Supplementary Figure 1 Experimental setup. Schematic of an optical setup for measuring a transmission matrix.

Transcription

1 satial light modulator (S) holograhic unit S lane shutter mirror camera Conjugated S lane (demagnified, 3 ) olarizer Sulementary Figure 1 Exerimental setu. Schematic of an otical setu for measuring a transmission matrix.

2 umber of iteration SR = Sulementary Figure ean number of iteration as a function of to ratio ( ) and SR. The total number of iterations decreases as increases. However, because large requires a large transmission matrix for a given, the comutation time for each iteration would roortionally increase.

3 Sulementary ote 1: Required conditions for the seckle-correlation scattering matrix. Since the rincile of the resent method relies on the mathematical moment relations of chaotic fields [Eq. ()], it would work, regardless of hysical roerties of diffuser (i.e. thickness, degree of scattering, absortion) or the size of transmission matrix (T) do not matter unless below two conditions are violated: t, t,, t, and y are randomly diffused (comlex Gaussian distributed) This is the reason why the diffusive layer is necessarily required. In the case of an otical lens, for examle, an outut light field is well-ordered that does not satisfy the condition 1 as well as Eq. (), unless the inut field is already randomized. In the case of a diffuser, in contrary, condition 1 can be satisfied unless the inut field is intentionally well shaed to make the outut field be ordered.. Orthogonality between the column vectors of a T, 1 q tt q r. This condition is essential to retrieve a unique and exact solution, i.e. a correct otical field image of a samle. This assumtion of orthogonality has been acceted for transort 1,, 3 through a diffusive layer q 4, 5, 6, 7, We note that regarding the condition, there exist (very few) correlated elements in a T 8. onetheless, even in such a situation, our method is still valid and can be successfully imlemented by simly changing the basis of interest. Once the transmission matrix is calibrated, we can freely transform the basis. Then, the unwanted correlation elements can be filtered out by simly selecting the eigenvectors (or left-singular vectors) u1, u,, u (instead of t 1, t,, t ) as the basis of interest, which ensures the condition. Since the eigenvectors of a diffusive layer are also randomly diffused in general, they also satisfy the condition 1. Thereby, we can reconstruct a hologram by constructing a Z matrix: where, q u, q 1 r Z u u y y u u y y, (1) q q r q r r in this case. Even if a T is not a full-rank matrix (i.e. rank < ), it is still ossible to imelement our method by filtering out the null sace vectors. However, in such cases, the effective number of inut basis would be decreased down to the rank of the transmission matrix.

4 Sulementary ote : Theoretical exectation of the intrinsic signal-to-noise ratio et us consider light transmission in a diffusive layer with a T matrix, T. Using singular value decomosition, we can rewrite the T as T = USV, where U is a matrix comosed of orthonormal outut basis vectors u1, u,, u ; V is a matrix comosed of orthonormal inut basis vectors v1, v,, v ; and S is a diagonal matrix, whose diagonal comonents are singular values whose elements satisfying s1 s s. Then, for an arbitrary inut field y Tx u, which imlies 1 u x v, transmitted light can be exressed as y 0 1 are the first and second term of Eq. (3), resectively. Since as t u ; therefore, 1. Here, we define ty and t Tk, t ty, which can also be decomosed where y u. The 1 u u 1 1 1, u u and are calculated as, () where coefficients, q q, q1 1 q q, q1 1, and, (3) are comlex-gaussian arbitrary values. The cross multilication terms can be neglected for a sufficiently large. Using the ambiguity of coefficients, we can simlify the and as, From 1 tt, 1 yy, tt, and similarly, 1.. (4) 1 yy. However, from yy ote that sans whole -dimensional outut sace,

5 while and sans subsace. Thereby, the intrinsic root-mean-squared signal-to-noise ratio can be calculated as follows: SR q q, intrinsic q q, (5) In summary, as the ratio / increases, the intrinsic noise from the second term of Eq. (3) diminishes. Since can be easily increased, for examle, by enlarging the camera ixel number, / can be increased, and thus the signal-to-noise ratio of this method, which would be beneficial for ractical alications.

6 [Sulementary ethods] Detailed imlementations of the roosed method. In this section, the detailed exerimental rocedure is exlained with a flow chart. The following diagram introduces the notation of our flowchart: X number of element : normal field form, : conjugated field form, : absolute square form, exlicit variable name We emloyed the exressions in ATAB for the notation of element-wise multilication (.) and division (./), to reresent seudo algorithms for the rocedure.

