SUPPLEMENTARY INFORMATION major advantage of the plasma-like scheme is the presence of the photonic...

SUPPLEMENTARY INFORMATIONdoi: 10.1038/nphoton.2010.31

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1

Supplemental Information for

Nonlinear Self-Filtering of Noisy Imagesvia Dynamical Stochastic Resonance

Dmitry V. Dylov and Jason W. Fleischer*

*E-mail: [email protected]

This PDF file includes: Supplemental Text Supplemental Figs. S1, S2, S3, S4 Supplemental References

© 2010 Macmillan Publishers Limited. All rights reserved.

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SUPPLEMENTARY INFORMATION doi: 10.1038/nphoton.2010.312

Supplemental Text: This Supplemental Information discusses the nonlinear coupling of coherent and incoherent light, and its

consequences, presented in the main text. It is divided into four parts. Part 1 covers the basic derivation of the

dispersion relation, showing explicitly how the nonlinear propagation of partially-coherent light can be treated as a

photonic plasma. Its main result is the derivation and explanation of the gain rate (2) in the text. Part 2 covers the

dynamics of a signal wave that is partially incoherent, while Part 3 describes interactions between a coherent wave

and a partially-coherent background. Exact analytical results show that the presence of any coherent component

eliminates thresholds for instability and define new characteristic length scales. Finally, Part 4 describes the

influence of signal mode structure on the coherent-incoherent dynamics.

1. Derivation of dispersion relation for spatially-incoherent light

We start by considering the paraxial approximation for beam propagation:

02

2 =Δ+∇+∂∂

⊥ ψψβψτn

zi (s1)

where ψ is the slowly-varying amplitude of the electric field, 02 nπλβ = is the diffraction coefficient for light of

wavelength λ in a medium with base index of refraction n0, and Δn is the time-averaged response of the nonlinear

medium. The Wigner transform, defined by

ζkζrζrζkr ⋅∗∞+

∞−−⋅+= ∫ i

NN ezzdzf ),2(),2(21),,( ψψπ

(s2)

obeys the Wigner-Moyal equation [S1, S2, 15, 20]

(s3)

where the arrows in the sine operator indicate that the spatial derivative acts on the index change and the momentum

derivative acts on the distribution.

To build intuition, we will consider first a purely incoherent spatial beam. In Section 2, we will generalize the

results to include contributions from coherent components as well.

A. Weak nonlinear coupling

To lowest order, Eq. (s3) becomes the radiation transport equation (1) used in the text [13, 20]:

(s4)

,021sin2 =

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂Δ+

∂∂⋅+

∂∂ f

krnf

zf

rs

rkβ

,0=∂∂⋅

∂Δ∂

+∂∂⋅+

∂∂

krrk fnf

zf β



SUPPLEMENTARY INFORMATIONdoi: 10.1038/nphoton.2010.31 3

Linearizing this for )](exp[)(),,( 10 zgxifkfzkxf −+= α gives:

(s5)

where we have assumed an inertial Kerr nonlinearity, τγ In =Δ with a phase-independent, time-averaged

response to the intensity I = I(x,y,z) = ∫ fdk and reduced the discussion to one transverse dimension. The similarity

of Eq. (s5) to the Vlasov equation from plasma physics led Fedele and Anderson to interpret the inhibition of

nonlinear growth by the statistics as a type of Landau damping [20]. However, they did not identify plasma-like

parameters or consider the potential for inverse Landau damping (wave growth), with no threshold, when the

underlying distribution is non-monotonic [22]. Both of these factors are crucial here.

From the experiment, we are concerned with the quasi-thermal, Gaussian distribution of light created by the

rotating diffuser [25] of Fig. 1: )2/exp()2()( 220

2/120 kkIkkf xx Δ−Δ= −π , where Δk = 2π/lc represents the spectral

spread for a beam with correlation length lc. For long-wavelength perturbations, we take )(/ xkg βα << and

expand the denominator of Eq. (s5), giving

(s6)

Next, we write g = gR + i gI and assume that the growth/decay rate |gI| << |gR|. Explicitly accounting for the

principal value and pole in the integral gives

( )2/31 22DPR gg λα+= (s7)

βαβ

γγαπ

gkxI

xkfIg

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂⋅≅ 00

2 (s8)

where βγ /0IgP = is an effective plasma frequency and PD gk /Δ= βλ is an effective Debye length [22]. Eq.

