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An inhomogeneous linear refractive index profile, such as that occurring in biological tissues, is shown to significantly alter stimulated Raman scattering (SRS) and coherent anti-Stokes Raman scattering (CARS) microscopy images. Our finite-difference time-domain simulations show that near-field enhancement and microlensing can lead to an increase of an object’s perceived molecular density by a factor of nine and changes in its perceived position by 0.4 μm up to 1.0 μm. Thus the assumption that SRS scales linearly and CARS quadratically with density does not always hold. Furthermore, the inhomogeneous linear index can cause false CARS and AM-SRS signals, even for a homogeneous nonlinear susceptibility.
© 2016 Optical Society of America
1. Introduction
Nonlinear optical imaging microscopes allow for label-free non-destructive imaging of biological processes in cells and tissues [1,2]. Two such techniques are stimulated Raman scattering (SRS) microscopy and coherent anti-Stokes Raman scattering (CARS) microscopy, which allow for video-rate [3–5] and hyperspectral imaging [6], where in the latter per-pixel spectral information is obtained. Using broadband CARS one can obtain high resolution 3D images of the entire fingerprint region [7], where most organic molecules have their identifying Raman peaks and ranges from 500–2000 cm−1. Meanwhile, SRS with its linear dependence on density and enhanced contrast has been used to obtain multispectral images at 30 frames per second allowing one to identify virtually any organic molecule, even when their vibrational spectra are very similar [5], including in the fingerprint region [8]. This is useful for biomedical applications such as tracking drugs as they move through tissue [9,10], detecting tumours [11], and the tracking of newly synthesized proteins as they are being produced [12].
For these imaging techniques to produce reliable images, it is crucial to understand the nonlinear image formation mechanisms, especially with regards to contrast and distortions. Due to the coherent nature of CARS, interference of the CARS signal with the signal from the nonresonant background (NRB) can occur — even when techniques such as FM-CARS are used — resulting in reduced contrast, spatial shifts [13] and spatial-spectral coupling, where distortions in the Raman spectrum as retrieved from the CARS spectrum were observed [14]. Unlike CARS, SRS does not have a comparable nonresonant process, but when the background is inhomogeneous in the nonresonant third-order non-linear susceptibility, a background signal can be generated on the same order of magnitude as the SRS signal [15].
In this paper, we find that the difference between the refractive index of an object and the background medium, even when Δn/n < 0.12, can cause significant distortions of SRS and CARS images due to near-field enhancement of the input laser beams and microlensing. Though the modest differences in the indices of refraction in biological material between the objects and the background medium are exploited in phase-contrast microscopy [16], these differences are ignored in most models for CARS and have not been explored in SRS. For CARS, it is known that such a refractive index mismatch in the surrounding medium can cause the epi-CARS signal to be masked by reflections of the forward CARS signal [17]. Microlensing through spheres on top of a glass plate has been investigated as well and shows photonic nanojets forming as well as diffraction rings in the far field [18]. The linear index can also cause shadows in the nonresonant background [19] though filtering techniques such as FM-CARS have not been investigated in that regard. In theoretical work in CARS, enhancements to the induced nonlinear polarization near the interface of a single dielectric sphere and a surrounding medium have been explored [20] as well as the near-field enhancements for two such spheres [21]. These last two works, however, did not not have the nonlinear effects directly incorporated in their FDTD simulations but instead manually obtained the induced polarization from the calculated input field and used the free space Green’s function to propagate the CARS signal into the far-field. Additionally, they only considered objects at the laser focus which does not represent a true measure of the strength of the far field CARS signal, due to interference effects with the background anti-Stokes signal [13].
We use FDTD simulations to show that significant changes of the measured SRS and CARS signal can occur in the far-field images depending upon the refractive indices, the shape of the Raman scatterer, and the position of the input laser focus. The enhanced near fields of the input beams around and within a single Raman-active sphere do not have a large effect on the magnitude of the CARS and SRS signal. They do, however, affect the perceived object position. Furthermore, when only off-resonant material is present, false SRS and CARS far-field signals occur due to inhomogeneities in the refractive index alone.
