Coherent transfer function of Fourier transformspectral interferometric coherent anti-Stokes

Raman scattering microscopy

Naoki Fukutake

Materials & Advanced Research Laboratory, Nikon Corporation, 10-1, Asamizodai 1-chome,Minami-ku, Sagamihara, Kanagawa 252-0328, Japan (fukutake.naoki@nikon.co.jp)

Received May 2, 2011; revised June 21, 2011; accepted June 21, 2011;posted June 22, 2011 (Doc. ID 146899); published July 27, 2011

We analyze the optical resolution of Fourier transform spectral interferometric-coherent anti-Stokes Ramanscattering microscopy, which extracts the complex amplitude of an image by using a spectral interferometric ef-fect. Image-formation formulas are presented that describe the properties of the image observed by the apparatus.The image-formation properties represented by the coherent transfer function are different depending on themode(transmission, reflection, etc.) of the microscopy. © 2011 Optical Society of America

OCIS codes: 180.6900, 180.4315, 110.0180, 110.2990.

1. INTRODUCTIONBiological tissues strongly scatter light in the visible range,making high-resolution deep imaging impossible for tradi-tional confocal microscopy. Nonlinear optical microscopy hasovercome this limitation by using near-IR excitation [1–10].The nonlinear optical microscopy is ideal for future in vivo

applications, because it is a noninvasive technique that doesnot damage the tissue. It provides higher quality images at agreater sample depth compared with traditional confocal mi-croscopy. Because of the nonlinear dependence on the exci-tation intensities, the signal is generated only at the focal spot,providing inherent optical sectioning capability. Furthermore,the three-dimensional (3D) optical resolution is enhanced byuse of a pinhole before a detector for a nonlinear confocalsystem [11]. A wide range of nonlinear optical microscopyhas been demonstrated, such as two-photon excited fluores-cence (TPF) [1–3], second-harmonic generation [4–6], third-harmonic generation [7,8], and stimulated-emission depletion[9,10] microscopy.

Coherent anti-Stokes Raman scattering (CARS) micro-scopy [12,13] has also been well-known as one of the non-linear imaging systems. The CARS microscopy provideschemically selective image contrast based on the intrinsic vi-brational modes of molecular species, avoiding the need forlabels. In addition, the CARS imaging systems also employsthe near-IR lasers to maximize imaging depth and minimizephotodamage to cells. The CARS microscopy has been apowerful tool for imaging deep within intact tissue, allowinga vast expansion of imaging potential for live cell research.Over the past few years, CARS microscopy has been demon-strated to have valuable applications to chemical imagingof biological systems. A variety of CARS microscopy havebeen developed and widely used to characterize complexmaterials [14–16].

In the CARS microscopy, two laser beams, the pump at fre-quency ωp and Stokes at frequency ωs, are temporally and spa-tially overlapped to generate anti-Stokes emissions at a

frequency of ωas ¼ 2ωp − ωs, such that the frequency differ-enceωp − ωs is tuned to match a particular Ramanactive vibra-tion frequency. The resonant CARS emission is several ordersof magnitude greater than that from spontaneous Raman scat-tering. The CARS amplitude is given by Aas ¼ χð3ÞA2

pAs, whereAp and As are the amplitudes of the pump and the Stokes field,respectively, and χð3Þ is the third-order nonlinear susceptibil-ity. The χð3Þ has vibrationally resonant and nonresonantcomponents. The CARS spectrum is different from its corre-sponding spontaneous Raman spectrum due to the nonreso-nant component, which complicates the spectral assignmentand causes difficulties in image interpretation. Since the imag-inary part of the resonant component corresponds to thespontaneous Raman spectrum, the nonresonant componentneeds to be suppressed. The interference effect between thetwo components permits the extraction of the resonant com-ponent by using the nonresonant component as a localoscillator. This technique is referred to as a Fourier trans-form spectral interferometric CARS microscopy (FTSI-CARSmicroscopy) [17].

