2.1.

Laser System

A home-built extended cavity laser was constructed, following the design of Cho, ²³ incorporating a $1\text{-}\mathrm{cm}$ Ti:Sapphire crystal optically pumped with $3.5\mathrm{W}$ from a solid-state $532\text{-}\mathrm{nm}$ diode laser (Coherent, Verdi). The cavity utilizes concave mirrors (CM1 and CM2) with a $10\text{-}\mathrm{cm}$ radius of curvature and 99.9% reflectivity for $700\text{to}900\mathrm{nm}$ . A fold mirror routes the beam for three round-trip passes between two multipass mirrors, which are concave with $2\text{-}\mathrm{m}$ radius of curvature and $81.25\text{-}\mathrm{cm}$ separation. The output coupler is 10% transmissive at $650\text{to}1150\mathrm{nm}$ . The total path length is $736.6\mathrm{cm}$ . Mode-locking is achieved by selecting for Kerr lensing with an adjustable intracavity slit, and the dispersion is controlled by two fused silica prisms with apex $63$ degrees, separated by $50.8\mathrm{cm}$ . The pulse train period is $50\mathrm{ns}$ , corresponding to a $20\text{-}\mathrm{MHz}$ pulse repetition rate with a $780\text{-}\mathrm{nm}$ center wavelength.

The oscillator is first set up for optimum continuous wave (cw) and mode-locked performance without the multipass mirrors [prism separation $48.25\mathrm{cm}$ , length of output coupler arm (from CM1) $135\mathrm{cm}$ ]. The multipass mirrors are inserted, and the now-extended cavity is optimized and the prisms are separated slightly more to compensate for the dispersion by air from the additional cavity length.

A commercial Ti:Sapphire laser (Coherent Mira), which generates nominally $50\text{-}\mathrm{fs}$ pulses at $800\mathrm{nm}$ with a repetition rate of $76\mathrm{MHz}$ was tuned to $780\mathrm{nm}$ for use as the standard for comparison. A flip mirror is used to switch between laser sources. Immediately after the flip mirror, before the cytometry system, 3% of the incident laser light is sent via a beamsplitter to a digital power meter for a reference power reading.

The laser sources are characterized both with a spectrometer as well as a commercial SHG-FROG device (Grenouille, SwampOptics) and software (VideoFROG, Mesaphotonics). The extended cavity reliably produces $80\text{-}\mathrm{fs}$ pulses with $12\text{-}\mathrm{nm}$ bandwidth and a 0.48 time-bandwidth product. Typical Mira pulse widths are $85\mathrm{fs}$ with $26\text{-}\mathrm{nm}$ bandwidths for a 1.0 time-bandwidth product and show significantly greater time-frequency structure than the extended cavity pulses.

2.2.

Two-Photon Flow Cytometer

The two-photon flow cytometry apparatus is illustrated in Fig. 1 . A capillary flow system was employed to emulate flow similar to what is encountered in in vivo measurements. Either the home-built extended cavity or the Mira laser beam is focused with a long-working distance objective (Olympus $40\times$ , 30% power transmission at $800\mathrm{nm}$ ) into a $100\text{-}\mu \mathrm{m}$ square capillary with a $100\text{-}\mu \mathrm{m}$ fused silica wall (WWP100375, Polymicro Technologies) through which the sample flows at a constant rate (usually 0.3 or $1.0\mathrm{mL}∕\mathrm{h}$ ) under the force of a syringe pump (KD Scientific). A dichroic mirror ( $700\text{-}\mathrm{nm}$ cutoff, Melles Griot) directs the laser into the microscope objective while transmitting the fluorescence collected through the same objective. A secondary dichroic mirror ( $620\text{-}\mathrm{nm}$ cutoff) and sharp cutoff interference bandpass filters ( $530∕35\mathrm{nm}$ for GFP, $660∕50$ for DiD, $525∕250$ for single-channel FITC or beads experiments) are used to separate the fluorescence from different fluorophores into two channels, which are detected with two distinct photomultiplier tubes (PMT, Hamamatsu HC7421-40) and recorded with two multichannel photon counting scalers (MCS, Stanford Research SR430), which are capable of time resolution as low as $2\mathrm{ns}$ and have a sharp cutoff in detection at $720\mathrm{nm}$ . The signals (number of photons counted per each $1.3\text{-}\mathrm{ms}$ bin of sampling interval) are sent to a computer for data analysis.