(s7) is a Bohm-Gross dispersion relation [S3] for nonlinear statistical light, showing that the statistical distribution

responds to perturbations via Langmuir-type waves. Growth or damping of these waves is a resonant process that

depends on the spectral shape of the distribution f0 and the relative mode matching between the perturbation and the

nonlinearity. For example, it is clear from Eq. (s8) that there are no growing modes if 00 <∂∂ xkf . In contrast,

non-monotonic backgrounds with regions of 00 >∂∂ xkf have a “non-equilibrium” source of free energy which can

drive instabilities. Classic examples of this are so-called “bump-on-tail” instabilities, well-known in plasma physics

[S3] and recently demonstrated by us in optics [22, 23].

( ) xxx dkkg

kfg∫

∞

∞− ⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+≈ 2

2

02

2

31 αββγα

01

0

=−∂∂

+ ∫∞

∞−x

xx kg

kf

dkαβ

αγ




A major advantage of the plasma-like scheme is the presence of the photonic Debye length, which serves as a

compound, characteristic length of system. A disadvantage is that the above expressions only hold in the limit of

weak growth rates, so that highly-nonlinear phenomena like modulation instability are not captured at this level of

approximation. On the other hand, the modes of Eqs. (s7-s8) can be used as a basis for stronger wave coupling [S4,

S5]. This is the approach used phenomenologically in the main text.

B. Strong nonlinear coupling

For stronger nonlinear coupling, higher-order terms from the sine operator in Eq. (s3) become important

( fn kr33 ∂⋅Δ∂ , etc.). For a homogeneous distribution f0 = f0(kx), the series represents a momentum translation and

gives the dispersion relation [20]

(s9)

This equation, also derivable directly from Eq. (s1), implicitly describes the evolution of the mutual coherence

function τψψ ),(),(* 21 zxzx . Exact analytic solutions are possible for coherent plane waves [ )(00 kkIf x δδ −= ]

and for incoherent beams with Lorentzian distributions [16] ( ) ( )2200 /)( kkkIkf xx Δ+Δ= π . This latter distribution

appears in many contexts of noise [1], but its diverging second moment 2xk makes it an unphysical choice here.

Nevertheless, using it in Eq. (s9) gives [16]

2

0 2ˆ~~~ ⎟⎠⎞

⎜⎝⎛−+−= αδαθα ng (s10)

where kkΔ=0θ is the diffraction angle characterizing the incoherence of the ensemble envelope, 00 nIn γδ = is

the fractional change of the refractive index due to nonlinearity, and “~” implies that the corresponding variable is

expressed in units of k (e.g. kgg =~ ). This formula shows clearly that the gain rate g of a perturbation mode α

results from a competition between statistical spreading (de-phasing) and nonlinear coupling. It correctly predicts

that intensity modulations will occur above a nonlinear threshold [16, 17] (for fixed beam statistics) and gives the

dominant spatial scale kxmax, obtained by taking the limit α→0. These features are generic and will hold for any bell-

shaped distribution. Other features, however, such as the growth rates of other modes [23] and the turnover with

nonlinearity [S6], depend on the particular statistical details of the incoherent beam.

( ) ( )∫

∞

∞− −−−+−=

x

xxx kg

kfkfdkαβ

αααγ 221 00




2. Nonlinear dynamics of a partially incoherent image

There are three possible configurations for the chart and the diffuser: (1) diffuser behind the chart, (2) diffuser after

the chart, and (3) diffuser separate from the chart. Only the last geometry, used in the main text, provides two

beams of relative coherence for signal-noise coupling. The first two geometries create an input beam which is

partially spatially incoherent, either by using statistical illumination to begin with (1) or by diffusing the object light

after illumination (2). Nonlinear dynamics of the former case have been considered in a variety of contexts [37, 38].

Here, we introduce spatial nonlinearity to the latter case, emphasizing that this is the typical scenario experienced

when imaging through turbid media.

