Two-dimensional random access multiphoton spatial frequency modulated imaging

: Spatial frequency modulated imaging (SPIFI) enables the use of an extended excitation source for linear and nonlinear imaging with single element detection. To date, SPIFI has only been used with ﬁxed excitation source geometries. Here, we explore the potential for the SPIFI method when a spatial light modulator (SLM) is used to program the excitation source, opening the door to a more versatile, random access imaging environment. In addition, an in-line, quantitative pulse compensation and measurement scheme is demonstrated using a new technique, spectral phase and amplitude retrieval and compensation (SPARC). This enables full characterization of the light exposure conditions at the focal plane of the random access imaging system, an important metric for optimizing, and reporting imaging conditions within specimens.


Introduction
Multiphoton microscopy (MPM) [1][2][3][4][5][6][7][8][9][10] is an important subset of emergent biomedical imaging techniques. The hallmark capability of MPM systems is the ability to image through scattering media with minimal damage to the specimen, with access to labeled and endogenous contrast mechanisms, while maintaining a high degree of spatial resolution at (or in some cases beyond) the diffraction limit of light. For a typical MPM system, operating with a near infrared (NIR) source in the 700-1300 nm range, this amounts to lateral spatial resolution at or below the micrometer-scale, at up to millimeter-scale depths within specimens [6,7]. Such qualities make the MPM family of techniques highly suited for biomolecular research and have allowed MPM systems to become the tool of choice for the study of live cellular-level dynamics deep within scattering tissue specimens [3][4][5]. Indeed, MPM systems, as a tool for biomolecular research, already boast an array of accomplishments. For example: MPM systems have been used to study calcium and sodium dynamics in living brain tissues [11], for imaging of in vivo angiogenesis and metastasis of cancer cells [12,13], for study of the progression of Alzheimer's disease via in vivo visualization of amyloid-beta deposits through a cranial window [14], and for dynamic, in-vivo, imaging of neuronal activity [15,16]. MPM techniques have found use in several adjacent areas of research besides microbiology and biomedicine, such as characterization of materials [17,18], or as a probe for phenomena such as exciton lifetime [19,20].
Presently, MPM prototype designs will generally address one or more of the following ambitions; advancement of at-depth performance in scattering media [1,2,6], reduction of image acquisition time [8,9,15,16], and maintaining or bettering diffraction-limited spatial resolution [7]. Additionally, one outstanding issue is the need for improved characterization of the light exposure conditions experienced at the specimen plane, ultimately related to the excitation pulse characteristics. Often, finding the most efficient excitation conditions amounts to a difficult balancing act: one must find the optimal parameter space (e.g., repetition rate, average power, pulse duration, etc.) which maximizes signal-to-noise (SNR), while simultaneously minimizing potentially perturbative exposure intensities harmful to delicate biological specimens [21]. It has been known that pulse characteristics play a major role in the efficiency of MPM signal generation for almost two decades, and that these characteristics can relate to photodamage [21]. However there are still major discrepancies in reported damage [22], and presently, measured intensity conditions are sparsely reported. An optical system designed to balance all of these concerns would function as a high impact research tool.
Here, we demonstrate a new two-dimensional, single element detection, linear and nonlinear imaging system. It employs multiple focal spots that can be randomly accessed. Assuming a linear scan strategy, N focal spots image a field-of-view N times more efficiently then a single focal spot [9,10]. Arbitrary control over foci placement can increase image acquisition (i.e., improve imaging efficiency) by focusing only on regions of interest. Additionally, the system introduced in this paper also provides a straightforward means of characterizing the temporal amplitude and spectral phase of the excitation source of each focal volume at the focal plane. These objectives are accomplished by (a) using a spatial light modulator (SLM) to enable diffraction-limited, multifocal random access MPM [8,23] (b) implementing two dimensional spatial frequency modulation for imaging, or SPIFI, to afford single element detection for extended source geometries [7,24], and (c) notably, includes a spectral phase and amplitude retrieval and compensation (SPARC) platform to perform in-situ pulse characterization of exposure conditions, for multiple focal spots on and off axis, directly at the image plane for the first time.