We then turn our attention to a pair of spheres and observe CARS and SRS signals can be an order of magnitude larger than what would be expected and object positions are shifted up to 1.0 μm. One sphere acts as a microlens, and creates a field enhancement inside the second sphere, leading to an enhanced CARS or SRS polarization. Microlensing from dielectric spheres on top of a sample has been used to create photonic nanojets to obtain superresolution CARS images [22], whereas the microlensing here is due to the internal objects present in the sample. This internal microlensing we found leads to an enhancement and shifted position in the image, suggesting a larger scatterer density than actually exists in the object. These effects depend on the numerical aperture (NA) of the collection lens and can be reduced, but not eliminated, by increasing the NA. This microlensing effect continues to create large signal enhancements when the spheres are moved several sphere radii apart, and causes the sphere closer to the laser source to be masked by the first one, up to a separation distance of 1.0 μm for two r = 0.5 μm spheres. Understanding these mechanisms is key to correctly interpreting CARS and SRS images.
2. Methods
We employ FDTD to study the impact on SRS and CARS images of refractive index mismatch between wavelength-sized spherical objects and a background medium. The input laser beams are Gaussian beams focused by a parabolic mirror. Intensities are measured in the far-field by taking the electric field at the domain boundary, extracting a particular frequency component using a direct Fourier transform and then using the fields at that frequency to calculate the far-field using a free-space Green’s function approach. This near-to-far-field transformation is an exact solution for a homogeneous background medium outside of the simulation domain and is equivalent to extending the simulation domain. Furthermore, the obtained far-fields are integrated over the surface of a collection lens so any diffraction effects are automatically accounted for. Inside the simulation domain, near fields are measured at the pump and Stokes frequencies through a discrete Fourier transform in time. The polarization for SRS at the Stokes (ωS) and pump (ωp) frequencies can be written as [23]
where is the frequency-dependent resonant third-order nonlinear susceptibility and ∊0 is the vacuum permittivity. The electric fields in the medium as a result of illumination by the input pump and Stokes beams are represented by E(ωp) and E(ωS) respectively. The polarization for CARS can be written as where ωAS is the anti-Stokes frequency and is the nonresonant third-order non-linear susceptibility, which we take to be that of a frequency independent Kerr medium. Our implementation of and automatically calculates both polarizations. We solve the non-magnetic 3D Maxwell equations in CGS units, using where χ(1) and are the first- and third-order instantaneous susceptibilities, respectively, and the polarization of the resonant medium is [24] where ★ denotes a convolution and Here, is the time-independent amplitude of the Raman susceptibility, ωR the resonant frequency of molecular vibrations, and γR the damping factor; ℱ−1 denotes the inverse Fourier transformation. The χR allows us to simulate the nonlinear polarizations for both SRS and CARS. This, along with the particular values of the nonlinear coefficients used, is described in section 3 of Popov et al. [13].When the difference ωp − ωS matches a resonance in the Raman spectrum of the molecule of interest, the induced polarization is very strong and becomes the dominant source for the measured signal. This is what allows for label-free imaging. For CARS the signal measured at ωAS does not come only from the induced polarization that scales with , but also from nonresonant processes ( ), which we model as a frequency-independent Kerr medium and are responsible for the NRB. It is generally much weaker than the CARS signal, but its effect increases as the size of the Raman-active object decreases, and becomes comparable to the wavelength of the input beams. Often frequency modulation of one of the input beams is used to reduce the effects of the NRB, thus we include FM-CARS in our studies. The SRS process leads to loss in the pump beam and gain in the Stokes beam which is measured at the pump or Stokes frequency. The change in signal is very small compared to the input beams, so an AM or FM lock-in technique is used to extract the SRS signal. Therefore, for SRS, we study both AM-SRS and FM-SRS.
Our simulation domain contains either one or two spheres in a background medium. The objects can have a linear refractive index either different from, or the same as, the background medium. Additionally, the background medium always has a homogeneous to simulate the NRB while the spheres can be made Raman-active by setting non-zero. Because we can turn on and off all of these individual properties, we can determine the consequence of each effect separately. We use λS = 800 nm and λp = 1042 nm beams in a medium with n = 1.33. The objects themselves can have n = 1.5 or n = 1.33. The entire simulation domain has a size of 12 × 12 × 12 μm with a resolution of 25 nm [15].
The input beams are Gaussian laser beam sources, tightly focused through a parabolic mirror of radius 500 μm with an NA of 0.95 and a ratio of beam waist to mirror radius of 0.4. The focal point is at the centre of the simulation domain. The far-field signal is collected by integrating over a transverse area in the far field corresponding to an NA of 0.6, except for simulations where we vary the NA from 0.1 to 0.6. For ease of calculation, we simulate the scanning of the laser over the sample by moving the object inside the simulation domain and keep the laser focus constant. As we only move along the laser propagation axis, we consider only normal incidence. The pump and Stokes beams propagate in the x direction and are polarized in the y direction. The objects are placed on the x-axis and their positions are taken to vary with x, including through the focal point of the lasers.
