Abstract
Polarization sensitive second harmonic generation (PSHG) provides additional information in intensity only SHG imaging. In particular, it offers the SHG effective orientation of the implicated SHG active structures. Assuming that those structures possess cylindrical symmetry, the supplementary contrast is based on the ratio of two non-vanishing, independent elements of the χ 2 tensor. This ratio is experimentally extracted by fitting pixel by pixel a theoretical model to the PSHG images and by finding the maximum frequency value or the mean of the consequent pixels' histogram. In the present study we show that the above χ 2 elements' ratio critically depends on the tilted-off the plane SHG active structures. We performed PSHG in different z-planes of a starch granule (presenting radially oriented amylopectin molecules, the SHG source in starch) and we found different pick values of the χ 2 elements' ratio histogram for each plane. By assuming a fixed value for the χ 2 elements' ratio, we present here a generalized three dimensional (3D) model that determines the 3D orientation of the SHG active structures.
INTRODUCTION
During the recent previous years advantages in instrumentation as well as in data acquisition methodology and analysis have facilitated second harmonic generation (SHG) microscopy as one of the most promising imaging modalities for future clinical applications. Among others, the newest technological evolvements include quantum dot lasers microendoscopes and deformable mirrors. SHG imaging is deservingly spurred by the scientific community since it enjoys inherent features that make it attractive for biomedical imaging. It is considered one of the minimally invasive deep penetration laser scanning microscopy techniques, because it usually uses infrared fs pulses to excite inside a submicron active volume endogenous SHG source molecules found in the majority of tissues. Collagen, muscle and microtubule based biological structures are the most common SHG contrast agents found in plethora in all mammals.
INTRODUCTION
Except the technological improvements in SHG imaging, parallel progress concerns the SHG signal interpretation from the biological samples and the maximum acquisition of the available information from the non-linear light matter interaction. Towards the elucidation of the morphological features of the SHG images, FFT 1 and pattern recognition based approaches 2, 3 have been proposed lately.
INTRODUCTION
Another approach for extracting additional information in SHG has been presented early in the history of SHG from biological samples and it was based on the modulation of the produced SHG signal from tendon collagen after rotating the incoming linear polarization 4 . This signal modulation was fitted into a theoretical model that was taken from the nonlinear optics of nonlinear crystals. Cylindrical symmetry was assumed and the ratio of two non-vanishing, independent elements of the χ 2 tensor has been calculated. This was the first demonstration of polarization sensitive SHG (PSHG) in a biological sample in 1986. Since then, this ratio has been calculated several times for collagen ~2 4-7 and muscle ~0.5 [5] [6] [7] [8] , while just recently it was presented for microtubules to be ~4 9 .
INTRODUCTION
*williamsbradley@example.net; phone 303.525.0448; fax 303.525.0448; icfo.es Next significant evolvement in PSHG imaging was the extraction of the so called helical pitch angle of the SHG source structures 5 . This was accomplished combining the PSHG χ 2 tensor elements' ratio and surface SHG physics. Among others, it was assumed that the SHG source is a helix with geometrical values of radius and pitch 5, 6 . Those structural parameters have already been calculated earlier using X-ray based experiments and computer simulations for the several constitute molecules of collagen and muscle. Then, correlation of the experimentally retrieved SHG effective orientation (helical pitch angle) value with the X-ray data, resulted pitch angles of ~45°5 -7 and ~63.5°5 -8 , respectively. The above pitch angles have been attributed to the one polypeptide helix of the collagen triple helix and to the a-helix of the myosin's tail 6 for collagen and muscle, respectively. In microtubules, the experimentally PSHG retrieved angle coincides with the diagonal of the rectangle containing the tubuling heterodimmers with respect to the long axis of the microtubule 9 .
