Patent ID: 12248133

DEFINITIONS

References in the specification to “one embodiment”, “an embodiment”, “an example embodiment”, etc., indicate that the embodiment described can include a particular feature, structure, or characteristic, but every embodiment may not necessarily include the particular feature, structure, or characteristic. Moreover, such phrases are not necessarily referring to the same embodiment. Further, when a particular feature, structure, or characteristic is described in connection with an embodiment, it is submitted that it is within the knowledge of one skilled in the art to affect such feature, structure, or characteristic in connection with other embodiments whether or not explicitly described.

The terms “a” or “an” are used to include one or more than one and the term “or” is used to refer to a nonexclusive “or” unless otherwise indicated. In addition, it is to be understood that the phraseology or terminology employed herein, and not otherwise defined, is for the purpose of description only and not of limitation. Furthermore, all publications, patents, and patent documents referred to in this document are incorporated by reference herein in their entirety, as though individually incorporated by reference. In the event of inconsistent usages between this document and those documents so incorporated by reference, the usage in the incorporated reference should be considered supplementary to that of this document; for irreconcilable inconsistencies, the usage in this document controls.

In the methods of preparation described herein, the steps can be carried out in any order without departing from the principles of the invention, except when a temporal or operational sequence is explicitly recited. Recitation in a claim to the effect that first a step is performed, and then several other steps are subsequently performed, shall be taken to mean that the first step is performed before any of the other steps, but the other steps can be performed in any suitable sequence, unless a sequence is further recited within the other steps. For example, claim elements that recite “Step (1), Step (2), Step (3), Step (4), and Step (5)” shall be construed to mean step (1) is carried out first, step E is carried out last, and steps (2), (3), and (4) can be carried out in any sequence between steps (1) and (5), and that the sequence still falls within the literal scope of the claimed process. A given step or sub-set of steps can also be repeated.

DETAILED DESCRIPTION

In the following description, the present invention is set forth as preferred examples. It will be apparent to those skilled in the art that modifications, including additions and/or substitutions may be made without departing from the scope and the spirit of the invention. Specific details may be omitted so as not to obscure the invention; however, the disclosure is written to enable one skilled in the art to practice the teachings herein without undue experimentation.

In optical microscopy of heterogeneous sample, both imaging resolution and depth are drastically degraded by the aberrations induced by the imaged sample and imaging system itself. The present invention provides methods and systems for accurately measuring and correcting the optical aberrations encountered in multiphoton microscopy. Specifically, a non-invasive method to accurately measure the aberrated electric-field (E-field) PSF of excitation laser inside the sample is provided, which enables identification and correction of the wavefront aberrations to achieve near-diffraction-limited resolution imaging.

Accordingly, an object of the invention is to identify the wavefront distortions of in vivo multiphoton microscopy, thereby providing AO corrections to system and/or sample induced aberrations. The principle is based on the measurement of the amplitude and phase of the E-field PSF. If the E-field PSF information is known, then the overall wavefront distortion is obtained by the Fourier transform of the E-field PSF and can be corrected by using a wavefront corrector. The optical system is arranged according to a conventional multiphoton microscopy with several important modifications: in addition to the excitation beam, a second optical beam has been provided, which is weaker than the excitation beam and modulated at a specific frequency f. The modulation could be either a phase modulation or an intensity modulation. The two beams are then combined and focused into the specimens to excite nonlinear optical signals (e.g. two- or three-photon excited fluorescence, second- or third-harmonic generation, coherent Raman scattering, et al), which is detected by a photodetector (e.g. photomultiplier tube, PMT)

Due to the large intensity difference between the strong and weak optical beams, the detected nonlinear optical signal is mainly contributed by the strong optical beam, which is constant with time. Meanwhile, this signal is modulated at frequency f and/or its harmonics due to the interference of these two beams. The amplitude and phase of the electric field of the weak optical beam at the focal point of the strong beam can be estimated by demodulating the detected signal using a lock-in amplifier. By scanning the weak optical beam spatially with respect to the strong optical beam and recording the electric field at a plurality of spatial positions, the profile of the E-field PSF can be re-constructed. This detected E-field PSF is the convolution of the true PSF of the weak optical beam with a δ-like function, where the δ-like function is a multiple order of the strong optical beam PSF (depending on the order of nonlinear interactions) multiplied by the object function (e.g., the distribution of signal source in the focal region). The phase of the Fourier transform of the detected E-field PSF is then added to a wavefront corrector (e.g. spatial light modulator or deformable mirrors). Moreover, the residual aberrations can be eliminated iteratively by repeatedly measuring the E-field PSF and updating the wavefront-shaping element.

