Resolution of spectral focusing in coherent Raman imaging

Mojtaba Mohseni; Christoph Polzer; Thomas Hellerer

doi:10.1364/OE.26.010230

1. Introduction

Coherent Raman Scattering (CRS) enables non-invasive imaging in cell biology offering a three dimensional view of the specimen without the need of labeling [1]. The chemically specific Raman contrast of the images is ensured by a spectral resolution of 15 cm⁻¹. Different approaches demonstrated that this requirement can be accomplished even with 150 cm⁻¹ broad fs-laser pulses: Coherent quantum control [2,3], time-gated multiplex coherent anti-Stokes Raman scattering (CARS) [4] and spectral focusing (SF) [5]. Among them, SF has recently received a lot of attention, since it cannot only be applied to CARS- [6–13], but also to stimulated Raman scattering (SRS) microscopy [14–18]. The implementation of SF is attractive for two reasons. Firstly, it allows the simultaneous use of a single laser source for the most popular imaging modalities in multimodal microscopy like two photon excited fluorescence (TPEF), second and third harmonic generation (SHG, THG) together with CRS [6–8]. Secondly, very fast spectral tuning is possible with SF by simply changing the time delay between the laser pulses. The high tuning speed is essential for hyperspectral imaging that combines high information content with high imaging speed [17,18].

Since its introduction for CARS microscopy, several groups have further investigated and developed the SF technique [6–20]. Chirping of the laser pulses that is required for SF, has been accomplished by material dispersion of glass blocks as the simplest way [9], by grating compressors [5], by acousto-optic crystals [18] or by employing spatial light modulators (SLM) as the most sophisticated way [19,20]. SF simplified CRS setups further by the usage of supercontinua from a PCF [6,13,14] or ultrabroad spectra from a single Ti:Sa that replaced the second laser source for the Stokes beam [8,10]. Finally, much faster tuning has recently been achieved by replacing motorized first-time delay stages by much faster acousto-optic crystals [18].

In this article, we focus on the spectral resolution achievable with SF and we present experiments for determining the exact value for the resolution in coherent Raman imaging. We demonstrate how third order dispersion, which to date has not been considered in studies of SF, deteriorates the achievable spectral resolution. To this end, we examine the dispersion up to the third order and its impact on CARS spectra. Furthermore, we estimate an upper limit for the resolution regarding common approaches and propose better schemes to overcome this barrier. As a resume, we present a practical guide to the implementation of SF in CRS microscopy.

2. Theory

Application of SF to CRS requires laser pulses that are chirped due to dispersion. In the following, we describe the theoretical background necessary to understand the spectral resolution enhancement in SF. Since vibrational spectra are typically depicted in wavenumbers $\tilde{ν}$ [cm⁻¹], we chose this as unit instead of v [Hz].

2.1 Second Order Dispersion

Dispersion acting on ultrashort laser pulses is treated in detail elsewhere [21]. Here, we give only a short summary. Dispersion is introduced by expanding the phase $φ$ of a broadband laser pulse around the central frequency $ω_{0}$ into the spectral domain:

φ (ω) = φ (ω_{0}) + \frac{\partial φ}{\partial ω} |_{ω_{0}} (ω - ω_{0}) + \frac{1}{2!} \frac{\partial^{2} φ}{\partial ω^{2}} |_{ω_{0}} {(ω - ω_{0})}^{2} + \frac{1}{3!} \frac{\partial^{3} φ}{\partial ω^{3}} |_{ω_{0}} {(ω - ω_{0})}^{3} + ...

The quadratic and cubic terms describe the group delay dispersion (

G D D : = \partial^{2} φ / \partial ω^{2}

) and third order dispersion (

T O D : = \partial^{3} φ / \partial ω^{3}

), respectively. They alter the frequency distribution of the laser pulse whereas the first two terms have no influence in this regard. Fourier transformation of the laser pulse leads to the time dependent phase whose derivative with respect to time yields the instantaneous central frequency

ω (t)

of the laser pulse for considering dispersion up to second order:

\frac{\partial φ (t)}{\partial t} = ω (t) = ω_{0} + 2 β \cdot t

The parameter

β

describes the linear slope of the central frequency referred to as linear chirp. The Fourier transformation relates the chirp parameter