Fig. 1

(a) Two-photon flow cytometry setup. LP: $760\text{-}\mathrm{nm}$ cutoff long-pass filter. SP: $700\text{-}\mathrm{nm}$ cutoff short-pass filter. Dichroic mirror cutoffs are $750\mathrm{nm}$ and $620\mathrm{nm}$ , top to bottom. BP: bandpass filters selected appropriately for the fluorophore of interest (BP1: $525∕250\mathrm{nm}$ for FITC or beads, $520∕35$ for GFP; BP2: $660∕50$ for DiD). (b) Typical CCD images of beads. (c) Blood flowing through the glass capillary. (d) Newton’s rings when the short-pass emission filter is removed. Capillary inner diameter is $100\mu \mathrm{m}$ .

Long-pass filtering of the excitation source ( $760\text{-}\mathrm{nm}$ cutoff) and short-pass filtering of the emission ( $700\text{-}\mathrm{nm}$ cutoff) are employed to minimize noise on the PMT detectors. When the short-pass filter is removed, backscattered laser light enables careful positioning of the objective for an optimal laser focal spot at the center of the capillary. Before each data acquisition, the diffraction pattern scattered off the capillary bottom inner surface is examined out-of-focal plane [Fig. 1d]. This ensures proper alignment of the switch mirror between laser sources to minimize aberrations.

The spot size radius $\left(w\right)$ at the focus of the two-photon cytometer is determined for each laser by matching a Gaussian to the CCD intensity image of backscattered laser radiation as well as the image of two-photon volume signal from a drop of concentrated Rhodamine dye.

2.3.

Peak Analysis Algorithm

A MATLAB program is used to extract fluorescent peaks above the background noise level from the multichannel photon counting scaler trace signals. The particle detection threshold in a channel is set as the maximum photon count per bin from control traces at the same power, plus one photon. This threshold is typically a few photons per $1.3\text{-}\mathrm{ms}$ bin for unlabeled cells flowing in PBS with $12\text{-}\mathrm{mW}$ laser power (at the objective focus). For a control run equal in length to the data run, this is generally a stricter requirement than just a mean and five or six standard deviations of the control set photon counts. The program scans the trace signals for fluorescent peaks above the detection threshold. Once a peak is located, the peak characteristics, including height (maximum fluorescent signal within the peak), width (FWHM or number of consecutive bins above the background threshold), and location (the index of the maximum bin) are stored. The beginning and end of a peak are determined by crossings of the background threshold value, set as the mean plus a standard deviation of photon counts from the control run at the corresponding power. A double-peak event is treated as a single event if the fluorescent signal between the two peaks does not fall below the background threshold value. Although resolution of up to $2\mathrm{ns}$ is achievable on the multichannel scalers, $1.3\text{-}\mathrm{ms}$ bins are chosen so that all photons from a single cell fall within the duration of a single bin, for maximal signal to noise. For two-channel measurements, the data from each channel are analyzed independently and combined by arrival time.

The flow rate of whole blood in the glass capillary is highly variable, which produces a more variable background than the corresponding experiment with cells flowing in PBS. Thus, analysis of whole blood data is performed adaptively in $1\text{-}\mathrm{kB}$ intervals ( $1.34\mathrm{s}$ of flow for $1.31\text{-}\mathrm{ms}$ bins). Initially, background mean $m_e$ and standard deviation ${\sigma }_e$ estimates are calculated from bins with less than half the maximum photon count. Estimated events are then extracted employing background threshold $m_e+{\sigma }_e$ and peak detection threshold value an additional 3 ${\sigma }_e$ higher. All bins of data outside estimated events are used to calculate the true background mean $m$ and true standard deviation $\sigma$ . A background threshold value $m+\sigma$ and peak detection threshold of $m+4\sigma$ are then used to find the actual events.

