One valuable feature of sine-GFM is its measurement speed. Compared to linear-GFM, the light power lost through the SLM is significantly lower for sine-GFM and, as a result, it allows measurements under shorter exposure time and higher acquisition rate. In fact, sine-GFM can acquire images as fast as the camera acquisition speed since there is no need for additional image processing. Therefore, sine-GFM is suitable for studying dynamic samples such as biological cells. Since sine-GFM measures the first-order derivative of the phase, it is very sensitive to fluctuations where there is a rapid change of the field. In particular, GFM can detect nanoscale motions of red blood cell (RBC) membranes. These fluctuations have been studied actively in the past few years both for their interesting dynamics, at the basic science level [

19–

24], and their potential for diagnosing disease at a single cell level [

25].

In previous studies, the dynamics of RBC membrane fluctuation has been studied in terms of the relationship between the spatial frequency,

*k*, and the mean-square displacement (MSD) of the RBC membrane,

*Δu*^{2}(

*k*) [

26].

In

Eq. (4),

*k*_{B} is the Boltzmann constant,

*T* the absolute temperature,

*κ* the membrane bending modulus,

*and σ* the

*apparent* tension coefficient. Later, it was found that the physical origin of the tension mode is the coupling between compression and bending modes [

23]. From the equation, it is expected that the MSD of the membrane depends on

*k*^{−2} at low spatial frequencies and on

*k*^{−4} at high spatial frequencies.

We used sine GFM to measure the spatial power spectrum,

_{}. The spatial power spectrum of the sine-GFM image is, to a good approximation,

_{}, as obtained by taking the modulus squared of

Eq. (1) and ignoring the sin

^{2} term. Thus, to obtain the MSD, we need to remove the (known) sinusoid.
shows this procedure and the result of RBC membrane dynamics under sine-GFM. Blood smeared on a glass slide was imaged using 100x, 1.4NA oil immersion objective. The image acquisition was performed with Andor iXon

^{+} EMCCD using 2-by-2 binning, which yielded an acquisition rate of 8 frames per second. shows one frame of the time-lapse stack of RBC sine-GFM images. In order to get a spatiotemporal power spectrum, we took the 3D power spectrum of our time-resolved images and resliced it to show the spatial frequency along the modulation direction (

*k*_{y}) and temporal frequency,

*ω*. This

*ω*-

*k*_{y} domain image, , contains full information about the spatiotemporal fluctuations of RBC membranes. By taking the average over

*ω*, the MSD modulated by the sine is obtained (). Since this sine function is due to the SLM filter, it is known and can be removed numerically via a simple division. shows the resulting MSD for normal discocyte RBCs, where the expected

*k*^{−2} and

*k*^{−4} power laws are obtained. We prepared a sample of osmotically swollen cells and studied the differences between the two fluctuation power spectra. shows the MSD for swollen,

*spherocyte* RBCs prepared by adding water to the blood smear. Both plots are obtained by averaging over 20 measurements and the error bars represent the standard deviation. In , we show a comparison between the dynamics of discocyte and spherocyte cells. Interestingly, we found that for spherocytes, the

*k*^{−2} power law behavior is shown to exist at high frequencies as well which is an indication that the bending mode is subdominant throughout the entire spatial domain. This finding is consistent with recent studies performed by quantitative phase imaging, where the spatial correlations associated with the membrane fluctuations appeared narrower for swollen cells.

Remarkably, our results show that useful information can be retried from nanoscale fluctuations using only a derivative of the phase and not the phase itself. This is quite a general result, i.e., similar information as in and can be obtained using a commercial DIC microscope. This type of measurement is much more easily implemented than a quantitative phase imaging method (see Ref [

1]. for a review of such methods). However, the price we pay for not using full (

*quantitative*) phase information is the lack of

*quantitative* statements we can make about the mechanical parameters of the membrane, e.g.,

*κ* and

*σ.* In other words, using the gradient images to calculate the power spectra, there is no method to normalize the curves (e.g., ). Thus, we cannot

*fit* the curve to extract quantitatively the parameters. If quantitative information is the goal, one should use instead quantitative phase imaging, as shown in Refs [

1,

20–

25].