β

to the GDD applied to the laser pulse in the following way:

β = \frac{2 G D D}{τ_{0}^{4} + 4 G D D^{2}} \approx \frac{1}{2 G D D} f o r τ ≫ τ_{0}

The application of GDD stretches the pulse width

τ_{0}

of the Fourier transform limited (FTL) pulse to the chirped width

τ

:

τ = τ_{0} \sqrt{1 + {[\frac{4 ln 2 \cdot G D D}{τ_{0}^{2}}]}^{2}} \approx 2.77 \frac{| G D D |}{τ_{0}} for τ ≫ τ_{0}

Interestingly, the same amount of GDD stretches shorter FTL pulses further than longer FTL pulses. Connecting Eq. (3) and (4) reveals that the product

τ \cdot τ_{0} \approx 4 \ln 2 | G D D | \approx 2 \ln 2 / | β |

is a direct measure of the chirp parameter

β

for

τ ≫ τ_{0}

. The consequences of dispersion for the laser frequency distribution in time are depicted in Fig. 1. The time-bandwidth product

Δ ω_{0} \cdot τ_{0} = 4 \ln 2 \approx 2.77

determines the area of the ellipses shown in Fig. 1 and therefore the bandwidth of a FTL laser pulse. The same relation

Δ ω \cdot τ = 2.77

holds true for the chirped pulse width

τ

and the instantaneous bandwidth

Δ ω

of the pulse. Consequently, stretching the pulse in time decreases the instantaneous bandwidth by the stretching factor. On the other hand, the overall bandwidth

Δ ω_{0}

remains unchanged (Fig. 1(b)).

Fig. 1 Frequency-time distribution, the so-called Wigner distribution, (a) for a Fourier-transform limited laser pulse and (b) the same laser pulse that is linearly chirped.

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2.2 Resolution in Coherent Raman Imaging

In coherent Raman imaging, two laser pulses called pump and Stokes with center frequencies $ω_{p}$ and $ω_{s}$ are overlapped in time for resonant enhancement of a non-linear signal by a Raman band at the vibrational frequency $Ω = ω_{p} - ω_{s}$ . The overall bandwidth $Δ ω = \sqrt{Δ ω_{S}^{2} + Δ ω_{P}^{2}}$ for the coherent Raman excitation is converted to wavenumbers using $Δ \tilde{ν} = Δ ω / 2 π c$ . It describes the bandwidth of the driving force $F ~ E_{p} E_{s}^{*}$ for the Raman active vibration from the nonlinear interaction of pump and Stokes electric fields, $E_{p}$ and $E_{s}$ . For the spectral resolution $Δ \tilde{ν}$ in CRS microscopy, Eq. (5) applies:

Δ \tilde{ν} = \frac{2 \ln 2}{π c} \sqrt{2 (τ_{p}^{- 2} + τ_{s}^{- 2})} = 20.8 ps \cdot {cm}^{- 1} \sqrt{τ_{p}^{- 2} + τ_{s}^{- 2}}

This equation is valid for either Fourier-transform-limited or equally chirped pulse widths

τ_{p}

of the pump laser pulse and

τ_{s}

of the Stokes laser pulse regardless of the time delay between both pulses. Comparison of Figs. 2(a) and 2(b) illustrates the improvement in resolution by use of SF. Previous work reported consistency between theoretical simulations based on this assumption and their experimental results in SRS as well as in CARS microscopy [18, 20].

Fig. 2 Coherent Raman excitation with (a) FTL laser pulses, (b) laser pulses with the same chirp, (c) laser pulses with different chirp, (d) frequency scan with time delay and (e) frequency scan with time delay in presence of TOD.

