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Third harmonic generation and microscopy, enhanced by a bias harmonic field
Optics Communications 457 (2020) 124660
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Third harmonic generation and microscopy, enhanced by a bias harmonic field Christian Stock a ,∗, Kaloyan Zlatanov b , Thomas Halfmann a a b
Technische Universität Darmstadt, Institut für Angewandte Physik, Hochschulstraße 6, 64289 Darmstadt, Germany Institute of Solid State Physics, Bulgarian Academy of Sciences, Tsarigradsko chaussée 72, 1784 Sofia, Bulgaria
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Keywords: Nonlinear microscopy Third harmonic generation Ultrafast nonlinear optics Dispersion
ABSTRACT We present strong enhancements of the signal yield in third harmonic generation microscopy by seeding the optically nonlinear sample with some intensity at the third harmonic wavelength, in addition to the driving fundamental beam. By applying a third harmonic bias pulse with a power of less then 0.1% of the fundamental beam, we boost the signal yield by more than a factor of 3000, compared to the conventional third harmonic signal from the microscopy sample. The signal enhancement is most pronounced at low laser intensity and/or weak nonlinear susceptibilities. This makes the concept particularly suitable to improve the signal-to-noise ratio from samples with weak signals, e.g., as typical for applications of nonlinear microscopy. Moreover, we demonstrate improved spatial resolution in beam propagation direction by more than an order of magnitude. This exploits the dependence of our approach for enhanced third harmonic generation from inevitable dispersion in the sample. The improvement is most evident in cases, where the sample geometry allows only weak focusing.
1. Introduction Coherent nonlinear microscopy (CNM) is a powerful tool for threedimensional imaging [1,2]. It utilizes frequency conversion processes of ultrashort laser pulses, strongly focused into a sample. The image is obtained by monitoring radiation generated by nonlinear frequency conversion in the laser focus, while the focus is moved through the sample. Examples for CNM processes are second harmonic generation (SHG), third harmonic generation (THG) or coherent anti-stokes Raman scattering (CARS) [3]. In contrast to conventional two photon excitation fluorescence (TPEF) microscopy [4], CNM does not require labeling and does not deposit energy in the sample. This makes CNM particularly interesting for microscopic imaging of sensitive biological samples [5,6]. While CARS requires two laser beams to tune to vibrational Raman resonances in molecules for efficient frequency conversion, SHG and THG apply a single off-resonant laser beam only. This makes the latter two techniques rather simple to implement. Typically, microscopic imaging by SHG and THG is implemented with a fixed-frequency laser far off resonances in the sample. However, such far off-resonant nonlinear interaction yields only low optical conversion efficiencies and hence, rather limited signal, signal-to-noise ratio, and contrast. There are several approaches to enhance the signal yield in CNM, e.g. using resonant enhancements of the nonlinear susceptibility [7–9] or dispersion to optimize phase matching conditions [10]. However, these approaches are limited to specific media
or require tunable laser systems to drive the nonlinear interactions in the vicinity of resonances. Moreover, near-resonant excitation deposits energy in the sample. This may damage the sample by absorption of the fundamental radiation, or lead to losses by re-absorption of the nonlinear signal. Recently, an alternative, promising route to enhance signals in CNM by so-called optical stimulation was introduced, and demonstrated for the case of SHG [11]. The basic idea is to start the nonlinear optical process at an already higher signal intensity level from the beginning on — rather than from zero signal intensity. Hence, in the case of SHG, coincident with the fundamental radiation some weak radiation at the second harmonic is introduced into the sample. This additional bias harmonic pulse enhances the conversion process to produce more photons in the SHG detection channel. The previous demonstration of optically stimulated SHG yielded signal enhancements beyond 104 in a biologically relevant medium. The signal enhancements are the largest for low input power and weak nonlinear susceptibilities. We note, that the concept of optical stimulation by a bias harmonic field could also be understood in terms of heterodyne detection, with both approaches reconciled in a quantum electrodynamical treatment [12]. However, experimentally they differ from each other. In heterodyne detection, a harmonic field is generated in the sample and then mixed with a reference field. Hence, the fundamental field remains uneffected by the mixing. In optical stimulation the interaction takes place in the sample,
∗ Corresponding author. E-mail address: [email protected] (C. Stock).
https://doi.org/10.1016/j.optcom.2019.124660 Received 24 May 2019; Received in revised form 24 September 2019; Accepted 27 September 2019 Available online 30 September 2019 0030-4018/© 2019 Published by Elsevier B.V.
C. Stock, K. Zlatanov and T. Halfmann
Optics Communications 457 (2020) 124660
Stimulated THG requires a second incident field at the frequency of the generated harmonic 3𝜔. In undepleted pump approximation the bias harmonic field is simply:
and the energy is transferred between fundamental and harmonic fields depending on the phase [13,14]. The aim of our work is to transfer the concept of optical stimulation to THG imaging, and also investigate the effect of inevitable dispersion in the medium. In the following we will use the short term ‘‘stimulated THG’’ for the conversion process enhanced by a bias harmonic field, and compare it to conventional THG (i.e., without a bias field). While SHG is possible in non-centrosymmetric media only, THG occurs in any medium of arbitrary symmetry [2]. Therefore, THG microscopy has a much broader range of applications compared to SHG. We note, that THG imaging with tightly focused laser beams is sensitive to interfaces, i.e., the third harmonic is emitted only when the laser focus intersects the interface between two media of different nonlinear susceptibilities. This makes THG microscopy a powerful tool to image the structure of heterogeneous samples without background signal from bulk material. Stimulated THG can be implemented as an extension to a conventional THG microscope, without the need of a tunable laser, a second laser system, or any other larger modification of the experimental setup. Moreover, as the approach provides larger enhancement with decreasing nonlinear susceptibility, the potential signal enhancement and its impact on THG imaging is expected to be even larger in THG compared to SHG. In the following we will present a theoretical treatment of stimulated THG for Gaussian beams. Afterwards, we discuss successful experimental implementation of the concept, with systematic measurements of the signal yield from a fused silica sample, indicating enhancements up to a factor of 3000. We will also demonstrate improvements of the spatial resolution along the optical axis by exploiting dispersion in stimulated THG.