Spectral phase and amplitude reconstruction and compensation (SPARC)
Well-established pulse measurement techniques are available [25][26][27][28][29][30], yet pulse measurements at the focal plane of the microscope are rarely reported. For efficient multiphoton excitation, most systems employ a dispersion compensation system. The intent of our design is to show that the dispersion compensation system can also serve as the pulse characterization tool, enabling measurements directly at the focal plane of the microscope. We denote this system as SPARC: spectral phase and amplitude reconstruction and compensation [31]. This simple, cost-effective integration facilitates more routine reporting of the specimen exposure conditions allowing researchers to better understand and compare the impacts of such exposure between biological systems [22]. Additionally, by providing a straightforward means of optimizing the pulse parameter space at focus, the SPARC technique could lead to improved implementation of expensive nonlinear microscopy platforms.
A SPARC system starts with a grating pair arranged for zero dispersion as shown in Fig. 1 (i.e., a Martinez pulse compressor [32,33]). A unity magnification telescope images the first grating onto the second, the gratings are placed a focal length, F, away from the entrance and exit of the telescope. Next, a variable-slit is placed at the intermediate focal plane between the two lenses of the system. This plane is where the beam is spectrally dispersed, and the individual wavelengths come to a focus. The variable-slit is set to be roughly the full width at half maximum (FWHM) of the dispersed beam diameter. The measurement takes place by scanning the slit across the entirety of the dispersed beam profile in steps dx, while the transmitted beam is focused by the objective lens into a frequency doubling crystal placed in the microscope sample plane. The subsequent spectrally resolved second harmonic generation signal (SHG) is measured per step in dx, and a two-dimensional density plot, called a SPARC trace, is built from indexing each measurement-per-step into a 2D matrix. Details for taking SHG measurements at high numerical aperture for pulse characterization can be found in [34]. Measurements were performed using a 100 µm thick KDP crystal cut for Type I second harmonic generation (SHG), and the signal was detected with an Ocean Optics HR4000+ spectrometer. The SPARC technique utilizes a pulse phase retrieval algorithm to determine both the spectral phase and temporal amplitude from the measured data set. The algorithm is discussed in detail in [31]. It begins by minimizing the metric between the input "guess" and the measured pulse, then a relaxed averaged alternating reflection (RAAR) algorithm [35] retrieves a directly sampled representation of the complex field function, from which the spectral, phase, and temporal pulse profiles can be computed.
An example of a SPARC measurement taken at the focal plane of a 0.65 numerical aperture (NA) optic (Zeiss A-plan 40x/0.65NA objective) is shown in Fig. 2. The measurement is shown in the upper left quadrant, and the retrieved data presented in the remaining panels. One of the strengths of SPARC is that the image of the slit is also retrieved. The slit profile can then be directly compared to the actual slit width, for example, adding an additional physical constraint that enables the user to assess the validity of the measurement. An accurate SPARC measurement will retrieve a clean slit image and fundamental spectrum that directly correspond to independent measurements of these same parameters. In this case for example, the slit width was measured to be 1.17 mm and the retrieved profile yielded a width of 1.18 mm.

Random access microscopy
A prominent design challenge for MPM systems has been finding a means of increasing image acquisition rate with minimal impact to the advantages of the point scanning MPM technique. The first method of increasing image acquisition was to extend the excitation geometry from a single point, to a line-cursor and the first video-rate MPM systems were achieved in this way [8]. In this manifestation of extended source multiphoton imaging, frame rates of 30 Hz were obtained using 50 mW of average power from an amplified, 100-fs Ti:sapphire system operating at a repetition rate of 250 kHz. Optical sectioning was improved with a detection slit, which also helped eliminate out-of-focus contributions due to scattering perpendicular to the slit, but could not discriminate against scattering along the slit length.
Multi-focal MPM (MF-MPM) designs introduced the use of sets of point-foci rather than extended excitation geometries such as a line, improving axial resolution and achieving higher frame-rates [8][9][10]15,16,23,[36][37][38][39]. MF-MPM systems still require rastering the beams; some systems hybridize the original point-scanning method while others employ any of a number of adaptive optical beam steering techniques [9,10,23]. However, a drawback to many of the many multifocal systems is the necessity of having to employ two-dimensional detection, such as a camera.
Random Access Multiphoton (RAMP) microscopy was developed with the purpose of optimizing the imaging efficiency [15,16,39,40]. In a RAMP system, an image is produced by scanning a point focus to a user-controlled local, "at random". Random beam positioning is induced by adjustment of an upstream element (e.g., an SLM) in order to place the focus anywhere within the excitation objective focal volume. Consequently, RAMP techniques can be adaptive, scanning "smarter." An adaptive routine in a RAMP-MPM can significantly reduce the time needed to explore the volume, by only devoting high pixel density at areas of significance (sparsing) where measurements of the targeted biological dynamics can be observed.

