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A.S. The and M.V., M.P. and M.P., and M.P., A.L., G.B. G.B., A.L., tools; and reagents/analytic G.B., data; new analyzed research; contributed A.L. designed and A.S. G.B. research; and performed M.V., G.B., contributions: Author owo orsodnemyb drse.Eal [email protected] or [email protected] Email: addressed. be may adam.scholefield@epfl.ch. correspondence whom To is h rgnlepsn pcrmcnb algorithmically proper- be absorption can spectrum color exposing Lippmann plate’s recovered. original a the the from with ties, reflected demon- not spectrum together we is the addition, plate, plate In given spectrum. Lippmann that, exposing a strate the from as same reflected the spectrum experimentally, and the theoretically of how both analysis show, end-to-end and complete process a the provide We regarding approach. misconceptions still the are there result, his sig- this for the of Despite nificance Physics photography. in in colors Prize reproducing Nobel of method 1908 the won Lippmann Gabriel Significance netepaehsbe xoe,i srmvdfo h liquid the from removed is it exposed, been has plate the Once light to exposure the (2), view of point chemical a From a n dmScholefield Adam and , y https://doi.org/10.1073/pnas.2008819118 . y raieCmosAttribution-NonCommercial- Commons Creative . y https://www.pnas.org/lookup/suppl/ a,1 | f7 of 1

APPLIED MATHEMATICS AB r = ρ exp(j θ). Here, ρ is the attenuation factor and θ is the phase shift. As we will show, the choice of reflective medium plays a sig- nificant role in the color reproduction. The latent image—that is, the silver density after exposure—is proportional to the power of the superposed waves inside the emulsion:∗

Z ∞  ωz   ωz  2 2 L(z) ∝ |A(ω)| exp j + r exp −j dω 0 c c Z ∞   2ωz  = |A(ω)|2 1 + ρ2 + 2ρ cos − θ dω. 0 c Here, we have assumed that, as depicted in Fig. 2, the reflected wave travels in the positive z direction and z = 0 at the mirror.

Development. As previously described and depicted in Fig. 2C, the development enhances the silver density of the latent image Fig. 1. Self-portrait of Gabriel Lippmann viewed under different illumina- differently at different depths. This can be modeled by a mul- tion. (A) Diffuse illumination. (B) Directed light whose incoming direction is tiplicative function W (z). This results in the following analysis the mirror image of the viewing direction with respect to the plate’s surface. operator, A, that maps the incoming power spectrum P(ω) = |A(ω)|2 to the developed silver density in the emulsion:

solution diffuses from the top surface of the emulsion toward Z ∞   2ωz  A{P}(z) = W (z) P(ω) 1 + ρ2 + 2ρ cos − θ dω. the emulsion–glass interface, leading to a weaker development c at deeper depths. This explains the decaying nature of the silver 0 density in Fig. 2C. Here, we have incorporated the unknown proportionality factor Fig. 2 also shows that, before viewing, the plate is turned between the latent image and the power of the superposed waves upside down with respect to the orientation used for exposure into W (z). In addition, W (z) = 0 when z is outside the finite and the glass surface is painted black. In addition, a prism is thickness of the emulsion and, due to the diffusion described attached to the emulsion surface with an index-matched glue— previously, it decreases with increasing depth. historically, Canada balsam was the adhesive of choice. To view the captured image, the plate is illuminated with a white light Viewing. The recorded silver density pattern can be thought of source. The prism separates the front surface reflection from as elementary partially reflective mirrors, whose reflectance is the desired emulsion reflection and the black surface mini- proportional to the silver density after processing. When the mizes unwanted back reflections. The desired reflection from developed plate is illuminated, the light reflected from the emul- the emulsion consists of scattered light from the silver particles sion can be modeled as the sum of the reflections from each of distributed throughout the depth of the emulsion; depending these elementary partially reflective mirrors. This Born approx- on the wavelength, the reflected light waves add more or less imation neglects all but first-order reflections. It is not hard to constructively or destructively. see that such a model is not physically realistic—in particular, it Although the reproduced colors can look accurate to the eye, assumes that all of the incoming light reaches each elementary if we examine the full spectrum reflected from a Lippmann plate partially reflective mirror. Despite this physical inconsistency, it and compare it to the original, we notice a number of incon- is a good model for Lippmann photography because it is accurate sistencies, many of which have never been documented even in in the regime of interest where each partially reflective mirror modern studies (3–6). By thoroughly understanding the nature of reflects a small amount of light. these inconsistencies, we are able to study invertibility; i.e., given Denote the silver density after processing by D(z) = A{P}(z) a spectrum produced by a Lippmann photograph, is it possible to and assume that the plate is illuminated with a flat spectrum with undo the distortions and reconstruct the original input spectrum. R ∞ wavefunction proportional to 0 exp(j ωt) exp(−j ωz/c)dω. Then, under the previously described first-order model, the Mathematical Formulation wavefunction of the light reflected from the emulsion at To understand this in more detail, consider the mathematical frequency ω is operators that model how the input spectrum is stored in the Z ∞ plate and how the developed plate synthesizes a color image (7). 0 0 0 Uω(z, t) ∝ D(z ) exp(j ωt) exp(j ω(z − 2z )/c) exp(j φ)dz . For simplicity, we give the equations for a single location of the 0 image, so that the only spatial variable is z—the depth through the emulsion. Here, φ is the phase shift introduced by the reflection from any We model light fields as pulsed plane waves traveling in of the elementary partially reflective mirrors. the direction perpendicular to the mirror. The one-dimensional Due to the extremely short temporal period of optical waves, (1D) wavefunction of a monochromatic light wave travel- we observe only their power spectrum (intensity). Therefore, we ing in the positive z direction, at time t, is U (z, t) = define the synthesis operator, S, as the operator that maps the A exp(j ωt) exp(−j ωz/c). Here, j is the imaginary unit, A silver density D(z) to the power spectrum of the reflected wave. is the complex amplitude, c is the speed of light, and ω Given the previous calculations, we have for any z and t is the single angular frequency of the monochromatic wave. Z ∞   2 For a polychromatic spectrum, the 1D wavefunction is an 2 0 2ωz 0 S{D}(ω) = |Uω(z, t)| ∝ D(z ) exp −j dz . infinitesimal sum of monochromatic wavefunctions: U (z, t) = c R ∞ 0 0 A(ω) exp(j ωt) exp(−j ωz/c)dω.

Exposure. During exposure, the incoming wave interferes with *In practice, the response of a photographic plate to light is only linear in a certain the reflected wave. The reflection from the mirror can be mod- regime of exposure times and intensities, the silver density reaching saturation after a eled as a multiplication with the mirror’s reflection coefficient certain point; its response is described by the so-called Hurter and Driffield curves (8).

2 of 7 | PNAS Baechler et al. https://doi.org/10.1073/pnas.2008819118 Shedding light on 19th century spectra by analyzing Lippmann photography Downloaded by guest on October 2, 2021 Downloaded by guest on October 2, 2021 hr h kernel the where this development, uniform for that, oper- shown synthesis be and operator, analysis can combined the It of composition ators. the of consists pipeline. Full of transform Fourier the of propor- density. value silver viewing content absolute the the frequency squared of with the physics to light the tional reflected that in see result we process equation, the at represented Looking is which wave, reflected the light (E creating the gray). illuminated, interfere, in is waves part plate partial the imaginary When These and (D) pattern. color effects. in recorded development part spectrum. the models ( C power (real that of pattern. window wavefunction interference layer spatial complex exposing infinitesimal a the a by each to by multiplied proportional on is be reflected density to partially silver assumed the is is and depth location’s developed spatial is that plate throughout the density silver The (B) sheet. 2. Fig. hdiglgto 9hcnuysetab nlzn ipanphotography Lippmann analyzing by spectra century 19th on light Shedding al. et Baechler uhwae hnwt ealcrflco,terfatv index is refractive reflection the reflector, the metallic Although surface a air. with emulsion than the ambient weaker let the much just with to contact is make approach liquid with simpler contact in a surface mercury, emulsion the putting by created ally Medium. Reflective of Choice Discussion we and as Results 9); (6, operation true. squaring not is a this to show, up reconstruction sometimes perfect achieved, that in is is example, misconception common For a invertibility. case, this the are regarding there plate, misconceptions thick still infinitely the an uni- in throughout and the development reflector Furthermore, metallic form in perfect ignored. a role of be case critical considered cannot commonly a therefore play and components process these spectral the All emul- we affect media, reproduction. properties section, reflective development next different and how thicknesses, the show sion In to modeling technique. this Lippmann’s use multispec- using reproducing data and capturing tral of process end-to-end the 3). (Fig. emulsion the of ness Here, ABCDE H hl hsfruaini o rva,i losu oflymodel fully to us allows it trivial, not is formulation this While Z (ω S h rgnlpwrsetu nieto n pta oaino h lt,md fa mlinlyro glass a on layer emulsion an of made plate, the of location spatial one on incident spectrum power original The (A) pipeline. photography Lippmann , Z ω (ω 0 = ) SA{ = ) ie u oeig h ulLpmn process Lippmann full the modeling, our Given r c P ∗ S (1 H }(ω Z Z − (ω SA (ω 0 Z exp(−2j ) 0 , ∝ − stefloigkre operator kernel following the is , ω