3𝑏 (𝑧) = 𝐴3𝑏 𝑒𝑖3𝜔∕(2𝑐)𝑛3 𝑧 𝑒𝑖𝜑 ,
i.e., the amplitude 𝐴3𝑏 remains constant, but the phase difference to the fundamental field changes with the optical path length 𝑛3 𝑧, and we also permit an initial phase 𝜑 with regard to the fundamental field. The bias field exhibits another term to be added to the wave equation (2), and consequently also adds to the solution in Eq. (3), yielding: 3 (𝐿) = 𝑖
1 + 𝑖𝜁 (𝑧)
{ } −𝑞𝑟2 ∕ 𝑤20 [1+𝑖𝜁(𝑧)]
𝑒
,
field. The last interference-like term 𝑃𝑠 ∝ 𝑃13 𝑃3𝑏 describes stimulated THG. The latter depends upon the phase 𝜗, which is equal to the phase 3𝜔𝑛 𝑧 of the bias harmonic field 𝛷(𝑧) = 2𝑐3 + 𝜑 (see Eq. (5)) plus a constant offset by the phases of the complex parameters 𝐽 (𝛥𝑘, 𝐿) and 𝜒 (3) . Thus, 𝜑 serves to control the direction of the energy flow, i.e., whether we get enhanced or suppressed THG emission. The relative strengths of conventional and enhanced THG are determined by the prefactor | |2 9𝜔2 |𝜒 (3) | | | 𝜅= |𝐽 (𝛥𝑘, 𝐿)|2 . 4𝑛31 𝑛3 𝜖02 𝑐 4 𝜋 2 𝑤40
1
Thus, we expect large enhancement for a strong bias harmonic field, but also for low conventional THG yield, e.g., low nonlinear susceptibility or low fundamental power. The latter is an important feature for realistic applications: The effect of stimulated THG is the largest, when the THG signal yield is low — which is very typical for nonlinear microscopy with its very low conversion efficiency. Moreover, we see that also the power ratio between the fields is of relevance. Assuming a constant total power 𝑃 = 𝑃1 + 𝑃3𝑏 available from our laser system, from Eq. (9) we find that the ratio 𝑃1 ∕𝑃3𝑏 = 3 yields the largest enhancement.
(1)
3. Experimental setup The experimental setup is schematically depicted in Fig. 1. A titanium sapphire laser oscillator (Spectra Physics, Tsunami), pumped by a DPSS system (Coherent, Verdi G7), provides pulses with a center wavelength of 810 nm, average output power of 1 W, pulse duration of 65(5) fs (FWHM), and repetition rate of 82 MHz (cf. Fig. 1). The fundamental radiation is frequency tripled in two beta barium borate (BBO) crystals by phase-matched SHG (Type I, 0.2 mm, 28.9◦ ) and subsequent sum frequency generation (SFG, Type II, 0.13 mm, 54.4◦ ) to provide the bias third harmonic beam at center wavelength of 270 nm and average output power up to 6 μW. As the conversion efficiency in the BBO crystals is very low, the beam profile of the fundamental beam is not distorted by the conversion stage. After the BBO crystals, the fundamental beam and bias harmonic beam are separated by a dichroic mirror (Layertec, #102854), and their intensity is varied by neutral density filters (Thorlabs, NDC-100C-2M, 2 mm thick). The beams propagate through a Mach–Zehnder-type interferometric setup involving a variable delay stage with piezo (Thorlabs, NFL5DP20S),
(2)
with the amplitude 𝐴31 of the driving fundamental field at carrier frequency 𝜔, the index of refraction 𝑛3 for the THG field, the third-order nonlinear susceptibility 𝜒 (3) , and the phase mismatch 𝛥𝑘 = 3𝑘1 − 𝑘3 , between the harmonic and fundamental field. For undepleted pump, and assuming negligible absorption of the generated harmonic, the solution of Eq. (2), after a propagation distance 𝐿 then reads 3 (𝐿) = 𝑖
3𝜔 (3) 𝜒 𝐽 (𝛥𝑘, 𝐿)𝐴31 2𝑛3 𝑐
(3)
with the phase-matching integral 𝐿
𝐽 (𝛥𝑘, 𝐿) =
∫0
𝑒𝑖𝛥𝑘𝑧 𝑑𝑧. [1 + 𝑖𝜁 (𝑧)]2
(8)
With this factor we define the enhancement by stimulated THG, assuming an optimal phase (sin 𝜗 = 1), as: √ 𝑃 + 𝑃𝑐 𝑃3𝑏 2 𝜂= 𝑠 =1+ √ . (9) 𝑃𝑐 𝑃3 𝜅
where 𝑤0 is the beam waist, 𝜁(𝑧) = 2𝑧∕𝑏 is a dimensionless propagation distance 𝑧 along the optical axis, normalized to the confocal parameter 𝑏 = 𝑘𝑤20 and 𝑞 is the harmonic order, i.e., 𝑞 = 3 in the case of THG. Inserting Eq. (1) in Maxwell’s wave equation yields the nonlinear wave equation for Gaussian beams 𝑑3 (𝑧) 3𝜔 (3) 3 𝑒𝑖𝛥𝑘𝑧 =𝑖 𝜒 𝐴1 , 𝑑𝑧 2𝑛3 𝑐 [1 + 𝑖𝜁 (𝑧)]2
(6)
with the power 𝑃1 of the fundamental field, the power 𝑃3𝑏 of the bias third harmonic field, and a phase 𝜗. The first term 𝑃𝑐 ∝ 𝑃13 in Eq. (7) describes conventional THG due to the fundamental field alone. The second term 𝑃3𝑏 is the offset due to the propagating bias third harmonic √