2.4.

Cell Preparation

Untransfected MCA-207 cells and MCA-207 cells stably transfected with GFP were kindly provided by Dr. Kevin McDonough at the University of Michigan. They were labeled with DiD (Invitrogen) according to the manufacturer’s protocol. After labeling, cells were resuspended in PBS or bovine whole blood with $10\text{units}∕\mathrm{mL}$ heparin (Pel-Freeze Biologicals) for two-photon flow cytometry. Cells resuspended in PBS were used for conventional flow cytometry.

2.5.

Conventional Flow Cytometry

Prior to flow cytometry analysis, stained cells were washed three times in phosphate buffered saline (PBS) containing 0.1% bovine serum albumin (BSA) and resuspended in $1\mathrm{ml}$ of washing buffer. The acquisition was performed on a Beckman-Coulter EPICS-XL MCL flow cytometer. Collected data were analyzed using Expo32 software (Beckman-Coulter, Miami, Florida).

2.6.

Calculations

Simple geometrical models are used to interpret the signals from the two different laser sources. We calculate the expected fluorescence emission for a cell flowing through the beam focus with different spatial distributions of fluorophore and varying incident laser powers as follows. Let us define a right-handed $x\text{-}y\text{-}z$ Cartesian coordinate system with $z$ the axial direction of the propagating laser beam and $x$ the direction of laminar fluid flow in the capillary. Let $B(x,y,z)$ be the square intensity of the laser beam at each point in space. We assume that this beam is Gaussian. Figure 2a shows an example of $B$ as a function of $y$ and $z$ at the $x=0$ plane (center of the capillary) for a beam waist radius $w$ of $2\mu \mathrm{m}$ ( $1∕{\mathrm{e}}^2$ point). Figure 2b shows surfaces of constant intensity in all three spatial dimensions for the same Gaussian beam.

Fig. 2

Calculation of expected signals. Here we have defined the right-handed $x\text{-}y\text{-}z$ coordinate system with $z$ the axial direction of the propagating laser beam and $x$ the direction of fluid flow in the capillary. $(x_0,y_0,z_0)$ is a variable for the location of the center of the fluorophore-containing object. (a) Magnitude of the beam squared-intensity function $B$ in the $x=0$ plane for a beam waist radius $w$ of $2\mu \mathrm{m}$ ( $1∕{\mathrm{e}}^2$ point). (b) Various level sets of $B$ in three spatial dimensions. (c) Instantaneous signal (arbitrary units) for a $10\text{-}\mu \mathrm{m}$ infinitesimally thick spherical shell centered at $(x_0,y_0,z_0)=(5,5,0)\mu \mathrm{m}$ in a Gaussian beam (waist $w=2\mu \mathrm{m}$ ). (d) Signal (arbitrary units) for a shell centered at $(x_0,y_0,z_0)=(x_0,5,0)\mu \mathrm{m}$ as a function of $x_0$ origin, which increases as the object flows in the $x$ direction. The net signal is the integral of the curve.

