1. Introduction

Noncollinear second harmonic generation (SHG) is an important nonlinear optical technique that provides new capabilities in the characterization of nonlinear materials. In the seek for the enhancement of surface SHG, a noncollinear autocorrelation scheme was firstly theoretically proposed by Gierulski et al. [1]. It was then experimentally developed by Muenchausen [2] and Provencher [3] by introducing the use of two noncollinear pump beams. Later on, Figliozzi [4] and Cattaneo [5–7] have exploited the capability of this experimental configuration to separately address the bulk and surface nonlinear responses. More recently, we developed a method based on the simultaneously variation of the polarization state of both fundamental beams, while the incidence angle is fixed, thus the noncollinear SH signal is represented as a function of the polarization states of both pump beams. The resulting polarization map displays a pattern which is characteristic of the investigated crystalline structure and offers the possibility to address several properties as the absolute values of the non-zero terms of the nonlinear optical tensor [8], the ratio between the different non-zero elements of the nonlinear optical tensor [9], the orientation of the optical axis [10], to name some. In particular since this method avoids sample rotation, it is extremely interesting for those conditions where the generated signal would be strongly affected by sample rotation angle, i.e. for samples which are some coherence lengths thick, when using short laser pulses, or for nano-patterned samples.

In the present work we measured noncollinear SHG arising from a film composed by an oriented Bacteriorhodopsin protein (BR) grown by electrophoretic deposition technique, onto a substrate covered by an ITO film [11]. At a fixed incidence angle, the polarization state of both the fundamental beams was systematically varied, while the generated signal has been characterized for different polarization states by rotating a suitable analyzer. Both experimental and theoretical results show that this method, which doesn’t require sample rotation, is an useful tool to put in evidence the presence of optical chirality as well as the multipole contributions to the nonlinear polarization, such as magnetic-dipole (at both ω and 2ω) induced nonlinear polarization.

2. Noncollinear second harmonic from a Bacteriorhodopsin film

The transmembrane protein Bacteriorhodopsin (BR) can be commonly found in the purple membrane of the Halobacterium salinarium, which naturally occurs in salt marshes. BR is one of the simplest known active membrane transport system [11], working as a light-driven proton pump, i.e. converting light energy into a proton gradient across the bacterial cell membrane. Interest in this protein stems not only from its unique photochemistry as a light-driven proton pump, but also from its potential as an active component of biomolecular device applications [12].

Each BR monomer contains a covalently bound retinal cromophore, which is responsible for the strong absorption in the visible as well as for its outstanding nonlinear optical response. The cromophore retinal axis spontaneously arrange with an orientation angle of 23 ± 4° with respect to the plane of the purple membrane, so to form an isotropic conical polar distribution around the normal [11], as shown in Fig. 1(a) . Trimers of BR proteins assemble in an hexagonal two dimensional lattice within the purple membrane, forming thereby a sort of crystalline structure, see Fig. 1(b). Finally, the resulting P3 symmetry structure of BR arises from consecutive stacking of the hexagonal lattice represented by the membrane sheets, see Fig. 1(c) [13–15].

 

Fig. 1 Geometry of the investigated film of BR containing purple membrane. (a) The retinal chromophores form a cone around the normal to the membrane plane, at an angle of ~23° with respect to the membrane plane. (b) BR is organized in trimers, i.e. three groups of 7 helices and a retinal cromophore (in white) buried inside the protein. (c) Upper view of the hexagonal arrangement of the BR timers in the purple membrane resulting in P3 symmetry class.

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One of the most intriguing properties of BR relies in its optical chirality [16], i.e. the lack of mirror rotation axes of any order. Molecular chirality is extremely important in biology, chemistry and material science, and is conventionally investigated by linear optical techniques. Chiral molecules, in fact, manifest circular dichroism, i.e. their light absorption is sensitive to the handedness of the circular polarization of light, and optical rotatory dispersion, i.e. they can rotate the plane of polarization of linearly polarized light. From the point of view of nonlinear optical characterization, both SHG and sum frequency generation (SFG), have been shown to be very sensitive to chirality [17,18] and three different methods have been developed. A second harmonic generation analog of optical rotatory dispersion, SHG-ORD, investigate those transitions allowed by the so-called chiral contributions to the nonlinear optical susceptibility, that are otherwise forbidden, as an $\widehat{s}$-polarized SH signal arising from a $\widehat{p}$-polarized pump beam. Alternatively, the optical chirality can be revealed by investigating the signal of linear difference (LD-SHG) due to the diverse nonlinear SH response arising from a pump beam that is linearly polarized at + 45° and −45°, respectively. Finally, considering the use of a circularly polarized pump beam, in SHG-CD a difference in the SH signal generated by circularly polarized light of different handedness is detected.