In this section, we will briefly discuss our simple theoretical model of THG enhanced by a bias harmonic field. As an extension to the related work in SHG [11] we consider focused Gaussian beams (which is more appropriate for realistic applications of CNM) and also include the inevitable effect of dispersion. We follow the standard theoretical treatment of harmonic generation with focused Gaussian beams [15]. A fundamental beam propagates through an optically nonlinear medium and drives THG. We assume an undepleted pump, i.e., the THG signal intensity shall be very small compared to the driving fundamental intensity. The Gaussian-shaped fundamental and THG beam are represented by: 𝑞 (𝑧)
3𝜔 (3) 𝜒 𝐽 (𝛥𝑘, 𝐿)𝐴31 + 𝐴3𝑏 𝑒𝑖3𝜔∕(2𝑐)𝑛3 𝐿 𝑒𝑖𝜑 . 2𝑛3 𝑐
From this total amplitude we calculate the total stimulated THG power as: √ 𝑃3 = 𝜅𝑃13 + 𝑃3𝑏 + 2 𝜅𝑃13 𝑃3𝑏 sin 𝜗, (7)
2. Theory
𝑞 (𝑟, 𝑧) =
(5)
(4) 2
C. Stock, K. Zlatanov and T. Halfmann
Optics Communications 457 (2020) 124660
Fig. 2. Variation of the enhancement in a CaF2 sample vs. temporal delay between fundamental pulse (𝑃1 = 100 mW) and bias harmonic pulse (𝑃3𝑏 = 4 μW). For better visibility we show only the upper half of the data (i.e., for positive values of the y-axis) here. The inset shows an enlarged section around zero delay, with regions of stimulated THG (+) and back conversion (−). To precisely determine the signal power we monitored the standard deviation 𝜎 of power fluctuations in small delay intervals. The standard deviation is directly proportional to the amplitude of a sinusoidal oscillation. Taking four signal periods with 20 samples per period leads to an error in the measured signal power of less than 1%. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 1. Experimental setup (BBO: two stage frequency tripling unit, ND: variable neutral density filter, BP: band pass filter, DS: delay stage, AL: aspheric lens, S: Sample, C: collimator lens, PMT: photo multiplier tube).
and are again overlapped prior to the microscope setup by another dichroic mirror (Layertec, #102854). The group velocity dispersion accumulated in both beam lines stretches the fundamental and third harmonic pulses to roughly 80 fs at the sample position. To provide equal pulse durations, we selected appropriate optics with matching dispersion in the beam lines. In the harmonic microscope the beams are focused with an aspheric objective lens (Edmund Optics, UV fused silica aspheric lenses, EFL = 10 mm) to a beam waist of 𝑤0 ≈ 6 μm (𝑧𝑅 = 140 μm) into the glass sample. The maximum average fundamental power in the sample is 240 mW. The maximal third harmonic beam power is 5 μW. For spatial overlap, we compensate the different focal length in fundamental and THG beam by a second focusing lens in the fundamental beam. The focus can be scanned in propagation direction (z) through the sample. The THG signal generated in the laser focus in the sample is collimated by a lens, separated by two dichroic mirrors and a band pass filter (Laser Components, 280 ± 25 nm BP) and detected on a photo-multiplier tube (Hamamatsu, R4220) Residual background due to SHG from the frequency tripling stage is blocked by a band pass filter (Laser Components, 280 ± 25 nm BP) before the sample. To determine the third harmonic signal generated in the sample, we modulate the fundamental beam with an optical chopper (Scitec Instruments, 310CD) at a modulation frequency of 30 kHz. This enables signal processing in a lock-in amplifier (Scitec Instruments, 450DV2) to separate the THG signal from the large background of the bias harmonic beam at the same wavelength. The experiments require a stable overlap of the fundamental and bias harmonic pulses in time and space. We check the spatial overlap of the laser foci with a CCD camera. The temporal overlap of the pulses is controlled by a delay stage in the interferometric setup (see Fig. 1), permitting a precision in the range of 10 nm, corresponding to a temporal delay of 0.03 fs, and a relative phase difference between the pulses of 0.23 rad with regard to the third harmonic frequency. This is sufficient to vary the relative phase such, that maximal enhancement (or suppression) of THG emission by bias pulses is possible. The phase stability is estimated as 𝜆∕50 over 10–50 ms required for one measurement point and 𝜆∕10 over 10 s required for a full measurement. Thus, the setup is interferometrically sufficiently stable, also without any active phase stabilization, as the data presented in the next section clearly demonstrate.
