compound speckles and their statistical and

Invited Paper

COMPOUND SPECKLES AND THEIR STATISTICAL AND DYNAMICAL PROPERTIES Steen G .Hanson,a Michael Linde Jakobsen,a René Skov Hansenb and Hal T. Yura.c a

Risø National Laboratory, Technical University of Denmark, Post Box 49, Roskilde, Denmark b University of Southern Denmark, Niels Bohrs Allé 1, 5230 Odense M, Denmark c Electronics and Photonics Laboratory, The Aerospace Corporation, Los Angeles, California, 90009 ABSTRACT Two issues will be treated in this presentation, both focusing on gaining a deeper understanding of dynamic speckles, aiming at the use for probing dynamical properties of scattering structures. The first issue to be addressed is the dynamics of speckles arising from illuminating a solid surface giving rise to fully developed speckle with two mutually separated beams. It will be shown that usually the speckle pattern will consist of larger speckles with an inherent fine structure, each of which will usually be moving with different velocity. Next, the dynamics of the speckle pattern arising from scattering off a diffuser as seen through a second static diffuser is analyzed. Here, it is shown that the second and static diffuser will act as a pivot point about which the speckles will move. This facilitates a scaling of the speckle displacement facilitating a very minute measurement of the displacement of the first diffuser.

1. INTRODUCTION Dynamic speckles have been investigated for a long period of time, and their dynamic properties for single scattering from solid surfaces are well understood, in case the structure is illuminated with a single beam. 1 Measurement systems relying on illuminating a solid object with two beams are widely used, e.g. in the Laser Doppler Velocimeter (LDV) 2 or in the Laser Time-of-Flight velocimeter (LTV) 3 . In case of the LDV, two beams are made to cross at the point of interest at the object, and the velocity is revealed in the difference in Doppler shift suffered for the two overlapping beams. The LTV relies on probing the delay time for light scattered off two mutually separated laser spots at the target, i.e. probing the time-lagged crosscovariance between the speckle patterns scattered off the two spots, followed by finding their mutual time-lag. The two illuminated spots are separately imaged onto two detectors, whereby no cross-talk between the scattered light is observed. Yet, another system relies on probing the light scattered off two incident spots on a target, here providing direct measurement of the rotational speed of the object, independent on the radius of rotation, distance to the object, and wavelength. 4 This method - as the LDV method - relies on coherently mixing the scattered light from the object, the difference frequency here being proportional to the instantaneous angular speed. Here, we will theoretically investigate the dynamical behavior of the speckle pattern arising when the object is illuminated with two spatially separated beams. Specifically, we will assume the two incident beams to have identical radii of curvature for the field and the same intensity width, although being spatially separated and having different incident angles. The theoretical finding will be supported by an experiment showing the basic features. In the second part of this report, we will treat the problem of “speckled speckles” not as defined by J.W. Goodman 5 where the combined intensity distribution is found from a surface giving rise to fully developed speckles, illuminated by a “speckled” intensity distribution. Here, we investigate the dynamics of the speckles from a moving scattering plate, when seen through a second scattering, static diffuser. It will be shown that the dynamics of the observed speckles behave with the static diffuser as the pivoting point. Thus, scaling of the observed speckle velocity with respect to the object velocity is possible, being it a magnification or a reduction. Besides, this investigation will be of importance for depicting bio-speckles, where scattering structures, i.e. blood flow, is examined through a static scattering structure, here the stratum cornea. The theoretical findings are supported by initial experiments. Finally, a short discussion and a conclusion will be offered.