3. Numerical results for a single Raman-active sphere
3.1. Single Raman-active sphere on resonance
In order to study the effects of object/background refractive index mismatch on SRS and CARS images, we first look at the magnitude of the electric field at the pump frequency, ωp, around and within a single spherical object as it moves through the focal point of the tightly focused pump laser source. The left plots of Fig. 1 show the undistorted pump field in the x–y plane for a single sphere with the same index of refraction as the homogeneous background medium with n = 1.33. The plots on the right show the field strengths when the sphere has a larger refractive index of n = 1.5. The difference in the refractive index between the sphere and the medium is modest, but distortions to the fields are clearly visible: The focal point is shifted and additionally the field strength is increased up to 53%. A ring can be seen in the top right and is similar to those found by Ferrand et al. [18]. The near-field plots for the Stokes beam are similar (not shown). Though we only show the pump beam here, in our FDTD simulations both the pump and Stokes beams are included. Since the polarization in Eqs. (1)–(3) scale as the cube of the pump/Stokes field, the maximum measured signal, i.e. the intensity, can potentially increase by a factor of 13. However, for that scenario the input fields would have to be enhanced uniformly everywhere inside the sphere, which is clearly not the case. From the top left and top right pump field plots in Fig. 1, we note that there is very little field enhancement inside of the sphere therefore only small effects on SRS and CARS images are expected. However, there is a strong focused spot in the background medium caused by the object which acts as a microlens.
Fig. 1 The field magnitude of the pump beam in a n = 1.33 medium focused at x = 6.0 μm is shown in the presence of a homogeneous refractive index (left) and for the case where it is distorted by the presence of a r = 1.0 μm sphere with n = 1.5 (right). The position of the sphere is indicated by the (red) circles, where a dotted line means the sphere is refractive index-matched, and a solid line means it is index mismatched.
In Fig. 2, we plot the simulated far-field intensity of the CARS/FM-CARS and AM-/FM-SRS signals generated from a r = 1.24 μm sphere with and without linear index mismatch to the background, as the object is moved through the laser focus along the laser propagation axis (x-axis). In all cases, we find that signal strength remains almost unchanged. However, the measured position of the sphere is shifted towards the collecting lens by 0.4 μm for CARS and FM-CARS and by 1.0 μm for AM-SRS and FM-SRS. From earlier work [13], we determined that the refractive index mismatch affects the perceived position of the object in CARS and FM-CARS for a r = 0.4 μm sphere, but there the dominant effect was due to interference with the NRB and the Gouy phase shift at the position of the object, leading to an order of magnitude larger shift than that caused by refractive index mismatch. Here the sphere is much larger so the effect of the NRB is much smaller in comparison as can be deduced from the signal strength of CARS and FM-CARS being nearly the same. As the NRB plays only a very small role for CARS and FM-CARS, and does not exist for AM-SRS and FM-SRS, the deviation in the measured position arises exclusively from the refractive index mismatch.
Fig. 2 CARS (top left), FM-CARS (top right), AM-SRS (bottom left) and FM-SRS (bottom right) far-field signals as a function of bead positions, x, along the laser propagation axis of a single r = 1.24 μm sphere with (blue filled squares) and without (red open circles) a linear index mismatch with the background medium (n = 1.33). In all cases, the laser focal spot is at x = 6.0 μm.
3.2. Single Raman-active sphere off resonance
Next we investigate the effect of near-field enhancements of the pump and Stokes field on the off-resonant SRS and CARS for a single sphere of radius r = 0.25 μm. The nonresonant susceptibility, , is taken to be the same for the sphere and the background and is either nonzero for the on-resonant case or zero for the off-resonant case. We measure the far-field signal at the SRS and CARS frequencies for those two cases, the results of which are shown in Fig. 3. In the CARS measurements (Fig. 3 top left), a large signal above the NRB baseline for a homogeneous nonlinear susceptibility is detected despite the lack of resonant material. Turning off the linear index mismatch leads to a flat uniform NRB (thick black dashed line) as expected. The strength of the off-resonant signal (orange open circles) is about half of that of the on-resonant case (filled blue squares). The fact that this occurs with a homogeneous shows that the patterns in the nonresonant background commonly seen in CARS measurements are not only the result of an inhomogeneous nonlinear susceptibility, but can also be caused by an inhomogeneous index of refraction. FM-CARS reduces these off-resonant signals, but a small signal remains (Fig. 3 top right). As expected the CARS and FM-CARS signal of the on-resonant sphere are not centred at the focal point of the laser source [13]. There is however also a shift in the off-resonant signals, which cannot be caused by interference as there is no resonant signal to interfere with. It is instead caused by the refractive index mismatch that causes an enhancement in front of the sphere due to microlensing. Therefore, the strongest nonresonant signal does not occur when the sphere is exactly in the focal point of the laser, but when the sphere is in front of the focal point.