Single-element detection of extended sources
Extended sources scan areas within the field-of-view in a parallel fashion (as opposed to a sequential fashion) resulting in a more efficient image capture per unit time interval. However, as stated earlier, extending the dimensions of the excitation beam generally necessitates a switch to a two-dimensional detector such as a camera. The net result is then a system that is susceptible to image degradation from any signal light that may be scattered in the specimen or along the imaging path back to the camera. Further, a reduction in sectioning can result when a line source is used compared to a point source. Creative new single element detection methods for MPM systems are enabling the use of multi-focal excitation or extended source excitation schemes for use in depth-resolved imaging within specimens where optical scattering is an issue [23,41]. For example, Ducros et al. have shown that by using a digital micromirror device (DMD) in conjunction with a SLM, they can excite multiple focal volumes from which fluorescence is all simultaneously detected by a single photomultiplier (PMT) [23]. The SLM is used to create multiple focal spots at any desired location within the field-of-view, while the DMD uses amplitude modulation on the excitation beam to encode a binary sequence specific to each focal spot. The signal from the detector is ultimately decoded by multiplying this signal with a sequence specific to the generated focal volumes.
Spatial frequency modulated imaging (SPIFI) [7,24] facilitates linear and nonlinear imaging processes with continuous extended sources, while remaining compatible with single element detection. In SPIFI, an extended source is created by focusing the excitation beam to a line by using a cylindrical lens. The line is then passed through a mask which modulates each pixel along the length of the line in amplitude at a different rate (see Fig. 3). The means of achieving the modulation can vary, yet most current systems rely on a transmissive modulation spinning disk based upon the Lovell mask design [42]. A schematic of a 1D SPIFI setup is given below in  The Lovell mask has a characteristic modulation function [24,42]: where the parameter R is the radial distance from the mask center, θ is the angular coordinate with-respect-to the center point of the mask, ∆k is the modulation frequency rate increase as a function of the radius position and k o is the modulation frequency offset (∆k and k o are parameters set by the mask geometry). The mask is spun at a constant angular frequency dθ dt = 2πf r , setting the rotation period of the mask, T m = (f r ) −1 . Examination of Eq. (1) shows that SPIFI adopts a coordinate-to-frequency mapping scheme, enveloped in the radial dependence of the frequency ∆kR, as a way to encode spatial information into a signal which is suitable for point-detection. The image data is recovered by computing the power-frequency spectrum from the signal light; the frequency indicates a particular pixel position, while the magnitude (integrated area of the temporal signal) indicates the contrast of the pixel value.
Equation (1) can be extended for our two dimensional, multi-focal case as follows: first we define a coordinate system relative to the intended field-of-view at the microscope focal plane (shown above in Fig. 4). The microscope field-of-view is indicated as the yellow shaded area in both Fig. 3 and Fig. 4. Note that although these figures show this plane on the mask, it is the same field-of-view which is ultimately image-relayed to the microscope focal plane. We define coordinate x c as an offset in the x direction of the center of the field-of-view relative to the mask center. Within this field-of-view, we define polar coordinates (ρ, ϕ) for a spot generated by the SLM within this field-of-view, where ρ indicates the relative radial coordinate and ϕ indicates the relative angular coordinate. The x,y cartesian coordinates of a given focus can be decomposed into: Using simple geometric analysis, the relation between coordinates (R, θ) and (ρ, ϕ) for a given focus, n i , can be written as: We know x c , ρ, θ and ϕ for a given point as defined in Fig. 4. The modulation frequency at a specific R i , denoted as f m,i , is given by This yields the modulation frequency for each focus: where we have defined the parameter κ = f r ∆k. These relations show how to avoid overlap of the modulation frequencies in order to maintain distinct points. As long as the focal spots occupy unique radial positions, R i , any number of focal spots can be used anywhere with the yellow zone. The chirp-rate parameter similarly defines the resolvable distance between two unique radii as δR i = (κT m ) −1 (see [24] for further details). Fig. 4 shows that the three focii are at the distinct radii, R 1 , R 2 , and R 3 respectively. Fig. 5(A) shows the modulation function for three focal spots, the detected signal which is an integration of the three individual signals, and the image of the focal spots (B), and the retrieved spots from the detected signal (C). Determining sampling rates for SPIFI are detailed in [24]. In general, optimized conditions for the highest spatial resolution are maintained by sampling the detected signal at rates greater than twice the carrier (modulation) frequency of the outermost radius of the SPIFI mask.