Eighth International Conference on Correlation Optics, edited by Malgorzata Kujawinska, Oleg V. Angelsky, Proc. of SPIE Vol. 7008, 70080M, (2008) 0277-786X/08/$18 doi: 10.1117/12.796870

Proc. of SPIE Vol. 7008 70080M-1 2008 SPIE Digital Library -- Subscriber Archive Copy

2. COMPOUND SPECKLES 6

The effect of illuminating an object with two or a multitude of spatially separated beams has been employed in a miniaturized optical system for monitoring speckle displacement. 7 Here we use spatially filtering velocimetry 8 where the dynamics of a bandpass filtered part of the speckle pattern is employed. We have successfully increased the signal strength by enhancing the Fourier components at the bandpass region by illuminating the object with several beams. We will theoretically investigate the speckle dynamics in case the object is illuminated with two spatially separated beams with the same width and the same radius of field curvature, but with different angle of incidence. In order to highlight the essential message, we will only treat the problem in one dimension in a scalar representation, and for free space propagation between the object and the plane of observation. Finally, we will assume the target to give rise to fully developed speckles, i.e. the surface roughness of the target exceeds the wavelength, and the lateral scale of the surface roughness is much smaller than the spot size of each of the spots. The dynamical properties of the various scales in the observation plane will be given with respect to speckle size and velocity as well as decorrelation. The setup is shown in Fig. 1, where the target and the associated illumination parameters are shown to the left. The spatial filtering detection system is shown to the right. Here, we will primarily analyze the speckle dynamics in the pplane. The spatial filtering setup probing the speckle pattern essentially consists of a linear prism array, where every second prism facet will direct the transmitted rays upward, while the rest will direct the rays downward. The pitch of the prism array is denoted /. A lens with focal length f will convert the exit angles from the array into position at the two detectors. The subtraction of the two detector signals will facilitate a bandpass filtering of the speckle spectrum. The detected frequency will equal the speckle velocity divided by /. Needless to say, the speckle size providing the most to the signal, will be the speckles of size slightly smaller than /. Larger speckles will cover an entire prism period, and thus provide an identical signal in the two detectors, resulting in a cancellation of the difference signal. Smaller speckles will give rise to speckle averaging and thus be responsible for a strong decrease in signal strength, as well.

VT4.

I. D

1(t)

I)

f

Fig. 1 Compound speckle field observed in the p-plane from a multitude of illuminated spots at the moving object in the r- plane. An array of facets with pitch / followed by a lens and two detectors facilitates the spatial bandpass filtering of the speckle pattern.

Consequently, our aim by properly designing the illumination of the target is twofold: first, increase the power in the speckle spectrum in the bandpass region of the receiver, and secondly to achieve a gearing of the speckle velocity with respect to the object velocity greater than unity in order to increase the sensitivity. We will specifically consider the case where the object is illuminated with two incident fields

Proc. of SPIE Vol. 7008 70080M-2

ª (r r 'r / 2) 2 ik (r r 'r / 2) 2 º U1,2 (r ) U 0 exp « B ik (r B 'r / 2) T » , 2 2R w0 ¬ ¼

(1)

with the mutual displacement 'r , the radius of curvature R, and the beam width(s) w0 . The small angle of incidence is rT . The Greens’ function for propagation between the object plane and the input plane (p-plane) for the array system omitting all unimportant constants is for free space given by: ª ik (r p) 2 º . exp « 2 L »¼ ¬

G (r , p )

(2)

The scattered field off the target is given by (3)

U ref (r, t ) K (r, t ) U inc (r )

where U inc (r ) U1 (r ) U 2 (r ) is the incident field, and the surface is assumed to give rise to fully developed speckle, which means that the ensemble average of the complex reflection coefficient K (r, t ) is given by

constant u G ª¬r -r c v t t c º¼ .

K (r, t )K (r c, t c)

(4)

Here we have assumed the magnitude of the reflection coefficient to be equal to unity, and only the scattered phase to be influenced during scattering. The velocity of the scattering surface is v. The space-time intensity covariance is given by *('p,W )

I (p, t ) I (p 'p, t W ) I (p, t ) I (p 'p, t W ) .

(5)

Assuming the scattered field to be a circular symmetric Gaussian process in that we have assumed a fully developed speckle field, the normalized space-time intensity covariance will be given by the absolute square of the normalized field correlation function, which is:

J ('p,W )

U (p, t ) U (p 'p, t W )

.