In multiphoton microscope, the incident light is normally focused by an objective lens to excite the nonlinear optical signal at the location of interest. The spatial variation of the light intensity in the focal plane is defined as the intensity point-spread function (IPSF). In analogy, the complex-valued electric field at the focal plane is regarded as electric field point-spread function (EPSF, where IPSF=|EPSF|2), and the nonlinear optical signal is proportional to the multiple orders of the illumination intensity. As for two-photon microscopy, the fluorescence emission is quadratically dependent on the intensity of excitation laser, that is, the fourth power of the amplitude of the electric field (|EPSF|4). The present invention is to obtain the aberrated E-field PSF from the nonlinear optical signals, thereby identifying and optically correcting the wavefront distortions.

In order to obtain the aberrated E-field PSF from the nonlinear optical signals, the system arrangement is based on the basic layout of a regular laser scanning microscope, with several important modifications according to one embodiment of the present invention (FIG.1). That is, in addition to the scanning beam, a unscanned second optical beam is provided, which is not scanned, but parked within the field-of-view. Then, increasing the intensity of one of the two beams with respect to the other leads the strong beam to become more point-like (due to the nonlinear excitation), and by scanning one beam against the other, i.e. scanning a point-like probe across the weak beam's E-field PSF. More specifically, the method provided in the present invention is to probe the weak aberrated beam with the strong beam, and can be achieved by using either phase modulation or intensity modulation.

The analysis of operation principle takes two-photon excitation as an example and can be easily adapted to other nonlinear microscopies. As for the two-photon microscopy, the fluorescence signal generated by the interference of the scanning and stationary beams at a scanning coordinate x is given by the equation (1):
I(x)∝∫|Estat(x′)+Escan(x′−x)|4O(x′)dx′(1)
where Estatand Escanis the complex-valued E-field PSF of the stationary and scanning beam and O(x′) is the real-valued object function related to the distribution of fluorophores in the focal plane. To estimate the wavefront distortions, Escanis firstly required to be solved from the equation (1). To achieve this purpose, one of the embodiments of the present invention is to modulate the phase difference of the two beams. The detailed operation principle, optical implementation and experimental results will be shown as follows:

To solve the equation (1), the term Estat(x′)+Escan(x′−x)4is expanded accordingly. For simplicity, it is denoted as a=Estat(x′) and b=Escan(x′−x). Then the algebraic expansion formula for |a+b|4, with a and b being complex-valued, is shown in equation (2) as:
c=|a+b|4=|a|4+2|a|2ab*+2|a|2a*b+2|b|2b*a+2|b|2ba*+a2b*2+a*2b2+4|a|2|b|2+|b|4(2)
If the phase of the scanning beam b is modulated linearly with time, i.e., b=b0ejωt, where b0=|b|, j=√{square root over (−1)} and ω=2πf is the modulation frequency. Then the equation (2) becomes as follows:
c(t)=|a|4+2|a|2ab0*e−jωt+2|a|2a*b0ejωt+2|b0|2ab0*e−jωt+2|b0|2a*b0eωt+a2b0*2e−2ωt+a*2b02ej2ωt+4|a|2|b0|2+|b0|4(3)
If only the components of frequency co is considered, then equation (4) is adopted:
{c(t)}(ω)=2|a|2ab0*e−jωt+2|a|2a*b0ejωt+2|b0|2ab0*e−jωt+2|b0|2a*b0jωt(4)
Assuming |a|2>>|b0|2, the last two items of the equation (4) can be ignored to obtain equation (5):
{c(t)}(ω)=2|a|2ab0*e−jωt+2|a|2a*b0ejωt(5)
Setting the ratio between the two beams such that |Escan| is much smaller than |Estat| (e.g., |Escan|2/|Estat|2<0.1), the equation (1) with components of frequency ω reads,
{I(x)}(ω)∝∫(2|a|2ab0*e−jωt+2|a|2a*b0ejωt)O(x′)dx′(6)
This signal I(x) is then demodulated by performing a lock-in amplifier measurement at frequency a. The two orthogonal outputs X (i.e., in-phase) and Y (i.e., quadrature) of the lock-in amplifier are:
X=∫I(t)coswtdt=∫2|a|2(ab0*+a*b0)O(x′)dx′(7)
Y=∫I(t)sinwtdt=∫2|a|2j(−ab0*+a*b0)O(x′)dx′
Therefore, the |a|2ab0* can be solved as