System
The newly developed random access, multifocal SPIFI system is shown in Fig. 6. The platform is divided into three subsystems for clarity; beam generation and dispersion compensation (green), spatial modulation (yellow) and the microscope imaging platform (blue). Fig. 6. System schematic: The microscope platform and relative layout of the components are displayed in the figure above. Conceptually, the system can be broken down into three sub-systems which are highlighted in green (pulse generation and dispersion compensation: SPARC system), yellow (spatial modulation with spatial light modulator (SLM)) and blue (microscopy platform with single element detection). G1&G2:1000 / mm diffraction gratings, AC1&AC2: F= 100 mm achromats separated by 2F, AC3: F = 75 mm achromat, AC4: F=400 mm achromat.

Pulse generation and dispersion compensation
This subsystem and the participating elements are highlighted in green in Fig. 6.

Laser oscillator
The All-Normal-Dispersion or ANDi oscillator [43] has a low manufacture cost, is relatively simple to construct, and does not require extensive maintenance. Our home-built system operates with the following parameters: 250 mW average power, 71 MHz repetition rate, 1050 nm central wavelength, pulse duration of 165 fs. The output spatial intensity profile diameter measures 1.8 mm (1/e 2 ).

SPARC as a dispersion compensation system
A feature of the ANDi laser oscillator is that the output pulse is positively chirped. In addition, a typical microscopy platform consists of multiple glass elements which add dispersion (the net dispersion can be wide ranging, anywhere from 5,000 fs 2 to >50,000 fs 2 ) that stretches pulses in the nominal 100-200 fs range to the picosecond range ultimately compromising the efficient generation of any nonlinear signals. As discussed earlier, the SPARC system is itself a compressor used for second order dispersion (SOD) correction. One may simply move the second grating away from the zero-dispersion point in order to impart negative SOD, and hence provide SOD compensation for the laser and the optical system. Figure 7 shows a diagram of the SPARC platform with the second grated shifted to provide negative SOD. Fine tuning the grating spacing which best compensates for subsequent dispersion then becomes an iterative process between adjusting the grating spacing, then measuring the pulse characteristics with a SPARC scan until the shortest pulse duration is achieved. Indeed, this procedure was followed to achieve the optimized pulse profile shown in Fig. 2.   Fig. 7. SPARC System as a Compressor: The SPARC system can be used to compensate for SOD by adjusting the position of the second grating. Pushing the grating away from the zero-dispersion point to F + ∆ adds negative SOD. Notably, this leaves the output spatially chirped.
Notably, used in this manner, the SPARC system will impart a spatial chirp to the profile to the beam, and will add to the net third order dispersion (TOD). The potential effects of the spatial chirp and TOD relevant to MPM imaging specifically are: a) residual TOD will reduce the efficiency of nonlinear signal generation [44], and b) if the spatial chirp is severe, clipping on the aperture of the system optics will also clip the edges of the spectrum, which impacts the pulse shape/duration. As such, in implementing the dispersion compensation system for the microscope and laser, it is reasonable to question using a spatially chirped beam in an imaging system. Advantages: when performing dispersion compensation, the efficiency of the dispersion compensation system is critical. Signal gains in terms of a shorter pulse width can be mitigated by losses in pulse energy through the compensation system. By using a single-pass grating system, a high transmission efficiency is readily achieved. Greater than 90 percent transmission from off-the-shelf components is now routine. Disadvantages: we are only correcting for SOD, the TOD of the SPARC system adds to the residual TOD of the glass already in the system. We have to watch out for spectral clipping, as the beam is spatially chirped. In our system, wavelength tuning is more complex and not as straight-forward as it is for systems such as developed by Akturk et al. [45].
A set of measurements were designed to evaluate the system performance under the spatial chirp condition. First, we present an image of a sample with endogenous second and third harmonic generation response (SHG and THG respectively) to show that the possibly mitigating effect of residual TOD is outweighed by the gain in shortening the pulsewidth [46]. The sample is a piece of EAGLE XG glass that has been systematically stressed to create cracked, subsurface structures. The images shown in Fig. 8 were only obtained after dispersion compensation: without the compressor, neither THG or SHG was able to be generated above the noise-floor of the PMT. This demonstrates that nonlinear signal generation was not degraded, but in fact, enhanced by the compensation system, despite residual TOD.
Next, we examine the impact of the spatial chirp on the system by assessing whether or not spectral clipping leads to a loss in the pulse spectral bandwidth. We measure the spectrum of the beam as it propagates through the system, the results of which are shown in Fig. 9. The measurements show that the spectrum indeed propagates through the entirety of the system without experiencing any loss in bandwidth due to the spatial chirp. Not that the spectrum at the focal plane is slightly attenuated at the lower wavelengths by the final focusing optics and that this feature is also captured in the spectrum recovered by SPARC in Fig. 2.   9. Evaluation of Spatial Chirp: The spectrum of the ANDI oscillator is measured at several points: the initial output (Blue), after the SPARC system (Lavender) and finally at focus, at the image plane (Mauve). The overlap of the traces shows that spectral clipping does not significantly impact the pulse bandwidth. In all three cases, a fiber spectrometer module (Ocean Optics HR4000 plus) was used to measure the spectrum.