U (p, t ) U (p, t )

(6)

In arriving at this expression, we have assumed the process to be stationary, i.e. the time averages do only depend on the time difference. Besides, we have assumed spatial stationarity. The total normalized field correlation function is thus given by

J ('p,W ) J 11 ('p,W ) J 12 ('p,W ) J 21 ('p,W ) J 22 ('p,W )

(7)

where we by inserting Eqs. 1-4 into Eq. 7 we get:

J ii ('p,W )

³ G(r, p) G (r vW , p 'p)U (r)U

i

J 12 ('p,W )

i

(r vW ) d 2 r , i 1, 2 ,

(8)

(r vW ) d 2 r

(9)

³ G(r, p) G (r vW , p 'p)U (r)U

and


1

2

J 21 ('p,W )

³ G(r, p) G (r vW , p 'p)U

2

(r ) U1 (r vW ) d 2 r

(10)

It can be shown that J 12 ('p,W ) and J 21 ('p,W ) for all practically purposes will vanish due to the lack of overlap between the two spatially separated incident beams. Only in case there is no decorrelation as the scattering structure moves from one spot to the next, will the overlap integral give a contribution, and in this case the correlation time will be long, and thus the spectral contribution will happen at a very low frequency. In the cases of interest in this context, this contribution can be neglected. Note that this indicates that the interference pattern arising from scattering off the two spots vanish as an ensemble average. The normalized intensity correlation function thus becomes: *('p,W )

2

2

J 11 ('p,W ) J 22 ('p,W ) 2 Re ª¬J 11 ('p,W )J 22 ('p,W ) º¼

.

(11)

Performing the indicated integrations, we find the normalized intensity correlation function to be: * ( 'p , W )

ª (vW ) 2 ('p vW (1 L / R )) 2 º 1 exp « 2 » 1 cos > 2('p 2vW )kT @ . 2 wS2 ¬ w0 ¼

(12)

We have for simplicity assumed that the two illuminating beams originate from th e plane of observation, i.e. 'r 2 LT . The first exponential depicts the speckles including their dynamic properties, in case the object was illuminated with just a single beam. The gearing of the speckle velocity with respect to the object velocity is given by (1 L / R ) and the decorrelation length is given by w0 . The speckle size ws is given by wS 2 L / kw0 . The expression in the last parenthesis is responsible for a fine structure embedded in the larger speckles, which here will move with twice the velocity of the larger speckles, in which they are embedded. This gearing factor is closely linked to the mutual angle of divergence between the two incident beams, which here corresponds to a synthetic radius of curvature being equal to L, which for the larger speckles would give a gearing of “two.” The extent to which of the two velocities in Eq. 12 will be probed, depends on the spatial filter in the receiver, cf. Fig. 1. The power spectrum of the speckles is given by the temporal Fourier transform of the intensity correlation function, which becomes: P (Z )

(13) 2

2

§ ª ZN Z1 º» exp ª« ZN Z1 º» ·¸ ª º ZN2 C ¨ 2 exp « exp « 2 2» ¨ « 1 E 2 (1 L / R )2 » « 1 E 2 (1 L / R ) 2 » ¸ ¬ 1 E (1 L / R ) ¼ ¬ ¼ ¬ ¼¹ ©

with the normalized spatial frequencies

ZN

Z kw0 v / L

and Z1

4 LT . w0

(14)

The Fresnel number is given by

E{

wS w0

2 L / kw02 .

The spectrum thus consists of a sum of three spectral contributions, the first of which stemming from a single spot, centered at the origin. Two equally displaced spectral contributions appear at each side of origin with a


(15)

displacement 'Z 4T kv . The extent to which a given speckle velocity will be probed, depends on the receiver transmission function, i.e. whether it has a spatial bandpass for the scale of the larger speckles or a scale corresponding to the fine structure. Figure 2 displays the speckle pattern as it appears during free space propagation, in case the object is illuminated with four spots placed in a quadrant. Here the fine structure of the speckles will be two orthogonal fringe patterns.

Fig. 2. Speckle pattern with 2-D fine structure from object illuminated with four spots placed in a quadrant.