Ef⁡(x)=X+jY=∫4⁢a2⁢a⁢b0*⁢O⁡(x′)⁢dx′∝∫Es⁢t⁢a⁢t⁡(x′)⁢|2⁢Es⁢t⁢a⁢t⁡(x′)⁢O⁢(x′)︸δ-like⁢⁢function·Es⁢c⁢a⁢n*⁡(x′-x)⁢dx′(8)

Considering the E-field PSF as an aberrated focus including a strong center and weak side lobes, the cubic term |Estat(x′)|2Estat(x′)O(x′) can be regarded as a highly peaked, δ-like function. The measured signal Ef(x) is thereby a convolution of the aberrated E-field PSF Escan* with the δ-like function. Therefore, Ef(x) can be used to approximate the true E-field PSF of the imaging system. Knowing the aberrated E-field PSF, the time reversal symmetry of optical propagation can be applied to correct for optical aberrations by phase conjugation. The desired correction pattern for the wavefront-shaping element is the two-dimensional Fourier transform of the measured E-field PSF showing in the equation (9).
{Ef(x)}={|Estat(x)|2Estat(x)O(x)}·{Escan(x)}*  (9)

The convolution kernel |Estat(x′)|2Estat(x′)O(x′) in the aforementioned analysis is not exactly a S function, therefore, the estimated E-field PSF will approximate, but not perfectly stand for the true E-field PSF. However, after applying the Fourier transform of the estimated E-field PSF on the wavefront shaper, the updated beam will nevertheless be closer to a diffraction-limited spot, which in turn means that the third power of its amplitude will more closely resemble a S function. Repeating the E-field PSF measurement process will obtain more accurate E-field PSF in each correction step, thereby eliminating residual aberrations in an iterative manner. It shows that the amplitude of the corrected E-field PSF will be taken to the third power after each correction step. Consecutive cubing of the corrected E-field PSF amplitude will finally turn any speckle pattern into a sharply peaked focus in a finite number of steps.

After 1stcorrection shown below, the electric field distribution at the focal plane will be,

E1⁢st⁢⁢corr⁡(x)=ℱ-1⁢{ℱ⁢{|Es⁢t⁢a⁢t⁡(x)⁢|2⁢Es⁢t⁢a⁢t⁡(x)⁢O⁡(x)}·ℱ⁢{Es⁢c⁢a⁢n⁡(x)}*︸1⁢st⁢⁢corr⁢⁢wavefront·ℱ⁢{EP⁢S⁢F⁡(x)}︸aberration}(10)
Because the Estatand Escanis equal to the E-field PSF EPSF(x) within a small lateral range defined by the the memory effect and the time reversal property of light propagation implies that{Escan}*·{EPSF(x)}=1, and the corrected E-field PSF in the 1stiteration is equal to,

E1⁢st⁢⁢corr⁡(x)=⁢ℱ-1⁢{ℱ⁢{|Es⁢t⁢a⁢t⁡(x)⁢|2⁢Es⁢t⁢a⁢t⁡(x)⁢O⁡(x)}}=|⁢Es⁢t⁢a⁢t⁡(x)⁢|2⁢Es⁢t⁢a⁢t⁡(x)⁢O⁡(x)=|⁢Es⁢t⁢a⁢t⁡(x)⁢|3⁢O⁡(x)⁢ejφs⁢t⁢a⁢t⁡(x)=EP⁢S⁢F⁡(x)⁢|3⁢O⁡(x)⁢ejφP⁢S⁢F⁡(x)(11)

Next, repeat this process and scan the corrected scanning beam (E1st corr) against the corrected stationary beam (E1st corr), and insert E1st corrinto the equation (1) to redefine Escanand Estat. Follow exactly the same analysis and update the 2ndcorrective wavefront to the wavefront-shaping element and the corrected PSF in the 2nditeration will be,

