In coherent antiStokes Raman scattering (CARS) microscopy reported until now, conventional microscope objectives are used, so that they are limited for introduction into a living body. Gradientindex(GRIN) rod lenses might be a solution for miniaturized microscope objectives for invivo CARS microscopy. However, due to the inherent large amount of chromatic aberration, GRIN rod lenses cannot be utilized for this purpose. CARS imaging catheter, composed of miniaturized microscope objective and fiber bundle, can be introduced into a living body for minimally invasive diagnosis. In order to design the catheter, we have to first investigate design requirements. And then, the optical design is processed with design strategies and intensive computing power to achieve the design requirements. We report the miniaturized objective lens system with diffractionlimited performance and completely corrected chromatic aberrations for an invivo CARS imaging catheter.
I. INTRODUCTION
In recent years, non linear optical (NLO) microscopy has been highlighted for visualizing the morphological details of living tissue which cannot be resolved by ultrasound or magnetic resonance imaging. NLO microscopy utilizes the signal from twophoton excitation fluorescence (TPEF)
[1

5]
,second harmonic generation (SHG)
[6

7]
, sum frequency generation (SFG)
[8

9]
, or coherent antiStokes Raman scattering (CARS)
[10

15]
, which can be combined with other techniques such as CARS microscopy with TPEF and SFG microscopy
[11
,
13]
. That’s why the combined(or multimodal) microscopy can visualize the subcellular complex structures which cannot be resolved well by single NLO microscopy. Different NLO imaging methods each have their distinctive advantages
[13]
: TPEF can be used to visualize proteins, ions with fluorescent labeling or specific autofluorescent structures; SHG and SFG are selective to noncentrosymmetric molecular assemblies such as collagen fibrils; and CARS is naturally sensitive to lipidenriched structures such as adipocytes. CARS microscopy is a more desirable tool compared with other microscopy,because it allows labelfree strong vibrational imaging. So, a multimodal NLO imaging system is preferably constructed based on CARS scattering microscope.
In order to generate CARS signal, we have to use two laser pulses with frequencies of ω
_{p}
for pump beam and ω
_{s}
for Stokes beam. So to speak, CARS signal is generated efficiently in a collinear excitation geometry with tight focusing and without time delay between pump (ω
_{p}
) and Stokes beam (ω
_{s}
). If the frequency difference, ω
_{p}
ω
_{s}
, matches molecularspecific frequency resonantly, a blueshifted and intense CARS signal is emitted at ωAS = 2ω
_{p}
ω
_{s}
. From this CARS signal, three dimensional labelfree images are obtained with submicrometer resolution
[10

15]
. In CARS microscopy reported until now, conventional microscope objectives are used, so that they are limited for introduction into a living body. Gradientindex (GRIN) lenses might be a solution to embody invivo catheter
[1

5
,
16

18]
. However, due to the inherent large amount of chromatic aberration, GRIN lenses cannot be utilized to focus two pulses with frequencies of ω
_{p}
and ω
_{s}
at a point. In order to insert CARS imaging catheter into a living body with minimal invasiveness,we need a miniaturized objective lens system like GRIN rod lenses without focus error and deviation for pump and Stokes beam. We will discuss the design of a miniaturized objective lens system joined with an imaging fiber bundle in the following sections.
II. THE STUDY FOR DESIGN REQUIRMENTS
 2.1. Tight focusing and wavelength
A typical schematic diagram of a laser scanning CARS(coherent antiStokes Raman scattering) microscope is presented in
Fig. 1
. As shown in
Fig. 1
, two pulses with different wavelengths are synchronously focused inside the sample, and then transformed to scattering light signals to visualize the morphological details such as forwarddetected CARS (FCARS), epidetected CARS (ECARS), twophoton excitation fluorescence (TPEF), and sum frequency generation(SFG) signals
[11]
.
From the viewpoint of optical design, two pulses for pump(ω
_{p}
) and Stokes (ω
_{s}
) beam have to be concentrated to a common focus with a chromatic aberrationfree condition.In fact, CARS signals are related to the induced thirdorder polarization
[19