The experimental setup is similar to that shown in Fig. 1 of the main text, though now the diffuser is placed in-

line with and immediately after the resolution chart. In this case, the chart is imaged while the diffuser is not. Its

role is to scatter light from the object, so that the chart features are diffused and unrecognizable. Figure S1a shows

the image of the resolution chart, without the diffuser in place, on the input face of the crystal while Fig. S1b shows

the input face with the diffuser added. Light exiting the crystal after linear propagation (not shown) has the same

diffuse pattern. As the nonlinearity is increased (Figs. S1c-f), a pattern emerges at the output face but with

completely different dynamics than that presented in the main text. Here, there is no separation between signal and

noise, so that the combined, partially incoherent beam experiences modulation instability instead of a photonic

beam-plasma instability. Consequences include a nonlinear threshold for MI, the disappearance of relative intensity

and coherence as tuning parameters, a stronger preference for the crystalline axis, poorer visibility, and a slower fall-

off after the signal peak. On the other hand, it is clear that nonlinearity can improve the signal output.

Optimal recovery of the noisy image will depend on many of the same parameters as presented in the text, such

as a matching of spatial scale with correlation length. Likewise, other methods may enhance the signal fidelity, such

as modulation of the signal via photon density waves [S7-S9] and the use of (digital) post-processing techniques.

These will be the subject of future work.

3. Coupling of coherent and spatially-incoherent light

In this section, we are concerned with the nonlinear coupling of coherent and spatially-incoherent light. The

presence of the coherent component immediately implies strong coupling, so that we will use the full dispersion

relation Eq. (s9). As mentioned above, the Gaussian distribution more accurately models the experiment and gives

the correct dynamics of the modes, but it can be examined only in approximate form. A Lorentzian distribution, on

the other hand, allows exact analytical solutions and also gives proper threshold behavior. Since we are primarily

concerned with threshold behavior here, we will focus our discussion on the latter distribution.

For a coherent input combined with a Lorentzian ensemble, the total distribution takes the form

(s11) ( ) ( ) ( )[ ] ,21

)(2 022

00 ∑

≥− −−++

Δ+Δ=

mmxmmxm

xx kkJkkJ

kkkI

kf δδπ




where the signal has been decomposed into its constituent plane waves. To begin, it is instructive to consider the

simplest case of one coherent plane wave with an intensity equal to that of the incoherent background, i.e. taking the

limit km = 0, J0 = I0. Eq. (s9) with (s11) plugged in then yields

(s12)

In the purely incoherent limit, 00 →J , Eq. (s12) correctly retrieves the threshold relation (s10). In contrast, for

signal intensity 00 ≠J , the gain 0~ ≥g for any intensity 0I of the statistical light (provided 2~2αδ ≥n , a condition

which is always satisfied for long wavelengths), implying that even for a very weak nonlinearity the mixture is

unstable to perturbations. That is, there is no threshold for instability, as in the purely incoherent case. Note also

that unlike incoherent MI, for which the growth rate (s10) is separable into coherent and incoherent contributions,

the dispersion relation for the mixture is intrinsically inseparable. Figure S2a shows all the three gain curves ( g~ ,

cohg~ , incohg~ ) drawn for two values of nδ : when the incoherent contribution by itself is stable ( 4107.1 −×=nδ ,

0~ <incohg ) and when it is unstable ( 4108 −×=nδ , 0~ >incohg ). In both cases the curve for g~ , given by the exact

formula (s12), lies between the coherent and incoherent curves, indicating graphically the coupling between the two

components.

Equation (s12) is valid only for a 50/50 mixture of coherent and incoherent light. To study an arbitrary mixture,

we varied the intensity ratio of coherent and incoherent components and studied Eq. (s9) numerically. The results,

presented in Fig. S2b, demonstrate that any amount of coherent component makes a mixture unstable, for any degree

of self-focusing nonlinearity. Experimental support for this behavior, obtained from the setup in Fig. 1 with the

chart replaced by a plane wave, is given in Fig. S3.

The only information content in the plane-wave “signal” is its intensity, and it is this parameter which

characterizes the length scale of the output modulations. As in the simpler case, it can be shown that there is no

threshold for instability. Experimental support for this is given in Fig. S3.

We emphasize again that the nonlinear response of the coherent signal is crucial to the propagation dynamics.

Interestingly, it was just this response (coherent self- and cross-phase modulations) that was explicitly neglected by

Mitra and Stark in their analysis of nonlinear information capacity [28]. While there is a danger in extrapolating

results from our limited samples to an ensemble of inputs required to analyze information capacity, it is safe to stress

the importance of a coherent seed for details of the capacity rollover. Further, as we argued in the text, the Gaussian

form of our input-output correlation suggests that it will carry over into the ensemble case as well [49].