Let $S(x,y,z)$ be the distribution function of fluorophores in an object centered at the origin. For GFP expressed uniformly throughout a cell centered on the origin, we can approximate $S$ as unity within a radius $R$ of the origin, zero elsewhere. For DiD, bound to the membrane, $S$ is modeled as a thin shell of thickness $t$ with outer radius $R$ . The object will not, in general, be centered at the origin. Let $(x_0,y_0,z_0)$ be a variable for the location of the center of the fluorophore-containing object. Thus, the distribution function of the fluorophore for the object situated in the defined $x\text{-}y\text{-}z$ coordinate system is $S(x-x_0,y-y_0,z-z_0)$ . It follows that the instantaneous net two-photon fluorescent signal from the object centered at $(x_0,y_0,z_0)$ is proportional to:

As an object in the capillary undergoes laminar flow within the $x\text{-}y\text{-}z$ coordinate system, its center position $(x_0,y_0,z_0)$ will be displaced in the $x$ direction, and thus the $x_0$ value increases directly with time. During this period, the signal will change. Figure 2d shows the anticipated instantaneous signal for a shell centered at $y_0=5\mu \mathrm{m}$ , $z_0=0\mu \mathrm{m}$ as a function of different points of $x_0$ origin, representing flow through the capillary. The double-peak event with broad base emphasizes the reason for choosing an experimental data collection bin width that will include all photons from the passing object. Note that the separation between the peaks will depend on the $z_0$ position of the shell, which defines the position of the object relative to the propagation axis $z$ of the laser beam. The total signal obtained from an object flowing through the laser focus is the integral of the instantaneous signal at each traversed position during the period of observation. Thus, we calculate the net observed signal from the object by summing over all the $x_0$ positions that the object traverses as its center translates in the $x$ direction during flow:

For each possible origin $(y_0,z_0)$ in the capillary cross section, we calculate the anticipated net signal from the flowing object (Fig. 3 ). Henceforth, “short channel” will refer to the simulated GFP signal from a uniform distribution of fluorophore over a $10\text{-}\mu \mathrm{m}$ radius sphere, and “long channel” will refer to simulated DiD signal from a uniform distribution of fluorophore over a $10\text{-}\mu \mathrm{m}$ radius, $0.5\text{-}\mu \mathrm{m}$ -thick shell. These assumptions of distributions are made from the known expression of GFP throughout the cytoplasm of the cell and the binding of DiD only to the membrane at the irregular surface of the cell. The beam waist, from the spot-size measurement, is assumed to have a radius $w$ of $2\mu \mathrm{m}$ . The signal level is scaled into a photon count by normalization to the brightest single event in the corresponding channel of the two-photon flow cytometry data. The simulated detection threshold can also be taken from the compared two-photon flow data.

Fig. 3

(a) Net signal (arbitrary units) from a $10.0\text{-}\mu \mathrm{m}$ -radius sphere or (b) shell $(t=0.5\mu \mathrm{m})$ with different origins $(y_0,z_0)$ in the capillary cross section, flowed down the $x$ direction. The beam propagates in the $z$ direction with a $2.0\text{-}\mu \mathrm{m}$ spot size.

3.1.

Dye Solution Signal Enhancement with Extended Cavity

We begin by checking the enhancement of the fluorescence signal in a dye solution, and indeed we find the expected approximately fourfold improvement for the extended cavity laser relative to the $76\text{-}\mathrm{MHz}$ laser. This enhancement is modestly exceeded in the $3.3\text{-}\mu \mathrm{M}$ solution of the biological dye fluorescein isothiocyanate (FITC), flowed through the capillary at $0.3\mathrm{mL}∕\mathrm{h}$ [Fig. 4a ]. The fluorescence signal is proportional to the square of the exciting power as expected.

Fig. 4

(a) Fluorescence emission under extended cavity or Mira laser pulses for a fluorescein-filled capillary, with power-law fits intersecting the ${10}^0$ axis at 7.6 (extended) or 1.4 (Mira) for a 5.7-fold enhancement. (b) Mean fluorescence of detected $2\text{-}\mu \mathrm{m}$ yellow-green microspheres with $20\text{-}\mathrm{MHz}$ or $76\text{-}\mathrm{MHz}$ oscillators, with power-law fits intersecting the ${10}^0$ axis at 32.7 (extended) or 12.4 (Mira) for a 2.6-fold enhancement. Best-fit power laws are 1.4 and 1.7 when only the brightest 100 events at each power (of $\sim 800$ ) are selected.

