1
Timothy R. Hillman 1 , Toyohiko Yamauchi 2 , Wonshik Choi 3 , Ramachandra R. Dasari 1 , Michael S. Feld 1 †
YongKeun Park 4, *, and Zahid Yaqoob 1, *
1 G. R. Harrison Spectroscopy Laboratory, Massachusetts Institute of Technology, 77 Massachusetts Ave.,
Cambridge MA 02139
2 Hamamatsu Photonics, Hamamatsu City, Japan
3 Department of Physics, Korea University, Seoul 136-701, Republic of Korea.
4 Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Republic of
Korea.
†
Deceased
*Correspondence:
Prof. YongKeun Park,
Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701,
Republic of Korea. Tel: (82) 42-350-2514, Email: yk.park@kaist.ac.kr
Dr. Zahid Yaqoob
G. R. Harrison Spectroscopy Laboratory, Massachusetts Institute of Technology, 77 Massachusetts Ave.,
Cambridge MA 02139, Tel: (1) 617-253-9774, Email: zyaqoob@mit.edu

Optical alignment
2
We describe in more detail the method for overcoming the challenging alignment and wavefront-matching issues in DOPC. The different light propagation paths are indicated with colored directional curves in Supplementary
Figs. 1-3.
The issue of the CCD1/SLM relative spatial displacement is addressed by incorporating a direct imaging relationship between them, the key to the sub-wavelength accuracy attained in the optical alignment. That is, the
SLM plane is directly projected onto CCD1 via the 4f telescopic geometry depicted in Supplementary Fig. 1(a).
Demagnification permits us to achieve near-maximum overlap between the rectangular active areas of the two electronic devices. It is, however, not necessary to achieve pixel-to-pixel correspondence. By writing a test pattern to the SLM [Supplementary Fig. 1(b)] and observing the image at CCD1 (with mirror M1 being necessarily flipped down, that is, not included in the beam path), we can determine the image transformation between the
SLM and CCD1 planes to sub-pixel precision. The contrast in this image is greatest when the SLM is operated in amplitude-control mode (instead of pure phase mode), which can be achieved by rotating both the illumination polarization state and Pz2 by 45°. Supplementary Figs. 1(c)-(d) illustrate the measured responses at CCD1 to the test pattern of Supplementary Fig. 1(b) for different input-beam polarization states. The correspondence between the patterns of crosses permits one-to-one mapping between the SLM and CCD1. In addition to determining the imaging relationship, it is also necessary to characterize the position-dependent phase response associated with the transformation between the two planes. Using phase-control-mode polarization settings, this is achieved by recording a hologram on CCD1 between the light reflected from the SLM (when a uniform write pattern is applied) and a plane wave (propagating along Arm 2, forward). The latter is generated by removing all of the optical elements in Arms 1 and 2. Supplementary Figure 1(e) shows the measured phase pattern that, when applied to the SLM, generates a plane phase front at CCD1 .

(a) SLM
BS1
L1
Pz1
Pz2
3
(b)
L2
CCD1
(c)
(e)
(d)
π
0
-π
BS2
Supplementary Figure 1 (a) Imaging path (between SLM and CCD1), indicated with a red arrow; (b) Test pattern written to SLM (writeable region 792×600 pixels); (c),(d) Measured responses at CCD1 to test pattern of
(b) for different input-beam polarization states, permitting the one-to-one mapping from SLM to CCD1 to be determined; (e) Correction phase pattern applied to SLM (color bar in radians) in order to generate a plane phase front at CCD1.
Description of the acquisition and reconstruction steps
The light paths for the sample and reference waves which interfere to generate the acquisition-step hologram, are depicted in red and green, respectively, in Supplementary Fig. 2. For the former, plane-wave illumination propagates along Arm 1 (forward direction). M3 is flipped down (out of the beam path), but the target is flipped up (into the beam path), so the beam acquires the profile of the amplitude mask. After reflection at BS3, the beam propagates along Arm 2 and impinges upon the turbid sample as a near-plane wave, passing through the turbid sample and experiencing multiple scattering before being incident on CCD1 (M1 is flipped down). The reference wave propagates along Arm 3 after reflection at the SLM. In order to match the reference/sample beam powers to obtain an interference pattern with maximum visibility, the polarizer pair Pz1–Pz2 is used as an attenuator. The angular orientation of Pz1 determines the optical power transmitted through the (fixed) horizontally aligned Pz2.

