Quantitative phase microscopy of red blood cells during planar trapping and propulsion

Here, we have combined quantitative phase microscopy and waveguide trapping techniques to study changes in RBC morphology during planar trapping and transportation.

phase contains information about the specimen. , are the spatial carrier frequencies of ( , ) interferogram along axes. In practical applications, it is envisaged that , and are , ( , ) ( , ) ( , ) slowly varying functions compared to the variation introduced by the spatial carrier frequencies , and .
The above intensity modulation can be rewritten in the following form for convenience The Fourier transform of Eq. 2 is given as follows: ( , ) = ( , ) + ( -, -) + * ( + , + ) (4) Electronic Supplementary Material (ESI) for Lab on a Chip. This journal is © The Royal Society of Chemistry 2018 The term is simply a background (DC) term at the origin in the Fourier plane. The term ( , ) corresponds to +1 order term contains information about the object and situated at . Similarly, is -1 order term situated at which carry complex ( + , + ) * ( + , + ) ( -, -) conjugate information about of the specimen. After applying Fourier filtering of zero and -1 order terms, Eq. 4 can be reduced into the following form: ( , ) = ( -, -) (5) The filtered spectrum is, first, shifted at the origin and then inverse Fourier transformed to retrieve the complex signal , ( , ) ( , ) = ( , ) = ( , )exp ( ( , )) (6) subsequently the wrapped phase map from the following expression 2, 3 : where and are the imaginary and real part ( , ) = tan -1 [ ( ( , )) ( ( , )) ] (7) of the complex signal. The reconstructed wrapped phase map lies between to . These discontinuities -+ are then corrected by minimum L P -norm two dimensional (2D) phase unwrapping algorithm 4 . The retrieved unwrapped phase map can be further used to calculate corresponding height map 'h(x, y)' by using the following expression 3 : where, is the difference between refractive indices of cell and outside medium. is the peak wavelength ∆ of the light source ( = 632 nm for He-Ne laser).

Waveguide trapping theory of RBC deformation:
The phenomenon of RBC deformation can be explained using the finite element numerical simulations of optical forces and optical pressure of RBC trapped on top of waveguide surface 5 . The optical force can be found by integrating the Maxwell stress tensor T over the cell surface S 6, 7 : where the outwardly directed unit normal on the surface is given by . It has been shown previously that ⃗ the determination of optical forces using local optical pressure instead of using Maxwell stress tensor approach results in smaller noise and use smaller computational time 5,8 . The force density on a nonmagnetic, isotropic dielectric body in an electromagnetic field is 9, 10 : with denoting the electric permittivity of vacuum, E the electric field and is the permittivity gradient. 0 ∇ It was reported that the net effect of the force density (i.e. pressure multiplied with surface normal) was to straighten the part of the cell that is overlapping with the evanescent field and make it parallel to the waveguide 5 . Although, the largest force component was to attract the RBC towards the surface. As the cell is an elastic body, the net effect of the optical forces (force density) is to press the cell downwards and to straighten the part of the cell that is interacting with the evanescent field and make it parallel to the waveguide. When this part of the cell is becoming straight, the rest of the cells deform and a change in the shape occurs. The elastic membrane of the RBC plays a crucial role in spreading the localized optical pressure over the entire cell 11 .

Influence of the trapping beam (i.e. IR light) on the phase measurement
We systematically studied the influence of IR wavelength on phase measurement results. The trapping experiments of RBCs with and without IR filter both are performed, which is inserted into the beam path just before the camera. Any significant change in the recovered phase maps of trapped RBC with and without IR filter is not observed. However, it is observed that the use of IR filter degrades the interferogram quality and subsequently reduces the spatial phase sensitivity of the system due to multiple reflections of laser light between top and bottom surfaces of IR filter. Therefore, insertion of IR filter into the beam path is not preferred.
The IR beam does not influence the phase measurement results because of the following reasons: (1) there is no coherence between IR (used for trapping), and red (used for phase imaging) because both the light sources emit statistically independent radiation. Therefore, IR beam would not interfere with He-Ne laser beam and subsequently does not influence the phase results, (2) in our proposed system; IR light beam is not directly reaching at the camera plane. The IR photons scattered due to the roughness of waveguide and specimens (RBC) and un-scattered photons are actually reaching at the camera plane, which are very less in numbers. To see these photons at the camera plane, the camera is run at maximum exposure and gain. However, the numbers of photons due to phase imaging laser (He-Ne laser @ 632 nm wavelength) are quite high, which increases the signal to noise (SNR) of interferogram and subsequently reduces the influence of IR light on the phase measurements of RBCs. In case of IR leaking to the camera, it will increase the DC level in the interferogram and results a decrease in the reconstructed phase sensitivity and (3) spectral response of the CMOS camera for IR wavelength (~1070 nm) is quite poor (quantum efficiency (QE) < 5%). However, QE of the camera is maximum at 600 nm wavelength. This again increases the SNR of recorded interferogram at 632 nm and reduces the effect of IR wavelength.
If other trapping wavelength (say 532nm) is used for all trapping experiments, then a narrow bandpass green filter would be needed into the beam path as the QE of the camera at 532 nm wavelength is quite high (~ 70 %). However, this may reduce the spatial phase sensitivity of QPM due to multiple reflections inside the green filter.

