = 170 μsec). Thus, after a time period that is proportional to the constant of 170 μsec the spatial coherence will be lost in this eample. At lower volumetric flow rates, longer time constant, at which the spatial coherence is preserved, is obtained (these can be at a range of several seconds). As mentioned before the coherence length equals to the product between the coherence time and the speed of light. 3. SPECKLE PATTERNS Speckle patterns are spatially random self-interference spots of coherent light that are generated when spatially coherent light is reflected off or transmitted through a rough surface. Figure presents an illustrative image of generated speckle pattern in green (wavelength of 53 nm) laser spot of light. Speckles accumulate themselves as spots, and thus they are basically a locally generated non-uniformity of laser beam intensity obtained along the plane perpendicular to the direction of propagation. The average power density remains the same, but the local power density is not uniform, having higher power density within the speckle spot, and lower power density around it. Fig. : Eperimental image of speckle pattern generated in green (wavelength of 53nm) laser beam. Inherent changes in the propagation of the speckle pattern will uniquely characterie each specific location in the volume of the speckle field [6]. The contrast of those random speckle spots is directly related to the spatial coherence of the Proc. of SPIE Vol J-3

4 light. For highly coherent illumination, the contrast is 100%, and it is reduced to ero (i.e., no speckles are generated at all) for fully spatially incoherent light. Therefore, as eplained above, when coherent light is passing through a biological tissue, the contrast of the speckles changes according to the eisting volumetric flow rate of the fluid moving through the tissue. For static tissues without any flow, the contrast will remain unchanged (not a common case). Increasing the flow rate reduces the time at which the contrast of the speckles is reduced. Mathematically speaking when speckle patterns are involved their statistics can be described by the amplitude of their transversal and the longitudinal correlation function. Let us assume a circular uniform intensity at the object plane, being the diffusive source that generates the secondary speckle. Then, the dependence of the value of the correlation function on the transversal and the longitudinal positions may be modeled as follows: the correlation function of the intensity between two points separated in the transverse plane by radial distance of s and which are located at distance from the diffusive object may be estimated as [6]: J1 ( πφs ) Γ Transversal () s = I 1+ λ π s Φ λ () where I is the mean of the intensity in the output plane, Φ is the diameter of the illuminating beam, J1 is the first kind Bessel function, λ is the optical wavelength and is the aial distance. With respect to the longitudinal etent of the speckle, the problem reduces to the calculation of the aial correlation function of the intensity between two points separated aially by a distance of Δ. In this case, with the same assumptions as in the transverse case in addition to the requirement that Δ will be small compared with the absolute distance, the correlation of intensity results with [6]: ( ) ΓLongitudinal ( Δ ) = I 1 sinc + Δ Φ 8λ (3) It is also important to distinguish between primary and secondary speckle patterns. Primary speckle patterns are generated by projection when the light passes through a ground glass or a diffuser, and then illuminates the detection system. Secondary speckle patterns are self-generated random patterns created due to the roughness of the illuminated surface from which the light is reflected towards the detection system. A biological tissue acts as such a diffusive and scattering medium as well. The speckle statistics depends upon the ground glass or the diffuser that are used to project the patterns (primary speckle), and on the surface characteristics on which they shine (secondary speckle). The parameters of the optical system can determine the dominance of each type (primary or secondary) of speckle patterns since the diffraction resolution limit of the camera is being proportional to the product between the optical wavelength and the F-number of the imaging lens (the ratio between the focal length and the diameter of the lens). That way, if the diameter of the imaging lens fits the diameter of the spot of light that illuminates the diffuser (that generates the primary speckle patterns), those speckle patterns will be limited by the diffraction resolution limit of the camera (determined by the F-number), while the secondary speckle patterns, which are smaller, will be partially filtered out (spatially) by the camera, since the F-number will be too large to contain their full spatial structure. Note that the change in the wavelength of light or in the wavelength of the generated speckle patterns can be created only due to non-linear process (as fluorescence) in which photons are being absorbed and regenerated via the non linear medium that is being illuminated. In linear medium, as regular tissues such as the human skin or the cornea, the wavelength of the speckle patterns is identical to the wavelength of the illuminating beam (in physical formulation, nonlinear medium, in contrast to linear one, can be defined as a medium in which the obtained dielectric polariation is not proportional to the electric field of the light, but rather to high power of the field (power of two and higher)). 4. COHERENCE AND SPECKLE IN PHOTOMEDICINE As previously discussed, coherence as well as speckle patterns are related to the distribution of the field of light (and mainly to its phase or the way it is varied with time and space). In the field of photochemistry, on the other hand, light must be absorbed before photochemical reactions can occur, and thus its intensity (the absolute value square of the field) rather than its field plays the major role. Therefore, the phase of light, which is a fundamental issue in coherence related effects such as speckle is irrelevant when irradiating tissues, e.g., in the field of LLLT. Proc. of SPIE Vol J-4

5 Moreover, as previously mentioned, due to fluid flows going through the biological tissue, the spatial coherence of the illuminating beam is lost, and the contrast of the secondary speckles is reduced quit quickly, depending on the volumetric flow rate of the fluid flowing through the tissue [5]. Therefore, at least in theory, spatially and temporally incoherent diodes can serve as light sources in photomedicine just as well as lasers (coherent sources). REFERENCES [1]. N. Ben Dov, G. Shefer, A. Irintchev, A. Wernig, U. Oron and O. Halevy, 1999, Low-energy laser irradiation affects satellite cell proliferation and differentiation in vitro, Biochim. Biophys. Acta 1448, []. K. C. Smith, 005, Laser (and LED) Therapy Is Phototherapy, Photomedicine and Laser Surgery 3, [3]. L. Hode, 005, The Importance of the Coherency, Photomedicine and Laser Surgery 3, [4]. J. W. Goodman, 000, Statistical optics, Wiley classic library. [5]. D. Filer, H. Duadi, R. Ankri and Z. Zalevsky, 011, Determination of Coherence Length in Biological Tissues, Lasers in Surgery & Medicine 43, [6]. L. Leushacke and M. Kirchner, 1990, Three dimensional correlation coefficient of speckle intensity for rectangular and circular apertures, J. Opt. Soc. Am. A7: Proc. of SPIE Vol J-5