## Abstract

We study second harmonic generation (SHG) from non-centrosymmetric nanocrystals under
linearly polarized (LP) and circularly polarized (CP) excitations. Theoretical models
are developed for SHG from nanocrystals under both plane-wave and focused
excitations. We find that the focused excitation reduces the polarization dependency
of the SHG signal. We show that the SHG response under CP excitation is generally
inferior to the average of LP excitations over all orientations. We verify the theory
by measuring the SHG polar responses from BaTiO_{3} nanocrystals with a
scanning confocal microscope. The experimental data agrees well with the theory.

© 2010 OSA

## 1. Introduction

Second harmonic generation (SHG) microscopy has been developed as a powerful nonlinear optical imaging tool for examining endogenous structures in biological samples [1–7]. SHG only takes place in a non-centrosymmetric environment, and provides the imaging contrast of specific endogenous biological structures, such as collagen, muscle, and microtubules in a mostly isotropic environment. In biological samples, the molecular structures and orientations determine the nonlinear susceptibility. As a result, the polarization dependent measurement of the SHG signal can be used to study the molecular structures of biological samples [3, 8–11].

While the endogenous SHG signal is attractive for label-free non-invasive imaging, exogenous SHG markers are also desirable due to the flexibility of having the SHG contrast from any target of interest. Recently, efficient SHG from non-centrosymmetric nanomaterials has been reported [12–29]. These nanoparticles emit coherent, non-bleaching and non-blinking SHG signal with a broad flexibility in the choice of excitation wavelength due to the non-resonant SHG process, showing great promises as imaging probes. We therefore refer to these SHG-active nanocrystals as “Second Harmonic Radation IMaging Probes (SHRIMPs).” The coherent SHG signal accommodates interferometric detection of SHRIMPs and therefore offers the benefits of high signal-to-noise ratio and nonscanning three-dimensional imaging [15, 21, 23]. By exploiting the flexibility in the selection of the excitation wavelength, SHRIMPs have also been demonstrated as deep imaging markers [22]. Furthermore, the polarization dependent SHG response of SHRIMPs has been explored at the single-nanocrystal level [12–14, 16, 18, 20, 23, 28], where the orientation of the SHRIMP can be determined in the far field by a polarization measurement.

Despite the merits offered by the polarization sensitive SHG response, it may complicate a spatial distribution measurement of the SHG active targets of different orientations. As a result, circularly polarized (CP) excitation has been frequently adopted as an alternative [30–32]. It is thus important to examine the SHG response under linearly polarized (LP) and CP excitations in SHG nonlinear microscopy. We show in this paper that the SHG response under a CP excitation is generally inferior to the average of the SHG responses under LP excitation over all orientations.

Laser scanning microscopy, such as scanning confocal microscopy, is the most popular SHG microscopy where a high numerical aperture (NA) objective tightly focuses the excitation beam to reach a high local intensity for efficient nonlinear phenomena to take place. The transverse and axial field components generated through tightly focusing the incident laser beam, known as the depolarization effect [33], can significantly modify the overall polarization dependent SHG response [34, 35]. For the detection, the SHG signal is either collected by the same objective in epi-geometry or by another objective in the transmission geometry. A similar depolarization effect should also be considered in the detection for an accurate estimation.

In this paper, we use barium titanate (BaTiO_{3}) nanocrystals to study the SHG
response under LP and CP excitations. We consider the depolarization effect introduced
by the use of a high NA objective, including a tightly focused beam for the excitation,
and also the collection efficiency of both transverse and axial SHG polarization
components. We measure the polarization dependent SHG response by a standard scanning
confocal microscope. Excellent agreement between the experiments and the theory is
observed.

## 2. Theoretical models of SHG from nanocrystals

#### 2.1 SHG from nanocrystals under a plane-wave excitation

We start our study with a nanocrystal under a uniform LP excitation using
BaTiO_{3} nanocrystals. The crystal structure of the BaTiO_{3}
nanocrystal is tetragonal, which belongs to symmetry class 4 [36]. Due to the crystal symmetry, the SHG response is determined
only by the orientation of the c-axis of the nanocrystal, and the rotation of the
nanocrystal around the c-axis has no influence on the SHG response. The orientation
of an object in a three-dimensional space can be defined by three Euler angles in an
Euler coordinate. To define the orientation of the c-axis in space, the degree of
freedom is reduced to two angles which can be described in a spherical coordinate.
The orientation of the c-axis of the nanocrystal can be uniquely defined by the
angles ${\theta}_{0}$and ${\varphi}_{0}$ in the spherical coordinate as shown in Fig. 1(a)
. The incident excitation propagates along the *Z* axis and the
excitation polarization angle *γ* can be rotated in the $XY$ plane by a half wave plate.