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APPLIED MATHEMATICS AB

Fig. 4. The effect of the reflective medium on color reproduction. (A) Simulation showing how two different phase shifts at reflection lead to different interference patterns (Left) and reflected waves (Right). In the plots of the reflected waves, the real (black solid) components are the same (up to the sign), but the imaginary (dotted) components differ significantly. This leads to different skewing of the reflected power spectrum (blue solid). The skewing is toward red for θ = 180◦ and toward blue for θ = 0◦.(B) Similar effects are observed with real mirrors, both in simulations (Top) and in experiments (Bottom). In particular, reflection from mercury skews colors toward red, and reflection from air skews colors toward blue.

Lippmann plate was exposed to this rainbow, with both types of were exposed to monochromatic light from a laser. In addition, reflector, and then developed and observed. The observation was we show the reflected spectra of the plates. The oscillations in performed by photographing the plate with a color-calibrated these spectra suggest that the plates have depths of 4.6 and 6.4 digital single-lens reflex camera. Comparing the experiment to µm, which approximately match the 4.2 and 5.8 µm measured on the model’s prediction, we observe the expected shifts of the the micrograph. In addition, the microscope images confirm the reflected spectra: For instance, green is rendered bluish with air nonuniform development curve. and yellowish with mercury. We also see that bright reds are Finally, in Movie S2, we provide a three-dimensional recon- more challenging to reproduce with air, and blues are harder struction of a similar plate from ptychographic X-ray tomogra- to represent with mercury. The dark patches in the measured phy at cryogenic temperature (12, 13). colors for wavelengths greater than 600 nm occur because the plate’s dyes are less sensitive to these wavelengths; we will further Spectrum recovery. As we have seen, SA 6 =I (I being identity) elaborate on the effect of dye profile when we look at spectrum and therefore the spectrum reflected from a Lippmann plate is a recovery. distorted version of the original. A natural and interesting ques- tion to ask is whether we can find a left inverse of SA and thus Thickness and Development of the Plate. Recall that the interfer- theoretically recover the original spectrum. ence pattern is multiplied by the window function W (z) to model Since the synthesis operator outputs a power spectrum, we the differing effects of the developer at different depths and lose phase information and it is not obvious that SA is invert- the finite thickness of the emulsion. Now, since the recorded ible (14), even in the purely theoretical sense where everything interference pattern stores low to high spectral frequencies as is exactly within our model. If we also measured the phase, the we move from the mirror side of the plate to the emulsion– combined operator SA would be linear and, as long as we took glass interface, a finitely thick emulsion results in high spectral enough samples and restricted to a finite spectral range such frequencies being clipped. Note that, by low and high spectral as the visible spectrum, the operator could be exactly inverted frequencies, we mean the slow and fast varying components of with another linear operator. However, because of the loss of the power spectrum, not blue and red colors. phase, we have to use a more advanced approach. In particu- The simplest example of this is if we assume homogeneous lar, inspired by algorithms from irregular sampling (15), we use development. In this case, W (z) is a rectangle window that is an iterative algorithm that exploits the prior information that zero for values of z outside the finite thickness of the emul- the true original spectrum must be a valid power spectrum. The sion and, as depicted in Fig. 5 A3, this results in both a algorithm enforces this by alternatively modifying the current smoothing of the reproduced spectrum and Gibbs ripples. These estimate so that it satisfies each of the following two proper- ripples provide a fingerprint of the truncation of the inter- ties: 1) When passed through the operator SA, it matches the ference pattern and the thickness Z of the emulsion can be measurements, and 2) it is real, positive, and does not con- accurately calculated from their period ∆ν in the frequency tain frequencies corresponding to depths outside the estimated domain: 2Z ∆ν = c. thickness of the emulsion. To enforce the second property, a As previously discussed, the development is not homogeneous nonnegative least-squares (NNLS) algorithm (16) is applied. across the emulsion, but instead is stronger closer to the surface. In terms of convergence, simulations have demonstrated that, To model this, W (z) should decay with depth. As depicted in in the noiseless case with an infinitely thick emulsion and known Fig. 5 A4 and as is well known from filter design theory (11), this and invertible dye and development profiles, the proposed damps the Gibbs ripples. algorithm converges exactly to the original spectrum. Further- To demonstrate these effects in practice, Fig. 5 also shows more, the algorithm is robust to noise and can be modified so electron microscope cross-section images of two plates which that, in the case of finitely thick emulsions and more general