the blue line shows a Gaussian fit of the envelope with a constant offset. The FWHM of the Gaussian fit is 100(5) fs, i.e., in good agreements with the fundamental pulse duration. When the pulses do not overlap, we get conventional THG, driven by the fundamental beam alone. When the pulses overlap, the THG signal either strongly increases or decreases, depending on the exact choice of the delay (i.e., the relative phase). The inset in Fig. 2, showing an enlarged section around zero delay, clearly reveals the sinusoidal interferometric oscillation pattern. We observe either THG enhancements of 𝜂 = 100 by stimulated THG, or THG suppression by back conversion to the fundamental wave. The measured oscillation period (0.79 fs) is smaller than the expected value without dispersion (a wavelength of 270 nm corresponds to an oscillation period of 0.9 fs), but also larger than the value of 0.69 fs calculated from the phase 𝛷(𝑧). This indicates, that our simple model overestimates the effect of dispersion, but gives the correct tendency. For the following measurements we set now the pulse delay to zero, i.e., such, that THG emission is enhanced by the bias harmonic pulses. We systematically vary the powers of fundamental beam and bias harmonic beam, and monitor the possible enhancement 𝜂. 4.1. Power dependencies of stimulated THG Fig. 3 shows the dependence of the THG signal power (and enhancement) vs. the power of the fundamental beam, while the power of the bias harmonic beam is kept fixed. As expected, conventional THG (red data points in Fig. 3) increases with third order in the fundamental power. As also expected from theory (see Eq. (7) stimulated THG increases in a lower order with the fundamental power. Hence, the enhancement is larger for lower fundamental powers. The simulations fit very well with the experimental data (compare solid lines√and dots
in Fig. 3), confirming dependencies 𝑃𝑐 ∝ 𝑃13 and 𝑃𝑠 ∝ 𝑃13 . We note, that in the calculation we had to reduce the effective bias power 𝑃3𝑏 by a factor of 1200 compared to the experimental value. This is mainly due to non-perfect overlap of the foci and additional averaging effects: As the different frequency components in the laser pulses experience different phase velocities, we get averaging over areas of THG enhancement and back conversion, which reduces the enhancement. Also residual chirps of different strengths in the two beamlines could cause the observed reduction of the signal enhancement In the range of
4. Results Fig. 2 shows the THG signal from the first surface of a 1 mm thick calcium fluoride slide, when we vary the pulse delay of the fundamental beam in the delay line. Gray dots show the experimental data points, 3
C. Stock, K. Zlatanov and T. Halfmann
Optics Communications 457 (2020) 124660
4.2. Improving spatial resolution by stimulated THG THG signal enhancements are of obvious relevance for harmonic microscopy, e.g., to increase the signal yield and image quality (e.g., as defined by the achievable image contrast or signal-to-noise-ratio). We will focus now on another potential application with relevance to nonlinear imaging. The approach uses inevitable dispersion in the sample to improve the spatial resolution in propagation direction (i.e., along the optical axis). We will discuss the concept in an instructive demonstration experiment, when we scan the laser focus in our THG microscope through a fused silica slide with a thickness of 1 mm. In order to demonstrate the increase in spatial resolution, we work with a large Rayleigh length of the microscope objective. Hence, we replace the aspheric lens in the objective by two plano-convex lenses (EFL = 150 mm, one for each beam). The beam waists increase now to (𝑤0 ≈ 20 μm, 𝑧𝑅 ≈ 1.5 mm) for the fundamental beam and (𝑤0 ≈ 8 μm, 𝑧𝑅 ≈ 240 μm) for the TH beam. We compare now the signal yield for stimulated THG and conventional THG, when we scan the laser focus across the sample (or, in our setup, the sample across the laser focus). We remember, that THG is emitted at the interfaces only. Hence, we expect to see two peaks (for the two interfaces at the front and back of the sample) in the plot of the THG signal vs. sample position. Fig. 5 shows the results of this measurement. In the case of conventional THG (see data points set as gray dots in Fig. 5(b)) we observe only a single peak. The conventional THG microscope cannot resolve the separation of the two interfaces, as the Rayleigh length of the focus is too large. For stimulated THG we must align now the pulse delay such, that the fundamental pulse and the bias harmonic pulse temporally perfectly overlap, i.e., with interferometric precision required for constructive coherent interaction. If we align for zero delay at the front interface, due to dispersion in the sample the pulses will move apart from each other. Hence, at the back interface the pulses will no more overlap. Therefore, we will observe large stimulated THG from the front surface only, but none from the back surface (see data points set as pink dots in Fig. 5(b)). If we align for zero delay between the pulses at the back surface, the situation is vice versa, i.e., large stimulated THG from the back surface and none from the front surface (see data points set as light blue dots in Fig. 5(b)). Hence, due to dispersion, the bias pulse enhances THG at only one interface at the time. As a consequence, stimulated THG permits resolution of the two interfaces now, i.e., the peaks of the two stimulated THG signals in Fig. 5(b) are clearly separated by 1.04(1) mm. We investigated now the effect of dispersion (or the pulse delay, rp.) on the stimulated THG signal in an extended measurement, when we monitor the stimulated THG power both vs. pulse delay as well as the sample position (see Fig. 5(c)). The optimal pulse delay for stimulated THG varies by 0.48 ps between the front and back surface of the sample (see green line in Fig. 5(c)). On the upper side of the dashed green line the bias harmonic pulse follows the fundamental pulse, on the lower side of the dashed green line the fundamental pulse follows the bias pulse. As the strongly elliptical regions of maximal THG signal indicate, the variation of stimulated THG signal with the pulse delay is much stronger than with the sample position (under conditions of weak focusing (NA = 0.02) and for ultra-short laser pulses). Thus, variation of the pulse delay enables a strong variation of the THG signal and enables improved resolution. Finally, we performed a measurement of the stimulated THG signal vs. optimal pulse delay, i.e., a scan along the green line in Fig. 5(c). Thus, we moved the sample across the laser focus and, for each position z, choose the optimal pulse delay 𝛿𝜏(𝑧) to maintain perfect pulse overlap and maximal THG signal. Fig. 5(d) shows the experimental results. From the optimal pulse delay and using values for indices of refraction and dispersion from literature [16], we calculated an equivalent propagation distance, which is given as an additional upper 𝑥-axis in Fig. 5(d). We fit the experimental data with the sum of two Gaussian functions. The calculated thickness of the sample is 1.015(1) mm.