Numeric analysis of the residual dispersion and compensator efficiency
To further analyze the impact of the residual TOD and the transmission efficiency of the compensation system, we performed the following numeric analysis. The measured spectrum at focus from Fig. 9 was used to numerically model the effect of TOD on the pulse-width. We used a residual TOD value of ∼100k fs 3 to generate the spectral phase, and the measured pulse spectrum, to calculate the pulse profile expected with residual TOD. The results of this calculation are shown in Fig. 10, showing good agreement with the pulse shape and width displayed in the measured and retrieved pulse profile of Fig. 2 from an in-situ measurement at focus: both the pulse width and slight asymmetry in the wings are consistent between the two. Figure 10 also reproduces the retrieved pulse amplitude of the bottom-right plot of Fig. 2 with error bars included. These were obtained using the Bootstraps method outlined in [47] and represent the error associated with the pulse retrieval algorithm.
Finally, we extend the argument for SPARC by considering the following. The combination of residual uncompensated dispersion and transmission losses on the generation of a second order nonlinear signal are examined in Table 1. In all cases we assume an 80-fs pulse duration of Gaussian pulse shape. We use a shorter pulse width model which is consistent with many systems used to image deep into specimens and show that the concept proposed here is still applicable in this domain. In the first row, the central wavelength is 1050 nm, 1000 l/mm gratings are used, and the SOD compensation is for 50,000 fs 2 of BK7 glass. The net dispersion calculation is through fourth order. In this case, the compressed pulse is not transform-limited and the net reduction in signal generation for a second order process in terms of the pulse shape is 76 % (compared to 100 % for a transform-limited Gaussian pulse). The transmission efficiency of a grating is 96 %, there are two gratings in the system. The net impact on the signal generation efficiency is then the product of these two factors, which in this instance is 65 %. This geometry leaves a spatially chirped beam. The last three rows in the table examine a pulse (80-fs) centered at 910 nm, for the case of 60,000 fs 2 . This is a wavelength and residual dispersion of interest to the community doing deep multiphoton imaging. Here two gratings are evaluated (1200 l/mm and 1851 l/mm) for use in the SPARC system, and the overall efficiency is compared to a prism compression system using SF57 glass. A transmission of 75 % is used based upon the prism compressor design presented in [45]. The extent of the spatial chirp can be quantified by the beam aspect ratio (BAR) [48]. A beam with no spatial chirp has a BAR of 1, when spatial chirp is present the BAR exceeds 1. The calculated BAR is given in Table 1 in all instances. The BAR is an important metric that quantifies the structure of the spatially chirped beam, and when included, enables a rigorous comparison across systems that use beams that exhibit spatial chirp. Table 1. Impact of pulse shape and compressor efficiency. The first row assumes an 80-fs Gaussian pulse at a central wavelength of 1050 nm, 50k fs 2 of SOD, efficiency is for SPARC. The last three rows assume an 80-fs Gaussian pulse at a central wavelength of 910 nm, for a SPARC system with gratings of different l/mm, 60k fs 2 of SOD. The final row is for a prism compressor.  Fig. 10. Numeric Analysis of Impact of TOD on Pulse Profile: (A) The measured pulse spectrum, and the calculated residual phase at the focal plane is used to simulate the pulse intensity envelope at the focal plane. The residual TOD results in a slight asymmetry. The simulated pulse profile compares extremely favorably to the measure profile (Fig. 2) (B) A bootstraps analysis [47] was performed on the measured, retrieved pulse profile to highlight the uncertainty in the measurement.