3.2.

Fluorescent Bead Signal Enhancement

We next investigate the enhancement of fluorescence in polystyrene beads. We flow a uniform suspension of $2.0\text{-}\mu \mathrm{m}$ yellow-green fluorescent spheres (Ex505/Em515) (Molecular Probes). The mean fluorescence of detected events shows significant sub-square law power scaling with both the $20\text{-}\mathrm{MHz}$ and the $76\text{-}\mathrm{MHz}$ lasers [Fig. 4b]. The difference in best-fit power laws further manifests itself as a less than fourfold fluorescence signal enhancement at low incident laser power.

We also confirm that the interaction length is consistent with single bead detection. The mean event duration is five $40.96\text{-}\mu \mathrm{s}$ bins (FWHM). At the $0.3\mathrm{mL}∕\mathrm{h}$ flow rate in the $100\text{-}\mu \mathrm{m}$ square capillary, this corresponds to a $2.4\text{-}\mu \mathrm{m}$ mean interaction length per $2\text{-}\mu \mathrm{m}$ bead. Thus, single fluorescent microspheres are resolved by a two-photon detection region with lateral dimensions on the order of a micron.

3.3.

Dual-Labeled Fluorescent Cells

To investigate the enhanced ability of the extended cavity laser to detect cells via expressed fluorescent protein, we flow dual-labeled (GFP-expressing and DiD-labeled) MCA-207 cells in PBS and count the number of detected events and mean brightness thereof. The DiD staining serves as a control for the number of cells flowing through the two-photon excitation region. We repeat the experiment in whole bovine blood to mimic the autofluorescent background of a noninvasive in vivo experiment.

The number of detected events in PBS and blood are plotted versus average incident laser powers for both sources ( $20\mathrm{MHz}$ or $76\mathrm{MHz}$ ) in Fig. 5 . The sensitivity of detection via GFP is determined as the fraction of total events in either channel that are also detected in the short wavelength channel (into which GFP fluoresces). We observe a sigmoidal (s-shaped) curve of detection sensitivity via GFP versus average power in both PBS and blood [Fig. 5c and 5d]. At high power, most cells are detected using either laser source. For maximal detection sensitivity near the detection threshold, this highly nonlinear nature implies a dramatic advantage for modest increases in fluorescent signal at low excitation power. We take as the detection threshold the minimum power where over 50% of the maximum fraction of GFP cells detectable is identified. In PBS, this is $2.4\mathrm{mW}$ for the extended cavity $20\text{-}\mathrm{MHz}$ oscillator and $6.4\mathrm{mW}$ for the Mira $76\text{-}\mathrm{MHz}$ oscillator. In blood, this value increases to $3.2\mathrm{mW}$ for the extended cavity but remains at $6.4\mathrm{mW}$ for the Mira, and these values are statistically significant (more than a standard deviation separation).

Fig. 5

The total number of events detected in two channels over six repetitions of a $13.4\text{-}\mathrm{s}$ two-photon flow cytometry measurement of dual-labeled MCA-207 cells in (a) PBS or (b) blood at varying powers. Error bars are $6\times$ the standard deviation of these measurements (because summed data over six repetitions is shown). The ratio of total counted events (in either or both channels) giving a detectable GFP signal are shown in (c) PBS and (d) blood.