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2.3 Time to Frequency Mapping

In SF, the coherent Raman spectrum results from the cross-correlation of the pump and Stokes laser pulse by mapping the time-axis of the cross-correlation to wavenumbers using the following calibration:

d \tilde{v} = \frac{d t}{2 π c \cdot G D D} = 5.3 ps \cdot {cm}^{- 1} \frac{d t}{G D D}

This is visualized in Fig. 2(d) and valid for linearly chirped laser pulses only. It cannot be applied to FTL pulses because their spectral components are all present at the same time as shown in Fig. 2(a). Additionally, Eq. (6) reveals the negative effect of the timing jitter between pump and Stokes pulses on the spectral resolution. For example, a timing jitter of dt = 100 fs corresponds to

d \tilde{v} = 6.6 {cm}^{- 1}

spectral jitter if the applied GDD amounts 80000 fs². Connecting the three Eqs. (4), (5) and (6) leads to the result that the following time step

Δ t = \sqrt{2 (τ_{0 p}^{2} + τ_{0 s}^{2})}

corresponds to the spectral resolution

Δ \tilde{v}

of SF and does not depend on the applied GDD. Here,

τ_{0 p}

and

τ_{0 s}

are the FTL pulse widths of the pump and the Stokes beam, respectively. Equation (7) implies that the optimum sampling step size of the cross-correlation that is given by the achieved resolution, is only determined by the FTL pulse widths.

Mismatch of Chirps

It has to be considered that the spectral resolution depends not only on each pulse separately but as well on the interplay between the two pulses. Therefore, the slope $2 β \approx G D D^{- 1}$ of the instantaneous frequency $ω (t)$ is important as well. Two chirped pulses have the same slope only if the corresponding products $τ \cdot τ_{0}$ are equal. Different slopes as shown in Fig. 2(c) deteriorate the spectral resolution that broadens by the additional term $Δ {\tilde{ν}}_{Δ β}$ :

Δ {\tilde{ν}}_{Δ β} = \frac{| Δ β |}{π c \sqrt{τ_{p}^{- 2} + τ_{s}^{- 2}}} \approx 10.4 ps \cdot {cm}^{- 1} \frac{Δ G D D}{τ_{m} G D D_{m}} for \frac{Δ G D D}{G D D_{m}} ≪ 1 a n d \frac{Δ τ_{0}}{τ_{m}} ≪ 1

where the approximation applies if the FTL pulse widths

τ_{0}

and the GDD for both pulses do not differ much. Here,

τ_{m}

is the mean value of the FTL pulse widths and

G D D_{m}

is the mean value of the GDD applied to both pulses. By contrast,

Δ τ_{0}

denotes the difference of FTL pulse widths, whereas

Δ G D D

denotes the difference of GDD that leads to the deterioration of the resolution. The additional term

Δ {\tilde{ν}}_{Δ β}

is dependent on the time delay

Δ t

between the pulses: the temporal overlap described by the expression

1 / \sqrt{τ_{p}^{- 2} + τ_{s}^{- 2}}

for

Δ t = 0

has to be replaced by the actual overlap for a time delay

Δ t \neq 0

in Eq. (8).

2.4 Third Order Dispersion

If only pulse broadening by GDD is considered, there is no limit for resolution enhancement by SF. However, most common chirping devices apply not only GDD but also TOD and higher order dispersion as well [22]. To exemplify the order of magnitude, Table 1 shows the calculated amounts of TOD that different devices apply together with a GDD of 80000 fs² to a laser pulse with center wavelength λ = 800 nm. This amount of GDD stretches a 100 fs pulse to 2.2 ps according to Eq. (4). Two laser pulses with different wavelengths but with the same stretched pulse width of 2.2 ps used as pump and Stokes lead to the resolution of 13.3 cm⁻¹ according to Eq. (5), a typical value used for coherent Raman microscopy.

Table 1. TOD of different chirping devices for applying GDD = 80000 fs² to a laser pulse at λ = 800 nm. Calculations were carried out according to [22] but with errors corrected.

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The theoretically derived amounts of TOD represent the lower limits for practical realizations because TOD is very sensitive to misalignments and may be much higher for actual setups of prism and grating based chirping devices. All devices exhibit a fixed ratio TOD/GDD that is highest for gratings followed by prisms and material dispersion. As a rule of thumb, microscope optics introduce a GDD of only about 8000 fs² whereas chirping devices employed for spectral focusing introduce a GDD of 80000 fs². Consequently, the TOD of the microscope is ten times smaller than that of the chirping device and therefore negligible. Increasing the applied dispersion with the chirping devices has two counteracting effects on spectral focusing. On the one hand, the applied GDD improves the resolution. On the other hand, however, the accompanying TOD decreases the resolution again. Furthermore, the non-linear slope of the laser frequency distributions changes the time to frequency mapping that becomes non-linear too, hence Eq. (6) no longer applies. Additionally, the resolution is affected for the same reason as depicted in Fig. 2(e). When scanning the Raman frequencies by means of a time delay, the tuning is no longer smooth but the Raman spectrum is disturbed by periodic spectral features.