INTRODUCTION
Another recent improvement in PSHG was the presentation of the generalized two-dimensional theoretical model which allowed the pixel by pixel fitting of the PSHG images to the model 8 . It follows the introduction from our group of the pixels histograms, or pixel values distributions, of the several parameters of the biophysical model. We have also utilized as source of information: i) the maximum frequency of the pixels histograms values and ii) the extraction of mean and standard deviation from the pixels' histograms [7] [8] [9] [10] . Moreover, we used Fast Fourier analysis (FF-PSHG) to extract very fast the available information 11 . The above ideas provisioned us to define metrics, for the conditions needed to demonstrate quantitative pixel resolution discrimination of collagen and muscle in the same image 7 . We note that the different SHG sources provide different polarization modulations, thus the PSHG based contrast in tissue is nowadays composed of 3 colors (collagen, muscle, microtubules).
INTRODUCTION
The 4th color or the new experimentally retrieved ratio of χ 2 tensor elements concerns the starch granules. In recent study we sliced the granule in the equator and we found χ 2 elements williamsbradley@example.net, 2σ=1.9 10 . This data provided a picked pixels histogram for the helical pitch angle with maximum williamsbradley@example.net° 2σ=9.3°. This pitch angle demonstrates amylopectin as the SHG source in starch 10 .
INTRODUCTION
All the above studies were performed assuming that the SHG active structures were contained in a plane (2D model). In the current work we demonstrate that the experimentally retrieved PSHG χ 2 elements' ratio critically depends on the tilted-off the plane SHG structures. We performed PSHG in different z-planes of a starch granule (starch granules poses a convenient radial structure, which allows the evaluation of the tilted-off the plane structures) and we found different pick values of the χ 2 elements' ratio histogram for each plane.
INTRODUCTION
In order to explicitly calculate the three dimensional orientation of the amylopectin molecules in starch we develop here the three dimensional model for PSHG (3D-PSHG) imaging in a way that introduces the effect of the tilted molecules to the χ 2 elements' ratio. Then, we assume that the amylopectin's structure does not change from point to point and we fix its χ 2 elements' ratio. Comparison between the fixed elements' ratio and the PSHG experimentally retrieved value permits the pixel resolution extraction of the tilted off the plane angle. This off-plane angle (and the width of its pixels' histogram) explains probable histogram broadening obtained for the χ 2 elements' ratio in PSHG measurements using the 2D model [7] [8] [9] [10] [11] . The importance of those results relays in the fact that pixels' histograms were proposed recently as a method to quantify structural information with application in diagnosis 12 .
MATERIAL AND METHODS

PSHG microscope
The setup is based on an adapted inverted microscope (TE2000-U, Nikon, Japan) with the scanning performed by a pair of galvanometric mirrors (galvos) (Cambridge Technology, UK). Two identical 60x objectives with numerical aperture (NA)=1.4, (Plain Apo-Achromatic, Nikon, Japan) were used for excitation and forward-collection of the SHG signal, respectively. We used a Kerr lens modelocked Ti:sapphire laser (MIRA 900f, Coherent, France), with pulses of 160 fs (measured at the sample plane 13 ) with a repetition rate of 76 MHz and was operated at a central wavelength of 810 nm. After the galvanometric mirrors, we placed a linear polarizer that was followed by a zero order λ/2 wave plate (QWPO-810, CVI Melles Griot) on a motorized rotational stage, which was rotated in steps to change the polarization at the sample plane. A telescope arrangement was used to ensure that a collimated beam was filling the back aperture of the objective lens. A short-wave-pass dichroic beam splitter (FF720, Semrock Inc) has been placed before this excitation objective. In the forward collection geometry a proper mount and detection unit was implemented. This unit contained the collection objective, a long-wave-pass dichroic beamsplitter (FF665, Semrock Inc), a BG39 filter, a 15nm FWHM band pass filter centered at 406nm (FF01-406/15-25, Semrock Inc) and a PMT (H9305-04, Hamamatsu, France). The objective lens was mounted on a micrometric 3D translational stage with tilt correction. A labVIEW interface program was written to control the raster scanning of the galvo-mirrors and the data acquisition (DAQ) card. Typical frame acquisition times for a single 500 x 500 pixels image were about 1-2s. Any effect on the depolarization of the fundamental beam introduced by the different optical components was assessed by measuring the extinction ratio of the fundamental incident light for every polarization. The laser source has a typical extinction ratio value of >500:1. This value drops to 63:1 after the dichroic mirror and to 25:1 when the 60x (NA)=1.4 objective is added. The measured mean power in the sample plane during our experiments was ~10mW.