E2⁢nd⁢⁢corr⁡(x)=ℱ-1⁢{⁢{ℱ⁢E1⁢st⁢⁢corr⁡(x)2⁢E1⁢st⁢⁢corr⁡(x)⁢O⁡(x)}·ℱ⁢{E1⁢st⁢⁢corr⁡(x)}*︸2⁢nd⁢⁢corr⁢⁢wavefront·ℱ⁢{E1⁢st⁢⁢corr⁡(x)}︸residual⁢⁢wavefront}=E1⁢st⁢⁢corr⁡(x)3⁢O⁡(x)⁢ejφstat⁡(x)=(EPSF⁡(x)3⁢O⁡(x))3⁢O⁡(x)⁢ejφPSF⁡(x)(12)
It is shown that by iteratively probing the E-field PSF and correcting the excitation beam, Ecorrwill converge towards its 3rdpower. Applying this nonlinear factor on any aberrated focus will help any dominant mode to prevail against other weaker side lobes and finally result in a near-diffraction-limited E-field PSF. In sum, the amplitude of Ecorrwill be taken to the power of three in each correction step for two-photon excitation and will converge faster if higher order of nonlinear process is adopted. For example, in the case of three-photon excitation, the convergence rate will be proportional to the 5thpower.

It is clear from aforementioned analysis that the number of correction steps required to converge towards a diffraction-limited E-field PSF will strongly depend on the original shape of the scattered E-field PSF. The presence of a dominant mode will allow us to converge faster, compared to a case when multiple modes have comparable intensities. Moreover, as can be seen from the equation (11) and equation (12), if the sample under the optical microscopy is sparse and includes tiny point-like fluorescent structures (i.e. O(x) is a δ-like function), the convergence rate will be faster than that of the uniform fluorescence case (i.e. O(x) is a uniform function).

FIG.1illustrates the first type arrangement for the phase modulation method according to one embodiment of the present invention so as to identify and correct for optical aberrations as well as biological imaging. The system comprises an interferometer arrangement that has an input beam splitter (BS1), which separates an incoming laser beam (e.g. from a femtosecond laser) into a first and a second optical beam. The input beam splitter (BS1) may include, for example, but is not limited to a polarization beam splitter, a non-polarizing beam splitter, other beam splitting devices, or any combination thereof. The first interferometer branch includes a phase modulator (PMOD) and a beam scanner (GM1). The phase modulator may include, for example, but is not limited to a piezo scanner, an acousto-optic modulator, an electro-optic modulator, or any combination thereof that can provide linear modulation of the phase delay of the first optical beam with respect to the second optical beam from the full range of −π to π as shown inFIG.2A. The phase modulator may also be driven by a function generator (FGC).

Referring toFIG.2B, it showed the desired modulation is achieved by the driving signal for the phase modulator. The beam scanner may include, for example, but is not limited to galvo scanning mirrors, resonant scanning mirrors, acousto-optic deflectors (AOD), other Micro Electromechanical System (MEMS) mirrors or any combination thereof that can provide two-axis scanning of the excitation beam. The first or second interferometer branch includes an optical delay line (not shown inFIG.1) to ensure that the two beams are overlapped in time after combination such that interference can occur. The two beams are then combined by using another beam splitter (BS2), including for example, but not limited to a polarization beam splitter, a non-polarizing beam splitter, or any combination thereof. The combined optical beam may pass through a linear polarizer (P) to ensure that the polarization direction satisfies the requirement of the wavefront corrector (WFC). The wavefront corrector can modify the received excitation light before directing it to the laser scanning microscope. The laser scanning microscope is arranged according to a standard configuration, which comprises a pair of scanners (GMx and GMy) for scanning the excitation beam in the X-Y direction, followed by the objective lens (OL) for focusing light into sample and one or more detectors such as photomultiplier tube (PMT) for detecting the nonlinear optical signals generated by the interaction of excitation laser with the sample.