In this paper, we address the mechanism for improving theoptical resolution of the FTSI-CARS microscopy after review-ing fundamental concepts of the nonlinear microscopy. Weshow that the optical resolution of the FTSI-CARS microscopycan be evaluated by using the coherent transfer function(CTF). We calculate the CTFs of the FTSI-CARS microscopyfor transmission mode, reflection mode, and three types (typeA, B, and C) of 4Pi modes.

2. IMAGE-FORMATION FORMULAS OFNONLINEAR OPTICAL MICROSCOPYA nonlinear optical microscopy system is shown schemati-cally in Fig. 1, in which the coordinate systems are given.The excitation beam (e.g., a laser pulse train of a single wave-length or laser pulse trains of multiple wavelengths super-imposed in time and space) is focused by an excitation systemonto a thick object, such as a biological sample, scanned in

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three dimensions. Although we assume in what follows thatthe sample-stage scanning is conducted instead of the laserscanning, it does not influence the optical resolution. Theemission from the polarization in the localized region is fo-cused by a collection system onto a pinhole immediately be-fore a detector. For simplicity, the first Born approximation isapplied to the interaction between the object and both the ex-citation beam and the emission. In this approximation, multi-ple scattering and depletion of the beam are neglected. Here,both the excitation and the collection systems are assumedto be rotationally symmetrical systems with no aberration.

Although most nonlinear processes can be described byeither the coherent or the incoherent phenomena, we unitedthe image-formation formulas of the coherent and the incoher-ent systems into a single expression for convenience of com-parison between the optical resolutions in the coherent andthe incoherent systems. The observed intensity Iðx0Þ of thesystem is a function of the sample-stage displacement x0 ¼ðx0; y0; z0Þ and is given by [11]

Iðx0Þ ¼ ∭ γðx1 − x2ÞχðiÞðx1 þ x0ÞχðiÞ�ðx2 þ x0Þuexðx1Þu�exðx2Þ

× ucolðxa − x1Þu�colðxa − x2ÞaðxaÞd3x1d3x2d3xa: ð1Þ

Here, γðxÞ represents the coherence properties of emission,which is a 3D function that acts as the degree of coherence[18] for the emission, χðiÞðxÞ corresponds to the object ampli-tude, ucolðxaÞ is the amplitude point spread function (aPSF)for the emission formed by the collection system, and aðxaÞis the 3D real function describing the size of the pinhole. InEq. (1), uexðxÞ ¼

Qij¼1 EjðxÞ, where EjðxÞ represents either

the electric field EjðxÞ of the jth excitation beam with awavelength of λj or the electric field complex conjugate E�

j ðxÞ.We take EjðxÞ to be EjðxÞ for absorption and E�

j ðxÞ for stimu-lated emission in the radiation-matter interaction. Note thatEjðxÞ is expressed by the aPSF of the jth excitation beamformed by the excitation system. The scalar wave approxima-tion is applied to the electric field to facilitate understandingof the optical resolution through the CTF.

Equation (1) reduces to that of either the coherent or theincoherent systems when γðxÞ is defined in the two followingcases. If γðx1 − x2Þ ¼ 1, then Iðx0Þ becomes the intensity dis-tribution for the coherent system, whereas if γðx1 − x2Þ ¼δðx1 − x2Þ, then Iðx0Þ becomes the intensity distribution forthe incoherent system, where δðxÞ is a 3D delta function. Inthe coherent system such as CARS microscopy, the complex

amplitude of the emission is proportional to χðiÞðxÞQij¼1 EjðxÞ.

In this case χðiÞðxÞ is the ith order nonlinear susceptibility. Theobserved intensity of the coherent system ICðx0Þ reduces to

ICðx0Þ ¼Z ����

ZχðiÞðxþ x0ÞuexðxÞucolðxa − xÞd3x

����2aðxaÞd3xa:

ð2Þ

In the incoherent system such as TPF microscopy, theobserved intensity IICðx0Þ becomes

IICðx0Þ ¼ZZ

jχðiÞðxþ x0Þj2juexðxÞj2jucolðxa − xÞj2d3xaðxaÞd3xa:ð3Þ

Note, although χðiÞðxÞ in Eq. (3) is not the nonlinear suscep-tibility, we use the identical symbol χðiÞðxÞ to that of thecoherent system as the object amplitude distribution also inthe incoherent system.