20]
,
A typical schematic diagram of a laser scanningCARS (coherent antiStokes Raman scattering) microscopethat realizes nondestructive molecular imaging fromforwarddetected CARS (FCARS) epidetected CARS(ECARS) twophoton excitation fluorescence (TPEF) orsum frequency generation (SFG) signals produced by twosynchronized pulses with frequencies of ω_{p} and ω_{s}. D.Mdichroic mirror.
where E
_{p}
(ω
_{p}
) and Es(ω
_{s}
) are the electric fields for pump and Stokes beams and E
_{p}
(ω'
_{p}
) is the second interaction with the pump beam in CARS process. The delta function means that the integration satisfies the condition ofω
_{as}
= ω
_{p}
 ω
_{s}
+ ω'
_{p}
. And the intensity of CARS signal can be expressed as
From Eq. (1) and Eq. (2), it can be known that CARS signal is proportional to the second power of pump beam intensity(I
_{p}
^{2}
) times the first power of Stokes beam intensity (I
_{s}
).Therefore, CARS objective lens system should satisfy diffractionlimited performance and chromatic aberrationfree condition to focus two different wavelength pulses into a point.
Meanwhile, in order to decide wavelengths of pump and Stokes beam, we refer the research of Conor L. Evans and coworkers
[11]
. They have used 817 nm for pump beam and 1064 nm for Stokes beam to obtain a combined CARS and TPEF tissue image. Since we expect multimodal non linear optical (NLO) imaging based on CARS microscope,we set design wavelengths to 817 nm and 1064 nm.
 2.2. Numerical aperture and clear aperture
Numerical aperture is used to specify the lightgathering power of an optical system, which can be expressed at object space (fiber side) or image space (sample side)
[21]
.The amount of CARS signal captured by an optical system is proportional to the square of numerical aperture at sample side (NA
^{2}
_{S}
) and optical resolution is proportional to the first power of numerical aperture (NA
_{S}
).
Therefore, in order to decide on an appropriate NA
_{S}
, we refer to the research of H. Wang and coworkers
[17]
.They have used the conventional microscope objective with needlelike bill, namely the MicroProbe Objective(MPO) lens from Olympus Inc. with an outer diameter of 1.3 mm and 0.5 NA
_{S}
for water immersion. Additionally,from the Olympus website, we could also find another type of MPO lens with an outer diameter of 3.5 mm and 0.7 NA
_{S}
. Hence, we can decide design requirements for numerical aperture and clear aperture reasonably: NA
_{S}
will range from 0.5 to 0.7 for water immersion; since we will use a fiber bundle with an outer diameter of 0.7 mm and an acceptance numerical aperture of 0.35, the diameter of clear aperture (CA) will range from 0.7 mm to about 2.5 mm.
 2.3. Field of view and magnification
The previously mentioned MPO lens has field of view of 220 ㎛ at sample side (FOV
_{S}
) and an outer diameter of fiber bundle of 700 ㎛ (0.7 mm). And many of reported CARS papers show that image areas are about 75㎛ or 100 ㎛ wide
[13