3. Materials and Methods

The optical setup for this study is shown in Fig. 3. The laser system is a femtosecond fiber laser (FemtoFiber Dichro Design, TOPTICA Photonics AG) that operates at a fixed repetition rate of 80 MHz. It emits two synchronized laser pulses with wavelengths centered at λ_p = 784 nm and λ_s = 1034 nm and pulse widths (FWHM) of $τ_{p} = 90 fs$ and $τ_{s} = 95 fs$ , respectively. Aim of this study is to understand how dispersion affects the spectral resolution in CARS microscopy based on spectral focusing. For this purpose, a spatial light modulator (Cambridge Research & Instrumentation) was used to change the spectral phase $φ (ω)$ of the FTL laser pulses in an arbitrary manner giving full control over GDD and TOD. In order to apply sufficient second order dispersion without pushing the spatial light modulator (SLM) to its limit, the pulses are pre-stretched to 0.7 – 0.8 ps by passing first two prism stretchers before entering the SLM. In our setup, we used two pulse stretchers but only a single SLM. The laser beams hit the liquid crystal array of the SLM at different locations enabling individual manipulation of their spectral phases $φ (ω)$ referenced to each center frequency. This was possible, because the array contained as many as 640 pixels enabling the division into two independently acting parts with a spectral resolution better than 0.3 nm/px for each wavelength. The optical setup included a 4f-configuration of the concave mirror to Fourier transform the time domain into the spectral domain at the location of the SLM. Therefore, the grating induced virtually no dispersion in conjunction with the concave mirror over the full spectral range. In our study, the SLM works in three ways: Firstly, it completely cancels out higher order dispersion induced by the strong dispersive elements like the prisms and the grating. Secondly, it applies additional GDD of both signs to stretch further or to recompress the laser pulses. Thirdly, it adds well-defined amounts of TOD.

Fig. 3 Setup (M, M1: mirror, D: dichroic mirror, R: retroreflector, G: optical grating, CM: concave mirror, L: lens, Ob: objective, F: filter, BBO: SHG crystal). The femtosecond laser pulses were chirped by a prism stretcher. The retroreflector induced a height offset. After the prism stretcher, the beam was reflected by the mirror M1 and D1 onto the optical grating. Thereafter, the dispersion of 3rd and higher orders was compensated using the SLM. The green line indicates the pump laser, the red line the Stokes laser. The SLM also allowed additional desired phase modulation. A FROG setup was used to analyze the modulated pulses. Subsequently, the laser pulses were coupled into a multiphoton microscope (MPM).

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A home-built FROG setup based on second harmonic generation was employed to characterize the laser pulses after the SLM. Therefore, not only the pulse width but also the phase $φ (t)$ of the dispersion modulated laser pulses could be measured accurately. The pump and Stokes laser pulses were coupled into a home-built multiphoton microscope. For laser scanning, a two-axis resonant galvo scanner (Thorlabs Inc.) allows for high speed scanning of the sample with a frame rate of 12.5 fps. The frames were averaged 20 times in order to achieve a good signal-to-noise ratio in the CARS image. A Nikon Apo TIRF oil immersion objective (100x, NA 1.49) focused the beams onto the sample. The objective was designed for correcting chromatic aberrations in the range of 435 – 1064 nm which ensured the spatial overlap of the excitation beams. The generated CARS signal was collected in forward direction by a Nikon NIR Apo lens (60x, NA 1.0) and detected with an InGaAsP-photomultiplier tube (PMT, Hamamatsu Photonics) after spectral filtering with two bandpass filters.