3. OPTICAL RESOLUTION OF NONLINEAROPTICAL MICROSCOPYWe move into the Fourier domain (spatial-frequency domain),which is informative when dealing with the optical resolution.The 3D Fourier transform of Iðx0Þ is

ZIðx0Þ exp½−i2πf 0 · x0�d3x0 ¼

ZTðf ; f − f 0ÞX ðiÞðf Þ

X ðiÞ�ðf − f 0Þd3f ; ð4Þ

with the transmission cross coefficient (TCC) given by

Tðf ; f − f 0Þ ¼ ∭ Γðf 0ÞAðf 2 − f 1ÞUcolðf 1ÞU�colðf 2Þ

× Uexðf 1 − f 0 − f ÞU�exðf 2 − f 0 − f þ f 0Þ

× d3f 0d3f 1d3f 2; ð5Þ

where X ðiÞðf Þ, Γðf Þ, Aðf Þ, Ucolðf Þ, and Uexðf Þ denote the 3DFourier transforms of χðiÞðxÞ, γðxÞ, aðxaÞ, ucolðxÞ, and uexðxÞ,respectively. The TCC describes the imaging properties ofthe system and provides the information on the optical reso-lution [18–20]. Equation (4) indicates that the image is formedby adding the interference of pairs of waves, and the interac-tion between each pair is weighed by the factor Tðf ; f − f 0Þ.The Fourier components of f and f − f 0 in the object contri-bute to the generation of the Fourier components of f 0 in theimage. Note that Ucolðf Þ is equivalent to the wave vector dis-tribution (3D pupil function) of the collection system for theemission, and

Uexðf Þ ¼�½pl ⊗ … ⊗ Pm�⋆½P�

n ⊗ … ⊗ P�o��ðf Þ; ð6Þ

where Pjðf Þ is the Fourier transform of EjðxÞ and is the 3Dpupil function for the jth excitation beam. The symbols ⊗and ⋆ denote a 3D convolution in the spatial-frequencydomain

ðg1 ⊗ g2Þðf Þ ¼Z

g1ðf 0Þg2ðf − f 0Þd3f 0; ð7Þ

Fig. 1. (Color online) Schematic of a nonlinear optical microscopysystem.

1690 J. Opt. Soc. Am. A / Vol. 28, No. 8 / August 2011 N. Fukutake

and a 3D cross correlation in the spatial-frequency domain

ðg1⋆g2Þðf Þ ¼Z

g1ðf 0Þg2ðf 0 − f Þd3f 0; ð8Þ

respectively.Equations (1) and (5) can be applied to a 4Pi mode as well

as the transmission mode and the reflection mode. Threetypes A, B, and C exist in the 4Pi mode [21]. In type A, thesample is coherently illuminated from both �z directionsby two opposing objective lenses, and the emission is col-lected from the þz direction and focused onto the confocalpinhole. In type B, the sample is illuminated from the þz di-rection, and the emission is collected from both directions andcoherently combined at the detector. In type C, the sample iscoherently illuminated from both directions, and the emissionis collected from both directions and coherently combined atthe detector.

We consider the 3D pupil functions of the excitation beamand the emission. The 3D pupil function is shaped like a partof a spherical shell. The Fourier transform of the 3D pupilfunction is equivalent to the aPSF. The aPSF of the jth excita-tion beam EjðxÞ is formed in the sample, and the aPSF of theemission ucolðxaÞ is formed on the pinhole. The example of the3D pupil function is depicted in Fig. 2. The 3D pupil functionUþðf Þ in Fig. 2 corresponds to the beam with the wavelengthof λ and the NA of n sin θ, propagating in the direction of þz,where n is the refractive index of the sample. The 3D pupilfunction U−ðf Þ corresponds to the beam propagating in thedirection of −z. In the transmission mode, the 3D pupil func-tion for the excitation beam corresponds to Uþðf Þ, and the 3Dpupil function for the emission also corresponds to Uþðf Þ, butthe wavelengths of the excitation beam and the emission maybe different. Likewise, in the reflection mode, the 3D pupilfunction for the excitation beam corresponds to Uþðf Þ,whereas that of the emission corresponds toU−ðf Þ. In the typeA 4Pi mode, the 3D pupil function for the excitation beamcorresponds to Uþðf Þ þ U−ðf Þ and that of the emission corre-sponds to Uþðf Þ. In the type B 4Pi mode, the excitation beamcorresponds to Uþðf Þ, and the emission corresponds toUþðf Þ þ U−ðf Þ. In the type C 4Pi mode, the excitation beamcorresponds to Uþðf Þ þ U−ðf Þ, and the emission also corre-sponds to Uþðf Þ þ U−ðf Þ.