Spatial phase sensitivity of QPM:
The smallest morphological changes that can be measured by the proposed technique are defined by the phase/height sensitivity of the system. The spatial phase/height measurement sensitivity of the system is quantified using standard flat mirror and waveguide substrate as a test specimen.
First, spatial phase sensitivity of the system is measured with a standard optical flat mirror (Size: 1"; Flatness: λ/10 @ 633 nm; Supplier: Thorlabs) placed under proposed QPM. Supplementary Fig. S1a illustrates the interferogram of flat mirror. The recorded interferogram is processed using Fourier Transform fringe analysis algorithm for the retrieval of phase and subsequently height map of flat mirror. The phase and height map of the standard flat mirror are depicted in Supplementary Figs. S1b and S1c. The spatial phase and height measurement sensitivity of the system thus obtained is found to be equal to 24.2 mrad and 1.2 nm. Next, experiment is performed with waveguide chip (see Fig.1c in the main text) for the measurement of spatial phase sensitivity and compared with the results obtained for flat mirror. Supplementary Figs. S1d -S1f depict the recorded interferogram, recovered phase and subsequently measured height variation of the waveguide chip. The spatial phase and height measurement sensitivity of the system in case of waveguide is found to be equal to 50.9 mrad and 2.6 nm respectively.  Table S1 shows the comparison of spatial phase and height measurement sensitivity while using flat mirror and waveguide as test specimens. The peak to valley (P-V) error in the phase/height measurement for waveguide is observed to be approximately twice than that for standard mirror. This could be due to the multiple reflections from the ~2 μm thick SiO 2 layer over the Si substrate (see Fig.1c in the main text). The P-V spatial phase sensitivity of the system during optical trapping is found to be equal to ~ 313 mrad, which corresponds to the height measurement accuracy of 16 -20 nm. The spatial phase sensitivity is observed to be less during optical trapping than that for standard flat mirror. This could be due to the formation of unwanted spurious fringes, which arise due to multiple reflections from several layers of waveguide structures, water-coverslip and air-coverslip interfaces. This is further confirmed experimentally by employing the system for the height measurement of 20 nm rib height waveguide. The 20nm rib height of optical waveguide is not visible, i.e., embedded with phase noise of the system, in the reconstructed RBCs phase map ( Supplementary Fig. S2a). However, a 50nm rib height introduces a constant waveguide phase into the RBC's phase maps as depicted in Supplementary Fig. S2b.
To measure the height measurement accuracy of the system during trapping, experiment is conducted on a 50 nm rib height waveguide. Supplementary Figs. S3a -S3c present the interferogram, recovered phase and subsequently measured height map of the 50 nm optical waveguide. The height of the waveguide is measured to be equal to 49.7 2.5 nm by employing the present system. Thus, the height measurement