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APPLIED MATHEMATICS A

B

Fig. 6. Spectrum recovery. (A) The recovery algorithm is verified using self-made plates. (Left) Photographs of the color checker and of the Lippmann plate of the color checker. (Right) The original (i), measured (ii), and recovered (iii) spectra. (B) The recovery algorithm is applied to two historical plates. (Left) Photographs of the plates. (Right) The measured and recovered spectra.

ultrafast lasers to locally modify the refractive index of sub- Materials and Methods strates such as silica (17). Since refractive index changes lead To calculate the information-theoretic limit for the number of samples to reflections, we can, at least in principle, print Lippmann-style given in the Introduction, we note that an emulsion of finite thickness can multispectral images at will. only record spectral frequencies up to a maximal value. In addition, the

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J. , stesedof speed the is −1 rnilesans erentielle ´ 140–147 36, (Cambridge fsodium of −1 76 9, 139, SPIE ees ´ of 7 .M ai,K ir,N uioo .Hro rtn aeudsi ls iha with glass in waveguides Writing Hirao, K. Sugimoto, N. Miura, K. Davis, M. K. 17. Hanson, R. Lawson, C. 16. Gr K. Feichtinger, G. H. 15. Baechler G. 14. 2 .Behe,M ahlk,A coeed .Lty LCAV/Lippman-photography. Latty, A. Scholefield, A. Pacholska, M. Lippman of Baechler, photos Multispectral G. Pacholska, 22. M. Scholefield, A. Latty, A. Baechler, G. 21. Bjelkhagen, I. H. 20. Wolf, E. Born, noise. of M. presence 19. the in Communication Shannon, E. C. 18. Holler M. 13. Holler M. 12. eue w itrclpae rmteMus the from plates recov- historical spectrum two used the we For l’Elys 6B, equally. de Fig. silver of reduce experiment would ery wavelength absorbed energy any assuming and at reflection measurements neglecting transmission plate, sensi- unexposed power dye an spectrometric This of profile. using sensitivity obtained dye was the for tivity input. corrected The as finally measured. spectrum was measured spectrum result the reflectance The but taking the run, described then emulsion–air and was previously an added algorithm with was as restoration prism plate developed A U08C was fixed. a plate not on The made was interface. checker reflective color a of image inci- oblique to due shift for. spectral corrected was The (12 beam (20). probing values the of literature dence which with plates, line the per- in of by parts 1.52 is unexposed about thickness on be for measurements to reflectance needed found forming is was measurement, which spectral gelatin, from laser of estimation the index to compared refractive blue, toward The shifted wavelength. has spectrum the in peak the why cec onainGatCRSII5 Grant Foundation Science ACKNOWLEDGMENTS. in deposited been have (22). measurements (https://doi.org/10.5281/zenodo.4256774) Zenodo spectral and (21) zenodo.4650243) Availability. camera. Data hyperspectral IQ Specim a mm with captured 1 were than such 6 less optics, Fig. is focusing in area custom measured some and the Optics that Ocean from spectrometer ES unknown. are manufacturing hi ieadflxblt swl sgvn sacs otercleto of collection their to 6. access and 1 us Figs. in giving shown are as which well of some as plates, Lippmann flexibility historical Mus Sigrist, and the Manuel and time Franck, Mathys, Tatyana their Nora Idargo, Martin, Sarah