The coherence properties of emission γðxÞ affect the opticalresolution of the system, which means that the coherent sys-tem and the incoherent system would show the different im-age-formation properties, even if other conditions, such as thewavelengths of the excitation beams and the emission or theorder of nonlinearity, were equal. For example, the transmis-sion and the reflection modes in the coherent system exhibitthe different optical resolution as described in Section 4,whereas the transmission and the reflection modes in theincoherent system show the identical optical resolution forthe following reason. In the incoherent system, since theemission propagates isotropically in all directions irrespectiveof the excitation-beam direction, unlike the coherent systemwhich has a connection with phase matching, the same imagesare formed by both the transmission mode and the reflectionmode. In other words, the optical transfer functions (OTFs) ofthe transmission and the reflection modes in the incoherentsystem are identical. This is evidenced by the fact that theOTF of the incoherent system is expressed by [11]

OTFðf 0Þ ∝Z

Að−f ÞfðUcol⋆U�colÞðf ÞgfðUex⋆U�

exÞðf − f 0Þgd3f ;ð9Þ

which remains unchanged irrespective of which mode (thetransmission mode or the reflection mode) is employed.

4. COHERENT TRANSFER FUNCTION OFFTSI-CARS MICROSCOPYHere, we focus on the coherent systems such as CARSmicroscopy. In the confocal system where aðxaÞ ¼ δðxaÞ,Eq. (2) becomes

IconfC ðx0Þ ¼ jCðx0Þj2 ≡����Z

χðiÞðxþ x0ÞuexðxÞucolð−xÞd3x����2; ð10Þ

and substituting Γðf 0Þ ¼ δðf 0Þ and Aðf 2 − f 1Þ ¼ 1 into Eq. (5)gives

TconfC ðf ; f − f 0Þ ¼ CTFðf ÞCTF�ðf − f 0Þ; ð11Þ

where CTFðf 0Þ ¼ ðUcol⋆UexÞðf 0Þ is the CTF. The Fouriertransform of the image amplitude Cðx0Þ is expressed as

ZCðx0Þ exp½−i2πf 0 · x0�d3x0 ¼ XðiÞðf 0ÞCTFðf 0Þ; ð12Þ

which is the product of the CTF and the Fourier transform ofthe object amplitude. In the confocal system (aðxaÞ ¼ δðxaÞ),the generation of the zeroth-order (nondiffracted) emissionfrom the dc component of χðiÞðxÞ is governed by the phase-matching condition CTFð0Þ ¼ R

Ucolðf ÞUexðf Þdf ≠ 0. Whenthere is overlap between Uexðf Þ and Ucolðf Þ, the zeroth-orderemission propagates in the direction of f that satisfies thecondition of Ucolðf ÞUexðf Þ ≠ 0. The microscopy systems thatsatisfy the phase-matching condition can observe the zeroth-order emission when the nonlinear susceptibility of the sam-ple is homogeneous. The image produced by the microscopy

Fig. 2. Cross section of 3D pupil function in a spatial-frequencydomain. The configuration is shaped like a partial spherical shell witha radius of n=λ. Uþðf Þ corresponds to the beam propagating in thedirection of þz, and U−ðf Þ corresponds to the beam propagating inthe direction of −z.

N. Fukutake Vol. 28, No. 8 / August 2011 / J. Opt. Soc. Am. A 1691

system that does not satisfy the phase-matching condition,such as the reflection CARS microscopy which we will referto later, becomes unnatural because of the absence of the dccomponent.