In all of our two-photon measurements of cells ex vivo in PBS, cell settling remains a complicating factor, limiting each sample to a few minutes at most of run time before recycling or replenishing the syringe pump with cells. The abnormally large count in the DiD channel at the $4\text{-}\mathrm{mW}$ Mira excitation point of Fig. 5a reflects this difficulty in controlling the flow of cells. Settling effects are abated significantly for cells suspended in the viscous whole blood solution. However, in exchange, even under constant force of the syringe pump, the flow rate of blood through the long thin capillary is highly variable, often appearing to stop altogether for brief periods of time. Fortunately, we are able to normalize for this with two detection channels, as has also been the case for in vivo applications.^{24, 25} That almost all cells passing through the two-photon excitation volume do appear in the DiD channel under all but the smallest excitation powers is corroborated by the data. First, conventional flow cytometry (Fig. 6 ) confirms in excess of 90% labeling efficiency of cells with DiD. Second, in rough calculation, the observed numbers are consistent with very high detection efficiency. The flowed concentration of cells is known to be near 1000 cells per $\mu L$ . Assuming a cell diameter of $10\mu \mathrm{m}$ and a two-photon excitation region with $x$ axis $2\mu \mathrm{m}$ and $z$ axis $10\mu \mathrm{m}$ [e.g., the 0.3 level curve, Fig. 2b], we find that on the order of $0.22\times 0.3$ , or 7% of the total $100\mu {\mathrm{m}}^2$ capillary cross-sectional area is interrogated. Therefore, expecting 70 interrogated cells per flowed $\mu L$ of fluid and calculating flow volumes on the order of $20\mu \mathrm{L}$ during the sampling period, we can confirm an expectation of about 1400 total events for each data point in Fig. 5. Thus, we assume that the DiD channel accurately represents the number of cells interrogated for almost all data points.

Fig. 6

Conventional flow cytometry, with the signal in the GFP channel on the ordinate and the DiD channel on the abscissa. (a) MCA-207 cells; (b) GFP-expressing MCA-207 cells; (c) MCA-207 cells labeled with DiD; and (d) dual-labeled MCA-207 cells.

In both blood and PBS, mean brightness of detected events (Fig. 7 ) shows a sub-square law scaling, with lower power-law fit in PBS (1.4) than in blood (1.9). The average brightness of detected GFP events is significantly greater in blood than in PBS, and this difference is statistically significant. In part, this is due to periods of congested flow during which particles dwell in the laser focus for longer periods of time. This congestion also causes a variable background and prevents the use of one standard background measurement to determine thresholds in blood, in contradistinction to PBS.

Fig. 7

Mean photons per event detected in the GFP and DiD channels for dual-labeled MCA-207 cells in (a) PBS or (b) blood with varying power. Error bars are a single standard deviation of these measurements. Power-law fits are near 1.4 for PBS, 1.9 for blood. The distribution of detected event brightness for the $\sim 8\text{-}\mathrm{mW}$ Mira point is shown in (c) for PBS and (d) for blood (combined data of several experiment repetitions). Note the detection thresholds of 5 photons (DiD) and 3 photons (GFP) in PBS, from a background measurement. Detection thresholds of 9 photons (GFP) and 147 photons (DiD) in blood were automatically determined by the processing algorithm, although the control file at the same power never exceeded 15 and 6 photons, respectively.

Representative individual channel histograms of detected event brightness are shown for $10\text{-}\mathrm{mW}$ incident power in Figs. 7c and 7d. In PBS, both individual channel histograms peak at the detection threshold, and the DiD channel has an additional peak (near three hundred photons for $10\text{-}\mathrm{mW}$ average exciting power). In contrast to those in PBS, dual-labeled cells detected in blood exhibit a single broad peak far above the detection threshold. The much higher adaptively determined detection threshold for DiD (147 photons for $10\text{-}\mathrm{mW}$ incident power from the Mira) than GFP (9 photons) occurs despite the significantly lower background in the long channel than the short channel on control blood measurements at the same power. This effect is due to biasing by many extremely bright events of DiD. Simply cutting the PBS histogram at these thresholds and spreading somewhat (due to piecewise data analysis) neatly accounts for the blood histograms; these histograms are asymmetric in the GFP channel to a much greater extent than the DiD channel, which continues to enjoy its peak far above threshold. Conventional flow cytometry confirms approximately two orders of magnitude variation in cell fluorescent brightness (Fig. 6).

3.4.