Two samples were examined in this study: iron-(II)-iodate (Fe(IO₃)₂) and 2-chlorobenzamide (C₇H₆CINO). Fe(IO₃)₂ crystallizes in a non-centrosymmetric lattice and exhibits a strong second order non-linearity. It therefore lends itself to non-resonant sum frequency generation (SFG) experiments. In this study, SFG is used to evaluate the chirp of the laser pulses by determining their widths at the focus of the microscope. By contrast, 2-chlorobenzamide features a very sharp vibrational band for the aromatic CH-stretch vibration which makes it a good sample for exemplifying SF in non-linear Raman microscopy experiments, in our case CARS microscopy. Figure 4(a) shows an example of the SFG images and Fig. 4(b) one of the CARS images. For microscopy purposes, a small amount of each powder was sandwiched between a cover glass and a microscopy slide and sealed with nail polish. A set of at least 30 images was taken at different time delays between the laser pulses of the two wavelengths. The collected images were then analyzed and evaluated by ImageJ [23] and MATLAB (MathWorks, Inc.), respectively. A region of interest was chosen in each set for calculating its mean grey value. The mean values were then plotted against time delay representing a cross-correlation of the laser pulses. For non-resonant SFG, the cross-correlation directly mirrors the linear chirp of the laser pulses whereas for CARS, the resonant excitation of the Raman band alters the non-linear signal. Therefore, the sample’s spectral properties are important. Figure 4(c) shows the spontaneous Raman spectrum of 2-chlorobenzamide. It exhibits a single Raman band centered at 3067 cm⁻¹ that has a linewidth of 8.5 cm⁻¹. The powder was measured directly on a glass plate with a Raman spectrometer equipped with a He-Ne laser (λ = 633 nm). For the measurement, a confocal LabRAM HR UV/VIS (HORIBA Jobin Yvon) Raman microscope (Olympus BX 41) with a SYMPHONY CCD detection system was used.

Fig. 4 (a) SFG microscopy image of the sample iron-(II)-iodate, (b) CARS microscopy image of the sample 2-chlorobenzamide and (c) Raman spectrum of the latter.

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4. Experimental Results and Discussion

4.1 Third Order Dispersion

The detrimental effects of TOD on the spectral resolution have been described above. Figure 5 displays the measured CARS spectra of 2-chlorobenzamide for different amounts of added TOD. For better comparison, the spectra are normalized. Exerting full control on the dispersion, the SLM of our setup applied TOD of different amounts, i.e. 0 fs³, 5·10⁵fs³ and 1.5·10⁶fs³, while keeping the GDD of 69000 fs² including that of the microscope optics. We stress that GDD and TOD have the same amount but are referenced to the individual center frequencies of the laser pulses.

Fig. 5 (a) CARS images of 2-chlorobenzamide taken at different time delays between the laser pulses. The intensity changes reflect the spectral scanning of the vibrational resonance. (b) Extracted normalized CARS spectra with varying amounts of TOD but the same amount of GDD = 69000 fs² leading to a chirped pulse width of about 2 ps. TOD is completely compensated (red); TOD equals 5 ⋅ 10⁵ fs³ (brown); TOD equals 1.5 ⋅ 10⁶ fs³ (green).

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Not surprisingly, the highest spectral resolution is achieved with total compensation of TOD by the SLM (red curve in Fig. 5) whereas in the other cases, a spectral broadening can be observed. The spectral widths are $Δ \tilde{ν} = 16.6_{} c m^{- 1}$ for $TOD = 0_{} f s^{3}$ , $Δ \tilde{ν} = {42.2}_{} c m^{- 1}$ for $TOD = 5 \cdot 10^{5} f s^{3}$ , and $Δ \tilde{ν} = 74.5_{} c m^{- 1}$ for $TOD = 1.5 \cdot 10^{6} f s^{3}$ . The latter two TOD values are matched to the exemplary value for grating pairs shown in Table 1 and three times this value. The highest value was chosen to demonstrate that TOD can alter the time to frequency mapping to a degree that the derived spectrum could lead to the misinterpretation of the vibrational spectra (green curve in Fig. 5(b)). Combining the experimental results shown in Fig. 5 with the calculations shown in Table 1 leads to the conclusion that for a GDD higher than 100 000 fs², the negative effects of TOD will cancel out the improvement in resolution for any of the listed chirping devices. For FTL pulse widths shorter than $τ_{0} = 100 f s$ , even smaller amounts of TOD will affect the resolution significantly because it acts on the much broader spectral bandwidth of the pulses. Avoiding the detrimental effects of TOD for such high amounts of GDD is not straightforward but requires significant experimental efforts, as for example, the addition of an SLM to the setup to fully compensate higher order dispersion. Another option would be the combination of grating and prism devices that cancel out TOD due to their opposite signs as was reported for the compression of sub-10 fs laser pulses [22]. A specially designed GRISM could even integrate both stretcher types into a single device [24–26].