7 [Ste 1: Transmission matrix calibration] 1. Preare reset individual basis as inut basis. k1, k,, k = total number of S ixels = number of inut basis, Each reset basis forms corresonding a seckle attern on a camera lane. 3. k t = number of outut otical modes basis (number of used camera ixels in current system) 0 i 3 e t R I 0, t i 3 e i 3 e R I 1, R t1, R t,, R t Transmission matrix t R Introducing a certain reference field, each seckle field was measured based on a hase shifting interferometry method. This is the transmission matrix. I, 4. R Blocking a beam from the S arm, the intensity seckle of the reference field was measured.

8 [Ste : Field retrieval using the SS method] 1. An unknown incident field forms a corresonding seckle attern on the camera lane x y. easure the diffused intensity seckle of the incident field yy 3. t t y y 1 q r Rt.../ Rt yy q R t t 1 q y y r r Rt../ 1 Rt q R yy In order to construct the Z matrix, each term of Eq.(1) is calculated 4. Z α α, 1 α By decomosing the Z matrix, the eigenvector of the largest eigenvalue can be obtained 5. 1 k x Using the coefficients, retrieve the incident field.

9 [(otional) odified GS algorithm] 1. First, use the retrieved field as an initial guess. x x n 1 n, k x f Again, using the inverse of transmission matrix, transform into the iteration stos when the converges. R t1, R t,, R t Transmission matrix Using the transmission matrix, transform the into 3. Ry n 1 n, k 4.. ex i Ryn 1 R y Ry n Build by relacing the amlitude art with measured, while conserving the angular art.

10 ATAB code for the numerical simulation % Field retrieval using seckle-correlation scattering matrix (SS), numerical simulation code. % Written by KyeoReh ee (Biomedical otics laboratory, KAIST, Korea), % for further information, lease refer our grou homeage: bmol.kaist.ac.kr %% Initialization gamma = 55; % to ratio. set this square number. only for #visualization. (related to line 9) SR = 100; % linear signal-to-noise ratio (SR) to mimic ractical noisy situations. %% Incident field load load('examle_inutfield.mat'); % load variable X, size(x) = [, 1] comlex double matrix. = numel(x); % set this square number. only for #visualization. = gamma; X = X./ norm(x); % normalize X for simlicity Xd = reshae(x,[sqrt(),sqrt()]); % D reshae. only for #visualization. %% Transmission matrix (T) and intensity seckle generation dis('t generate start!'); tic; T = raylrnd(sqrt(1//),,).ex(1i random('unif',-i,+i-es('double'),,)); % transmission matrix generation. T_rac = awgn(t,sr,'measured','linear'); % add noise uon reset SR. dis(['t generate end!... Elased time: ', numstr(toc),'s']); Y = T X; % diffused field y generation. Y_rac = awgn(y,sr,'measured','linear'); % add noise uon reset SR. Y_racd = reshae(y_rac,[sqrt(),sqrt()]); % D reshae. #for visualization. Iyy = abs(y_rac).^; % measured intensity seckle single shot. Iyyd_osamd = abs(fftshift(ifft(ifftshift(... adarray(fftshift(fft(ifftshift(y_racd))),[sqrt(),sqrt()]))))).^; % oversamle Iyy. only for #visualization. %% Field retrieval using SS method dis('initial guess start!'); tic; sigma = mean(abs(t_rac).^,1); % normalization factor, sigma (see Eq. ()). Zmatrix = (T_rac'(remat(Iyy-mean(Iyy),[1,]).T_rac))./(sigma'sigma); % Z matrix calculation. [Xretv0,~] = eigs(zmatrix,1); % Field retrieval by selecting the eigenvector of the largest eigenvalue. dis(['initial guess end!... Elased time: ', numstr(toc),'s']); Xcorr0 = X'Xretv0/norm(X)/norm(Xretv0); Xretv0 = Xretv0ex(-1iangle(Xcorr0)); Xretv0d = reshae(xretv0,[sqrt(),sqrt()]); % calculate correlation between X and Xretv0. % global hase correction. only for #visualization. % D reshae. only for #visualization. %% #Visualization figure(1), set(1,'units', 'ormalized', 'OuterPosition', [ ]); sublot(43),imagesc(abs(xd),[min(abs(x)),max(abs(x))]),axis image; title('incident amlitude'); colorbar sublot(44),imagesc(angle(xd)),axis image; title('incident wavefront'); colorbar sublot(47),imagesc(abs(xretv0d),[min(abs(x)),max(abs(x))]),axis image; title('retrieved amlitude'); colorbar sublot(48),imagesc(angle(xretv0d)),axis image; title(['retrieved wavefront, Corr. = ', numstr(abs(xcorr0))]); colorbar sublot(,4,[1,,5,6]), imagesc(iyyd_osamd,[0,max(iyyd_osamd(:))0.5]), axis image; title('measured intensity seckle'); colorma(sublot(,4,[1,,5,6]),'gray'); ause(0.1);