(a)
4
Target
SLM
BS1
L1
Pz1
Pz2
(b)
L2
CCD1
(c)
BS3
L3
BS2
L4
Sample
Supplementary Figure 2 (a) Reference- (green) and sample-wave (red) paths for digital hologram acquisition; (b) Phase ramp written to SLM (linear color scale from 0 to 2π) for off-axis holography; (c)
Sample hologram recorded at CCD1.
The reconstruction light path, depicted in Supplementary Fig. 3, incorporates: reflection at the SLM, at which point the beam acquires the phase-conjugate profile; propagation along Arm 3; reflection at mirrors M1 (flipped up) and M2; then reverse propagation along Arm 2. The target pattern should be recovered following reverse transmission through the turbid sample. An image can be formed at either of two locations, corresponding to the two output ports of BS3. The first location (not depicted) is terminated with a beam stop and is currently unused.
For the second, corresponding to reflection at BS3 (and depicted in the figure), the amplitude target is flipped down, mirror M3 is flipped up, and an image is formed at CCD2 after partial reverse propagation along Arm 1.
Granting that SLM control permits the phase-conjugate wave to faithfully be delivered to the CCD1 plane, we note that the difference between the forward- and reverse-propagation paths through the turbid sample

5 corresponds to the sub-paths M1-CCD1 and M1-M2, respectively. The reason for placing M1/M2 at the same output port of BS2 as CCD1, rather than at the unused port, was to minimize the differences between these subpaths.
SLM
CCD2
BS1
L1
M3
Pz1
Pz2
L2
M2
BS3 L3 BS2
L4
M1
Sample
Supplementary Figure 3 (a) Reconstruction path in red, terminating at CCD2; (b) Simulated reconstruction of region of USAF target; (c) Representative SLM write pattern utilized to generate the reconstruction of (b).
Hybrid off-axis/phase-stepping approach for recording the hologram of highly scattering speckle field
When recording the hologram, the flexibility of the SLM can be used to perform either phase-stepping or offaxis holography. In the former case, several exposures are taken between which the reference-wave’s phase is varied by known increments. This is impractical for our setup, because the phase between the reference and

6 sample waves fluctuates rapidly due to their path differences. Off-axis holography is also inadequate due to the limited space-bandwidth product it affords to the reconstruction. The desired “+1-order” term must be separated from the “0-order” term and the twin “-1-order” term in the spatial frequency space. Thus, we utilized a hybrid off-axis/phase-stepping approach, where the phase increments between successive holograms were assumed to be random or unknown, as described in the following three paragraphs.
Let the sample-wave complex amplitude distribution (at CCD1) be denoted U ( , ) . For the -th phase step
S
(of total), the reference-wave distribution will be U ( , ) exp
R
   i exp
 x
 y

, where the second factor on the right-hand side describes the phase ramp used in off-axis holography, and the amplitude has been normalized to unity. The phase factor is
 i