4.2 Second Order Dispersion

However, a somewhat lower GDD of 69000 fs² applied to a 90 fs laser pulse leads to the excitation bandwidth of 14 cm⁻¹ that is best suited for coherent Raman microscopy because it fits the typical Raman linewidths [10]. Therefore, the next experiment explores the resolution for lower amounts of GDD and negligible amounts of TOD validating Eq. (5). We investigated three cases. In the first case (A), the dispersion of all optical components like the prism stretcher and the microscope optics is completely compensated by the SLM at the focus of the microscope (GDD is zero). In the second case (B), the overall amount of applied GDD is 45000 fs² leading to chirped pulse widths of 1.3 ps. In the third case (C), the amount of applied GDD is 69000 fs² leading to chirped pulse widths of 2 ps. Both SFG and CARS measurements were performed. Whereas the SFG experiments reflect the properties of the laser pulses only, the CARS data additionally show the effect of the interaction with a typical vibrational resonance. The normalized SFG cross-correlations for the three cases are plotted in Fig. 6(b). The fitted widths of (A) $τ_{c c} = 134_{} fs$ , (B) $τ_{c c} = 1930_{} fs$ and (C) $τ_{c c} = 2940_{} fs$ agree very well with Eq. (4) taking the expression $τ_{c c} = \sqrt{τ_{S}^{2} + τ_{P}^{2}}$ for the cross-correlation into account.

Fig. 6 (a) SFG images of iron-(II)-iodate at different time delays between the two laser pulses. (b) Corresponding normalized cross-correlations of the laser pulses: The blue curve shows case (A) of FTL laser pulses. The black and red curves show cases (B) and (C) where both laser pulses are chirped to about 1.3 ps and 2 ps, respectively.

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Exchanging the sample on the microscope allows to measure CARS cross-correlations for the three cases as well. These are normalized and shown in Fig. 7(a). One striking fact is the ten times narrower width of the CARS data as compared to the SFG correlations. In spectral focusing, the change in time-overlap sweeps over the Raman frequencies $Ω$ that are probed which is illustrated in Fig. 2(d). The narrower widths thus result from the fact that the frequency dependence of the Raman resonance is much stronger than the change in temporal overlap of the two pulses responsible for the shape of SFG cross-correlations. The widths of the chirped cases (B) and (C) in Fig. 7(a) do not vary much because of the GDD-independent time step size according to Eq. (7). On the other hand, the shape of both curves differ because for case (B) there are more non-resonant contributions to the CARS process than for case (C).

Fig. 7 CARS cross-correlations of 2-chlorobenzamide extracted from the intensity microscopy image stacks (not shown) for different laser pulse widths. Cases A (blue), B (black) and C (red) demonstrate the improvement in spectral resolution. a) plotted against time delay b) plotted against Raman shift. The inset shows the Raman spectrum of the sample.

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In Fig. 7(b), the CARS spectra are presented by mapping the time-axis of the cross-correlations to wavenumbers using Eq. (6) except for the case (A) where GDD equals zero and time to frequency mapping is not possible as mentioned above. The absolute position of the highest peak of curve (B) and (C) is referenced to the spectral location of the Raman band at 3067 cm⁻¹. Figure 7(b) shows the improvement in spectral resolution with increasing amounts of GDD. The experimental parameters are listed in Table 2 together with the theoretical resolution according to Eq. (5).

Table 2. Overview of experimental parameters and results and comparison with theoretical values.

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The experimental resolution is determined by the deconvolution $Δ \tilde{ν} = \sqrt{Δ {\tilde{ν}}_{c}^{2} - Δ_{R}^{2}}$ of the CARS spectral width $Δ {\tilde{ν}}_{c}$ and the independently measured Raman linewidth $Δ {\tilde{ν}}_{R} = 8.5_{} c m^{- 1}$ of the sample. There is excellent consistency between experiment and theory. We want to emphasize that the different shapes of the curves are due to varying non-resonant contributions to the CARS process and not to remaining higher order dispersion as it is the case for the spectra shown in Fig. 5.