Assuming the shape of the nanocrystal is spherical and the size is small compared
with the wavelength of excitation, the nanocrystal can be considered as a Rayleigh
particle. Following our previous approach and ignoring the material birefringence
[23], the electric field inside the
nanocrystal can be found to be in-phase and uniform as ${{\rm E}}_{P}{}^{\left(\omega \right)}=\left[3{\epsilon}_{m}/\left({\epsilon}_{P}+2{\epsilon}_{m}\right)\right]{{\rm E}}_{m}{}^{\left(\omega \right)}$ [37], where ${{\rm E}}_{m}{}^{\left(\omega \right)}$ is the incident electric field in the surrounding medium in the
absence of the particle, and ${\epsilon}_{P}$ and ${\epsilon}_{m}$ are the linear permittivities of the particle and the surrounding
medium respectively. The electric field ${{\rm E}}_{P}{}^{\left(\omega \right)}$ at the fundamental frequency *ω* is then
decomposed into three orthogonal components along the three axes in the crystal
frame, i.e. ${{\rm E}}_{P\text{\hspace{0.17em}}}{}^{\left(\omega \right)}={E}_{CX}{\widehat{e}}_{CX}+{E}_{CY}{\widehat{e}}_{CY}+{E}_{CZ}{\widehat{e}}_{CZ}$, where${\widehat{e}}_{CX}$, ${\widehat{e}}_{CY}$, and ${\widehat{e}}_{CZ}$ are unit vectors in the crystal frame as shown in Fig. 1 (a). The SHG polarizations along the three
crystal axes are related to ${{\rm E}}_{P}{}^{\left(\omega \right)}$ by

**d**is the second-order susceptibility tensor of the bulk BaTiO

_{3}crystal. The values we used in the simulation are d

_{15}= −41×10

^{−9}esu, d

_{31}= −43×10

^{−9}esu, and d

_{33}= −16×10

^{−9}esu [36].

Because of the subwavelength particle size, the electrostatic approximation holds, and the induced SHG polarizations are uniform inside the particle. By assuming also that the size of the particle is much smaller than the SHG wavelength, the SHG polarizations within the particle can be regarded as three orthogonal SHG dipoles with the amplitudes proportional to the strengths of the polarizations. These three orthogonal SHG dipole moments radiate like antennas at the SHG frequency. The total SHG radiation power ${\text{W}}_{0}$ can be found as [23]

*γ*(with respect to the

*X*axis as shown in Fig. 1(a)) when the nanocrystal is orientated at ${\theta}_{0}$ = 10, 50, and 90 degrees.

In nonlinear microscopy, the SHG signal is usually collected by a microscope objective. Since the SHG radiation is generally not a simple spherical wave, we further consider the collection efficiency provided by the objective to obtain an accurate estimation of the polarization dependent SHG response of a nanocrystal. The collection efficiency is determined by the overall far-field SHG intensity radiation pattern of the three orthogonal dipoles within the cone angle introduced by the objective. To calculate the collection efficiency, the three orthogonal dipoles are first projected back into the$XYZ$ lab frame, namely ${P}^{\left(2\omega \right)}={P}_{X}{\widehat{e}}_{X}+{P}_{Y}{\widehat{e}}_{Y}+{P}_{Z}{\widehat{e}}_{Z}$, where ${\widehat{e}}_{X}$, ${\widehat{e}}_{Y}$, and ${\widehat{e}}_{Z}$ are unit vectors in the lab frame. Each of the three new defined SHG dipoles radiates SHG field in the form of dipole radiation pattern [38]. Therefore, the SHG electric field radiation pattern in spherical coordinates can be related to ${P}^{\left(2\omega \right)}$ as:

It is worth noting that η is a function of the cone angle
*Ω*, the nanocrystal orientation, and the excitation
polarization. We consider two cases where the SHG signal is collected by a high NA
microscope objective (NA 1.2 water-immersion, *Ω* = 64.46
degrees) and a low NA microscope objective (NA 0.3 water-immersion,
*Ω* = 13.04 degrees). Taking into account η for the
collected SHG power, we plot the detected polarization dependent SHG responses for
these two cases in Fig. 1(c) and (d)
respectively. From Fig. 1(b), (c) and (d), it
is obvious that higher NA detection gives a response closer to the total SHG signal.
A substantial difference between Fig. 1(b)-(d)
takes place when ${\theta}_{0}$ is small. This is because a stronger axial dipole component (${\text{P}}_{\text{Z}}$) appears when ${\theta}_{0}$ is small and the objective has a lower collection efficiency of the
axial dipole than the transverse dipoles (${\text{P}}_{X}$ and ${\text{P}}_{Y}$).

The orientation of a BaTiO_{3} nanocrystal at the time of measurement under a
LP plane-wave excitation can be assumed to be random and equally likely to be in any
orientation in space. In a spherical coordinate system, the probability density
function of a random orientation is a joint probability distribution of the angles
*θ* and *ϕ*, which should lead to an
equal probability of orientation within every unit solid angle. This requirement
results in a probability density function of ${f}_{\mathrm{\Theta},\mathrm{\Phi}}\left(\theta ,\varphi \right)=\mathrm{sin}\theta /4\pi $. Note that the probability density function is not a uniform
density function in the two angles *θ* and
*ϕ*. With this assumption we can readily calculate the mean
and the standard deviation of the measured SHG signal. The relative standard
deviation (i.e. the standard deviation divided by the mean) of the measured SHG
signal is calculated to be: 23.7% for the case of the total SHG signal detection,
28.5% for the case of NA 1.2 water-immersion objective detection, and 40.7% for the
case of NA 0.3 water-immersion objective detection. The relative standard deviation
of the signal increases significantly when the NA of the collection objective
decreases. The effect of NA on the relative standard deviation of the SHG signal
reflects the sensitivity of the collected SHG signal to the detection geometry, which
is caused by the change in the SHG radiation pattern due to the combination of the
nanocrystal orientation and the polarization dependent SHG response.

The relative standard deviation of the SHG signal under a LP excitation can be reduced if the excitation polarization can be rotated in the excitation plane ($XY$ plane) within the time of measurement. When the excitation polarization rotates at an angular frequency much smaller than the optical frequency, it excites the nanocrystal in all polarization directions (still limited in the $XY$ plane) and the polarization dependent SHG intensity due to the orientation variance in ${\varphi}_{0}$ is averaged out. We calculate the relative standard deviation of the measured SHG signal under this rotating LP plane-wave excitation. We found the relative standard deviation is reduced to: 9.5% for total SHG signal detection, 11.9% for NA 1.2 water-immersion objective detection, and 22.6% for NA 0.3 water-immersion objective detection.

We further calculate the SHG response under a CP excitation. A CP excitation can be
resolved into two perpendicular LP excitations, of equal amplitude, and in phase
quadrature. Therefore, we can calculate the SHG response of a nanocrystal under a CP
excitation based on the model established above for the LP excitation. The SHG
response is plotted as a function of the nanocrystal orientation in Fig. 2
. Due to the symmetry of the crystal structure, the SHG response from a
BaTiO_{3} nanocrystal is not sensitive to the handedness of CP
excitations. In Fig. 2, we find that SHG
signal monotonically decreases to zero when ${\theta}_{0}$ decreases from 90 degrees to 0 degrees. This is the result of the
interference effect between the two perpendicular LP excitations in quadrature
resolved from the CP excitation and also the change of the effective nonlinear tensor
due to the crystal orientations. Since the total SHG signal from the nanocrystal
oriented at small ${\theta}_{0}$ under a LP excitation is considerable as shown in Fig. 1(b), the vanishing SHG signal of the
nanocrystal oriented at small ${\theta}_{0}$ under a CP excitation is therefore mostly due to the interference
effect.