Fig. 3 Far-field signals for anti-Stokes (top left), FM anti-Stokes (top right), AM pump (bottom left) and FM pump (bottom right) as a function of bead position x. This is for a single r = 0.25 μm sphere (n = 1.5) index mismatched with a background medium (n = 1.33) and a collecting lens NA of 0.6. The solid blue squares are for a resonant sphere and the orange open circles are for a nonresonant one. The top plots are expressed in units of the far-field NRB signal from bulk. The index-matched off-resonant signal, i.e. the NRB, is indicated by the black dashed line. The bottom plots have been scaled such that the peak value of the solid line for each sphere is one. In all cases, the laser focal spot is at x = 6.0 μm.
In Fig. 3 (bottom left), we also observe a clear signal in off-resonant AM-SRS where we would expect no signal at all. Like for CARS, it is induced by inhomogeneous χ(1) due to microlensing. In previous work [15], we showed that an inhomogeneous nonlinear susceptibility generated a background signal in off-resonant AM-SRS. This was due to the fact that the different beam intensities between the Stokes beam being on and off caused a difference in the induced refractive index through the Kerr effect via n = n0 + n2I in combination with the inhomogeneous n2. This in turn led to a NA-dependent difference in the collected signal in such a way that increasing the NA of the collecting lens decreases the nonresonant signal. Here we have a homogeneous χ(3) everywhere and we find that the nonzero off-resonant signal is caused by the refractive index mismatch between the sphere and the background medium. Additionally, the off-resonant signal now depends on the position of the sphere because the near-field enhancements depend on object position.
4. Numerical results for double spheres
The situation is very different in the case of multiple spheres as is evident from the near-field plots at the pump frequency for two touching r = 1.0 μm spheres shown in Fig. 4. The plots show two spheres in a background medium illuminated by a tightly focused laser beam. In the left plots, the spheres have the same refractive index as the background while in the plots on the right, the spheres are index-mismatched. In the latter, the sphere closest to the laser source acts as a microlens, resulting in an enhancement of the pump field strength up to 51%. Even though the enhancement is slightly less than for the single sphere, the location of the maximum enhancement is now inside the second sphere. As a result, the nonlinear induced polarization in the sphere closest to the collecting lens increases significantly and becomes the dominant source for the measured SRS and CARS signal. As the spheres decrease in size, the enhancement effects also decrease. However, they remain significant. We find that there is always an enhancement inside the second sphere due to the microlensing effect for spheres with radii as small as r = 0.2 μm, where an enhancement of up to 26% in pump field strength can be observed.
Fig. 4 The field magnitude of the pump beam in a n = 1.33 medium focused at x = 6.0 μm is shown in the presence of a homogeneous refractive index, configuration I, (left) and for the case of two r = 1.0 μm spheres, configuration II (right). The position of the sphere is indicated by the (red) circles, where a dotted line means the sphere is refractive index-matched, and a solid line means it is index mismatched. The field magnitudes for a single index-mismatched sphere, configuration III, are similar to those plotted on the right side of Fig. 1.
To gain deeper insight into the effects of object shape on SRS and CARS far-field measurements, we compare the far-field signal for three configurations that have the same volume of Raman-active medium and thus the same total number of scatterers. The first configuration I is two touching spheres that have the same refractive index as the background medium (n = 1.33). The second configuration II has the same shape as the first, but the refractive index of the spheres (n = 1.5) does not match the background medium. Finally, we consider configuration III, which is a single larger index-mismatched sphere with a volume equal to the combined volume of the two spheres in I or II. The near-fields look similar to those depicted on the right side of Fig. 1. Since each configuration has the same total volume, one would expect roughly the same signal strength from each. When interpreting SRS and CARS images, it is commonly assumed that the measured signal scales linearly (for SRS) or quadratically (for CARS) with density. However, our simulations show that this need not be the case.