14
,
17
,
20]
. Therefore, if we set FOV
_{S}
to 220 ㎛, magnification (M) will be 0.314 (=220/700). From geometrical optics, magnification (M) equals the ratio of numerical apertures (NA
_{F}
/NA
_{S}
), so that NAF will range from 0.16 to 0.22. Conclusively, we can confirm the consistency of design requirements that a fiber bundle with an acceptance sine angle of 0.35 makes the CARS signals pass into each of the fiber cores, namely 30,000 fiber cores .
 2.4. Telecentricity and working distance
In a telecentric system, one or both pupils are located at infinity and the principal ray enters or leaves the system parallel to the optical axis
[21]
. In order to maximize the amount of light returned from epidetected CARS into 30,000 fiber cores, we need to keep telecentricity at the fiber side.
The previous MPO lens has the working distance of 200㎛ which may not be optimized for CARS microscopy. In comparison, usual GRIN lenses have the working distance of tens of micrometers and will suffer less from sample scattering. Note that, in order to make an appropriate amount of scattering signals, working distance has to be determined carefully in epidetected CARS imaging. If we refer to the previous mentioned Evans’s study
[11]
in which CARS tissue imaging is seen up to the depth of 100 ㎛, we can set working distance to 100 ㎛.
 2.5. Total track length
CARS imaging catheter has to move freely without restriction inside blood vessels. In order to achieve this free movement, we need to minimize total track length(TTL) from object to image. From the research of K.Carlson and coworkers for invivo fiberoptic confocal reflectance microscope with an injectionmolded plastic miniature objective lens
[22]
, we can get information about TTL which is 17.5 mm or 20 mm. Therefore, it is reasonable to set the requirement of TTL between 10 mm and 20 mm.
III. OPTICAL DESIGN
To satisfy the design requirements of section II, we need initial lens data and strategies. As an initial data
[22

24]
for optimization, we present the usual microscope objective with magnification of 40 X and NA
_{S}
of 0.7 at
Fig. 2
. This lens system also has effective focal length(EFL) of 4 mm and an aperture diameter of 8 mm. Note that total track length (TTL) from object to image is 195 mm but track length (TL) from first lens surface to image surface is only 39 mm. Then, in order to satisfy the requirements of total track length (TTL) and clear aperture(CA), we have to minimize TTL first. That’s why the minimized TTL leads the minimized CA. Here, we use an idea which inserts one or two lenses between object and first lens surface, in order to reduce TTL of 195 mm down to onetenth or onetwentieth. That’s why the lens(es)can collimate the laser beam leaving from the fiber bundle,
Conventional microscope objective lens system withmagnification of 40 × and NA_{S} of 0.7. This system also haseffective focal length (EFL) of 4 mm and an aperturediameter of 8 mm. Note that total track length (TTL) fromobject to image is 195 mm but track length (TL) from first lenssurface to image surface is only 39 mm.
so that the lens system goes to the structure of minimized TTL. And we hope to move the aperture stop manually to different positions to fulfill the condition of telecentricity in the optimization process.
Because CARS (coherent antiStokes Raman scattering)is a nonlinear effect which is proportional to the square of induced thirdorder polarization, the signal is generated only at a focal point. Therefore, CARS objective lens system has to satisfy diffractionlimited performance and chromatic aberrationfree condition for pump beam of 817 nm and Stokes beam of 1064 nm. To reach the aberrationfree conditions, we will find various kinds of glass pairs through the optimization process for each doublet. And we make up a merit function (ψ) which is composed of the square sum of error functions (
φ_{i}
) representing the amount of aberration errors, as shown in Eq. (3) to Eq. (6)
[23]
,
where w
_{i}
, e
_{i}
, and t
_{i}
denote the weighting factors for the algorithm of damped least square, aberration error, and target value respectively (usually zero value for aberration error).F
_{i}
, W
_{i}
, R
_{i}
denote field, wavelength, and ray. Δx
_{i}
and Δy
_{i}
denote transverse ray aberration on image plane in which z is the optical axis. First, in order to minimize merit function(ψ), we focus on the correction of Seidel thirdorder aberrations,firstorder axial and lateral chromatic aberration, chromatic spherical aberration, and chromatic coma aberration
[25]
.That’s why we know from a design knowhow that Seidel thirdorder aberrations and firstorder chromatic aberrations are most important for diffractionlimited performance.And then we proceed to the optimization process with the correction of a general ray following Snell’s law. In general ray optimization, we permit the sample surface to have a curvature for correcting field curvature. That’s why a curved image surface is the most powerful and easiest way to correct an inherent field curvature aberration. This can be easily found at confocal reflectance microscopy or optical coherence tomography in which gradedindex (GRIN)lenses are used for microscope objective
[1