The relative standard deviation of the measured SHG signal under a CP plane-wave excitation is found to be 38 ± 3% for the cases ranging from the total SHG signal detection to low NA detection (NA 0.3 water-immersion objective). It is worth noting that the relative standard deviation of the SHG signal under CP excitation is much greater than that under the rotating LP excitation described previously. The increase in the relative standard deviation shows that the nanocrystal has different SHG response under a rotating LP excitation and a CP excitation. It is also interesting to notice the small variation (±3 %) in the relative standard deviation when the NA of the collection objective changes. This implies that, under a CP excitation, the SHG radiation pattern does not vary a lot as the nanocrystal orientation changes. In fact, the radiation pattern is always dominated by the transverse dipoles (${\text{P}}_{X}$ and ${\text{P}}_{Y}$), which is less sensitive to the NA of detection.

#### 2.2 SHG from nanocrystals under a tightly focused excitation

In nonlinear scanning microscopy, the excitation is tightly focused and scanned
across the sample to form an image. The depolarization of the LP excitation being
tightly focused by a high NA objective has been studied [33]. The depolarization effect gives rise to new excitation
polarizations at the focus which then participate in the SHG. As a result, the
depolarization may change the SHG polarization response significantly due to the
nature of SHG (as described in Eq.
(1)). We simulate the SHG response of a nanocrystal in a scenario of a
scanning microscope. It is convenient to introduce spherical polar coordinates as
shown in Fig. 3(a)
. A LP (*X*-polarized) plane-wave excitation propagating in the
*Z* direction of 812 nm wavelength is tightly focused by a NA 1.2
water-immersion objective and the beam waist is at *Z* = 0. The
focused field at the beam waist can be written as [33]:

*n*.

The magnitudes and the phases of the three perpendicularly polarized fields at the beam waist${{\text{\Epsilon}}_{i}|}_{Z=0}$, $i=X,Y,Z$, are plotted in Fig. 3(b)-(d) and (e)-(g) respectively. Besides the field at the original polarization (i.e. ${\left|{\text{\Epsilon}}_{X}\right|}_{Z=0}$), a considerable amount of axial component ${\left|{\text{\Epsilon}}_{Z}\right|}_{Z=0}$appears which will participate in the SHG process.

In a scanning microscope, the calculated complex excitation patterns described in Eq.
(5) are scanned across a nanocrystal and a scanning image of a nanocrystal is formed.
The pixel size in Fig. 3(b)-(d) is 60
´ 60 nm^{2}, corresponding to a scanning step size of 60 nm. Assuming
the nanocrystal is much smaller than the focused spot of the excitation, while the
excitation patterns is scanned across, it will pick up the local excitation fields
calculated at each pixel as a plane-wave excitation and emit SHG signal as described
in the previous section. Under this assumption, the theoretical SHG scanning image
can be obtained by calculating the SHG signal from the nanocrystal pixel-by-pixel
based on the excitation patterns. The finite size of the nanocrystal in reality would
make the measured SHG response deviate from this theoretical estimation. More
discussions on the validity of this assumption can be found in the Discussion
Section. We integrate the SHG intensity over the whole scanning image to represent
the SHG response of a nanocrystal at certain orientation and under a specific
excitation polarization using a scanning microscope.

The theoretical polarization dependent SHG response of a nanocrystal under a tightly
focused excitation (NA 1.2 water-immersion objective) is plotted in Fig. 4
where the nanocrystals orientated at ${\theta}_{0}$ = 10, 50, and 90 degrees are considered. Similar to the analysis of
plane-wave excitation, we calculate the total SHG signal and also the signal
collected by NA 1.2 and NA 0.3 water-immersion objectives, as shown in Fig. 4(a), (b), and (c) respectively. We find
Fig. 4(a)-(c) have a similar behavior as
Fig. 1(b)-(d), i.e. the decrease of the
collected SHG signal when the NA decreases at small ${\theta}_{0}$, which is due to the collection efficiency of the objective. We
also find that a tightly focused beam results in a slightly different SHG polar
response from a uniform excitation: where the uniform excitation gives a weak SHG
signal, such as ${\theta}_{0}$ = 90 degrees and *γ* = 0 degrees, the
tightly focused excitation gives a stronger SHG signal due to the depolarization
effect. In other words, the depolarization effect induces new excitation
polarizations, which results in an averaging effect in the SHG polar response.