First we consider the effect of the refractive index mismatch by comparing configurations I and II, which have identical shapes. In Fig. 5, we plot the far-field intensity for two r = 1.0 μm spheres as they are moved along the propagation axis for configurations I and II. The differences between the far-field intensities are substantial. First, the magnitude of the signal for configuration II (filled blue squares) has nearly doubled for every vibrational technique compared to the index-matched case I (open red circles). Because the induced nonlinear polarization is a third-order process, the field enhancements inside of the second sphere cause the induced nonlinear polarization to be greatly enhanced which creates a significantly larger signal. Second, the perceived position of the object in the image created by the index-mismatched spheres II is shifted towards the collecting lens by almost a micrometer. This is because the induced polarization in the second sphere is much larger than that of the first causing the total signal of the configuration to shift in the direction of the second sphere.
Fig. 5 CARS (top left), FM-CARS (top right), AM-SRS (bottom left) and FM-SRS (bottom right) far-field signals as a function of position along the laser propagation axis of two touching r = 1.0 μm spheres with (blue filled squares) and without (red open circles) a linear index mismatch. The green filled triangles are the signal for a single index-mismatched sphere of r = 1.25 μm. The single sphere has the same volume as the two r = 1.0 μm spheres combined.
To determine the effect of the distribution of scatterers, i.e. object shape, we next consider the far-field intensity of configuration II versus III, which have the same refractive index-mismatch with the background medium. In Fig. 5, we see that for each technique the signal enhancement effect is even more pronounced with an intensity enhancement of up to a factor of four for the double spheres (filled blue squares) versus the same volume single sphere (filled green triangles). This is partly because the single index-mismatched sphere does not contain an enhanced field within it, whereas the two spheres in configuration II do. Additionally, because the focal spot is wider along the propagation direction than it is in the perpendicular direction, the scatterers shaped as two touching spheres experience a higher field strength on average than a single large sphere. This indicates that far-field signal strength does not directly correlate with scatterer density as is commonly assumed, but is highly dependent on object shape.
We perform the simulations for configurations I, II, and III for a range of radii from 0.4 μm to 2.0 μm, and extract the largest far-field signal for each configuration and radius. We do this for each vibrational technique as well as for collecting lens NA’s of 0.1, 0.3, and 0.6. The ratios of these peak values provide an enhancement factor which quantifies the effects of shape and refractive index mismatch. In Fig. 6, we plot the ratio of the signal strength for configuration II to I, which shows the enhancement due to refractive index-mismatch between the objects and the background medium. When using a collecting lens of NA = 0.1 (left plot), a factor of up to six in far-field signal strength is measured for all of the techniques. For larger collection NA’s (middle and right plot) the enhancement is reduced, but still can be over a factor of two depending on the size of the spheres. There is a small difference between CARS and SRS signal enhancement which is due to the fact that the polarization for SRS in Eq. (2) scales with E(ωp)|E(ωS)|2 while for CARS it scales with E2(ωp)E★(ωS). The difference in wave length of the pump and Stokes field causes the near-field enhancements to be different.
Fig. 6 Far-field enhancement factor calculated as the ratio between the far-field signal intensity from two index-mismatched touching spheres of configuration II to that of two index-matched ones of configuration I. Blue filled circles are CARS, red filled squares are FM-CARS, green empty circles are AM-SRS, and orange empty squares are FM-SRS. That is, the top two lines are CARS/FM-CARS and the bottom two are AM-SRS/FM-SRS. From left to right a collection lens NA of 0.1, 0.3, and 0.6 is used.
In Fig. 7 we show the ratio of the peak far-field intensity strength of configuration III to II, where all objects have an index of refraction that is different from the background medium. A factor of up to seven in signal strength is measured between the double and single sphere for AM-SRS and FM-SRS. For CARS and FM-CARS, the enhancement is up to a factor of nine. The SRS enhancements are similar for both AM-SRS and FM-SRS, but the CARS enhancements differ between CARS and FM-CARS for small radii. That difference is due to the NRB. When the spheres are small, the CARS signal is dominated by the NRB for configurations II and III. For larger spheres, the CARS signal becomes more dominant than the NRB, which is why the CARS and FM-CARS enhancements plots in Figs. 6 and 7 converge to the FM-CARS enhancements where the NRB is largely filtered out. The difference between the SRS and CARS signal is larger compared to that in Fig. 6. The shape determines the areas where the far-field signal is generated by the induced nonlinear polarization. As the focal spot size of the incoming pump and Stokes laser sources is different, due to the different wavelengths, the SRS and CARS far-field signals will be even more affected through the different dependence on the Stokes and pump field illustrated earlier in Eqs. (2) and (3).