4.3 FROG Measurements

To proof the last point, we characterized the laser pulses with a FROG (Frequency Resolved Optical Gating) setup independently. Figure 8(a) shows a FROG trace and the corresponding autocorrelation of the chirped laser pulse at wavelength 1034 nm. Here, the SLM also adds GDD to the pre-stretched laser pulse as it compensates all higher order dispersion at the same time. The purely quadratic phase indicates the absence of higher order dispersion. The initially 95 fs long laser pulse is stretched by a factor 21 to 2.0 ps. As another example, Fig. 8(b) shows the pre-stretched pulse at wavelength 784 nm after its recompression to the Fourier-limit accomplished by the SLM that is indicated by the flat phase.

Fig. 8 FROG traces and autocorrelations of (a) the chirped Stokes pulse (λ = 1034 nm, τ = 2.0 ps) and (b) of the recompressed pump pulse (λ = 784 nm, τ = 90 fs).

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4.4 Dispersion of the Microscope

Finally, we want to stress that the dispersion of the microscope optics influences the resolution in spectral focusing as well. The GDD of the microscope optics differs for pump and Stokes pulses leading to different chirp parameters according to Eq. (3). This deteriorates the resolution according to Eq. (8). The simplest way of determining the GDD of the microscope accurately is the measurement of SFG cross-correlations in conjunction with a variable chirping device. Figure 9 shows the full width at half maximum of 15 cross-correlations as a function of the absolute value of the applied GDD on the laser pulses with the chirping device. In Fig. 9(a), the GDD applied on the 784 nm pulse is fixed whereas the amount of negative GDD applied on the 1034 nm pulse varies as indicated by the x-axis. Figure 9(b) shows the analog measurement with interchanged wavelengths. The width of the cross-correlation is $τ_{cc} = \sqrt{τ_{p}^{2} + τ_{s}^{2}}$ . This function is used for curve fitting the data by inserting Eq. (4) for one width of the pulses and leaving the other constant. The minimum of the regression curve determines the induced GDD of the microscope optics for the respective laser pulse independently of the other laser pulse. As a result, the microscope optics introduce a GDD of 4513 ${fs}^{2}$ and of 3035 ${fs}^{2}$ on pump (784 nm) and Stokes (1034 nm) pulses, respectively.

Fig. 9 GDD optimization of the microscope optics for both wavelengths 1034 nm and 784 nm at the location of the sample. (a) The pulse of the wavelength 784 nm is compensated while the GDD of the other pulse is varied. (b) GDD variation at 784 nm while the phase of the other pulse is compensated.

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Ignoring the difference in GDD would not result in a resolution of $Δ \tilde{ν} = 14.3 c m^{- 1}$ for case (C) but in $Δ \tilde{ν} = 16.8 c m^{- 1}$ that is slightly worse. The difference of $2.5 c m^{- 1}$ is calculated by inserting $τ_{m} = 92.5 fs, Δ G D D = 1500^{} {fs}^{2}$ and $G D D_{m} = 68250 {fs}^{2}$ into Eq. (8). For compensation of the measured microscope GDD, a phase profile corresponding to the negative amount of the measured GDD was applied on the SLM. Thus, the spectral phase is zeroed at the focus of the microscope due to the superposition. To ensure the proper function of the setup and to double-check the results for the measured GDDs, the cross-correlation between the laser pulses with compensated and uncompensated GDDs was measured. The results are shown in Fig. 10. The red curve shows the cross-correlation without dispersion compensation of the microscope optics whereas the blue curve shows the cross-correlation of fully compensated pulses that is significantly narrower. The measured width $τ_{cc} = 134 fs$ of the compensated laser pulses corresponds very well to the theoretical value for FTL-pulses $τ_{cc} = \sqrt{95 {fs}^{2} + 90 {fs}^{2}} = 131 fs$ . The width for the uncompensated laser pulses is $τ_{cc} = 212 fs$ .

Fig. 10 SFG cross-correlations for compensated (blue) and uncompensated laser pulses (red).