A comprehensible model to describe the effect of chirality is founded on the electron path in a chiral molecule [19]: as the electrons of chiral molecules are displaced from their equilibrium by the application of an electromagnetic field, they are forced to move along helical-like paths. This process gives rise to an induced magnetic dipole moment of the molecule in addition to the electric dipole moment, therefore chiral molecules may respond to both the electric and magnetic component of the field.

Within this context, we chose BR since it has a well established structure [11] and its crystal structure is known with a high degree of accuracy. Furthermore, the orientational averages connecting the retinal chromophore to the protein cage has been reported as the main responsible of SHG optical rotatory dispersion (ORD) in BR [16].

We performed measurements on an oriented BR film prepared by using the asymmetric electrostatic interaction of the surface charge of the membrane fragments with a charged support surface. An electrophoretic deposition technique was employed to grow a 4 µm thick oriented film onto a substrate covered by a 60nm thick ITO film. The resulting BR films, composed by ~800 purple membrane layers (of 5nm thickness each) were characterized in terms of homogeneity, optical and electrical properties.

Noncollinear SHG measurements were carried out by means of a noncollinear scheme working in transmission, whose geometrical configuration is shown in Fig. 2 . The output of a mode-locked femtosecond Ti:Sapphire laser system tuned at λ = 830 nm (76 MHz repetition rate, 130 fs pulse width) was split into two beams of about the same intensity, while the temporal overlap of the incident pulses was controlled with an external delay line.

 

Fig. 2 Scheme of noncollinear second harmonic generation.

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The polarization of both beams was varied systematically, by means of two identical rotating half wave plates. The sample was placed onto a rotation stage which allowed the variation of the sample rotation angle, α, with a resolution of 0.05 degrees. The two beams, after passing two collimating lenses, were sent to intersect onto the surface of the BR film with the angles β and γ = −β = 3°, measured with respect to α = 0 (laboratory frame). For a given α ≠ 0°, the corresponding incidence angles of the two pump beams (in the sample frame) result to be α₁ = α−β and α₂ = α−γ, respectively (see the inset of Fig. 2). In the reported measurements, the rotation angle α was fixed to = − 40°.

Considering two pump beams, tuned at ω₁ = ω₂ = ω, having two different incidence angles, with respect to surface normal, α₁ and α₂, and different polarization state, ${\phi }_1$and ${\phi }_2$(defined with respect to the y-z plane), their interaction with a noncentrosymmetric material, produces a nonlinear polarization oscillating at the frequency ω₁ + ω₂. Given the wave vectors’ conservation law, the generated noncollinear SH beam is emitted nearly along the bisector of the aperture angle between the two pump beams. This beam was then collected with an objective and focused on to a monomodal optical fiber coupled with a photon counting detector. A set of optical low pass filters were used to further suppress any residual light at ω₁ and ω₂, while an analyzer allowed to select the desired SH polarization state.

Any set of experimental measurements was obtained by systematically rotating the two half-wave plates between −90° and + 90° with a step of 4°. The two beams were impinging onto the BR film with α₁ = −37° and α₂ = −43°, i.e. the two pump beams have an aperture angle of 6° with respect to each other.

The experimentally measured $\widehat{s}$-polarized SH signal and $\widehat{p}$-polarized SH signal are shown in Fig. 3(a) and Fig. 3(b), respectively. The average input power was 125 mW, corresponding to an intensity level of some MW/cm². As already mentioned, the signal is given as a function of the polarization angles, ${\phi }_1$and ${\phi }_2$, of the two pump beams. From the obtained experimental results it is possible to retrieve some information related to the optical chirality of the investigated film. In particular, the presented experimental results show a slight asymmetry, with respect to the corresponding –theoretical- polarization map arising from P3 symmetry. In fact, considering the $\widehat{p}$-polarized signal, in absence of optical chirality the maximum signal would be located in correspondence of ${\phi }_1$and ${\phi }_2$ = 0°, i.e. both pumps beams have to be $\widehat{p}$-polarized. In contrast, the pattern shown in Fig. 3(b) presents a maximum that is somewhat shifted towards the negative quadrant, i.e. both ${\phi }_1$and ${\phi }_2$are <0. It’s worth to note that a however small value of the chiral components, as for instance |d14| = |d25| = 0.1·d33 [20], is still responsible for an observable effect onto the polarization chart. The deviation of maximum signal position was experimentally evidenced with good repeatability, and, although of small size, can be interpreted as the effect of the chiral components of the nonlinear optical tensor. Likewise, the sign of the optical chirality can also be retrieved from the polarization map of $\widehat{p}$-polarized SH signal. The position of the maximum signal is in fact determined by the sign of chiral components, i.e. inverting the sign of the chiral components would result in a shift of the maximum signal towards positive ${\phi }_1$and ${\phi }_2$.