The output current of the detector is converted to voltage by a current amplifier (Amp) and the signal is then fed into a lock-in amplifier to extract the amplitude and phase of the excited optical signal at the modulation frequency or its harmonics. The synchronization signal from the function generator is used as the reference signal for lock-in detection. The output signals of the lock-in amplifier are digitized by a data acquisition device (DAQ) and are further processed in a computer. In one embodiment of the present invention, the system is configured in such a way that the first optical beam has much lower intensity than the second optical beam (e.g., 1:10). Therefore, the second optical beam may be regarded as a δ-like function and can be used to probe the E-field PSF of the first optical beam. More specifically, in the aberrations identification process, the paired scanner GMx and GMy are fixed to a specific position such that the second optical beam parks at a fixed region to generate nonlinear optical signal. Then the scanner (GM1) in the first interferometer branch is scanned over a small field of view such that the first optical beam can be spatially displaced with respect to the second optical beam for a plurality of positions. A set of measurement of the output signals of the lock-in amplifier is carried out during the scanning of the first optical beam and is further processed in the computer to derive the E-field PSF of the first optical beam.

A first arrangement includes a wavefront corrector to compensate for the aberration of both two beams. In the aberrations correction process, a phase pattern derived from the Fourier transform of the identified complex-valued E-field PSF is updated on the wavefront corrector to correct the aberrations. The wavefront correction device may include, for example, but is not limited to a liquid crystal spatial light modulator, a digital micro mirror device, a deformable mirror or any combination thereof.

In one embodiment, the wavefront correction device can be placed at the second interferometer branch to correct the strong beam only, similar to an open loop configuration.

In another embodiment, the wavefront correction device may be placed at the plane conjugated to the objective pupil and/or any turbid layer inside or above the sample.

For the sample-conjugated wavefront correction, a tunable liquid lens can be placed on the objective pupil-conjugated plane to adjust the focal plane of the microscope system while keeping the conjugation of the wavefront correction device with the turbid layer. In addition, the aberrations of the imaging instrument can be corrected with a pupil-conjugated wavefront correction device by using a pre-calibrated lookup table that record the system aberrations at each focal plane.

The measurement for identifying and correcting aberration may be performed several times by iteratively measuring the complex-valued E-field PSF and updating the correction pattern to further eliminate the residual aberrations. According to the aforementioned description, the convergence rate is proportional to the 3rdpower of the previous correction step for two-photon excitation. Preferably, three iterations have been performed to compensate for the optical aberrations induced by biological samples according to one embodiment of the present invention.

A second arrangement for the phase modulation method according to one embodiment of the present invention aims to slightly shift the frequency of one or both optical beams by using acoustic optical effect. The linear phase modulation of an optical beam from −π to π is equivalent to the frequency shift of the optical wave, therefore, an optical frequency shifter is used to provide accurate phase modulation. The optical frequency shift can be generated by using the 1stdiffraction order of the acoustic optical modulator (AOM) or acoustic optical frequency shifter (AOFS). However, because the diffraction efficiency of acoustic optical effect is low for small frequency shift (f<20 MHz), two AOFSs with acoustic frequency of fo+f/2 and fo−f/2 are applied in the arrangement, where fois the center frequency designed for the AOFS and f is the frequency of modulation. For example, in order to generate a 10 MHz phase modulation, two AOFSs with frequencies of 75 MHz and 85 MHz are applied, respectively. The two AOFSs can be both placed at the same interferometer branch or one AOFS at each branch according to one embodiment of the second type arrangement for the phase modulation in the present invention (AO1and AO2shown inFIG.3). Here, the reference signal for the lock-in amplifier is the difference frequency of the two driving frequencies for the two AOFSs.

In addition, one single AOM is also sufficient to generate two diffraction beams with a slight frequency difference according to another embodiment of the second type arrangement for the phase modulation in the present invention. For example, if the AOM is modulated with a cosine function with frequency f i.e. f(t)=cos(2πf), the acoustic wave to drive the AOM will be
facoustic(t)=cos(2πf)cos(2πfo)=½ cos(2π(fo−f))+½ cos(2π(fo+f))  (13)
where fois the center frequency of AOM. This will generate two acoustic wave with frequency fo−f and fo+f, so the 1stdiffraction order of the AOM will have two output beams corresponding to these two frequencies and the frequency difference of these two beams is 2f. Referring toFIG.4, the AOM is regarded as a phase modulator as well as a beam splitter. The reference signal for the lock-in amplifier is the second harmonic frequency of the voltage signal applied to the AOM driver. In addition, the excitation laser for nonlinear optical microscopy is normally a periodic ultrashort pulse beam, which is benefit to relax the requirement for the phase modulation or frequency shift. For example, if the repetition rate of the light source is fl, a desired phase modulation of frequency f can be generated by shifting the optical wave frequency by fl+f or fl−f. Considering the optical beam with optical wave frequency fo, after inducing a frequency shift of fl+f the wave function becomes