Model Calculations

Modeled calculations help interpret unique features in the dual-labeled cell experimental results. In particular, we find that the different geometry of fluorophore distribution can account for the histogram of detected event brightness and explain the sigmoidal shape of number of detected events versus average incident power, as well as the sub-square law scaling of mean brightness.

Calculations on a sphere and shell predict, for each possible origin in the capillary cross section, the signal level in the GFP channel [Fig. 3a, sphere] or DiD channel [Fig. 3b, shell]. The calculated distribution of signal in these two channels (Fig. 8 ) essentially maps the area between various level sets of the curves in Fig. 3. For a sphere, there is a large peak in events near the minimum signal level, and a second minor peak at moderately low signals, whereas for a shell, the primary peak is at slightly higher signals than the minimum, and there is a pronounced second peak at higher signals, in reasonable agreement with the experimental results for dual-labeled cells in PBS [Fig. 7c]. This heterogeneity of distributions manifests itself as a lack of correlation between the intensities of both channels on a scatter plot [Fig. 3c], which is confirmed experimentally [Fig. 3d].

Fig. 8

Signal intensity distributions extracted for (a) sphere (GFP channel); (b) shell (DiD channel); and (c) scatter plot between the two channels. (d) The scatter plot of GFP versus DiD detected the events in Fig. 7d, under $8\text{-}\mathrm{mW}$ average excitation power from the Mira.

The sigmoidal curve for the fraction of cells detectable via the weak fluorescent protein signal is realized theoretically by assuming a signal that varies with squared laser power from zero to a set maximum and a fixed minimum threshold for detection. For example, by assuming heuristically from experimental results that the Mira can generate up to 1000 photons from a DiD shell but only 100 from a GFP sphere, and also assuming background-determined detection thresholds of 10 and 4 photons, respectively, the curve of circles in Fig. 9a is derived. Assuming fivefold higher signals for extended cavity excitation [from Fig. 4a], but the same detection thresholds, yields the left-shifted curve of crosses in Fig. 9a. In Fig. 9b, an analogous calculation demonstrates that even with a GFP signal precisely as strong as the DiD signal and a single-photon detection threshold, the sigmoidal curve shape of detectability will be preserved for the same geometrical reasons. For no detection thresholds, including one single photon, will 100% of detected events be found in the sphere channel. Effectively, the detectable fraction is the area of the $y_0\text{-}{z}_0$ plane in Fig. 3 above a certain finite signal level $S$ , which with increasing incident power falls at a lower level relative to the maximum $S$ . Nonlinearities are inherited from this signal landscape, leading to the observed subquadratic scaling of the two-photon signal.

Fig. 9

(a) Simulated fraction of total detectable events identified in the GFP channel for a fixed detection threshold of 4 photons for GFP and 10 photons for DiD, at varying laser powers (signal levels). The maximum signal produced at the highest incident laser power from the Mira is 100 photons for GFP and 1000 photons for DiD (circles), or $5\times$ higher maximum signal strength for the same incident laser power from the extended cavity (crosses). (b) Simulated fraction of total detected events appearing above the threshold in the DiD (circles) or GFP (lines) channel for a 3000-photon maximum achievable signal at highest incident laser power and a fixed detection threshold of 1, 5, 20, or 50 photons.

Figure 10 shows the anticipated GFP sphere (solid line) and DiD shell (open circles) fluorescent signal power-dependence based on the same heuristically determined maximum signals, in excellent qualitative agreement with the observed curves in Figs. 7a and 7b. Detected solid sphere median brightness also shows a strong sub-square law fit as observed in the original fluorescent microsphere experiments [Fig. 4b].

Fig. 10

(a) Simulated median brightness of detected GFP spheres (lines) or DiD shells (circles) for Mira or an extended cavity laser source. Calculation parameters are as in Fig. 8. (b) Notice that the best-fit power curve on a log-log scale exhibits a slope of 1.25 (1.29) for spheres with the extended cavity (Mira).