Comparison of strip and rib waveguides:
The propagation losses as a function of waveguide width for the strip and rib waveguide structures have been extensively studied 12,13 . It has been reported that the propagation losses are higher for narrow strip waveguide than for rib waveguide structures due to the roughness of the etched sidewalls 13,14 . In the present study, a comparative analysis of the strip and rib waveguide geometries was performed to obtain the best fit for the waveguide trapping and QPM integrated setup. First, a comparative study of the strip (H = 220 nm) and rib (H = 20 nm) waveguide having different widths (w = 2.5, 3, 4, 5, 10 μm) was made. Supplementary Figs. S4 and S5 show the total phase value, i.e., RBC + waveguide phase, for two different waveguide geometries. It is evident from the phase images shown in Supplementary Fig. S4 (a -e) that for the strip waveguides (H=220 nm), phase reconstruction of the RBCs are noisy at the edges of the waveguide, this was due to its abrupt change in the phase values at the edges. The edges of the waveguide suffers from sidewall roughness due to the fabrication process (ion-beam etching) 15 . Moreover, the sidewall roughness is random in nature along the length of the waveguide. Strip waveguide has thickness of 220nm, while the rib height is only 20 nm for rib waveguides. Consequently, there is far less sidewall roughness and smaller abrupt phase change at the edges for a rib waveguide as compared to a strip waveguide. Therefore, rib waveguides do not suffer from all the phase errors at the edges of the waveguide, thus producing a smooth RBC phase map as shown in Supplementary Fig. S5 (a -e).Rib waveguides were therefore preferred over strip waveguide for this work. Two straight red lines in the phase images were drawn to show the position of the rib waveguide, which otherwise is not visible due to small rib height of 20 nm. Next, the effect of waveguide height on the phase images was studied for a given width of the waveguide (of 2.5 μm). Supplementary Fig. S6 (a -e) shows the phase images of RBCs on waveguides of different heights (H = 8, 20, 50, 150, 220 nm). It was depicted from Supplementary Fig. S6 (a) and (b) that there was negligible influence on the RBC phase images by the waveguide with rib height of 8 -20 nm, while the phase artifacts started to appear when waveguides with rib height above 50 nm were used. Consequently, for all the studies in this work rib waveguide with rib height of 20 nm was chosen.

Defocus correction algorithm:
Before the measurement of various RBCs morphological parameters, a focus correction algorithm is employed for the accurate phase measurement. Focus correction algorithm is implemented by taking the Fourier transform of the complex signal ' ' to obtain an angular spectrum 16,17 .
where, and are the transform variables in terms of direction cosines , of the wave vector with respect to x, y axis and wavelength . The angular spectrum is then propagated along ( , ;0) the propagation direction 'z-axis' to obtain the field at different plane situated at a distance 'z' by using the following expression 17 : where, is the freespace transfer function which is defined as follows 17 : Then, inverse Fourier transform is done to obtain complex field ' ' in spatial domain at z distance: is the calculated complex field at a distance z. This can be further used to calculate the accurate phase/height maps related to specimens during their holographic phase imaging if the accurate defocus distance is known.
To calculate the defocus distance the complex field ' ' obtained from the Fourier transform method  (15) where μ is the mean of the amplitude of the complex field at distance d. The sharpness curve is formed by plotting the amplitude variance as a function of reconstruction distance. The minimum of the sharpness curve gives information about the defocus distance or true focal plane of the objective lens. Finally, the complex field ' ' is propagated by above calculated defocus distance using the steps given above to ( , ;0) accurately calculate the phase/height maps of the specimens.

Influence of defocus on the RBCs phase images:
Supplementary Fig. S7 clearly depicts the influence of defocus on the extracted RBC phase map. The recorded interferogram correspond to human RBCs is shown in Supplementary Fig. S7a. As can be seen, if we employ Fourier transform based phase recovery algorithm without focus correction, then defocused RBC phase information is reconstructed as illustrated in Supplementary Fig. S7b, which greatly influence the measurements related to RBC's morphological parameters. Therefore, it is mandatory to accurately quantify amount of defocus present in recorded interferograms.
Supplementary Fig. S7c is the sharpness curve depicting the variation of amplitude variance ' ' (calculated using Eq. 15) as a function of propagation distance 'z'. It can be envisaged from Supplementary  Fig. S7c that minimum of sharpness curve lies at z = + 8 µm, which gives information about true focal plane of the objective lens. The defocused RBC phase information is further refocused by propagating the complex field from the reconstruction plane to the focal plane of the objective lens (i.e., d = +8 µm) using angular spectrum propagation approach. After digital refocusing of the complex field, nice RBC phase map having perfect donut shape is reconstructed (Supplementary Fig. S7d).

Quantification of cell's surface area:
The total surface area 'S' of the RBC's membrane is the sum of all the area elements 'A 1 ' and projected area 'A 2 ' as illustrated in Supplementary Fig. S11, which is calculated by utilizing Eq. 20.
The area element of the cell's top surface 1 = 1 1 (16) The sum of all the area elements 1 = ∑ = ∑ (17) The projected area element of the cell's top surface on plane -2 = 2 2 (18) The sum of all the projected area elements The total surface area 'S' of RBC's membrane: Thus, 3D cell's information is utilized for all the calculations related to morphological parameters such as surface area, volume, surface area to volume ratio and sphericity.