E⁡(t)=⁢E0⁢ej⁡[2⁢⁡(fo+f+fl)]·∑i=-∞+∞⁢⁢δ⁡(t-ifl)=E0⁢ej⁡[2⁢⁡(fo+f)]·∑i=-∞+∞⁢⁢δ⁡(t-ifl)·ej⁡[2⁢⁡(fl)]=E0⁢ej⁡[2⁢⁡(fo+f)]·∑i=-∞+∞⁢⁢δ⁡(t-ifl)(14)
This is equivalent to the linear phase modulation of frequency f. This phenomenon is similar to the aliasing effect in the sampling process when a high-frequency signal is sampled with a low sampling rate. In general, flis approximately 80 MHz for the commonly used two-photon excitation laser and a commercially available 70 or 90 MHz AOM or AOFS can be used to generate a 10 MHz phase modulation. Referring toFIG.5andFIG.6, it illustrates the optical system by using this approach according to other embodiments of the second arrangement for the phase modulation in the present invention. Here, the reference signal for the lock-in amplifier is the difference frequency of the laser pulse synchronization frequency and acoustic frequency of the AOM or AOFS.

Example 1

A feasibility study to image the fluorescent beads through a thinned mouse skull is performed to evaluate the efficacy based on the arrangement for the phase modulation method according to one embodiment of the present invention. Here, the 50-μm-thickness skull was used as the aberration source and 2-μm-diameter fluorescent beads were used to evaluate the performance of AO correction. The arrangement of the optical system is shown inFIG.4and the driving signal for AOM (center frequency fo=80 MHz) is modulated by a cosine signal of frequency f=10 MHz, which results in two 1storder diffraction beams with frequency shift of 70 MHz and 90 MHz respectively (equation (13)). The imaging results are shown inFIG.7A to7D. Without AO correction, the fluorescence image was severely blurred by the optical aberrations induced by the skull as shown inFIG.7A. In order to measure the wavefront distortions, the strong beam at the brightest region has been parked as indicated by the cross inFIG.7Aand then scan the weak beam in a small field of view of 20×20 μm2to capture the complex E-field PSF by using the method described above. Referring toFIG.7B, the diffraction-limited resolution has been recovered after three iterations. The corrective wavefront applied to the wavefront correction device is displayed inFIG.7C. These results demonstrate that the method for identifying and correcting optical aberrations in the present invention can effectively correct the aberrations and improve the fluorescence intensity by more than 30 folds (FIG.7D).

Example 2

Moreover, an in vivo two-photon imaging of cortical neurons and dendritic spine structures in a Thy1-GFP mouse through a thinned skull window of approximately 50-μm thickness has been conducted to illustrate the potential applications of the present invention for high-resolution biological imaging. The results are shown inFIG.7E to7H. As shown inFIG.7E, the GFP labelled dendrites at 200 μm below the skull were significantly distorted and the spines were invisible without AO correction. After AO correction, the spines can be clearly resolved as indicated by the arrow heads inFIG.7Fand the two-photon excited fluorescence signal is also increased tremendously. These results show the method in the present invention can be used to improve the performance of brain imaging and facilitate neuroscience research in living animals.

In addition to phase modulation, another method has been developed by modulating the intensity of the excitation beam and deriving Escanwith a two-phase scheme. Basing on the equation (2), if the intensity of the scanning beam b has been modulated in such a way that b=b0cos wt, where ω=2πf is the modulation frequency. Then the equation (2) becomes
c=|a+b|4=|a|4+2|a|2ab0*coswt+2|a|2a*b0coswt+2|b0|2b0*(coswt)3a+2|b0|2b0(coswt)3a*+a2b0*2(coswt)2+a*2b02(coswt)2+4|a|2|b0|2(coswt)2+|b|4(coswt)4(15)
The component of frequency ω is,
{c(t)}(ω)=2|a|2ab0*coswt+2|a|2a*b0coswt+2|b0|2b0*(coswt)3a+2|b0|2b0(coswt)3a*(16)
Assume |a|2>>|b0|2, the last two items can be ignored and get
{c(t)}(ω)=2|a|2ab0*coswt+2|a|2a*b0coswt(17)
Setting the ratio between the two beams such that |Escan| is much smaller than |Estat| (i.e., |Escan|2/|Estat|2<0.1), the equation (1) with component of frequency ω reads,
{I(x)}(ω)∝∫(2|a|2ab0*coswt+2|a|2a*b0coswt)O(x′)dx′(18)
Next, the signal I(x) is demodulated using a lock-in amplifier at frequency ω, the R (i.e. magnitude) output of the lock-in amplifier is recorded as,
R=∫I(t)coswtdt=∫|a|2(ab0*+a*b0)O(x′)dx′(19)
If the phase difference between two beams has been changed to 0 and π/2 (i.e. bπ/2=b0ejπ/2) respectively, then