5]
. Codev of version 10.1 is used in optimization process.
As a result, we present the miniaturized microscopic optical system for CARS imaging catheter with fiber bundle in
Fig. 3
and summary table for design requirements in
TABLE 1
. We can confirm that all the predefined design requirements are satisfied. In section IV, we will discuss optical performance successively.
The miniaturized microscopic objective for CARSimaging catheter with fiber bundle in which total track length(TTL) is considerably reduced to 10.4 mm by the first lens.Compare with Fig. 2. And this objective lens system has EFLof 2.975 mm.
The summarized data for design requirments. WL denotes wavelength NAF and NAS denote numerical aperture at fiber side and sample side CA denotes the diameter of clear aperture FOVF and FOVS denote field of view at fiber side and sample side M denotes system magnification TEL denotes telecentricity which is defined by the angle between principal ray and optical axis WD denotes working distance and TTL denotes overall length from fiber to sample.
The summarized data for design requirments. WL denotes wavelength NAF and NAS denote numerical aperture at fiber side and sample side CA denotes the diameter of clear aperture FOVF and FOVS denote field of view at fiber side and sample side M denotes system magnification TEL denotes telecentricity which is defined by the angle between principal ray and optical axis WD denotes working distance and TTL denotes overall length from fiber to sample.
IV. OPTICAL PERFORMANCES AND DISCUSSION
We know that the most important thing is to get rid of monochromatic and chromatic aberrations, because coherent antiStokes Raman scattering (CARS) signals are generated only at a focal point and proportional to the third power of laser beam intensity, namely I
_{p}
^{2}
times I
_{s}
. The image quality is shown in
Fig. 4
for tangential (yz) and sagittal planes(xz). Before judgment of the image quality, we need to derive a key criterion for optical performance. The most frequently preferred criterion is the radius of the Airy disk which decides whether an optical system is diffractionlimited from what the transverse ray aberrations are inside the disk or outside. It is known that the radius of the Airy disk (R) can be calculated simply from Eq. (7)
[21]
,
where, λ denotes wavelength and NA
_{S}
denotes the numerical aperture at the sample side. For pump beam of 817 nm, R equals 0.71 ㎛. For Stokes beam of 1064 nm, R equals 0.93 ㎛.
Then, let’s look at the image quality for three different types of chart
[21
,
23]
. First, longitudinal spherical aberration(LSA) represents the difference of focus position according to onaxis rays leaving from the center point of the fiber bundle through different heights of entrance pupil. In LSA chart, 0.0 of horizontal axis means a reference imaging point, namely the best imaging point, on optical axis and 1.0 of vertical axis means the normalized height of entrance pupil. LSA chart shows that the maximum focus error is +0.27 ㎛ for pump beam of 817 nm and +0.28
The image quality charts which are longitudinalspherical aberration (LSA) astigmatic field curve (AFC) andpercent distortion. In charts 2.5E4 mm means 0.00025 mm or0.25 μm. T1 or S1 means the curved focus loci of tangentialplane (yz) or sagittal plane (xz) for Stokes beam of 1064 nmand T2 or S2 means for pump beam of 817 nm.
The explanation of the astigmatic field curve. Solidand dashed lines denote curved image surface and curvedfocus locus respectively. WI water immersion.
㎛ for Stokes beam of 1064 nm. And the maximum focus deviation between pump and Stokes beam is given to only 0.11 ㎛. These let us know that onaxis monochromatic and chromatic focuses are corrected perfectly. Second, astigmatic field curve (AFC) represents focus loci made by sequential offaxis points corresponding to the heights of the fiber bundle. Refer to
Fig. 5
. In
Fig. 4
, T1 or S1 means the curved focus loci of tangential plane (yz) or sagittal plane(xz) for Stokes beam of 1064 nm and T2 or S2 means for pump beam of 817 nm. AFC chart shows that the maximum focus deviation between pump and Stokes beam is given to be 0.19 ㎛ at the end of the image height. These also let us know that AFCs are corrected perfectly just like in LSA chart. Third, a data point on the distortion curve represents a position error between the center point of distributed rays on the curved image surface left from an object point and a reference point by magnification relationship.The important thing is to check the difference of the center point between pump and Stokes beams on the curved image surface. As shown in the distortion curve,there is no deviation, almost 0.005 % at the end of image height, which means the difference of 0.02 ㎛ from the definition of distortion
[21]
,
where, H'
_{p}
denotes the intersection height of principal ray through the center of the entrance pupil on sample plane and h' denotes the paraxial image height from the same object point. As a result, we can summarize that the designed optical system has the coincidence of focus points for pump and Stokes beam. In order to describe in detail, we will treat the performances further for such factors as spot diagram and point spread function.
Spot diagram and geometric encircled energy
[21
,
23]
are presented in
Fig. 6
and
Fig 7
. From
Fig. 6
with the scale bar of 0.2 ㎛, we can ensure the degree of coincidence for spot distributions between pump and Stokes beams. And from
Fig. 7
, we can see that the 100 % of encircled energy corresponds to diameter of circle of 0.22㎛ for onaxis and 0.60 ㎛ for the end of image height,
Spot diagram which shows plots of ray intersectionswith the curved image surface. The scale bar of 0.200E03mm means 0.2 μm. Red and blue marks denote Stokes andpump beam respectively.
Geometric encircled energy in which the circlediameter means a percentage of the enclosed energy. The6.0E04 mm means 0.6 μm.
namely 0.11 mm or 110 ㎛. This means perfect diffractionlimited performance if we remember that the radius of the Airy disk equals 0.71 ㎛ for pump beam of 817 nm and 0.93 ㎛ for Stokes beam of 1064 nm.
Fig. 8
and
Fig. 9
show point spread function (PSF) and diffraction encircled energy
[21
,
23]
. In
Fig. 8
, we can see aberrationfree PSFs with Strehl ratio of 0.999 for onaxis and 0.976 for the end of image height. And, unlike
Fig. 7
,we can see that the diameter of the circle approaches infinity for the 100 % of encircled energy in
Fig. 9
. That’s why PSF computes the characteristics of the image of point object (source) including the effects of diffraction. Since CARS intensity signals are treated at FWHM (full width at half maximum) which corresponds to the diameter of circle at 50 % encircled energy
[10