Fig. 3. Dependence of the average THG signal power vs. average fundamental power, while keeping the bias harmonic power fixed at 𝑃3𝑏 = 4.8μ W. Note the double logarithmic scale. Red dots indicate the contribution from conventional THG. Blue dots represent the contribution of stimulated THG. Solid lines show simulations based on ′ Eq. (7), with 𝜒 (3) = 3⋅10−23 𝑉 2 ∕𝑚2 and 𝑃3𝑏 = 4 nW. Green dots indicate the enhancement factor calculated from the THG data (with the dashed green line based on Eq. (9)). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 4. Dependence of the average THG signal power vs. average bias harmonic power ′ (𝑃3𝑏 = 𝑃3𝑏 ∕1200), while keeping the average fundamental power fixed at 𝑃1 = 53 mW. Note the double logarithmic scale. The color code is the same as in Fig. 3.
our laser powers, the maximal enhancement by bias harmonic pulses is around 𝜂 ≈ 3000, i.e., a signal gain of more than three orders of magnitude. Even for the largest possible fundamental power in our setup (i.e., with already strong conventional THG, hence, the least advantageous case for stimulated THG), the enhancement still is 𝜂 ≈ 20. Our above conclusions are confirmed by a second measurement, where we measure the dependence of the THG signal power (and enhancement) vs. the power of the bias harmonic beam, while the power of the fundamental beam is kept fixed (see Fig. 4). As the fundamental power does not vary now, conventional THG remains constant (see red data points in Fig. 4). Stimulated THG increases with the power of the bias harmonic beam (see blue data points in Fig. 4). 0 (trivial case) and The simulation confirms the dependencies 𝑃𝑐 ∝ 𝑃3𝑏 √ 𝑃𝑠 ∝ 𝑃3𝑏 (compare solid lines and dots in Fig. 4). We note, that the observable enhancements in our setup are limited by the maximal available bias harmonic power in the range of 5 μW, the maximal fundamental power around 240 mW, and the noise level of our detection setup, which requires a minimum fundamental power of 10 mW. As discussed in the theory section above, large THG enhancements are expected for weak fundamental beam and sufficiently intense bias harmonic beam. The optimal power ratio of 𝑃1 ∕𝑃3𝑏 = 3 would require a bias harmonic power of 3 mW, which we cannot achieve in our setup, or a fundamental power of only 15 μW, which is not enough to provide THG signal above the detection limit in our setup. Hence, in the measurement of Fig. 3 we applied the maximal available bias power. Nevertheless, already at the resulting far-from-optimal power ratio 𝑃1 ∕𝑃3𝑏 > 2000 we obtain a large enhancement of 3000. When we extrapolate this finding on the basis of Eq. (9), THG enhancements by six orders of magnitude should be possible for 𝑃1 ∕𝑃3𝑏 = 3. 4
C. Stock, K. Zlatanov and T. Halfmann
Optics Communications 457 (2020) 124660
Fig. 5. (a) Geometry of laser foci and sample. (b) Average THG signal power vs. sample position relative to the laser focus. To precisely determine the signal power we apply the same approach as described in the caption of Fig. 2. Gray dots show conventional THG, magenta and cyan dots show stimulated THG with optimized delay at the two interfaces. (c) Average THG signal power vs. sample position and pulse delay. Red color indicates large THG signal, blue color indicates low THG signal. The pink and light blue lines indicate the data already discussed above in Fig. 5. The green line depicts the variation of the optimal (relative) pulse delay across the sample. (d) Average THG signal power vs. optimal pulse delay (or equivalent propagation distance in the sample). Experimental data (green dots) and fit with a double-Gaussian function (black dashed line).
5. Conclusion
With the Rayleigh criterion we determine a resolution limit of 240 μm. Comparing this number to the resolution limit of 3 mm for conventional THG (as deduced from Fig. 5(b)), we find that stimulated THG yields an improvement of the spatial resolution along the optical axis by more than an order of magnitude. We note, that the improvement depends upon the pulse duration and the group velocity dispersion in the sample. To get a resolution of 2 μm, which is easily obtainable with tighter focusing, we would need a pulse duration around 1 fs (under the same experimental conditions). If possible, tighter focusing remains the simplest way to improve the spatial resolution for ultra-short pulses. Hence, the discussed approach with a bias field offers an advantage only in situations, when tighter focusing is not possible, e.g., if the sample geometry does not permit lenses close to the sample surface. As an interesting perspective, it would be possible to implement the approach also without moving the laser focus across the sample, as long as the Rayleigh length is not much shorter than the sample thickness. Application of a fast piezo-electric actuator in the delay line, with a scan range larger than the delay in the sample, would permit to shift the overlap through the sample in milliseconds (or less). This enables a fast analysis of a sample. Because of the negligible dispersion in air, the optimal delay is independent of the sample position. Therefore this method is also tolerant to small movements of the sample (in air), i.e. in the range of the Rayleigh length. Finally we note, that during the experiments on stimulated THG we discovered an additional phaseindependent THG signal, which also varied with the temporal overlap between fundamental and bias pulse in the sample. The signal is generated also inside homogeneous media and permitted large enhancements up to 105 . The effect is not yet understood and still under investigation in our laboratory.