4. Comments on the development of the resolution chart

For a signal with many modes, such as the resolution chart, the dynamics becomes considerably more complicated.

Internal wave mixing is more likely, and there are more mode-matching opportunities with the statistical

background. The Lorentzian approach in Section 3 can be applied, but the resulting equations require numerical

( )[ ] ,~4~2

~

2

~~ 2122

022

02

0 αδθδθαδα

θα

−+++−+−= nnng




solution and do not give much insight. On the other hand, the Gaussian approximation used in Section 1 does give

intuition into the multi-mode dynamics. For this, we assume that each mode of the coherent signal creates a local k-

space disturbance of the incoherent noise. For simplicity, we approximate the total incoherent background as a

central Gaussian distribution (initial noise) with M additional Gaussian beams, each with the same statistics

(correlation length lc) but positioned at different spatial frequencies (angular separations) δk01, δk12, δk23, …, δkM-1 M:

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

Δ= ∑

=

Δ

−−

Δ− −M

j

kkk

jk

k

x

jjxx

eIeIk

kf1

2

)(

200

2

21

2

2

21)(

δ

π (s13)

Following the same derivation as in Part 1, we find that the photonic plasma frequency now depends on the total

intensity ∑ == M

j jtot II0

of the individual intensities Ij, while the photonic Debye length becomes [23]

(s14)

In this formula, it is clear that each additional mode contributes to the overall phase diffusion, with a weight

determined by its relative intensity and spectral separation distance.

Simulation results of these composite parameters in the gain formula (s8) are shown in Fig. S4, both for fixed

base correlation length lc and for fixed nonlinearity δn. Qualitatively, the shift in the peak is the same as in the

experiment, though there is no change in its amplitude, an invariance we attribute to the simple choice of modal

components. For more realistic signals, the initial spectrum will not give Gaussian perturbations of equal amplitude,

correlation length, or modal spacing, and their relative contributions will add up and evolve due to nonlinear

coupling/synchronization during propagation. Nevertheless, the signal-noise dynamics will still follow the basic

principles outlined here.

⎟⎟⎠

⎞⎜⎜⎝

⎛+Δ= ∑

=

−M

j j

jj

totD I

kIk

1

21

21 δγβ

λ




Supplemental References

S1. Wigner, E., On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40, 749-759 (1932).

S2. Moyal, J.E., Quantum Mechanics as a Statistical Theory Proc. Cambr. Phil. Soc. 45, 99-124 (1949).

S3. Krall, N. A. & Trivelpiece, A. W., Principles of Plasma Physics. (McGraw-Hill, 1973).

S4. Popel, S. I., Tsytovich, V. N., & Vladimirov, S. V. Modulational Instability of Langmuir Wave-Packets, Phys. Plasmas 1, 2176-2188 (1994).

S5. Picozzi, A., Towards a nonequilibrium thermodynamic description of incoherent nonlinear optics, Opt. Expr. 15, 9063-9083 (2007).

S6. D. Kip, M. Soljacic, M. Segev, S. M. Sears, D. N. Christodoulides, (1+1)-Dimensional modulation instability of spatially incoherent light, J. Opt. Soc. Am. B 19, 502-512 (2002).

S7. Oleary, M.A., Boas, D.A., Chance, B., & Yodh, A.G., Refraction of Diffuse Photon Density Waves. Phys. Rev. Lett. 69, 2658-2661 (1992).

S8. Tromberg, B.J., Svaasand, L.O., Tsay, T.T., & Haskell, R.C., Properties of Photon Density Waves in Multiple-Scattering Media. Appl. Opt. 32, 607-616 (1993).

S9. Fishkin, J.B. & Gratton, E., Propagation of Photon-Density Waves in Strongly Scattering Media Containing an Absorbing Semiinfinite Plane Bounded by a Straight Edge. J. Opt. Soc. Am. A 10, 127-140 (1993).