Spatial and temporal modulation
Once the beam has been generated and dispersion compensation added, the next subsystem is responsible for producing an excitation source of multiple focal spots at random locations in the two-dimensional field-of-view of the microscope, then implementing the frequency multiplexing of the SPIFI technique to render such a geometry to be compatible with single element detection. The SLM subsystem, and its relevant components, is highlighted in yellow in Fig. 6. The SPIFI system is highlighted in blue in Fig. 6 [49,50]. After exiting the SPARC subsystem, the beam passes through lenses AC3 (Thorlabs AC254-75-B), and AC4 (Thorlabs AC254-400-B) separated by the sum of their focal lengths. This results in a magnification factor of 5.3, and increases the beam diameter to 9.6 mm (1/e 2 ). This magnification was chosen as it significantly overfills the SLM array aperture. The SLM configuration is described as "phase-mode," in which the diffraction pattern occurs because of relative phase-shifts between various points across the wavefront. The output of the SLM is imaged to the SPIFI mask by AC4. For this geometry, a simple binary phase grating is written to the SLM producing a variable array of foci. An SLM can be used to place multiple foci at arbitrary polar coordinates (ρ, ϕ), as defined by Eqs.
(2)-(4), in the lateral image plane. Varying the pixel density (SLM pixels per stripe of the grating) will adjust the radial coordinate (top images) and varying the angle of the grating stripes sets the angular coordinate. The duty cycle of the written grating dictates the power distribution between the generated foci. The axial resolution was characterized at 1050 nm for each of the focal spots following the procedure detailed in [51]. The resolution measured 6.6, 6.7 and 7.2 µm respectively which compares favorably to the diffraction-limited value of 6.5 µm. The lateral resolution was measured to be 1 µm which is the diffraction-limited value. Finally it is important to note that in order to perform efficient multiphoton imaging we are limited to three focal spots by the power of the fiber laser. In the general case, a higher average power laser source can drive more focal volumes simultaneously.
Manipulating pulses with a spatial chirp on an SLM has previously been demonstrated by Sun et al. [52]. What is particularly significant in that work is the demonstration and evaluation of multiple focal spots in three-dimensions. In their system, as in ours, the beam is spatially chirped across the SLM. Sun et al. used a source centered at 1040 nm, 350 fs pulses with a repetition rate that could be varied from 50 kHz to 1 MHz. These pulses are energetic enough that they were able to measure TPEF emission and machine glass substrates with two-dimensional 5x5 arrays of spatially chirped beams that result in simultaneously spatial and temporally focused (SSTF) spots [48]. Significantly, they show how pulse front tilt can be exploited to eliminate cross talk between the SSTF beams as they are shifted closer together. Using a Modified Weighted Gerchberg-Saxton method they demonstrate how to optimize the focal volumes for these applications in all four-dimensions: space and time.
Additionally, the spatial chirp can be exploited to optimize the SSTF focal volume passively. In Fig. 11 we first show how, with femtosecond laser pulses, the SLM grating impacts the final focal spot [53]. Using physical optics propagation calculations for our system, it is predicted that the diffracted orders will exhibit a small amount of lateral color that results in an asymmetric Airy pattern at the focal plane of the microscope. The aysmmetry is along the diffraction plane, and is opposite in sign relative to the diffracted orders. The aysmmetry in the +1 order is opposite in direction to the -1 order for example. This is verified experimentally, in the center image of Fig. 11. The experimental images are slightly over-exposed in terms of the central lobe, in order to be able to track the distortions of the side lobes easily. The spatial chirp can be aligned such that it is opposite to that created by the SLM for a particular set of orders (positive or negative). In the final image, we have rotated the spatial chirp such that it compensates for the residual lateral color in the positive order beam for example. The direction of the spatial chirp was optimized by placing a dove prism in the beam prior to the SLM. In this case, it is serendipitous that the BAR that results from the SPARC system to compensate for the SOD also provides a reasonable spatial correction as shown here. Fig. 11. Using the Spatial Chirp as a Spatial Correction: (A) A physical optic propagation of the focal spot at the focal plane, using Zemax lens tracing analysis software for the system shown in Fig. 7. (B) Measured focal spot revealing asymmetry. (C) Spatial chirp is oriented to correct for asymmetry.