We investigated THG enhanced by a bias harmonic field in microscopic demonstration samples, driven by ultra-short laser pulses at the fundamental wavelength and a small amount of bias radiation at the third harmonic wavelength. When the relative phase between the driving laser pulses is varied, the third harmonic yield from the sample either experiences strong suppression or enhancement. The enhancements are largest for small nonlinear susceptibility or low fundamental pulse power, i.e., the quite typical situation of low signal yield in nonlinear microscopy. At an appropriate relative phase and with a bias harmonic power of less then 0.1% compared to the fundamental beam, we boost the THG yield by a factor of 3000 compared to conventional THG imaging with a fundamental beam only. From the data we extrapolate potential signal enhancements of six orders of magnitude at an optimized power ratio of 1/3 between bias harmonic beam and fundamental beam. We performed numerical simulations, based on a simple model for harmonic generation of focused Gaussian beams, also taking dispersion of the microscopic sample into account. The simulations fitted very well to the experimental data and confirmed, e.g., the power dependencies of the THG signal and the effect of dispersion. Finally, we exploited the inevitable dispersion in a sample to improve the spatial resolution along the optical axis by stimulated THG imaging. In the approach an appropriate delay between fundamental pulse and bias harmonic pulse is used to compensate dispersion and temporally overlap the two pulses at an interface. Such, we demonstrated an resolution improvement by more than an order of magnitude. The effect is most evident when the sample geometry allows only moderate focusing. The demonstrations shall pave the way for stimulated THG 5
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Optics Communications 457 (2020) 124660
microscopy, enabling larger signals and image contrast, as well as less energy deposition in samples of small nonlinear susceptibilities.
[2] R. Carriles, D.N. Schafer, K.E. Sheetz, J.J. Field, R. Cisek, V. Barzda, A.W. Sylvester, J.A. Squier, Rev. Sci. Instrum. (ISSN: 1089-7623) 80 (2009) 081101. [3] C.L. Evans, X.S. Xie, Annu. Rev. Anal. Chem. (ISSN: 1936-1327) 1 (2008) 883. [4] P.T.C. So, C.Y. Dong, B.R. Masters, K.M. Berland, (2000) pp. 399–429. [5] D. Yelin, Y. Silberberg, Opt. Express 5 (1999) 169. [6] N. Olivier, M.A. Luengo-Oroz, L. Duloquin, E. Faure, T. Savy, I. Veilleux, X. Solinas, D. Debarre, P. Bourgine, A. Santos, et al., Science (ISSN: 0036-8075) 329 (2010) 967. [7] G.O. Clay, A.C. Millard, C.B. Schaffer, J. Aus-der Au, P.S. Tsai, J.A. Squier, D. Kleinfeld, J. Opt. Soc. Amer. B (ISSN: 0740-3224) 23 (2006) 932. [8] S.-P. Tai, C.-H. Yu, T.-M. Liu, Y.-C. Wen, C.-K. Sun, 2007 Conference on Lasers and Electro-Optics (CLEO) 2, (2007) p. 1. [9] C.-F. Chang, C.-H. Yu, C.-K. Sun, J. Biophotonics (ISSN: 1864-0648) 3 (2010) 678. [10] C. Stock, K. Zlatanov, T. Halfmann, Opt. Commun. (ISSN: 00304018) 393 (2017) 289. [11] A.J. Goodman, W.A. Tisdale, Phys. Rev. Lett. (ISSN: 10797114) 114 (2015) 1. [12] C.A. Marx, U. Harbola, S. Mukamel, Phys. Rev. A (ISSN: 10502947) 77 (2008). [13] A.J. Goodman, W.A. Tisdale, Phys. Rev. Lett. (ISSN: 10797114) 116 (2016) 2015. [14] Y. Gao, A.J. Goodman, P.C. Shen, J. Kong, W.A. Tisdale, Nano Lett. (ISSN: 15306992) 18 (2018) 5001. [15] R.W. Boyd, Nonlinear Optics, second ed., Academic Press, San Diego, ISBN: 0-12-121682-9, 2003. [16] I.H. Malitson, J. Opt. Soc. Amer. 55 (1965) 1205.
Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments The authors thank J. F. Kinder, F. Cipura and P. Ackermann (Technische Universität Darmstadt) for valuable discussions. This project received funding by the Deutsche Forschungsgemeinschaft (3791/131), and the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 641272. References [1] S. Yue, M.N. Slipchenko, J.-X. Cheng, Laser Photonics Rev. (ISSN: 18638880) 5 (2011) 496.
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Third harmonic generation and microscopy, enhanced by a bias harmonic field Christian Stock a ,∗, Kaloyan Zlatanov b , Thomas Halfmann a a b
Technische Universität Darmstadt, Institut für Angewandte Physik, Hochschulstraße 6, 64289 Darmstadt, Germany Institute of Solid State Physics, Bulgarian Academy of Sciences, Tsarigradsko chaussée 72, 1784 Sofia, Bulgaria
ARTICLE
INFO
Keywords: Nonlinear microscopy Third harmonic generation Ultrafast nonlinear optics Dispersion
ABSTRACT We present strong enhancements of the signal yield in third harmonic generation microscopy by seeding the optically nonlinear sample with some intensity at the third harmonic wavelength, in addition to the driving fundamental beam. By applying a third harmonic bias pulse with a power of less then 0.1% of the fundamental beam, we boost the signal yield by more than a factor of 3000, compared to the conventional third harmonic signal from the microscopy sample. The signal enhancement is most pronounced at low laser intensity and/or weak nonlinear susceptibilities. This makes the concept particularly suitable to improve the signal-to-noise ratio from samples with weak signals, e.g., as typical for applications of nonlinear microscopy. Moreover, we demonstrate improved spatial resolution in beam propagation direction by more than an order of magnitude. This exploits the dependence of our approach for enhanced third harmonic generation from inevitable dispersion in the sample. The improvement is most evident in cases, where the sample geometry allows only weak focusing.
