Reflection phase microscopy using spatio-temporal coherence of light

This document provides supplementary information to "Reflection phase microscopy using spatio-temporal coherence of light , " https://doi.org/10.1364/OPTICA.5.001468 . W e discuss how the combination of temporal gating and spatial decorrelation generates a single gating which is narrower than each alone. We also describe the formation of the reflection signal from the inter-cellular membrane and its interpretation. In addition, we discuss how to eliminate the system vibrations from the cellular membrane fluctuations. Finally, we provide the MSD plots for the cellular membrane fluctuation as a supplemental analysis.


Theoretical model of the system
In this section, we discuss the theoretical analysis of the spatiotemporal coherence gating in our reflection phase microscope. Following the derivation of 3D transfer functions in Refs. [1,2], the 1D transfer function can be expressed as a function of wave vector kz, and angular frequency ω as: where c is speed of light, and P(kz, ω) is axial aperture function defined in Eq. (2) of the main manuscript. Then the 2D Fourier transform of Eq. (S1) with respect to kz and ω gives us the complex line-spread function of our system as: where zs is the axial position of sample mirror and τR is the arrival time. Note that τR = 0 results in the Eq. (1) in the main manuscript.
Now, let us consider the two experimental scenarios -without and with dynamic speckle illumination as shown in Fig. S1(a) and S1(c), respectively. Without dynamic speckle illumination, the axial aperture function becomes a delta function where = is the optical path length delay. The above equation is identical to the Fourier transform of source spectrum ω 2 S(ω). Therefore, by measuring the axial response of the system without the dynamic speckle illumination, we can calibrate the spectrum of the light source as shown in Fig. S1(b). The resulting multi-peak spectrum illustrates the spectral complexity of the supercontinuum laser source. Finally, we can obtain the theoretical expectation of axial response of the system with the dynamic speckle illumination by substituting the source spectrum obtained in Fig. S1(b) into Eq. (S2). As shown in Fig.S1(c)-(d), the experimental result agrees extremely well with the theoretical model for both the amplitude and the phase.

Sche
When p approa always backgro and r measur clean b middle  Fig. 4 in the m clear membrane ed this measure were in the rang e variation in th asma membrane on phase due to red in a reflectio eme for the phas performing tim ach, phase noise a concern. In ound without a remove the p rement such pha background, esp e of the sample.
. Interpretation membrane to rane, PR: plane of efractive index the first term is nd is of no inter ality. The second ission and the re dex of a cell is ve e index differenc maller than nc [6 be negligible co of the first surf uctuation of the Eq. (S5) can be w 2 c n k φ ≈ validity of this a nt of the plasma he focus on the t phase of the ligh in Ref. [5]. assumption, we membrane fluc top plasma mem ht reflected from rom the phase embrane, we hout any ambig on phase measu embrane fluctua value is in the sa scribed in the m veral cells and a 40 nm in RMS. T phase due to fl mes smaller tha the nuclear mem ements in an int he separate opt ion measureme e used as refere However, in fficult due to the he imaging focu n phase. Lc: dista ce from PR to the second se, which is a n be dropped terms are the , respectively. of the culture y of the order y, the second e third term, ciently larger ce of the cell. mately as (S6) performed a ctuations. For mbrane of the m the surface, fluctuations obtain the guity, which is urements. We ations as 18.1 ame order of main text. We all measured This confirms uctuations of an that of the mbrane when terferometric tical paths is ent, a blank ence to trace a reflection e absence of a us lies in the ance from the o the nuclear To solve this problem, we tilted the slide glass so that a portion of the slide glass came in the field of view together with the cells [7]. The schematic is presented in Fig. S3(a). The tilting angle was about 9.2 degrees with respect to the horizontal plane. With this configuration small portion of the glass slide can be simultaneously imaged in the field-of-view as shown in Fig. S3(b). Since the phase change in the background region is only caused by the pathlength variations between sample and reference arms, we can use it as a reference region to track and remove the system phase noise. As shown in Fig. 4(e) in the main text, the phase noise measured in our system was typically tens of nanometers. This is comparable to the nuclear motions, and thus can significantly corrupt the reflection phase measurements. After removal of the phase noise, the system fluctuation reduced to 1.3 nm, which is the final sensitivity of our reflection measurements.

Mean square displacement of the phase fluctuations
As a metric of the quantification of fluctuation, we also calculated the mean square displacement (MSD) of the phase variation presented in Fig. 4 of the main text. The MSD is defined as [4,8,9] [ ] where t denotes the time average and τ is the time delay. The result is shown in Fig. S4. For the background fluctuation measured in the blank area, the MSD remains almost zero with no noticeable variation. In contrast, the MSD of plasma membrane gradually increased during the early time delay and later reached a plateau. On the other hand, the MSD for the nuclear membrane shows more dramatic change over time. It increased faster than that of the plasma membrane and showed a couple of humps. The rate of increase slowed over time.  Fig. S4. MSDs for the cell presented in Fig. 4 in the main text. Blue, red and black lines represent the MSDs for the nuclear membrane, plasma membrane, and background fluctuations, respectively.