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Supplementary Material for
Photon delocalization transition in dimensional crossover in
layered media
Sheng Zhang*, Jongchul Park, Valery Milner, Azriel Z. Genack
Physics Department, Queens College, The City University of New York, Flushing,
New York 11367, USA
In this supplementary section we describe the measurement of the speckle pattern as a
whole (Section A) as well as the determination of the coherence length (Section B) and
spatial spread (Section C) at the sample output. Finally, we describe 1D computer
simulations of wave propagation (Section D).
A. Measurements of speckle patterns
Accurate measurements of the intensity pattern at the output surface of layered samples
are essential for determining the nonuniformity of the sample as well as the degree of
spatial coherence of the wave within the sample, which, in conjunction with the spread of
the wave, determines the nature of transport. In single glass slides, the interference
pattern gives the orientation and angle of the wedge between the surfaces of the slide.
The field at the output surface of the sample can be measured by imaging the output with
a lens onto a CCD camera or by scanning an optical fiber leading to a photodiode over
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the output surface. Though, rapid measurements can be made with the CCD, the intensity
of highly collimated light such as the light transmitted through a small number of slides,
is modulated on a fine scale by interference in reflection from the CCD chip and the glass
window in front of the CCD. Examples of maps of transmitted intensity obtained using
the CCD for single slides are shown in Fig. S1.
Fig. S1 Examples of CCD image of the fringe patterns for single slides.
Thinner fringes with uniform spacing are present in all images as a result of
interference within the CCD assembly. Measuring the spacing of the thicker fringes, a,
yields the wedge angle between the two faces of each slides, which is in a direction
perpendicular to the fringes. The spacing of the thicker fringes in different slides varies
over a wide range, though the wedge angles of the slides are always nearly parallel to the
sides of the slide. The wedge angle corresponding to the three slides measured in Fig. S1
are 2.0  10-3, 1.1  10-3 and 8.8  10-4 rad respectively.
The narrow fringes caused from the scattering within the CCD camera assembly,
are still clearly visible for a stack of two slides (examples are seen in Fig. S2), but
become progressively less visible and finally disappear in CCD images for thicker
samples. This is because the modulation is washed out by the broader angular
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distributions of transmitted light through thicker samples. Some curvature of the wave
pattern results when the slides are pressed together by a retaining ring. Examples of
images for 2 slides are shown in Fig. S2, while examples for 20, 40 and 80 slides are
shown in Fig. S3.
Fig. S2. Examples of CCD image of speckle patterns for 2 slides.
Fig. S3. Examples of CCD image of speckle patterns for samples with (a) 20; (b) 40 and
(c) 80 slides.
To measure the near-field image in a small number of slides without instrumental
interference fringes, we scanned a cleaved optical fiber over the sample. The end of the
fiber was placed close to the sample output surface and translated in the plane with use of
a micrometer motion stage. The resolution of this image is determined by the mode field
diameter of the fiber, the distance of the fiber from the sample surface, b, and the
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divergence angle, θ, of the output wave. In our measurement, a single mode fiber with
mode field diameter of 8 µm and numerical aperture of 0.13 is used and b = 2 mm. Since
the divergence angle at the output of the sample is less than 0.2°, the resolution
achievable in samples of one or two slides is better than 15 µm, which is significantly
smaller than the speckle size. The speckle patterns shown in Figs. 1(a) and 1(b) for one
and two slides, respectively, were obtained by scanning a fiber over the sample, while
speckle patterns for thicker samples shown in Figs. 1(c) and 1(d) are CCD images.
B. Measurement of coherence length d 
In a sample in which the field is Gaussian, the field correlation length can be readily
obtained from the short-range component of the intensity correlation function. In the
present case, however, the analysis of intensity correlation is complex because of strong
localization effects and because the underlying disorder influences field statistics on the
scale of the transverse spread of the wave. We therefore chose to obtain the field
correlation length from a determination of the field correlation function in the near field,
x, y   E x, y E * x  x, y  y  from measurements in the specific intensity.
~
Since the far-field angular distribution of the field, E k x , k y  , is the Fourier transform of
the near-field spatial distribution,