Fig. 7 Far-field enhancement factor calculated as the ratio between the far-field signal intensity from two touching spheres of configuration II to that of a single sphere with equivalent volume of configuration III. Blue filled circles are CARS, red filled squares are FM-CARS, green empty circles are AM-SRS, and orange empty squares are FM-SRS. That is, the top two lines are CARS/FM-CARS and the bottom two are AM-SRS/FM-SRS. From left to right a collection lens NA of 0.1, 0.3, and 0.6 is used.
The large far-field signal enhancements due to the structure of the refractive index (Fig. 7), or due to the complex shape (Fig. 6) make it appear as if there is more of the molecule of interest present than is actually there. For a collecting lens NA of 0.6, the enhancements are clamped to a factor of two to four. However, this is still an appreciable amount, especially when considering the modest mismatch in refractive index between the spheres and the background medium. The reason that the far-field intensity enhancement depends on the NA is that the radial intensity pattern of the SRS and CARS far-field signals is wider in cases I and III than it is for case II. This can be seen in Fig. 8 where we plot the width of the far-field CARS intensity radiation pattern of configurations I, II and III as a function of object position along the laser propagation axis. The radiation pattern for two index-mismatched spheres II is narrower due to the microlensing of the second sphere. While the effect of this lensing on the image can be reduced by capturing as much of the radiation pattern as possible, signal enhancement effects cannot be reduced further by increasing the NA of the collecting lens. This is because the effects due to near-field enhancements cannot be eliminated. The total irradiated power is itself increased and the enhancements are not just a diffraction effect.
Fig. 8 Comparison of the far-field CARS intensity signals for r = 1.5 μm index-matched double spheres (red open circles) of configuration I, index-mismatched double spheres (blue filled squares) of configuration II, and single index-mismatched sphere of double volume (green filled triangles) of configuration III. Plotted is the divergence of the signal on the collecting lens. The divergence is taken as the width of a Gaussian function in units of angle fitted to the far-field intensity distribution.
The far-field SRS and CARS signal enhancements also occur when the spheres are no longer touching. They fall off very slowly as a function of distance between the spheres. In Fig. 9, we show the enhancement of SRS and CARS as a function of distance, as well as the FM-SRS signal for various separations as a function of object position along the laser propagation axis for two r = 0.5 μm spheres (top) and two r = 1.0 μm spheres (bottom). For an edge-to-edge distance between 2.0 μm of the two spheres, the far-field enhancement effect in the left plots is still very noticeable. A factor of two in signal enhancement for SRS and a factor of more than three for CARS is observed for both the r = 0.5 μm and r = 1.0 μm spheres. The microlensing of the first sphere makes the pump beam field more narrow and enhanced over a long range as can be seen, for example, in the first two plots of the pump beam enhancement for a single sphere in Fig. 1. Thus the spheres need not be touching for a significant modification of SRS/CARS imagery. For the same reason, we see in the lower right plot of Fig. 9 that the far-field FM-SRS signal of the second r = 1.0 μm sphere completely masks that of the first to such an extent that the first sphere is invisible in the far-field signal for all separation distances considered. Even for the r = 0.5 μm spheres (top right), the second sphere only becomes distinguishable from the first when they are more than two sphere radii apart, even though the spot size of induced third-order polarization is significantly smaller than that. More complicated structures with multiple differently shaped objects with different refractive indices in the sample will have more complicated effects on the SRS and CARS signal.
Fig. 9 Left: Enhancement (the ratio of the maximum far-field signal intensity for two separated spheres and a single sphere with the same total volume) versus edge-to-edge separation distance of the two spheres. Top left is for two r = 0.5 μm spheres and one r = 0.63 μm sphere. Bottom left is for two r = 1.0 μm spheres and one r = 1.25 μm sphere. The signal was collected with a collection lens with an NA of 0.6. Blue filled circles are CARS, red filled squares are FM-CARS, green empty circles are AM-SRS, and orange empty squares are FM-SRS. That is, the top two lines are CARS/FM-CARS and the bottom two are AM-SRS/FM-SRS. Right: FM-SRS far-field intensity for two r = 0.5 μm spheres (top) and two r = 1.0 μm spheres (bottom) as a function of position along the laser propagation axis. Different curves represent different sphere separation distances as indicated in the legends. Intensity was normalized to that of a single sphere.