SPIFI mask
The three foci are temporally multiplexed via the SPIFI technique, with a transmissive Lovell [42] mask responsible for the amplitude modulation in time. The pattern was fabricated on a 54 mm diameter, 3 mm thick, circular fused silica substrate [54] using our in-house femtosecond laser micromachining facility. The pattern uses a ∆k of 0.42 1/mm following Eq. (1). This ∆k was empirically found as providing optimum performance for the imaging optics and beam sizes upstream from the mask. The mask is mounted on a motor (Faulhaber, 2057S012BK1155) and the rotation rate is controlled by a matching speed controller (Faulhaber, MCBL3006). The mask can be spun from 50 to 405 Hz. It is the mask rotation period that establishes the pixel dwell time, which for this range of rotational speeds means the dwell time was varied from 20 ms down to 2.5 ms. The present system significantly oversamples [37] and future mask versions will be optimized to capture fewer cycles and further decrease the pixel dwell time.

Microscopy platform
The final subsystem of the platform resembles a point-scanning laser imaging system, and is shown highlighted in blue in Fig. 6. Following the SLM subsystem, the SPIFI mask is image relayed to the focal plane of the microscope by lens AC4 (Thorlabs AC508-400-B) and the objective lens (Zeiss A-plan 40x/0.65NA). This provides unity magnification at the back of the objective lens, where the beam diameter has been set by the size of the SLM reflective aperture of 7.6 mm (1/e 2 ). This value matches the objective back aperture value of 7.5 mm. The objective lens focuses into the sample. The sample is placed in a XYZ translation stage (Newport LTA-HS linear actuator, ESP301 motion controller and driver). The generated signal is collected in transmission with a custom built collection objective, COHNA [55] and/or collected in the epi-direction. A PMT (Hammamatsu R7400U) detects the signal and is coupled into a data-acquisition module (NIDAQ USB 6341). One may raster the beams purely by adjusting the SLM, or, create a fixed multi-focal geometry and use the stages to raster the sample with respect to the beams. We have used both methods with equal success. The images shown in this article were in general obtained by the latter method. The system components and data acquisition all takes place via a custom LabVIEW VI.

SPARC measurements
The first set of measurements demonstrate the utility of the SPARC technique by characterizing the temporal profile of the three foci individually (Fig. 12). The measurements shown in Fig. 12, in conjunction with the spatial measurements discussed previously, to our knowledge represent a first for two-dimensional, multifocal random access microscopy: a full spatio-temporal characterization of each of the individual spatially chirped foci on and off axis. The FWHM for each pulse profile is listed in the legend. Note that R1, the pulse exhibiting distortion, corresponds to lateral color that results in an asymmetric spatial Airy pattern discussed in Sec. 3.2.1