Three dimensional (3D) theoretical model PSHG intensity from an arbitrary oriented molecule with cylindrical symmetry
Several biophysical models for interpreting the PSHG contrast have been proposed in the past [4] [5] [6] [7] [8] [9] [10] [11] [12] . Here, we are based on the model described in Ref. [8] , since it allows for a pixel by pixel analysis without relying on the use of polarization analysis in detection or any sample alignment or rotation. Specifically, we assume that the laboratory coordinates system is the X-Y-Z. The laser is propagating along the laboratory Z-axis and its linear polarization is rotating in the X-Y plane in an angle α with the Y-axis. Then the electric field can be written in terms of the lab coordinates as:
Three dimensional (3D) theoretical model PSHG intensity from an arbitrary oriented molecule with cylindrical symmetry
. The cylinder's coordinates are chosen in such a way that the y-axis is contained in the X-Y plane and the z-axis coincides with the principal axis of the cylinder. With this geometry, the relation between the lab and the cylinder's frames is given by two angles, δ and φ, resulting in the following change of base: 
Three dimensional (3D) theoretical model PSHG intensity from an arbitrary oriented molecule with cylindrical symmetry
We then write Eq. (5) in the lab coordinates system using: $ $ˆˆĉ os cos cos sin sinŝ in coŝˆŝ in cos sin sin cos 
Three dimensional (3D) theoretical model PSHG intensity from an arbitrary oriented molecule with cylindrical symmetry
In the 2-D case, 2 π δ = and the above Eq. (10) 
Three dimensional (3D) theoretical model PSHG intensity from an arbitrary oriented molecule with cylindrical symmetry
which is the same formula used earlier [6] [7] [8] [9] [10] [11] [12] . We have chosen to maintain the ratio d 31 /d 15 in the above equations, in order to experimentally check whether both, Kleinman's symmetry conditions and the cylindrical symmetry assumption hold.
"Single axis-molecule" approximation
Now we are assuming dominant hyperpolarizability β (2) ννν along axis-ν in a (µ,ν,ξ molecular coordinates system) [5] [6] [7] [8] [9] [10] [11] . The χ (2) tensor is then described in the molecular frame using: 
"Single axis-molecule" approximation
Where B, is the PSHG experimentally retrieved anisotropy parameter of the SHG source structures. The angle θ e is now correlated to the angle of a single helix relative to the symmetry axis. A single helix is described by its radius R and its pitch, P. Then, the helical pitch angle θ e is extracted using, 
Attributed physical meaning to the experimentally retrieved anisotropy parameter B
Equation (12) provides the key experimentally retrieve anisotropy parameter B that is used for estimating the helical pitch angle θ e of the SHG source. Therefore, the objective is to retrieve the parameter B. For that we can use the 2D approach and use Eq. (11) to fit the PSHG experimental data taken from the equator of a starch granule. For the 2D analysis purposes, Eq. (11) 
RESULTS-DISCUSSION
To experimentally prove the concept outlined above, we use starch as a model sample. Starch is a major polysaccharide in the food chain and is lain down in the form of granules. Starch granules, although nature-made, possess extensive degree of crystalinity and non-centrosymmetric properties that give rise to strong nonlinear second harmonic generation (SHG) signal. The SHG signal originates from the amylopectin molecules 10 which are oriented radially with their nonreducing chain ends pointing towards the outer surface of the granule. That makes starch convenient sample to apply our 3D biophysical model. Rotation of the linear polarization reaching the granule usually provides two SHG hemispheres which are rotated with sin 2 dependency 15 . This behavior is characteristic of a radial structure.