3. SPECKLED SPECKLES. The setup for evaluating the influence of observing a moving diffuser situated behind a static diffuser is shown in Fig. 3. Generally, the space between the moving and the static diffuser can consist of a general optical system, e.g. described within the ABCD- framework, and so can the space between the static diffuser and the observation plane. In this article we will address the simpler setup in which the two areas consist of free space, in order to stress the application. The distance between the first and second diffuser is L1 and the distance between the second diffuser and the observation plane is L2 . Here, again, we want to derive the space-time intensity covariance in the observation plane:

* > p; p 'p@ { I ( p) I '( p 'p) I ( p) I '( p 'p)

U ( p) U '*( p 'p)

where primed variables indicates variables after a displacement of the first diffuser.


2

,

(16)

Intermediate optical ABCD systems

Displacement, r 'r

p

G1(r, q)

G2 (q, p) q

L1

L2

Diffuse moving object

Diffuser, static

Detector array

Fig. 3. Setup for analyzing speckled speckles. The dynamic of a moving diffuser is analyzed as being seen through a second, static diffuser.

The field in the observation plane is given by tracing the field from the r-plane via the q-plane to the p-plane: U ( p)

³

G2 (q, p) exp > i M 2 (q) @ ³ G1 (r, q ) U (r )d 2rd 2q.

(17)

Here the phase perturbation in the q-plane is given by exp > i M 2 (q) @ , and the stochastic field just after scattering off the moving diffuser in the r-plane is given by U1 (r ). The intensity covariance can thus be written: * > p; p 'p @

³ ³

* 1

G1 (r1 , q1 ) G (r2 , q 2 ) U (r1 ) U *' (r2 ) G2 (q1 , p) G2* (q 2 , p 'p) exp ª¬ i M 2 (q1 ) M 2 (q 2 ) º¼ d 2r1d 2r2 d 2 q1d 2 q 2

2

(18)

.

This integral can be calculated, based on the assumption that the first diffuser has a surface roughness larger than the wavelength, but a finite lateral scale, rc , and the second diffuser gives rise to fully developed speckle, which demands that the phase is delta-correlated across the surface. This indicates that the field in the observation plane will obey circular symmetric Gaussian statistics, as already anticipated in writing Eq. 16. For free space, we have the two Greens’ functions: 2 2 ª ª r q º q p º G1 (r, q) v exp « ik » and G2 (q, p) v exp « ik » . 2 L1 »¼ 2 L2 »¼ «¬ «¬

(19)

The intensity covariance becomes ª § L12 ª 8( L1'p L2 'x) 2 º k 2 rs2 C ('p, 'x) exp « » exp « ¨ 2 2 2 2 rc 4 L2 «¬ © L2 rs ¬ ¼

· 2º ¸ 'p » , »¼ ¹

(20)

where the spot radius of the incident beam on the first diffuser is rs , and the induced displacement of the first diffuser is 'x , and the difference coordinate in the p-plane is 'p in the x-direction.


Three parameters are of special interest here, namely: x

The gearing

{ 'p / 'x

x

The decorrelation length in the p-plane (obtained for 'x

x

The speckle size (obtained for 'x

L2 / L1 2 2 s 2 2

2 1

§ L k r · ¸ 2 2 4L ¹ © L2 rs

'pL1 / L2 ): ¨

0 ):