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5. Conclusion

Spectral focusing enables coherent Raman imaging with high spectral resolution. Common chirping devices apply not only GDD but also TOD to the laser pulses, which limits the resolution. The negative effects on 100 fs FTL pulses are significant for GDD amounts greater than 100 000 fs² using common chirping devices like gratings, prisms and glass blocks. The achievable spectral resolution is therefore limited to 10 cm⁻¹. For a better resolution, we propose to use GRISM or SLM setups that are able to compensate TOD.

The implementation of SF in CRS microscopy and its integration into a multimodal microscope is straightforward and does not require advanced laser diagnostic tools like FROG (Frequency Resolved Optical Gating), SPIDER (Spectral Phase Interferometry for Direct Electric-field Reconstruction) or MIIPS (Multiphoton Intrapulse Interference Phase Scan). On the contrary, we showed that taking SFG images at varying time delays for measuring a cross-correlation of the laser pulses is sufficient to determine the chirp parameter given by the product $τ \cdot τ_{0}$ and to investigate the dispersion of the microscope. Finally, the achieved spectral resolution and the absence of significant TOD can be verified with the CRS cross-correlation of a suitable sample like 2-chlorobenzamide that has a single sharp Raman line in the popular 3000 cm⁻¹ region and that is cheap and easy to handle.

Funding

Bayerisches Staatsministerium für Bildung und Kultus, Wissenschaft und Kunst (SERAFIM).

Acknowledgments

The authors would like to thank Andreas Zumbusch and Annika Enejder for fruitful discussions. The authors are grateful to Stefan Datz for measuring the Raman spectrum of 2-chlorobenzamide and Matthias Handloser for providing the iron-(II)-iodate sample. The authors appreciate the support of TOPTICA Photonics that lended the fiber laser and SLM.

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Chirping device	TOD [fs³]
Grating pair: distance x = 5.261 cm, groove spacing d = 1.2 µm	-4.5·10⁵
Brewster prism pair (SF57): tip-to-tip distance x = 4.9 m	2.1·10⁵
Material SF57 glass: length L = 36.6 cm	5.3·10⁴

Case	applied GDD [fs²]	Pump/Stokes pulse width [fs] Eq. (4)	SFG width [fs]		CARS spectral width [cm⁻¹] from Fig. 7(b)	resolution [cm⁻¹]
Case	applied GDD [fs²]	Pump/Stokes pulse width [fs] Eq. (4)	theor.	exp.	CARS spectral width [cm⁻¹] from Fig. 7(b)	experiment	Equation (5)
A	0	90/95	131	134	-	-	318
B	45000	1390/1310	1910	1930	23.6	22.0	21.8
C	69000	2120/2010	2920	2940	16.6	14.3	14.3

Chirping device	TOD [fs³]
Grating pair: distance x = 5.261 cm, groove spacing d = 1.2 µm	-4.5·10⁵
Brewster prism pair (SF57): tip-to-tip distance x = 4.9 m	2.1·10⁵
Material SF57 glass: length L = 36.6 cm	5.3·10⁴

Case	applied GDD [fs²]	Pump/Stokes pulse width [fs] Eq. (4)	SFG width [fs]		CARS spectral width [cm⁻¹] from Fig. 7(b)	resolution [cm⁻¹]
Case	applied GDD [fs²]	Pump/Stokes pulse width [fs] Eq. (4)	theor.	exp.	CARS spectral width [cm⁻¹] from Fig. 7(b)	experiment	Equation (5)
A	0	90/95	131	134	-	-	318
B	45000	1390/1310	1910	1930	23.6	22.0	21.8
C	69000	2120/2010	2920	2940	16.6	14.3	14.3

Resolution of spectral focusing in coherent Raman imaging

Abstract

1. Introduction

2. Theory

2.1 Second Order Dispersion

2.2 Resolution in Coherent Raman Imaging

2.3 Time to Frequency Mapping

Mismatch of Chirps

2.4 Third Order Dispersion

3. Materials and Methods

4. Experimental Results and Discussion

4.1 Third Order Dispersion

4.2 Second Order Dispersion

4.3 FROG Measurements

4.4 Dispersion of the Microscope

5. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (10)

Tables (2)

Equations (8)

Optics Express