Supplemental Figures:

b

f

d

c

a

eb

f

d

c

a

e

Figure S1 | Nonlinear development of a diffused image. An image of the resolution chart, a, is scattered

directly through a ground-glass diffuser, creating a blurred input into the crystal, b. Unlike the case considered in

the main text, there is no evolution of the signal for weak nonlinearity (δn = Δn/n0 = 1.3x10–4), c. Above the

nonlinear threshold, intensity modulations form and expose the main features of the chart. d: δn = 1.7x10–4, e:

δn = 2.0x10–4, f: δn = 2.3x10–4. Scale bar in b is 50 μm.




10

0 0.005 0.01 0.0150

1

2

3

4

5

6

0.01 0.0150.00500

5

4

3

2

1

6

0 0.01 0.02 0.03 0.04

0

10

20

30

40

0.01 0.02 0.03 0.040

40

30

20

10

0

A

B

δn = 8×10-4

δn = 1.7x10-4

J0 / I0 = 0.01 / 0.99

J0 / I0 = 0.5 / 0.5

J0 / I0 = 0.99 / 0.01

Wavenumber α / k

Wavenumber α / kG

ain

g(m

m–1

)G

ain

g(m

m–1

)

Coherent

IncoherentMixture

0 0.005 0.01 0.0150

1

2

3

4

5

6

0.01 0.0150.00500

5

4

3

2

1

6

0 0.005 0.01 0.0150

1

2

3

4

5

6

0.01 0.0150.00500

5

4

3

2

1

6

0 0.01 0.02 0.03 0.04

0

10

20

30

40

0.01 0.02 0.03 0.040

40

30

20

10

0

0 0.01 0.02 0.03 0.04

0

10

20

30

40

0.01 0.02 0.03 0.040

40

30

20

10

0

A

B

δn = 8×10-4

δn = 1.7x10-4

J0 / I0 = 0.01 / 0.99

J0 / I0 = 0.5 / 0.5

J0 / I0 = 0.99 / 0.01

Wavenumber α / k

Wavenumber α / kG

ain

g(m

m–1

)G

ain

g(m

m–1

)

Coherent

IncoherentMixture

Figure S2 | Theoretical growth rate for mixture of a coherent plane wave with a partially-incoherent

background. a, Analytical curves, given by Eq. (s12), for the case when the coherent intensity J0 is equal to the

incoherent intensity I0. The curve for the mixture is solid, for the coherent plane wave is dotted, and for the

partially-incoherent beam is dashed. b, Numerical results for the case when the intensity ratio J0/I0 is varied, for

fixed nonlinearity δn = 1.7x10-4. The gain rate for the mixture always lies between the coherent and incoherent

cases and is always positive.

a

b




11

B E

C F

05 / 95 05 / 95

10 / 90 10 / 90

A D0 / 100 0 / 100

B E

C F

05 / 95 05 / 95

10 / 90 10 / 90

A D0 / 100 0 / 100

Figure S3 | Experimental results of nonlinear coupling between coherent and incoherent components.

Shown are output pictures after 1 cm of propagation for, a–c, a coherent plane wave and, d–f, a coherent cosine

pattern coupling with a partially-coherent background. In each case, the background has a correlation length

lc=110μm and the nonlinearity is fixed at δn = 1.7x10–4 (below the modulation instability threshold for the purely

incoherent beam). Insets show the coherent / incoherent intensity ratio for each frame. Scale bar in a is 100 μm.

a

b

c

d

e

f




12

Gai

ng

(mm

–1)

B

100

M =1

M =2

M =100

M =...

Correlation length lc (μm)200 3000

0.00

0.02

0.04

0.06

A

Gai

ng

(mm

–1)

M =1

M =2

M =100

M =...

Coupling strength δn (x10 – 4)

0.00

0.02

0.04

0.06

6 120 93 15G

ain

g(m

m–1

)

B

100

M =1

M =2

M =100

M =...

Correlation length lc (μm)200 3000

0.00

0.02

0.04

0.06

A

Gai

ng

(mm

–1)

M =1

M =2

M =100

M =...

Coupling strength δn (x10 – 4)

0.00

0.02

0.04

0.06

6 120 93 15

Figure S4 | Theoretical growth rates for a multimode signal. a, Dependence of gain rate on coupling strength

δn as the number of modes M is increased. The correlation length is fixed at lc=110μm. b, Dependence of gain

rate on correlation length lc. The coupling strength is fixed at δn = 1.7x10–4.

a

b


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