5. Conclusion
We have shown that the near-field enhancements due to the refractive index mismatch can play a large role in SRS and CARS images. Objects act as internal microlenses. Due to the third-order nature of the nonlinear SRS and CARS processes, these field enhancements are cubed in the polarization term, which causes large distortions in the far-field.
Even for the modest differences in the refractive index, such as those occurring in biological material, far field signal enhancements of an order of magnitude can occur and objects can appear shifted on the order of micrometers, even using FM techniques. These results break the assumption that far-field intensity scales linearly with the number of scatterers for SRS or quadratically for CARS.
Further, linear index mismatch on its own can cause significant non-uniform off-resonant CARS and AM-SRS signals. The latter is especially noteworthy as AM-SRS should be background free. These off-resonant signals were greatly reduced in FM-CARS and FM-SRS, indicating the importance of using frequency-based filtering methods, such as hyperspectral analysis or FM techniques.
The objects studied in this paper are simple examples of inhomogeneous samples. In nature, an inhomogeneous sample will have a much more complicated structure, both in shape and refractive index profile, which can result in a complex pattern of near-field enhancements. This in turn would introduce complicated distortions into the SRS and CARS signal. No technique can remove all of these effects. Therefore the only way to account for them when studying SRS and CARS imagery, is to be aware of the range of the value for the refractive index throughout the sample and the length scale at which it changes. We have shown here that n = 1.5 objects embedded in a n = 1.33 background can lead to an order of magnitude higher far field signal; a larger mismatch would be expected to give even more significant enhancement.
Funding
Natural Sciences and Engineering Research Council of Canada (NSERC); Canada Research Chairs (CRC); Southern Ontario Smart Computing Innovation Platform (SOSCIP); the Canadian Foundation for Innovation (CFI); Ministry of Research and Innovation Ontario.
Acknowledgments
We would like to thank Sharcnet for the computational resources that made this article possible (NRAP1007/NRAP1352, up to 256 cores per simulation on orca).
References and links
1. J.-X. Cheng and X. S. Xie, “Vibrational spectroscopic imaging of living systems: An emerging platform for biology and medicine,” Science350 (2015). [CrossRef] [PubMed]
2. W. Min, C. W. Freudiger, S. Lu, and X. S. Xie, “Coherent nonlinear optical imaging: Beyond fluorescence microscopy,” Annu. Rev. Phys. Chem. 62, 507–530 (2011). [CrossRef] [PubMed]
3. C. L. Evans, E. O. Potma, M. Puoris’haag, D. Côté, C. P. Lin, and X. S. Xie, “Chemical imaging of tissue in vivo with video-rate coherent anti-stokes raman scattering microscopy,” P. Natl. Acad. Sci. USA 102, 16807–16812 (2005). [CrossRef]
4. B. G. Saar, C. W. Freudiger, J. Reichman, C. M. Stanley, G. R. Holtom, and X. S. Xie, “Video-rate molecular imaging in vivo with stimulated raman scattering,” Science 330, 1368–1370 (2010). [CrossRef] [PubMed]
5. Y. Ozeki, W. Umemura, Y. Otsuka, S. Satoh, H. Hashimoto, K. Sumimura, N. Nishizawa, K. Fukui, and K. Itoh, “High-speed molecular spectral imaging of tissue with stimulated raman scattering,” Nat. Photonics 6, 845–851 (2012). [CrossRef]
6. A. F. Pegoraro, A. D. Slepkov, A. Ridsdale, D. J. Moffatt, and A. Stolow, “Hyperspectral multimodal cars microscopy in the fingerprint region,” J. Biophotonics 7, 49–58 (2014). [CrossRef]
7. C. H. Camp Jr, Y. J. Lee, J. M. Heddleston, C. M. Hartshorn, W. R. Hight, J. N. Rich, J. D. Lathia, and M. T. Cicerone, “High-speed coherent raman fingerprint imaging of biological tissues,” Nat. Photonics 8, 627–634 (2014). [CrossRef]