To evaluate the averaging effect due to the tightly focused excitation, we calculate the relative standard deviation of the SHG signal as described previously. The relative standard deviation is calculated to be: 19.6% for the case of the total SHG signal detection, 23.6% for the case of NA 1.2 water-immersion objective detection, and 33.5% for the case of NA 0.3 water-immersion objective detection. The smaller relative standard deviation shows that a tightly focused excitation can reduce the variance of the polarization dependent SHG signal.

We calculate the relative standard deviation of the SHG signal with rotating LP excitation when it is tightly focused. The relative standard deviation is: 11.0% for total SHG signal detection, 11.7% for NA 1.2 water-immersion objective detection, and 17.3% for NA 0.3 water-immersion objective detection. The averaging effect due to the rotating LP excitation is again obvious.

We further calculate the SHG response of a nanocrystal under a tightly focused CP excitation. Analogous to the plane-wave excitation, the CP excitation is first resolved into two perpendicular LP excitations of equal amplitude and in phase quadrature, and then depolarized through tightly focusing respectively. The excitation field patterns of a tightly focused CP can be found, and therefore the SHG response can be calculated. The normalized SHG response is plotted as a function of the nanocrystal orientation in Fig. 5 , which is significantly different from Fig. 2. While the normalized SHG response drops from 0.3 to 0 as ${\theta}_{0}$decreases from 30 degrees to 0 degrees for the plane-wave excitation (in Fig. 2), it remains at around 0.3 for the tightly focused excitation (in Fig. 5). This is because the SHG polarization induced by the tightly focused CP excitation does not cancel out completely due to the depolarization effect.

We also calculate the relative standard deviation of the SHG signal under a tightly focused CP excitation. The relative standard deviation is calculated as ~27±1% for the cases ranging from the total SHG signal detection to low NA detection (NA 0.3 water-immersion objective). The much smaller relative standard deviation compared with the CP plane-wave excitation (38 ± 3%) again shows that a tightly focused excitation can reduce the variance of the polarization dependent SHG signal. The small range (±1%) of the relative standard deviation is also consistent to the case of CP plane-wave excitation.

We summarize the relative standard deviation of the SHG signal from a
BaTiO_{3} nanocrystal under different excitation geometry (plane-wave and
tightly focused excitations) of different polarizations (LP, rotating LP and CP) and
also for different NA of the detection in Table
1
. It is clear to see that the focused excitation and the high NA of the
detection can reduce the effect of polarization dependent SHG signal. The difference
between the rotating LP excitation and CP excitation is obvious. In the case of CP
excitation, both for the plane-wave and focused excitations, the relative standard
deviation of the SHG signal is not sensitive to the NA of detection. However, the
relative standard deviation of the SHG signal is usually greater under CP excitation
than under LP excitation. The CP excitation only provides lower relative standard
deviation in the signal when the NA of detection is low.

## 3. Experimental results

We used a standard scanning confocal microscope (Leica SP5) to excite and to detect SHG
signal from individual BaTiO_{3} nanocrystals. The X-ray diffraction pattern
(data not shown) confirms the crystal structure is tetragonal which is
non-centrosymmetric and allows for efficient SHG without further treatment. Isolated
nanocrystals were deposited on an indium-tin-oxide (ITO) coated glass slide and then
immersed in water for the confocal microscope measurement with a water-immersion
objective. Figure 6
is a typical scanning electron microscope (SEM) image of the nanocrystals
prepared on an ITO coated glass slide. It shows that the nanocrystals are nearly
spherical in shape and around 90 nm in diameter. It also shows that most of the
nanocrystals on the glass slide are isolated single nanocrystals. The excitation light
source was a Ti:Sapphire oscillator (Chameleon Ultra II, Coherent) generating 140 fs
laser pulses at 812 nm wavelength and 80 MHz repetition rate. The average excitation
power is approximately 15 mW. The excitation was tightly focused by a 63x NA 1.2
water-immersion objective and the SHG signal was collected by the same objective in an
epi-geometry. The SHG signal was detected by a photomultiplier (R6357, Hamamatsu) and
the ambient light was rejected by a narrow bandpass optical filter centered at 406 nm
with 15 nm bandwidth. The excitation polarization is controlled by a half-wave plate or
a quarter-wave plate placed in the excitation beam before it enters the confocal
microscope.