 

Fig. 3 Noncollinear second harmonic signal experimentally measured as a function of the polarization state of the first pump beam, ϕ₁, and the second pump beam, ϕ₂ (arbitrary units). Polarization state of the analyzer is set to (a) $\widehat{s}$, i.e. ϕ = 90° and (b) to$\widehat{p}$, i.e. ϕ = 0°, respectively. Sample rotation angle was fixed to α = −40°.

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3. Reconstruction of experimental results

Following these considerations, we implemented a theoretical model which includes different terms other than the electric-dipole, in order to fix the experimental data.

In general, in fact, the nonlinear optical polarization arising at 2ω, subsequent to the application of an electromagnetic field tuned at ω, is composed by several terms and can be written as follows [18,21,22]:

As well as for the electric-dipole, also the magnetic-dipole interactions, at ω and at 2ω, can be represented by a third-rank tensor, specific for the symmetry class as well as for the interaction itself.

The first term in the Eq. (1) is proportional to the nonlinear electrical susceptibility tensor, ${\chi }_{ijk}{}^{(2)eee}$, and to the magnetic-dipole interaction at ω, via the tensor${\chi }_{ijk}{}^{(2)eem}$:

In evaluating Eq. (1), it must be taken into account that the quadruple contribution, $\overrightarrow{Q}(2\omega )$, is typically negligible with respect to the other contributions, and is eventually included in the${\chi }_{ijk}{}^{(2)eem}$ tensor if it comes from the surface.

For what concerns the electric-dipole tensor, the point group symmetry P3 of BR is noncentrosymmetric, thus its second order susceptibility tensor, ${\chi }_{ijk}{}^{(2)eee}$, has the following nonvanishing components [20]${\chi }_{xxz}^{(2)eee}={\chi }_{yyz}^{(2)eee}$, ${\chi }_{zxx}^{(2)eee}={\chi }_{zyy}^{(2)eee}$ and ${\chi }_{zzz}^{(2)eee}$, along with the piezoelectric contraction. Two additional nonzero components of the nonlinear susceptibility tensor, ${\chi }_{xyz}^{(2)eee}=-{\chi }_{yxz}^{(2)eee}$, determine the so-called chiral contribution to the nonlinear optical response, since they appear only if molecules lack of mirror symmetry.

The contribution arising from the ${\chi }_{ijk}{}^{(2)eem}$ tensor is usually only a surface-like contribution, while it becomes a bulk contribution only if a two beams excitation is performed [22]. Furthermore, it has already been reported that the nonlinear magnetization becomes detectable in the presence of optical chirality [22,23].

The full expression of the generated SH power in the noncollinear scheme, P_2ω, as a function of the sample rotation angle α, as well as propagation angles and polarization states of both fundamental and generated beams, including the effect of absorption, trough the extinction coefficient at the fundamental, $k_{\omega }$, and at the second harmonic frequency, $k_{2\omega }$, can be written as:

Where the function $F(\alpha )=F\left({\text{A}}_{\text{1}}{\text{,A}}_{\text{2,}}{\text{t}}_{\text{ω}}^{\text{1}},{\text{t}}_{\text{ω}}^{\text{2}},{\text{T}}_{\text{2ω}}\right)$ comprises the dependence on the fundamental beams transverse areas onto sample surface (${\text{A}}_{\text{1}}{\text{,A}}_{\text{2}}$), the Fresnel transmission coefficients for the fundamental fields at the input interface (${\text{t}}_{\text{ω}}^{\text{1}}({\alpha }_1,{\phi }_1)$and ${\text{t}}_{\text{ω}}^{\text{2}}({\alpha }_2,{\phi }_2)$) as well as the Fresnel transmission coefficient for the noncollinear SH power at the output interface ($T_{2\omega }(\alpha ,\phi )$). On the other side, the effect of wavelength dispersion and optical absorption on the polarization chart is included in the term $\Phi (\alpha )$, that consists of the phase factor related to the internal angles of the beams [24] as well as an additional phase term determined by the imaginary part of the refractive index [24,25]. The power of the incident fundamental beams is taken into account in the term $\text{P}\left({\text{α}}_{\text{1}},{\text{α}}_{\text{2}}\right)={\text{P}}_{\text{ω1}}^{}\cdot {\text{P}}_{\text{ω2}}^{}\cdot e^{-2\left({\delta }_1+{\delta }_2\right)}$, which also includes an attenuation factor given by [24]:

Being the internal propagation angles of the fundamental ($\alpha {\text{'}}_1$ and $\alpha {\text{'}}_2$) and generated ($\alpha \text{'}$) beams calculated via Snell’s law.

Finally, ${\chi }_{eff}^{\text{(2)}}(\alpha )$ represents the effective susceptibility tensor, that contains the three different second order nonlinear optical tensors, all depending on the polarization state of both pumps and generated beams and on the fundamental beams incidence angles, α₁ and α₂:

It’s worth to note that the magnetic dipole ${\chi }_{ijl}^{(2)}{{}^{eem}}_{}^{}$tensor contributes twice to the total ${\chi }_{eff}^{\text{(2)}}(\alpha )$ being the product ${\chi }_{ijk}^{(2)eem}\cdot E_j(\omega )\cdot B_k(\omega )$ of Eq. (2) variant when exchanging the two pump beams, due to the noncollinear scheme that we employed. As a consequence, ${\chi }_{ijk}^{(2)eem1}\cdot {\overrightarrow{E}}_1\cdot {\overrightarrow{B}}_2$ and ${\chi }_{ijk}^{(2)eem2}\cdot {\overrightarrow{E}}_2\cdot {\overrightarrow{B}}_1$ determine two different contributes [21–23].

The experimental curves were fully reconstructed using the analytical expression for the effective second order optical nonlinearity in noncollinear scheme. Dispersion of the BR refractive index, n(λ), was taken from reference [26], i.e. n_ω = 1.526 and n_2ω = 1.549. Preliminary spectrophotometric analysis also indicates that birefringence results negligible. The pumping wavelength of 830 nm falls within the low optical absorption of BR, thus we retrieved the extinction coefficients of k_2ω = 0.002 and k_ω = 5∙10⁻⁴ from the linear transmittance spectra. Given all these data, the coherence length for the second harmonic generation process has been estimated to be of the order of 10 μm (for an incidence angle of −40°).

As already mentioned, concerning the electric-dipole tensor, ${\chi }_{ijk}^{(2)eee}$, we assumed the nonlinear coefficients values reported in Ref. [20], including the chiral elements, while components of the magnetic-dipole tensor, ${\chi }_{ijk}^{(2}{}^{)eem}$, as well as those of the nonlinear magnetization, ${\chi }_{ijk}^{(2)}{}^{mee}$, were treated as fitting parameters. The same, above mentioned, nonvanishing elements, typical of P3 symmetry group, are in fact present in the other two nonlinear tensors, while the magnetic-dipole tensors may also contain additional terms, such as ${\chi }_{zxy}^{(2)eem}$ [19,23]. Although the number of tensors’ elements for chiral molecules can be even larger [18] we have limited our fitting parameters to these terms. Being the electric-dipole still the main contribution to the SHG process, we used as starting values for the simulations those given in Ref [20]. for the electric-dipole parameters, and took into account also the imaginary-valued terms [17,23]. Furthermore, in the present analysis, the Kleinmann symmetry rules have been taken into account only for the electric-dipole terms.

In this way we could evaluate the ratios among the different nonlinear optical coefficients, and fix some relations among them. In Table 1 we report the resulting values of the nonlinear optical coefficients for the three tensors, all being normalized to the highest of them, i.e. the electric-electric dipole ${\chi }_{zzz}^{(2)eee}$.

Table 1. Values of the Nonlinear Tensor Elements Normalized to ${\chi }_{zzz}^{(2)eee}$_.

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One of the most intriguing feature of the present analysis, is that each different nonlinear optical tensor included in the Eq. (5) gives rise to a polarization map having its own pattern, including the two terms ${\chi }_{ijk}^{(2}{}^{)eem1}$ and ${\chi }_{ijk}^{(2}{}^{)eem2}$. In other words, the four terms given in Eq. (5) have their own dependence on the polarization state of the tow fundamental beams, that is peculiar of the nonlinear mechanism involved. Although the nonlinear optical tensor elements are almost the same, as well as the given experimental conditions, they present a completely different dependence on the polarization state of the input beams as well as on the polarization state of the output beam. It’s worth to note that these differences arise from the use of the nonlinear magnetic sources in the calculation of the ${\chi }_{eff}^{\text{(2)}}(\alpha )$.