Reduction to a 2-D case: equatorial PSHG imaging of wheat granules
In the following we present PSHG analysis acquired from Sigma-Aldrich (S5127, unmodified) wheat starch granules. In earlier work 10 we have performed PSHG imaging close to the equator of a granule, where we observed maximum SHG signal. Because of the starch's radial geometry, the amylopectin molecules close to the edge of the equator of the granule are arranged almost perpendicular to the direction of the laser propagation. In our experiments the incoming linear polarization was rotated clockwise between 0°-160°, in steps of 20°. Then, the 9 images were fed to the algorithm (Eq. and therefore, the cylindrical symmetry hypothesis and the Kleinman's symmetry conditions do hold in starch.
Three dimensional analysis: PSHG out of the equator
Here, in order to better examine the effect of the tilted of the plane filaments to the B parameter we run the algorithm (Eq. 15) in the same data, but with the A parameter fixed to 1.
Three dimensional analysis: PSHG out of the equator
For the whole granule, we found a highly peaked pixels' histogram centered at B=3.66 with ΔB=1.53. Less free parameters in the model resulted in better bell shaped pixels' values histogram for B parameter than before (Kleinmans/cylindrical conditions hold). In Fig. 1 we show the histograms formed after keeping pixels with coefficient of determination r 2 >95%.Then we moved ~5µm up from the equatorial plane and we performed PSHG analysis using Eq. (15) . For the whole granule, we found a new highly peaked pixels' histogram centered at C=3.34 with ΔC=1.42. Using the mean values for B=3.66 and C=3.34 and Eq. 16, we acquire δ=45.87°.
Three dimensional analysis: PSHG out of the equator
We note that in Eq. 16 the angle δ is not defined for B=3. Using this value (B=3) and Eq. (12) we find SHG effective orientation θ e =39.23°. This value coincides with the so called "the magic angle" 16 of Eq. (12) An angle value in which Eq. (12) diverges for maximum disorganization.
Three dimensional analysis: PSHG out of the equator
Once the SHG source structure has been identified in a biological sample (earlier reports, or present experiments), one can use its structural values (pitch and radius) acquired using other techniques (i.e. X-ray scattering or PSHG), and can calculate the χ 2 elements' ratio that corresponds to those values. Then he can define a fixed B parameter in Eq. (16) and obtain the tilted-off the plane angle δ for every pixel of the image. In that case, any experimental errors of the B parameter are translated into angle δ values. Thus, there is no discrimination between the actual angle δ and the possible experimental errors [17] [18] that are introduced to the C parameter.
Three dimensional analysis: PSHG out of the equator
The PSHG analysis presented here was performed in more than 10 granules and in all the cases the results were consistent with the values presented (data not shown). In particular, the SHG signal was maximum close to the equator of the granule. This plane was used for extracting the B parameter in the granules. Then, moving away from the equatorial plane always resulted C values smaller than the previous B value.
CONCLUSIONS
SHG imaging shows great promise to be applied in the clinic. Among its advantages, its coherent nature dictates a polarization dependency of the produced harmonics. Polarization sensitive SHG imaging offers additional information and means of contrast than intensity only SHG. In particular it provides an experimentally retrieved structurally dependant anisotropy parameter. In the present study we have examined the effect of the tilted-off the plane SHG active structures to this experimentally retrieved χ 2 elements ratio. We have developed a three dimensional PSHG model and we have applied it in starch granules. Starch granules possess a convenient radial architecture of SHG active structures which permits the calculation of their tilted-off the plane angle. We performed PSHG in different z-planes of a starch granule and we found different pick values of the χ 2 elements' ratio histogram for each plane. By assuming a fixed value for the elements' ratio, we determined the 3D orientation of the SHG active structures.
CONCLUSIONS
18. Gusachenko, I., Latour, G. and Schanne-Klein, M.-C., "Polarization-resolved Second Harmonic microscopy in anisotropic thick tissues," Opt. Express 18, 19339-19352 (2010) .
ACKNOWLEDGMENTS
This work is supported by the Generalitat de Catalunya 2009-SGR-159, the Spanish government grant TEC2009-09698 and FIS2009-09928. Authors also acknowledge The Laser Lab Europe (optobio) and the Photonics4Life networks. This research has been partially supported by Fundació Cellex Barcelona.

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