Multiphoton images
Finally a series of measurements demonstrates the capabilities of the two-dimensional, Random Access Multiphoton or RAMP-SPIFI method over a broad range of specimen types, imaging modalities and applications. In all of the following the average power of the individual focal spots was in general limited to 35 mW per focal spot.
First, a sample with high two photon excitation fluorescence (TPEF) efficiency and with a well defined structure that was characterized independently was selected for the task of calibrating the system (e.g., verifying the offset between the focal spots)as shown in Fig. 13. A Chroma slide (p/n 92001) which emits TPEF within the 600-650 nm band, was laser-etched [54] to have a series of five point stars embedded in its surface. A binary phase mask with a pixel spacing of 9 pixels/line, and a duty cycle of 69% was written to the SLM to create the 3-foci excitation geometries shown in Fig. 5(R1,R2,R3). In this instance three full images offset by the relative focal spacing result. Again, all three images are captured simultaneously. The two-dimensional capability is demonstrated by writing a grating which rotates the focal spots by 45 degrees within the field-of-view (as in Fig. 5(R1',R2',R3') for example). Again, a binary phase mask with a pixel spacing of 9 pixels/line, with grooves rotated 45 degrees, and a duty cycle of 69% was written to the SLM to create the tilted, 3-foci excitation geometry. The sample of cellulose fibers are stained with Rhodamine B. The white light image in Fig. 14(A) shows the fiber orientation and the field-of-view selected for imaging. A TPEF image with the SLM off (so only one focal spot is used) was first captured to verify the field-of-view (Fig. 14,  (B)). Next two TPEF images taken by the outermost focal spots are shown (the central focal spot image is omitted for clarity) in Fig. 14. (C) and (D). Finally, image slices for these same two focal spots are shown along the axial direction or XZ plane (Fig. 14, (E) and (F)).
Next SHG and THG signals from the muscle fibers from a mouse are used to generate image contrast. Figure 15(A) is the white light image, (B) and (D) are the SHG and THG images, and the final image is a composite of the white-light, SHG and THG images (C). Here the focal spots are shifted such that three separate images each cover a different field-of-view of the specimen, so that the final composite image is taken as efficiently as possible -all images acquired simultaneously. The raw image data is presented, with each pixel representing the peak amplitude of the signal at each spatial location. No attempt to normalize the intensity of the respective fields-of-view from the individual foci was made so that the regions traced out by the different focal patterns is apparent.

Conclusion
A new nonlinear imaging platform has been demonstrated for the first time with multiple novel advances. First, a compact (in this case 40 cm total path length), efficient (>92%), in line compensation system that also can be used to characterize the pulse amplitude and phase at the focal plane of the microscope has been demonstrated. In particular, we have shown that we could fully characterize, in space and time, each of the individual focal spots. Our system will improve the general understanding of biological response to exposure conditions by providing a simple method for characterizing the focal condition at the sample [22], and aid in enabling optimizing nonlinear signal generation especially in deep two-photon imaging applications. Significantly, once the SPARC system is in place, there is a Fourier plane available to help with higher order dispersion correction [30,44]. This could be achieved with a fixed mask or an SLM placed at this plane for example. Second, we have shown that SPIFI can be extended into two-dimensions with essentially the same mask design used for the original light sheet systems. [24]. A multifocal system is used in this case. As long as the individual focal spots are within the field-of-view back projected onto the SPIFI mask and are at a unique distance R i (as each radius gives a distinct frequency tag) from the center of the mask, all the individual focal spots can be distinguished. We were limited to three focal spots in this case by the low average power laser used as the excitation source. Finally, we have demonstrated that the spatial chirp imparted to the beam profile by the SPARC platform is minimally disruptive to a nonlinear microscopy system and, in fact, can be used to mitigate chromatic aberration imparted by an SLM.
Future work involves adding a fiber amplifier to the system [56] to increase the average power of the system. Field et al. [7] used this type of system with continuous line geometries on the order of 100 µm by 1 µm at the focal plane. This indicates we can extend the number of focal spots from three to tens. It should also be noted that the extension to three dimensions is plausible, as the SLM is capable of offsetting foci in the z-plane. In addition, we can repeat the SPIFI pattern in multiple zones for new mask designs that will reduce the oversampling [37] and further decrease the pixel dwell time. Conservatively, by increasing the number of focal spots from 3 to 10 and using a linear scan strategy a desired field-of-view will be sampled >3 times more efficiently then what is presently done. Next, by repeating the SPIFI pattern around the disc the pixel dwell time can be pushed into the microsecond range. For our 50 MHz repetition rate system, microsecond pixel dwell times ensures 50 to 100 pulses per pixel.
The SPIFI system is compatible with many existing multifocal geometries and is inexpensive to implement. All that is required is placing the SPIFI mask at an intermediate focal plane between the optics used to generate the foci, and the actual microscope focal plane. This will transform a multifocal microscope into a multifocal system compatible with single element detection.