The FTSI-CARS microscopy is one of the interferometrytechniques that acquire the image originating only from theresonant component. Note that the CARS emission is com-posed of the resonant component and the nonresonant com-ponent. Figure 3 shows the example of the schematic for the4Pi (type C) FTSI-CARS microscopy. In Fig. 3, ωp and ωs arethe angular frequencies of the excitation beams (the pumpbeam and the Stokes beam), respectively, and ωas is theangular frequency of the CARS emission. The Stokes beamhas a broad spectrum, whereas the pump beam is assumedto be quasi-monochromatic, resulting in a broad CARS spec-trum with high angular frequency resolution. In an experimen-tal setup, for example, a portion of the output from the lasersource is used for a seed laser to generate a supercontinuumin a photonic crystal fiber. The fundamental of the lasersource and the supercontinuum are used as the pump beamand the Stokes beam, respectively. Utilizing the fact that theresonant component of the CARS emission is a complex quan-tity while the nonresonant component is real, the FTSI-CARSmicroscopy extracts the vibrational spectrum equivalent tothe spontaneous Raman scattering, as described below. Therequirement is the condition that the nonresonant component,which plays a role as the local oscillator in the spectral inter-ferometry, be much larger than the resonant component. Thiscondition is satisfied in typical experiments. The CARS inten-sity observed through the confocal FTSI-CARS microscopy isa function of the sample-stage displacement x0 and the angularfrequency of the Stokes beam ωs and is given by

IconfC ðx0;ωsÞ ¼����Z

fχð3ÞNRðωsÞ þ χð3ÞR ðxþ x0;ωsÞg

× uexðx;ωsÞucolð−x;ωsÞd3x����2

≈ jCNRðωsÞj2 þ 2RefCRðx0;ωsÞgCNRðωsÞ; ð13Þ

with

CNRðωsÞ ¼Z

χð3ÞNRðωsÞuexðx;ωsÞucolð−x;ωsÞd3x

CRðx0;ωsÞ ¼Z

χð3ÞR ðxþ x0;ωsÞuexðx;ωsÞucolð−x;ωsÞd3x; ð14Þ

where χð3ÞR and χð3ÞNR are the vibrationally resonant and nonre-sonant third-order susceptibilities, respectively, and we as-sumed that the χð3ÞNRðωsÞ is spatially homogeneous. The firstterm in Eq. (13) stems from the nonresonant componentand is typically a smooth function of ωs. The second termin Eq. (13) is expressed as the real part of the resonantcomponent multiplied by the real smooth function of ωs. Weapproximate χð3ÞNRðωsÞ and

Ruexðx;ωsÞucolð−x;ωsÞd3x by con-

stants, which means that CNRðωsÞ becomes a real constantCNR. Therefore, the observed intensity IconfC ðx0;ωsÞ at each po-sition x0 is represented as the real part of the resonant spec-trum on top of a flat background

IconfC ðx0;ωsÞ ¼ C2NR þ 2CNRRefCRðx0;ωsÞg: ð15Þ

The Fourier transform of IconfC ðx0;ωsÞ from the angular-frequency domain to the time domain is composed of adelta function δðx0; tÞ at the time zero and the symmetricamplitudes ARðx0; tÞ þ A�

Rðx0;−tÞ with respect to the time zero:

FTðIconfC ðx0;ωsÞÞ ≈ C2NRδðx0; tÞ þ CNRfARðx0; tÞ þ A�

Rðx0;−tÞg;ð16Þ

with

ARðx0; tÞ ¼ FT½CRðx0;ωsÞ�: ð17Þ

Here, FT denotes the Fourier transform from the angular-frequency domain to the time domain.