R0=∫|a⁢|2⁢(a⁢b0*+a*⁢b0)⁢O⁡(x′)⁢d⁢⁢x′⁢⁢R2=∫|a⁢|2⁢(-a⁢b0*⁢j+a*⁢b0⁢j)⁢O⁡(x′)⁢d⁢⁢x′(20)
And |a|2ab0* can be isolated as

Ef⁡(x)=R0+j·R2=2⁢∫a2⁢a⁢b0*⁢O⁡(x′)⁢d⁢⁢x′∝∫|Es⁢t⁢a⁢t⁢(x′)⁢|2⁢Es⁢t⁢a⁢t⁡(x′)⁢O⁢(x′)︸δ-like⁢⁢function·Es⁢c⁢a⁢n*⁡(x′-x)⁢d⁢⁢x′(21)
Similarly, the correction pattern here is the Fourier transformation of Ef(x) and the convergence rate is the same as the aforementioned phase modulation method disclosed in the present invention. Although the analysis is based on a cosine modulation, other modulation functions such as on-off modulation are also applicable.

Referring toFIG.8, it illustrates the arrangement for the intensity modulation approach according to one embodiment of the present invention. The intensity of the first optical beam of the interferometer is modulated by an intensity modulator (IMOD), including for example, but not limited to acoustic optical modulator (AOM), electro optical intensity modulator (EOM), photoelastic modulator (PEM), optical chopper, or any combination thereof. The modulation signal can also be an on-off square wave or cosine wave function. The second interferometer branch includes a phase stepper (PS) to induce a 0 or π/2 phase delay by using a piezo stage or an electro-optical phase modulator. In general, the phase stepper may be configured in the first branch or outside the interferometer. Two measurements of the corresponding magnitude output of the lock-in amplifier are recorded for each spatial displacement of the first optical beam with respect to the second optical beam to obtain R0and Rπ/2. Then the complex-valued E-field PSF is derived by the equation (21). The procedure of aberration identification and correction is identical to the phase modulation approach described above.

Example 3

A performance evaluation is conducted to evaluate the efficacy based on the arrangement for the intensity modulation method according to one embodiment of the present invention. Here, the 50-μm-thickness skull was used as the aberration source and 2-μm-diameter fluorescent beads were used to evaluate the performance of AO correction. The arrangement of the optical system is shown inFIG.8, where a photoelastic modulator is configured to modulate the intensity of the weak beam and a MEMS mirror is used to alter the phase difference of the two beams. The imaging results are shown inFIGS.9A to9D. The fluorescence image was severely blurred by the optical aberrations induced by the skull without AO correction shown inFIG.9A. Referring toFIG.9B, the diffraction-limited resolution has been recovered by the identifying and correcting aberration methods after three iterations. The corrective wavefront applied to the wavefront correction device is displayed inFIG.9C. The fluorescence intensity is improved by more than 25 folds shown inFIG.9D. Similar to the phase modulation scheme, intensity modulation method in the present invention can also efficiently measure and correct the wavefront distortions and recover diffraction-limited imaging performance.

In summary, the present invention can be easily integrated into a standard multiphoton microscope for in vivo biological imaging. The performance of the present method is evidenced by both the in vitro and in vivo sample analyses and the results demonstrate that the present invention can effectively compensate for the aberrations and restore near-diffraction-limited imaging resolution.

The foregoing description of the present invention has been provided for the purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise forms disclosed. Many modifications and variations will be apparent to the practitioner skilled in the art.