~
E k x , k y    E x, y  exp i k x x  k y y  dxdy ,
(1)
the field correlation function is then the Fourier transform of the average k-vector
2
~
distribution for an ensemble of random samples, I k x , k y   E (k x , k y )
(S1),
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

x, y    I k x , k y  exp i k x x  k y y  dk x dk y .
(2)
We take the correlation length [S2, S3] of the speckle pattern, d  , to be twice the length
in which the real part of the spatial field correlation function decays to half its maximum
value. The correlation length was determined from measurements made with a collimated
incident beam (500-m diameter) with small angular divergence (0.046°). Measurements
of <I(x)> for different values of L are shown in Fig. S4(a). The angular width of
transmitted radiation is seen to be much greater than the divergence of the incident beam
but small enough that the reflection coefficient at individual interfaces is essentially
constant. The real parts of the corresponding correlation functions, (x), are shown in
Fig. S4(b). Correlation lengths defined as twice the length in which Re{(x)} decays to
0.5 are plotted in Fig. 3(b).
Fig. S4 (a) Far-field measurements of ensemble averaged angular distributions along the
x-direction for an incident Gaussian beam with waist diameter of 500 m. (b) Real part of
the field correlation functions obtained from the Fourier transforms of the specific
intensity in kx.
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C. Measurement of transverse spread of wave  
A collimated Gaussian beam with a waist of diameter of 300 m was used in the
measurements, This beam is narrow enough that the broadening of the beam on the
output surface can be discerned and broad enough that the angular divergence of 0.077
deg does not contribute significantly to the beam spread. A direct measurement of
I in ( x, y ) and  I out ( x, y) is made by imaging the wave at the output on a CCD camera.
Integrating over y, for example, gives, I in x    I in x, y  dy. Results for I in x  and
I out x  are presented in Fig. S5.
Fig. S5 Intensity distributions
I in x 
and
I out x 
. The incident Gaussian beam with
waist diameter of 300 m is narrow enough that the lateral spread of the wave can be
determined for L  30 , but is too broad for accurate measurements in thinner samples.
D. 1D simulations
Wave propagation in one-dimensional layered media can be modelled using the transfer
matrix method which relates the incident and reflected wave on one side of a material to
the same on the other side [S4]. The transfer matrix used to describe a single layer i,
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which includes a face separating two type of media with constant refractive indices (ni
and ni+1) and with thickness of the layer (with index ni) di can be expressed as,
 1
 *
t
mi   i
 ri

 ti

ri* 

t i* 
,
1 

ti 
(3)
Where ti and ri are given in terms of interfacial transmission and reflection coefficients
and phase factors determined by the layer thickness di,
ti 
2 ni ni 1
exp ini kdi ,
ni  ni 1 
(4)
n  ni 1  exp 2in kd .
ri  i
i
i
ni  ni 1 
The square magnitudes of ti and ri, are the transmission and reflection coefficients of
intensity, respectively, and the phases are the same as the corresponding fields. The
matrix for the sample of L layers of given index, and hence 2L space-face elements, is,
M 2L
 1
 *
t
  12... 2 L
r
 12... 2 L
 t12... 2 L

r12* ... 2 L
*
t12
... 2 L
1
*
t12
... 2 L



  m1 m2 ...m2 L ,


(5)
from which the transmission through the sample, T  t12... 2 L , can be calculated.
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Simulations using the refractive index 1.523 for glass and 1 for air are made for
10000 random configurations. The thickness of glass slides and air gaps vary in the
ranges 125-135 m and 5-10 m, respectively. The results for the scaling of average
transmission are shown in Fig. 2(a).
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References
[S1] I. Freund, D. Eliyahu, Phys. Rev. A, 45, 6133 (1992).
[S2] J. W. Goodman, Statistical Optics, (Wiley, New York, 1985).
[S3] I. Freund, N. Shvartsman, Phys. Rev. E, 51, 3770 (1995).
[S4] M. V. Berry and S. Klein, Eur. J. Phys. 18, 222 (1997).