The FTSI method converts the real part of the complexspectral amplitude RefCRðx0;ωsÞg to the original complexspectral amplitude CRðx0;ωsÞ. The imaginary part of the com-plex spectral amplitude ImfCRðx0;ωsÞg correlates with thespontaneous Raman spectrum, which we desire to obtain. Be-cause of causality, there should be no CARS emission beforethe excitation takes place at the time zero. Namely, the timeevolution of the resonant CARS emission, ARðx0; tÞ, has zeroamplitude before the time zero. If we extract the amplitudesafter the time zero (t > 0) from all the amplitudes in the timedomain, and inverse Fourier transform it, then we recover theoriginal complex resonant component

CRðx0;ωsÞ ∝ FT−1½H0ðtÞFT½IconfC ðx0;ωsÞ��: ð18Þ

Here, FT−1 stands for the inverse Fourier transform fromthe time domain to the angular-frequency domain, andH0ðtÞ ¼ f0ðt ≤ 0Þ; 1ðt > 0Þg is the Heaviside step function.

Fig. 3. Example of 4Pi (type C) confocal FTSI-CARS microscopy.A coherent supercontinuum generated by injecting pump pulses intoa photonic crystal fiber is used as the Stokes beam. A part of the beamfrom a laser source is used as the pump beam.

1692 J. Opt. Soc. Am. A / Vol. 28, No. 8 / August 2011 N. Fukutake

The Fourier transform of CRðx0;ωsÞ from the real-spacedomain to the spatial-frequency domain is represented as

ZCRðx0;ωsÞ exp½−i2πf 0 · x0�d3x0 ¼ Xð3Þ

R ðf 0;ωsÞCTFðf 0;ωsÞ;ð19Þ

where X ð3ÞR ðf 0;ωsÞ is the Fourier transform of χð3ÞR ðx0;ωsÞ [see

Eq. (12)]. In the case of the CARS microscopy, CTFðf 0;ωsÞ ¼ðUcol⋆UexÞðf 0;ωsÞ is equivalent to (see Eq. (6))

CTFðf 0;ωsÞ ¼ fUcol⋆ð½Pp ⊗ Pp�⋆P�sÞgðf 0;ωsÞ; ð20Þ

where Ppðf 0Þ is the 3D pupil function for the pump beam andPsðf 0;ωsÞ is the 3D pupil function for the Stokes beam. TheCTF changes depending on which wavelength in the resonantCARS spectrum is used when the image is produced. Note thatCRðx0;ωsÞ provides the complex amplitude of the image, andits imaginary part ImfCRðx0;ωsÞg is the desired image corre-sponding to the spontaneous Raman image.

Figure 4 shows the CTFs of the transmission mode, the re-flection mode, and three types (type A, B, and C) of the 4Pimodes in FTSI-CARS microscopy. For the purpose of graspingthe nature of the CTFs for the five modes, we approximated

the wavelengths of the pump beam, the Stokes beam, and theCARS emission by the identical wavelength λ. One can see thatthe five modes exhibit the different optical resolutions fromone another. The CTF of the transmission mode lies in the cen-ter of the spatial-frequency domain, while the CTF of the re-flection mode is shifted in the direction opposite to the wavevector of the excitation beam. The reflection mode detectsonly the emission traveling in the direction opposite to the ex-citation-beam propagation (reflection side), resulting in theshifted CTF in the direction of −f 0z. Since the zeroth-orderemission generated by the dc component of the object ampli-tude χð3ÞR ðx0;ωsÞ travels to the transmission side (in the samedirection as the excitation-beam propagation), the reflectionmode cannot detect the zeroth-order emission, resulting inthe lack of the dc component in the image amplitude. In thereflection mode, however, a portion of the higher-order emis-sion generated by the nonuniformity of the object amplitudemay travel to the reflection side. For example, the dots distrib-uted in the sample or the interfaces perpendicular to the op-tical axis generate the emission reaching to the detector in thereflection mode. It is because the dots and the interfaces con-tain the Fourier components (spatial-frequency components)represented as the CTF of the reflection mode in Fig. 4(b). Ifthe Fourier transform of the object amplitude lies over thenonzero region of the CTF, the image can be formed. In gen-eral, when the object has a rough structure, the transmissionmode is advantageous in comparison with the reflectionmode. In this case, the CARS-emission irradiance in the trans-mission mode is higher than that of the reflection mode. Incontrast, when the object has a fine structure, the reflectionmode also can form the image, resolving the Fourier compo-nents that lie over the nonzero region of the CTF. Although thereflection mode suffers from the disadvantage of not resolvingthe smooth plane whose normal is perpendicular to the opticalaxis as is the case with the dc component, or in other words,the image produced by the reflection mode is constrained bythe shifted CTF to a certain extent, it is of advantage in in vivo