1. Introduction Coherent nonlinear microscopy (CNM) is a powerful tool for threedimensional imaging [1,2]. It utilizes frequency conversion processes of ultrashort laser pulses, strongly focused into a sample. The image is obtained by monitoring radiation generated by nonlinear frequency conversion in the laser focus, while the focus is moved through the sample. Examples for CNM processes are second harmonic generation (SHG), third harmonic generation (THG) or coherent anti-stokes Raman scattering (CARS) [3]. In contrast to conventional two photon excitation fluorescence (TPEF) microscopy [4], CNM does not require labeling and does not deposit energy in the sample. This makes CNM particularly interesting for microscopic imaging of sensitive biological samples [5,6]. While CARS requires two laser beams to tune to vibrational Raman resonances in molecules for efficient frequency conversion, SHG and THG apply a single off-resonant laser beam only. This makes the latter two techniques rather simple to implement. Typically, microscopic imaging by SHG and THG is implemented with a fixed-frequency laser far off resonances in the sample. However, such far off-resonant nonlinear interaction yields only low optical conversion efficiencies and hence, rather limited signal, signal-to-noise ratio, and contrast. There are several approaches to enhance the signal yield in CNM, e.g. using resonant enhancements of the nonlinear susceptibility [7–9] or dispersion to optimize phase matching conditions [10]. However, these approaches are limited to specific media
or require tunable laser systems to drive the nonlinear interactions in the vicinity of resonances. Moreover, near-resonant excitation deposits energy in the sample. This may damage the sample by absorption of the fundamental radiation, or lead to losses by re-absorption of the nonlinear signal. Recently, an alternative, promising route to enhance signals in CNM by so-called optical stimulation was introduced, and demonstrated for the case of SHG [11]. The basic idea is to start the nonlinear optical process at an already higher signal intensity level from the beginning on — rather than from zero signal intensity. Hence, in the case of SHG, coincident with the fundamental radiation some weak radiation at the second harmonic is introduced into the sample. This additional bias harmonic pulse enhances the conversion process to produce more photons in the SHG detection channel. The previous demonstration of optically stimulated SHG yielded signal enhancements beyond 104 in a biologically relevant medium. The signal enhancements are the largest for low input power and weak nonlinear susceptibilities. We note, that the concept of optical stimulation by a bias harmonic field could also be understood in terms of heterodyne detection, with both approaches reconciled in a quantum electrodynamical treatment [12]. However, experimentally they differ from each other. In heterodyne detection, a harmonic field is generated in the sample and then mixed with a reference field. Hence, the fundamental field remains uneffected by the mixing. In optical stimulation the interaction takes place in the sample,
∗ Corresponding author. E-mail address: [email protected] (C. Stock).
https://doi.org/10.1016/j.optcom.2019.124660 Received 24 May 2019; Received in revised form 24 September 2019; Accepted 27 September 2019 Available online 30 September 2019 0030-4018/© 2019 Published by Elsevier B.V.
C. Stock, K. Zlatanov and T. Halfmann
Optics Communications 457 (2020) 124660
Stimulated THG requires a second incident field at the frequency of the generated harmonic 3𝜔. In undepleted pump approximation the bias harmonic field is simply:
and the energy is transferred between fundamental and harmonic fields depending on the phase [13,14]. The aim of our work is to transfer the concept of optical stimulation to THG imaging, and also investigate the effect of inevitable dispersion in the medium. In the following we will use the short term ‘‘stimulated THG’’ for the conversion process enhanced by a bias harmonic field, and compare it to conventional THG (i.e., without a bias field). While SHG is possible in non-centrosymmetric media only, THG occurs in any medium of arbitrary symmetry [2]. Therefore, THG microscopy has a much broader range of applications compared to SHG. We note, that THG imaging with tightly focused laser beams is sensitive to interfaces, i.e., the third harmonic is emitted only when the laser focus intersects the interface between two media of different nonlinear susceptibilities. This makes THG microscopy a powerful tool to image the structure of heterogeneous samples without background signal from bulk material. Stimulated THG can be implemented as an extension to a conventional THG microscope, without the need of a tunable laser, a second laser system, or any other larger modification of the experimental setup. Moreover, as the approach provides larger enhancement with decreasing nonlinear susceptibility, the potential signal enhancement and its impact on THG imaging is expected to be even larger in THG compared to SHG. In the following we will present a theoretical treatment of stimulated THG for Gaussian beams. Afterwards, we discuss successful experimental implementation of the concept, with systematic measurements of the signal yield from a fused silica sample, indicating enhancements up to a factor of 3000. We will also demonstrate improvements of the spatial resolution along the optical axis by exploiting dispersion in stimulated THG.