Pauline Ferloni, Sophie thank Sandrin, we least, Carole not but Maynes, Last lasers—we collaboration. Pau this ultrafast of Bel- with future the Yves modification about and material excited are to Gateau, us Julien introducing Ricca, for Ruben At louard thank PSI. at Coher- we Source laboratory, the Light Galatea at Swiss the the performed of were beamline Scattering measurements X-ray X-ray Small-Angle ent The S2. Movie in rendering, shown and reconstruction tomography three- X-ray the ptychographic M created dimensional and measurements, Elisabeth X-ray Holler, performed (PSI) sample, exper- Institute the Mirko prepared Scherrer and Paul particular equipment, the in acknowledge time, and and their thank we with addition, generous In tise. extremely who EPFL, been at all Group Laboratory have Environmental Central Syl- the Pierson, and Yann Coudret, chemists the vain and team; microscopy Prandoni; Microscopy Imaging; Electron Paolo for HySpex Holography; feedback, thank F Ultimate also his Polytechnique from We Ecole for Gentet discussions. Yves Fournier fruitful Design; Jean-Marc many Quantum the and and photographs mak- advice, the Lippmann to us of introducing for ing Alves Filipe thank We imaging.” multispectral o h pcrmrcvr loih oto xeieto i.6 Fig. of experiment control algorithm recovery spectrum the For l onws pcrlmaueet eecpue sn FLAME-S-XR- a using captured were measurements spectral pointwise All etscn laser. femtosecond 1995). Mathematics, Applied Applications and Mathematics Process. (2018). eoo tp:/o.r/058/eoo4574 eoie oebr2020. November 7 Deposited https://doi.org/10.5281/zenodo.4256774. Zenodo. 2021. March 31 Deposited https://doi.org/10.5281/zenodo.4650243. Zenodo. plates. Processing Light of Diffraction and Interference (1949). 10–21 temperatures. cryogenic and (2017). room at surements e ohpae r oee ihapim u h eal ftheir of details the but prism, a with covered are plates Both ee. ´ 8945 (2019). 4839–4854 67, MYAtMgah aocY stage. crYo Nano tOMography OMNY—A al., et OtclSine,Srne-elg d ,1995). 2, ed. Springer-Verlag, Sciences, (Optical MYPNavraiesml odrfrtmgahcmea- tomographic for holder sample versatile PIN—a OMNY al., et ue eouinpaertivlfrsas signals. sparse for retrieval phase resolution Super al., et oei vial nZnd (https://doi.org/10.5281/ Zenodo in available is Code ivrHld eodn aeil o oorpyadTheir and Holography for Materials Recording Silver-Halide p.Lett. Opt. rnilso pis lcrmgei hoyo Propagation, of Theory Electromagnetic Optics: of Principles ed ´ cei,Ter n rcieo reua sampling. irregular of practice and Theory ochenig, ¨ hswr a enspotdb h ws National Swiss the by supported been has work This ovn es qae Problems Squares Least Solving rl eLuan’ EF’)ItricpiayCentre Interdisciplinary (EPFL’s) Lausanne’s de erale ´ 7913 (1996). 1729–1731 21, 0–6 (1994). 305–363 1994, ◦ rmnra naradtu 8 thus and air in normal from 822 FmoipanDgtlti for twin “FemtoLippmann–Digital 180232, Esve,2013). (Elsevier, https://doi.org/10.1073/pnas.2008819118 2 h yeseta mgsused images hyperspectral The . le,adMca dtclwho Odstrcil Michal and uller, ¨ e.Si Instrum. Sci. Rev. e.Si Instrum. Sci. Rev. SceyfrIdsra and Industrial for (Society rc nt ai Eng. Radio Inst. Proc. ed l’Elys de ee ´ EETas Signal Trans. IEEE PNAS ◦ nemulsion) in 113701 88, 043706 89, Wavelets: | efor ee ´ an A, f7 of 7 37, ee ´

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