experiments.The 4Pi modes improve the optical resolution along the op-

tical axis, as a result of the interference of the excitationbeams from both sides in type A, the interference of the CARSemissions propagating to both sides in type B, or both effectsmentioned above in type C. In type A, the interference be-tween the two aPSFs of the excitation beams from the bothsides produces the standing wave in the sample, resulting in anarrower aPSF in the z direction. This means improvement ofthe optical resolution along the optical axis. Type A also con-tains the following phenomenon: the pump beam and theStokes beam propagating from the opposite directions gener-ate the CARS emission, which travels in the same direction asthe pump-beam propagation. In type B, the interference of theCARS emissions radiating to the transmission side and the re-flection side result in the combined CTF of the transmissionmode and the reflection mode, which is exactly expressed asthe sum of the CTFs of both modes. In type C, the most super-ior optical resolution is achieved by the combination of bothadvantages of types A and B. The area of the nonzero region ofeach island in the CTF (e.g., type C of 4Pi mode has fiveislands) increases at the identical location, with the increasingNAs of the microscope objectives.

Fig. 4. Cross sections of the coherent transfer functions for f 0y ¼ 0.(a) Transmission mode. (b) Reflection mode. (c) 4Pi mode (type A).(d) 4Pi mode (type B). (e) 4Pi mode (type C). The wavelengths ofpump, Stokes, and CARS emission are assumed to be approximatelyequivalent to λ, and the NAs of the excitation system and the collec-tion system are 0.9 (n ¼ 1).

N. Fukutake Vol. 28, No. 8 / August 2011 / J. Opt. Soc. Am. A 1693

In the reflection mode, if χð3ÞNRðxÞ were spatially homoge-neous, the emission of the nonresonant component, whichis supposed to be used as the local oscillator, would travelonly in the same direction as the excitation beam becauseof the phase-matching condition. In practice, however, sinceχð3ÞNRðxÞ has a distribution to a certain degree, the emission ofthe nonresonant component travels also in the direction op-posite to the excitation-beam propagation. Hence, one can ex-pect that the reflection mode of the FTSI-CARS microscopyfunctions as well as the transmission mode and the 4Pi modes.Interestingly, the image amplitudes obtained through any ofthe five modes (transmission, reflection, and three 4Pi modes)can be deconvoluted with the CTFs by using Eq. (19), which isthe simple relation between the Fourier transforms of the im-age amplitude and the object amplitude. The CTF of generalconfocal CARS microscopy also can be evaluated as well asthat of the confocal FTSI-CARS microscopy. In other words,the optical resolution of the general CARS microscopy, whichobtains jCNR þ CRðx0; wsÞj2 rather than CRðx0; wsÞ as the im-age, can also be discussed by using the CTF. The generalCARS microscopy cannot always extract only the resonantcomponent, and it cannot always deconvolute the image withthe CTF. If the χð3ÞR ðxÞ image is acquired by using some tools,such as the FTSI method, the image can be deconvoluted withthe CTF.

5. CONCLUSIONWe have dealt with the optical resolution of the confocal FTSI-CARS microscopy by using the CTF. This apparatus extractsthe complex amplitude of the CARS spectrum for each voxel.The complex amplitude of the image produced by the arbi-trary wavelength of the CARS spectrum has the simple rela-tion with the object amplitude through the CTF, which allowsthe deconvolution for the image amplitude. We have evaluatedthe optical resolution of the FTSI-CARS microscopy for trans-mission mode, reflection mode, and three types (type A, B,and C) of 4Pi modes. The transmission mode and the reflec-tion mode show the different optical resolution unlike the in-coherent system, and the 4Pi modes improve the opticalresolution along the optical axis.

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