Figure 7
shows a typical SHG confocal image of the nanocrystals under a LP excitation,
where the sample was prepared in a similar way as for the SEM measurement. The SHG
signal from individual nanocrystals shows great contrast. The background SHG from the
ITO/water interface is relatively weak. The pixel size in Fig. 7 is 60 ´ 60 nm^{2}. The full width at half
maximum (FWHM) of the SHG imaging spot size of the BaTiO_{3} nanocrystal is
about 300 nm, which matches well with the diffraction limited spot size at the SHG
wavelength based on a tightly focused excitation beam described in the previous section.
The SHG intensity of individual nanocrystals varies due to the size-dependent and also
the polarization-dependent SHG signal. Based on the SHG efficiency of BaTiO_{3}
nanocrystals described in Ref. 23, we estimate
the average power of the SHG signal from the BaTiO_{3} nanocrystals in our
measurement is approximately 10 −100 pW.

We measured the polarization dependent SHG response from individual nanocrystals by
rotating LP excitation with a half-wave plate. One SHG confocal image was captured for
each excitation polarization direction. The excitation polarization was rotated from 0
to 180 degrees with a 10-degree angular step size. We calibrated the excitation power at
the sample position as it varies about 5% when the excitation polarization changes due
to the polarization dependent response of the confocal microscope. The SHG response of
individual nanocrystals was found by integrating the SHG signal within the bright spot
in the confocal image, while the background SHG from the ITO/water interface was
subtracted. We measured the polarization dependent SHG response of 39 nanocrystals.
Figure 8
shows two representative polar diagrams of the SHG response of BaTiO_{3}
nanocrystal as a function of the excitation polarization. From the measured responses,
we can find the orientations of the nanocrystals by fitting with theoretical calculation
(corresponding to the results shown in Fig. 4
(b)). The fitting of the orientation of the nanocrystal is unique because each (${\theta}_{0}$,${\varphi}_{0}$) pair gives a different polar response except the ambiguity between ${\varphi}_{0}$and ${\varphi}_{0}+180$degrees. The measured responses agree well with the theoretical
calculation. In Fig. 8, the two nanocrystals were
found oriented at ${\theta}_{0}=70\pm 5$degrees, ${\varphi}_{0}=35\pm 5$degrees and at ${\theta}_{0}=50\pm 5$degrees, ${\varphi}_{0}=115\pm 5$degrees respectively. The 10 degrees resolution of the fitting is due
to the accuracy of the measurement. We did the fitting to all 39 nanocrystals, and
various orientations of the nanocrystals (${\theta}_{0}$ from 30 to 80 degrees) were observed from the measurement.

We further measured the SHG response of the same 39 nanocrystals under a CP excitation by replacing the half-wave plate with a quarter-wave plate at a proper orientation. The CP excitation intensity was kept the same as the LP excitation on the sample position. All the nanocrystals were observed under a CP excitation. Following the identical image processes, we found the SHG response of the nanocrystals under a CP excitation.

We compared the measured SHG response of a nanocrystal under a CP excitation with that
under LP excitations of the same intensity. Since the SHG response depends on the
excitation polarization *γ* under LP excitation, we use the
average SHG response over the angle of excitation polarization
*γ* from 0 to $\text{2\pi}$ for the comparison. Specifically, we define ${\text{\rho}}_{\text{CP/LP}}$ as the ratio of the SHG response under a CP excitation to the average
SHG response under LP excitations of the same excitation intensity:

*γ*.

From the measured SHG responses of the nanocrystals under CP and LP excitations, we obtained the ratio ${\text{\rho}}_{\text{CP/LP}}$ for each of the measured 39 nanocrystals. The values of the ratio ${\text{\rho}}_{\text{CP/LP}}$ are plotted with the corresponding fitted nanocrystal orientations ${\theta}_{0}$ for all 39 nanocrystals in Fig. 9 . We also plot the theoretical calculation of ${\text{\rho}}_{\text{CP/LP}}$ in Fig. 9, which is based on the model described in Section 2.2, considering the tightly focused excitation and the collection efficiency provided by the objective. We found the experimental data agrees with the theoretical calculation. In Fig. 9, it is clear that the SHG response of a nanocrystal under CP excitation is not simply an average of the SHG responses under LP excitations over the excitation polarizations (otherwise the curve should be a flat line at value of 1).