2

N
1
 i
, where

is a random variable with unknown i distribution. (The first term of the right-hand side of the preceding equation is assigned via the SLM write pattern, which assures that

will vary between the recordings even if i

does not.) i
Then the recorded-hologram intensity profile is equal to i
       i
T

    i
*

 
, (1)
Where
  
U
S
  2 
1 ,

( , )
 s
( , ) exp
 x
 y

, and
T x y

U
*
S
( , ) exp

 x
 y

, and the coefficient
 i
  i exp
  i
; its modulus accounts for the possible variation in coherence between the two beams over the series of N exposures.
The three terms of Eq. (1) are partially separated in spatial-frequency space, a consequence of off-axis holography. Thus, a region for which the second term dominates can easily be identified [see Supplementary Figs.
4(a),(b)]. If, without loss of generality, the coefficient

is assigned the value 1, then the relative values of the
1 measured signal in this region over the N exposures permits the values

2
, ... ,

to be recovered. Thus, for
N each point ( x,y ), we have three unknowns ( DC , T
+
, and T
-
), and N equations/measurements. We solve for the unknowns using the method of least squares for an over determined system, thereby isolating the desired term

7
U
S
 
. In practice, we used the value N = 6, and ignored the contribution of DC in Eq. (1), removing one unknown to simplify the calculation. The result is shown in Supplementary Fig. 4(c).
Supplementary Figure 4 (a) Spatial-frequency-space representations of the multiple phase-stepped holograms recorded according to Eq. (1); (b) Average (real amplitude basis) of the representations of part
(a); (c) Recovery of the desired “+1-order” reconstruction term using the hybrid off-axis/phase-stepped procedure. In (a),(b), the “±1-order” terms are indicated with a yellow rectangular border and the “zeroorder” term with an elliptical red border. A region for which the “+1-order” term dominates is shaded green.
The target can be flipped either up or down during the reconstruction stage. If flipped up, it provides a control mechanism for ensuring the precision of the alignment: the correctly reconstructed signal should match the target profile. Indeed, this process was utilized to provide further fine-tuning of the phase response between the SLM and CCD1. For an arbitrary patterned (transparent/opaque) target, the diffracted wave profile was acquired by
CCD1 in the absence of a scattering sample. At the reconstruction stage, an additional two-dimensional correction quadratic phase factor was applied to the SLM complex phase profile. In the absence of a scattering sample, the polynomial coefficients were adjusted in order to maximize the optical power transmitted through the target.
These parameters could then be universally applied to other turbid media and target sample combinations. Thus, utilizing the correction parameters acquired in this manner, DOPC may be readily performed.

8
Correlation length for reconstruction pattern written to the SLM
In Fig. 3, we demonstrated that the quality of the reconstructed image is rapidly degraded when the SLM pattern is translated by merely a few pixels. Here, we present support for that observation by measuring the complex autocorrelation function:

SLM
 m , n where
 
1 
U
SLM

,

*
SLM
 m
 
,
  n

,
# of pixels
U
SLM
 
represents the unit-amplitude complex phasor imparted by the [ m,n ]-th pixel. The modulus of

SLM represents the degree of similarity between the generated wavefront and a translated version of itself. We calculated

for all pairs
SLM
   n

satisfying r
   
2

1 2

5 .
1
0.8
0.6
0.4
0.2
0
0 5
0
1 2 3
Pixel translation
4
100 200 300
Distance at the sample plane [nm]
400 500
Supplementary Figure 5 Plot of measured moduli of the complex autocorrelation function,

SLM
2
, versus pixel-shift magnitude

r, for integer values of row and column pixel translations. The best-fit Gaussian curve has 1/e correlation width 1.24 pixels.

Supplementary Figure 5 shows plots of the data pairs
 r
9
SLM

and a best-fit zero-centered-Gaussian curve to the result. From the curve, we establish that the 1/ e correlation width of the pattern is approximately 1.2 pixels.
This is consistent with the observation of Fig. 3 that the reconstructed image information is essentially lost after translation by two pixels, and the measured values of

and visibility reported in the figure. We note that the autocorrelation operation corresponds to an inner product between a complex amplitude distribution and a spatially translated version of itself. Thus, after considering the Discussion section of the main text, h µ G
SLM
2
.
The correspondence is limited by the measurement errors inherent in calculating

.