In order to enlighten this feature, each term of the Eq. (5) has been separately calculated so to put in evidence its own pattern. In Fig. 4 we show, for the $\widehat{s}$-polarized SH output, the polarization chart corresponding to the electric-dipole ${\chi }_{eff}^{\text{(2)eee}}$ (see Fig. 4(a)), to the magnetic-dipole contributions ${\chi }_{eff}^{\text{(2)eem1}}$ and ${\chi }_{eff}^{\text{(2)eem2}}$ (see Fig. 4(b) and Fig. 4(c)) and the nonlinear-magnetization, ${\chi }_{eff}^{\text{(2)mee}}$ (see Fig. 4(d)). The corresponding polarization charts, calculated for $\widehat{p}$–polarization of the SH beam are reported in Fig. 5 . The multipolar response of a given sample, if present, can be evidenced in the overall polarization map, which is a superposition of the four considered nonlinear optical sources.

 

Fig. 4 Calculated polarization chart of the different nonlinear tensors responsible of $\widehat{s}$-polarized SHG generated by BR (a) ${\chi }_{eff}^{\text{(2)eee}}$, (b) ${\chi }_{eff}^{\text{(2)eem1}}$, (c)${\chi }_{eff}^{\text{(2)eem2}}$, (d)${\chi }_{eff}^{\text{(2)mee}}$.

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Fig. 5 Calculated polarization charts of the different nonlinear tensors, corresponding to $\widehat{p}$-polarized SH generated by BR (a) ${\chi }_{eff}^{\text{(2)eee}}$, (b) ${\chi }_{eff}^{\text{(2)eem1}}$, (c)${\chi }_{eff}^{\text{(2)eem2}}$, (d)${\chi }_{eff}^{\text{(2)mee}}$.

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By assuming the data given in Table 1, there is almost an order of magnitude of difference between the nonlinear magnetization coefficients and magnetic dipole if compared with electric dipole contributions. For both the output polarizations, we find that the weight of magnetic dipole sources results weaker that the corresponding electric-dipole. Most importantly, it is possible to observe that by taking into account the lonely ${\chi }_{eff}^{\text{(2)eee}}$contribution (Fig. 4(a) and Fig. 5(a)) is not possible to fully reconstruct the experimental data.

According to Eqs. (3) and (5), the superposition of these four terms give rise to the overall SHG polarization chart, which was experimentally detected. The SH power for $\widehat{p}$- and $\widehat{s}$-polarization state, was therefore analytically calculated as a function of the polarization state of both fundamental beam, i.e. by systematically varying ϕ₁ and ϕ₂. It’s worth to say that any small change of the parameters give in the Table 1, strongly affect the polarization map of the single contribute as well as their sum.

Interestingly, the overall effect of all the nonlinear effective sources (see Eq. (5)), for the $\widehat{s}$-polarized and $\widehat{p}$-polarized signals respectively, allows to retrieve the experimental data, as shown in Fig. 6 . Specifically, the maximum value for the$\widehat{p}$ -polarized SH signal is located in the negative quadrant of the polarization chart, as found experimentally.

 

Fig. 6 Second harmonic intensity as a function of the polarization state of the first pump beam, ϕ₁, and the second pump beam, ϕ₂, calculated for a 4 μm BR slab, including the effect of linear absorption. The polarization state of the analyzer is set to (a) $\widehat{s}$ and (b) $\widehat{p}$, respectively.

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Without losing generality, we may conclude that these results show that the nonlinear ellipsometric method that we have employed allows to put into evidence, for a chiral molecule like BR, the presence of magnetic-dipole contributions to the quadratic nonlinear response.

4. Conclusions

In conclusion, we investigated both experimentally and theoretically the second order nonlinear optical properties of a BR film with a noncollinear ellipsometric method. We show that fundamental information on the origin of the nonlinear optical response arising from BR can be recovered from the polarization chart of the generated signal as a function of polarization states of both pump beams. The polarization scanning method adopted is shown to be a valid and sensitive tool to probe the presence of magnetic-dipole contributions to the nonlinear optical response, otherwise difficult to evidence.