15
,
21]
, we can say that FWHM is to be 0.72 ㎛ for onaxis and 0.73 ㎛ for offaxis end. However, CARS signal is actually proportional to the third power of laser beam intensity, namely I
_{p}
^{2}
times I
_{s}
. Therefore, if we take FWHM for the cube of PSF, we can say that FWHM is to be submicrometer
Point spread function (PSF) which displays theaberrated diffraction structure with the polychromatic wavelengthsat a specified field point. Note that the size of eachfigure is 0.00127 by 0.00127 mm namely 1.27 by 1.27 μm.SR Strehl ratio.
Diffraction encircled energy in which the circlediameter means a percentage of the enclosed energy. 1.1E02mm denotes 11.0 μm.
resolution of 0.40 ㎛ for both onaxis and offaxis end.
V. CONCLUSION
In order to design the miniaturized microscope objective for coherent antiStokes Raman scattering (CARS) imaging catheter with a fiber bundle, we have first investigated the design requirements such as tight focusing, wavelength,numerical aperture, clear aperture, field of view, magnification,telecentricity, working distance, and overall length.From the study of these design parameters, we could build successful design strategies and proceed to optimization to satisfy diffractionlimited performance without focus deviations between pump and Stokes beam. Conclusively, we could present the miniaturized microscope objective for CARS imaging catheter with NA
_{S}
of 0.7, clear aperture of 1.8 mm, and TTL from object to image of 10.4 mm.
Additionally, we can say that the criteria of optical evaluation for the design of CARS optical system are established from this paper.
Acknowledgements
This work has been supported by 2010 Hannam University Research Grant.
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