11 %% (otional) odified Gerchberg-Saxton algorithm iterax = 1000; % max iteration number to revent infinite loo. itertor = 10^-5; % convergence criteria. dis('inverse T calculation start!'); tic; [U,S,V]=svd(T_rac,'econ'); % economic svd for inverse T calculation. sv = diag(s); Tinv_rac = V(remat(1./sv,[1,]).U'); % inverse T calculation. clearvars U S V dis(['inverse T calculation end!... Elased time: ', numstr(toc),'s']); dis('odified Gerchberg-Saxton iteration start!'); tic; Xcorr_iter = zeros(iterax,1); % archives correlation changes during iteration. Xiter = Xretv0; % set retrieved field for the initial guess. Xcorr_iter(1) = Xcorr0; for itr = 1:1:iterax Yiter = T_rac Xiter; % transform incident field into corresonding diffused field sace. Yiter = sqrt(iyy).ex(1iangle(yiter)); % relace amlitude art with measured intensity. Xiter = Tinv_racYiter; % return to the incident field sace. Xcorr_iter(itr+1) = X'Xiter/norm(Xiter); % correlation calculation. if abs(xcorr_iter(itr+1)) - abs(xcorr_iter(itr)) < itertor % iteration ends when the criterion is satisfied. break; end end dis(['odified Gerchberg-Saxton iteration end!... Elased time: ', numstr(toc),'s']); Xiter = Xiterex(-1iangle(Xcorr_iter(itr))); Xiterd = reshae(xiter,[sqrt(),sqrt()])./ norm(xiter); % global hase correction. only for #visualization. % D reshae. only for #visualization. %% #Visualization figure(), set(,'units', 'ormalized', 'OuterPosition', [ ]); sublot(41),imagesc(abs(xd),[min(abs(x)),max(abs(x))]),axis image; title('incident amlitude'); colorbar sublot(4),imagesc(angle(xd)),axis image; title('incident wavefront'); colorbar sublot(45),imagesc(abs(xiterd),[min(abs(x)),max(abs(x))]),axis image; title('retrieved amlitude'); colorbar sublot(46),imagesc(angle(xiterd)),axis image; title(['retrieved wavefront, Corr. = ', numstr(abs(xcorr_iter(itr+1)))]); colorbar sublot(,4,[3,4,7,8]), lot(abs(xcorr_iter(1:itr+1))), title(numstr(itr)) ause(0.1);

12 Sulementary References 1. Pooff S, erosey G, Fink, Boccara AC, Gigan S. Image transmission through an oaque material. ature Communications 1, 81 (010).. Bertolotti J, van Putten EG, Blum C, agendijk A, Vos W, osk AP. on-invasive imaging through oaque scattering layers. ature 491, 3-34 (01). 3. Katz O, Heidmann P, Fink, Gigan S. on-invasive single-shot imaging through scattering layers and around corners via seckle correlations. ature Photonics 8, (014). 4. Shi Z, Genack AZ. Transmission Eigenvalues and the Bare Conductance in the Crossover to Anderson ocalization. Physical Review etters 108, (01). 5. Vellekoo I, osk A. Universal otimal transmission of light through disordered materials. Physical Review etters 101, (008). 6. Yu H, Park JH, Park Y. easuring large otical reflection matrices of turbid media. Otics Communications 35, (015). 7. Yu H, et al. easuring large otical transmission matrices of disordered media. Physical review letters 111, (013). 8. Pooff S, erosey G, Carminati R, Fink, Boccara AC, Gigan S. easuring the Transmission atrix in Otics: An Aroach to the Study and Control of ight Proagation in Disordered edia. Physical Review etters 104, (010).