## 4. Discussion

In the theoretical calculation, we assume the size of the nanocrystal is much smaller than the focused spot. In the experiment, the size of the nanocrystals was around 90 nm in diameter. In comparison, the tightly focused spot size using an NA 1.2 water-immersion objective at 812 nm wavelength is about 480 nm FWHM transversely, which is more than 5 times greater than the particle size. In the axial direction, the depth of focus of the excitation is about 1 μm FWHM which is more than 10 times greater than the particle size. These dimensions support our assumption of the electrostatic approximation. The good match between the measured and calculated polarization dependent SHG responses shows the theoretical calculation is able to provide reasonable estimation. It also suggests our simple theoretical model in which the local excitation field of the nanocrystal is assumed to be a plane-wave, is valid in our experiment. Furthermore, it has been reported that for the particle size smaller than 150-200 nm in diameter under a tightly focused excitation at 945 nm wavelength, it is reasonable to use the single dipole approximation for the SHG emission [18]. Therefore, we believe the single dipole approximation is also valid for 90 nm diameter particle under the 812 nm wavelength excitation as in our scenario. In cases where we need to find a more accurate solution (i.e. for larger particles), one would need to calculate the excitation field inside the nanocrystal under a tightly focused excitation.

It is interesting to consider the SHG response due to the abrupt 180 degrees phase change in the tightly focused excitation pattern at the beam waist as shown in Fig. 3 (e)-(g). During the scanning, when the nanocrystal is at the boundary of the abrupt phase change, we will have an out-of-phase excitation on its two sides. The plane-wave excitation approximation will not hold in this situation. However, the magnitude of the excitation field is always weak at these boundaries of abrupt phase change. Therefore, the abrupt phase change in the excitation pattern should have little effect on the overall SHG response. Furthermore, the Gouy phase shift of the focusing in the axial direction should also have little effect on the SHG response because the nanocrystal is small compared with the depth of focus of the excitation.

We note that it has been reported an extra ellipticity in the excitation polarization may be introduced from the scanning system and the dichroic mirror due to their polarization sensitive reflective properties [9,39,40]. However we have not observed a substantial ellipticity polarization introduced to the excitation in our imaging system. The degree of polarization (DOP) is measured to be between 0.92 and 0.98 for all the LP excitations. The theoretical calculation shown in Fig. 9 is under the assumption that DOP is equal to 1. We roughly estimate the overall ellipticity effect in our system with the averaged value of DOP as 0.95 by using a corresponding elliptical polarization as the excitation in the calculation. By taking into account the ellipticity in the excitation, we plot the theoretical ratio ${\text{\rho}}_{\text{CP/LP}}$ as a function of nanocrystal orientation ${\theta}_{0}$ in Fig. 10 , along with the experimental data. The two theoretical calculations for DOP as 0.95 and 1 show similar behaviors, and they reasonably agree with the experimental data. Slightly more derivation is observed at small ${\theta}_{0}$ which may be due to the imperfect measurement. Therefore, we believe the ellipticity effect in the excitation polarization in our system is not significant.

## 5. Conclusion

We studied the SHG response from BaTiO_{3} nanocrystals under various
excitations. Theoretical models were developed to describe the SHG from nanocrystals
under both plane-wave and tightly focused excitations. Based on our studies, we found
the depolarization effect of the excitation caused by the high NA objective can have
substantial effect on the SHG signal. We studied the effect of NA of the microscope
objective in the SHG signal collection. Low NA detection is sensitive to the SHG
radiation pattern of the nanocrystal and therefore the polar response can be very
different from the total SHG signal. We also compared the SHG signal under CP and LP
excitations. While the CP excitation can be used as an alternative choice of excitation
for SHG microscopy, we show that the SHG response under CP excitation is generally
inferior to the average of LP excitations over all orientations. To verify our
theoretical models, we measured the polarization dependent SHG responses from
BaTiO_{3} nanocrystals with a scanning confocal microscope. A good agreement
between the theoretical calculation and experimental data was observed. The complete
knowledge of the polarization dependence of the SHG response from nanocrystals will be
necessary in applications where SHRIMPs are used as imaging probes for position and
rotation detection.

## Acknowledgements

The authors thank Dr. Paul Bowen at EPFL for providing the BaTiO_{3}
nanocrystals. This project is supported by the National Center of Competence in Research
(NCCR), Quantum Photonics.

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