8. X. Zhang, M. B. J. Roeffaers, S. Basu, J. R. Daniele, D. Fu, C. W. Freudiger, G. R. Holtom, and X. S. Xie, “Label-free live-cell imaging of nucleic acids using stimulated raman scattering microscopy,” ChemPhysChem 13, 1054–1059 (2012). [CrossRef] [PubMed]
9. L. Wei, F. Hu, Y. Shen, Z. Chen, Y. Yu, C.-C. Lin, M. C. Wang, and W. Min, “Live-cell imaging of alkyne-tagged small biomolecules by stimulated raman scattering,” Nat. Methods 11, 410–412 (2014). [CrossRef] [PubMed]
10. D. Fu, J. Zhou, W. S. Zhu, P. W. Manley, Y. K. Wang, T. Hood, A. Wylie, and X. S. Xie, “Imaging the intracellular distribution of tyrosine kinase inhibitors in living cells with quantitative hyperspectral stimulated raman scattering,” Nat. Chem. 6, 614–622 (2014). [CrossRef] [PubMed]
11. M. Ji, D. A. Orringer, C. W. Freudiger, S. Ramkissoon, X. Liu, D. Lau, A. J. Golby, I. Norton, M. Hayashi, N. Y. R. Agar, G. S. Young, C. Spino, S. Santagata, S. Camelo-Piragua, K. L. Ligon, O. Sagher, and X. S. Xie, “Rapid, label-free detection of brain tumors with stimulated raman scattering microscopy,” Sci. Transl. Med. 5, 201ra119 (2013). [CrossRef] [PubMed]
12. L. Wei, Y. Yu, Y. Shen, M. C. Wang, and W. Min, “Vibrational imaging of newly synthesized proteins in live cells by stimulated raman scattering microscopy,” P. Natl. Acad. Sci. USA 110, 11226–11231 (2013). [CrossRef]
13. K. I. Popov, A. F. Pegoraro, A. Stolow, and L. Ramunno, “Image formation in cars microscopy: effect of the gouy phase shift,” Opt. Express 19, 5902–5911 (2011). [CrossRef] [PubMed]
14. A. M. Barlow, K. Popov, M. Andreana, D. J. Moffatt, A. Ridsdale, A. D. Slepkov, J. L. Harden, L. Ramunno, and A. Stolow, “Spatial-spectral coupling in coherent anti-stokes raman scattering microscopy,” Opt. Express 21, 15298–15307 (2013). [CrossRef] [PubMed]
15. K. I. Popov, A. F. Pegoraro, A. Stolow, and L. Ramunno, “Image formation in cars and srs: effect of an inhomogeneous nonresonant background medium,” Opt. Lett. 37, 473–475 (2012). [CrossRef] [PubMed]
16. von F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica 1, 689–704 (1934). [CrossRef]
17. N. Djaker, D. Gachet, N. Sandeau, P.-F. Lenne, and H. Rigneault, “Refractive effects in coherent anti-stokes raman scattering microscopy,” Appl. Opt. 45, 7005–7011 (2006). [CrossRef] [PubMed]
18. P. Ferrand, J. Wenger, A. Devilez, M. Pianta, B. Stout, N. Bonod, E. Popov, and H. Rigneault, “Direct imaging of photonic nanojets,” Opt. Express 16, 6930–6940 (2008). [CrossRef] [PubMed]
19. M. Weinigel, H. G. Breunig, M. E. Darvin, M. Klemp, J. Röwert-Huber, J. Lademann, and K. König, “Impact of refractive index mismatches on coherent anti-stokes raman scattering and multiphoton autofluorescence tomography of human skin in vivo,” Phys. Med. Biol. 60, 6881 (2015). [CrossRef] [PubMed]
20. C. Liu, Z. Huang, F. Lu, W. Zheng, D. W. Hutmacher, and C. Sheppard, “Near-field effects on coherent anti-stokes raman scattering microscopy imaging,” Opt. Express 15, 4118–4131 (2007). [CrossRef] [PubMed]
21. J. Lin, H. Wang, W. Zheng, F. Lu, C. Sheppard, and Z. Huang, “Numerical study of effects of light polarization, scatterer sizes and orientations on near-field coherent anti-stokes raman scattering microscopy,” Opt. Express 17, 2423–2434 (2009). [CrossRef] [PubMed]
22. P. K. Upputuri, Z. Wu, L. Gong, C. K. Ong, and H. Wang, “Super-resolution coherent anti-stokes raman scattering microscopy with photonic nanojets,” Opt. Express 22, 12890–12899 (2014). [CrossRef] [PubMed]
23. R. W. Boyd, Nonlinear Optics (Academic, 2008), 3rd ed.
24. M. Fujii, M. Tahara, I. Sakagami, W. Freude, and P. Russer, “High-order fdtd and auxiliary differential equation formulation of optical pulse propagation in 2-d kerr and raman nonlinear dispersive media,” IEEE J. Quantum Elect. 40, 175–182 (2004). [CrossRef]