L2 /

1/ 2

L12 k 2 rs2 L12 2 4 rs2 rc

Of importance here is the possibility of scaling the speckle velocity around the pivoting point situated at the static diffuser. Secondly, the speckle size will increase as the distance between the two diffusers is reduced, until it reaches the value 2 L2 / krs as will be found for a single diffuser as the two diffusers merge. But it is of importance to realize that high displacement accuracy can be obtained as L1 is decreased. Had the speckle size increased inversely proportional to L1 , the measuring accuracy would not have been increased by reducing L1 . Finally, the decorrelation length in the r-plane becomes 2 L1 / krs for L1 o 0, which means that the available displacement that can be measured, is inevitably decreased as the gearing increases. Measurements supporting the indicated gearing have been performed. The crucial parameter, namely the speckle size as a function of L1 is shown in Fig. 4 together with a best fit to the theoretical curve for L2 500mm, O =0.6328ȝm, rs 660ȝm and rc 1.2ȝm. It should be emphasized that the lateral scale rc in this case does not strictly reflect a structural parameter for the dynamic diffuser, but merely restricts the angular spread of light scattered off this diffuser. In case rc goes to zero, the diameter of the intensity pattern at the second diffuser would go to infinity, and thus the speckle size will go to zero. In other words, the finite lateral scale introduced here remedies the error we introduce by having violated the Fresnel approximation with respect to beam propagation between the two diffusers. Speckle size at a distance of 500 mm Speckle size

Speckle size [µm]

Projected speckle size

150

100

50

Distance between diffusers [mm] 0

4,0

5,0

6,0

7,0

8,0

9,0

10,0

Fig. 4 Speckle size as a function of L1 for L2=500 mm.


11,0

12,0

3. CONCLUSION Two issues have been treated in this article, namely the fine structure of speckles arising from exposing the dynamic object with two spatially separated coherent beams. It has been shown theoretically - and partly experimentally - that a coarse speckle pattern arises with the same dynamical properties as would appear, had the object been illuminated with only one of the beams. Due to illumination with two beams, the coarse speckle pattern will contain a fine structure that usually moves with a different velocity. In the extreme, setups can be found, where the two patterns will move in opposite directions. This effect has here been applied in order to shape a speckle spectrum with a desired spectral shape, thus increasing the signal power in spatial filtering velocimetry, and as a “fringe benefit” increase the gearing” of the speckle velocity. Secondly a theoretical investigation of “speckled speckles” has been performed, partly supported experimentally. By investigating scattering from a dynamic diffuser through a second diffuser, it has been shown that a large gearing of the observed speckle pattern can be achieved, resulting in an increased sensitivity. The accompanying speckle size has been shown not to increase beyond a certain limit when the gearing is deliberately increased by closing the gap between the two plates. On the other hand, in some applications it might be desirable to have a gearing below unity in order to probe larger deflections. Formulas for depicting the speckle size, the gearing and the decorrelation properties have been given.

ACKNOWLEDGEMENT We acknowledge financial support from the Danish Council for Technology and Innovation under the Innovation Consortium CINO (Centre for Industrial Nano Optics). Further, we want to recognize the support from OPDI Technologies A/S.

REFERENCES 1

Joseph W. Goodman, Speckle Phenomena in Optics, Roberts & Company, Englewood, Colorado, 2006. L. E. Drain, The Laser Doppler Technique (Wiley, Chichester, 1980). 3 L. H. Tanner, “A particle timing laser-velocimeter,” report (Department of Aeronautical Engineering, Queens University of Belfast, Belfast, 1972). 4 S. G. Hanson and B. H. Hansen, “Laser-based measurement scheme for rotational measurement of specularly reflective shafts.” In: Fiber optic and laser sensors 12. Conference on fiber optic and laser sensors, San Diego, CA (US), 25-27 July 1994, Proc. SPIE 2292 143-153 (1994) 5 J.W: Goodman, Statistical Optics, John Wiley & sons, New York, NY, 1985. 6 S.G. Hanson, M. L. Jakobsen; H. C. Petersen, C. Dam-Hansen, J. Stubager, “Miniaturized optical speckle-based sensor for cursor control.” In: Speckles, from grains to flowers. Nîmes (FR), 13-15 Sep 2006, SPIE 6341) 6 p. (2006) 7 M. L. Jakobsen and S. G. Hanson, “Lenticular array for spatial filtering velocimetry of laser speckles from solid surfaces.” Appl. Opt. 43 , 4643-4651 (2004) 8 Y. Aizu and T. Asakura, Spatial filtering velocimetry: Fundamentals and applications, (Springer-Verlag, Berlin Heidelberg 2006) 2