3𝑏 (𝑧) = 𝐴3𝑏 𝑒𝑖3𝜔∕(2𝑐)𝑛3 𝑧 𝑒𝑖𝜑 ,
i.e., the amplitude 𝐴3𝑏 remains constant, but the phase difference to the fundamental field changes with the optical path length 𝑛3 𝑧, and we also permit an initial phase 𝜑 with regard to the fundamental field. The bias field exhibits another term to be added to the wave equation (2), and consequently also adds to the solution in Eq. (3), yielding: 3 (𝐿) = 𝑖
1 + 𝑖𝜁 (𝑧)
{ } −𝑞𝑟2 ∕ 𝑤20 [1+𝑖𝜁(𝑧)]
𝑒
,
field. The last interference-like term 𝑃𝑠 ∝ 𝑃13 𝑃3𝑏 describes stimulated THG. The latter depends upon the phase 𝜗, which is equal to the phase 3𝜔𝑛 𝑧 of the bias harmonic field 𝛷(𝑧) = 2𝑐3 + 𝜑 (see Eq. (5)) plus a constant offset by the phases of the complex parameters 𝐽 (𝛥𝑘, 𝐿) and 𝜒 (3) . Thus, 𝜑 serves to control the direction of the energy flow, i.e., whether we get enhanced or suppressed THG emission. The relative strengths of conventional and enhanced THG are determined by the prefactor | |2 9𝜔2 |𝜒 (3) | | | 𝜅= |𝐽 (𝛥𝑘, 𝐿)|2 . 4𝑛31 𝑛3 𝜖02 𝑐 4 𝜋 2 𝑤40
1
Thus, we expect large enhancement for a strong bias harmonic field, but also for low conventional THG yield, e.g., low nonlinear susceptibility or low fundamental power. The latter is an important feature for realistic applications: The effect of stimulated THG is the largest, when the THG signal yield is low — which is very typical for nonlinear microscopy with its very low conversion efficiency. Moreover, we see that also the power ratio between the fields is of relevance. Assuming a constant total power 𝑃 = 𝑃1 + 𝑃3𝑏 available from our laser system, from Eq. (9) we find that the ratio 𝑃1 ∕𝑃3𝑏 = 3 yields the largest enhancement.
(1)
3. Experimental setup The experimental setup is schematically depicted in Fig. 1. A titanium sapphire laser oscillator (Spectra Physics, Tsunami), pumped by a DPSS system (Coherent, Verdi G7), provides pulses with a center wavelength of 810 nm, average output power of 1 W, pulse duration of 65(5) fs (FWHM), and repetition rate of 82 MHz (cf. Fig. 1). The fundamental radiation is frequency tripled in two beta barium borate (BBO) crystals by phase-matched SHG (Type I, 0.2 mm, 28.9◦ ) and subsequent sum frequency generation (SFG, Type II, 0.13 mm, 54.4◦ ) to provide the bias third harmonic beam at center wavelength of 270 nm and average output power up to 6 μW. As the conversion efficiency in the BBO crystals is very low, the beam profile of the fundamental beam is not distorted by the conversion stage. After the BBO crystals, the fundamental beam and bias harmonic beam are separated by a dichroic mirror (Layertec, #102854), and their intensity is varied by neutral density filters (Thorlabs, NDC-100C-2M, 2 mm thick). The beams propagate through a Mach–Zehnder-type interferometric setup involving a variable delay stage with piezo (Thorlabs, NFL5DP20S),
(2)
with the amplitude 𝐴31 of the driving fundamental field at carrier frequency 𝜔, the index of refraction 𝑛3 for the THG field, the third-order nonlinear susceptibility 𝜒 (3) , and the phase mismatch 𝛥𝑘 = 3𝑘1 − 𝑘3 , between the harmonic and fundamental field. For undepleted pump, and assuming negligible absorption of the generated harmonic, the solution of Eq. (2), after a propagation distance 𝐿 then reads 3 (𝐿) = 𝑖
3𝜔 (3) 𝜒 𝐽 (𝛥𝑘, 𝐿)𝐴31 2𝑛3 𝑐
(3)
with the phase-matching integral 𝐿
𝐽 (𝛥𝑘, 𝐿) =
∫0
𝑒𝑖𝛥𝑘𝑧 𝑑𝑧. [1 + 𝑖𝜁 (𝑧)]2
(8)
With this factor we define the enhancement by stimulated THG, assuming an optimal phase (sin 𝜗 = 1), as: √ 𝑃 + 𝑃𝑐 𝑃3𝑏 2 𝜂= 𝑠 =1+ √ . (9) 𝑃𝑐 𝑃3 𝜅
where 𝑤0 is the beam waist, 𝜁(𝑧) = 2𝑧∕𝑏 is a dimensionless propagation distance 𝑧 along the optical axis, normalized to the confocal parameter 𝑏 = 𝑘𝑤20 and 𝑞 is the harmonic order, i.e., 𝑞 = 3 in the case of THG. Inserting Eq. (1) in Maxwell’s wave equation yields the nonlinear wave equation for Gaussian beams 𝑑3 (𝑧) 3𝜔 (3) 3 𝑒𝑖𝛥𝑘𝑧 =𝑖 𝜒 𝐴1 , 𝑑𝑧 2𝑛3 𝑐 [1 + 𝑖𝜁 (𝑧)]2
(6)
with the power 𝑃1 of the fundamental field, the power 𝑃3𝑏 of the bias third harmonic field, and a phase 𝜗. The first term 𝑃𝑐 ∝ 𝑃13 in Eq. (7) describes conventional THG due to the fundamental field alone. The second term 𝑃3𝑏 is the offset due to the propagating bias third harmonic √
In this section, we will briefly discuss our simple theoretical model of THG enhanced by a bias harmonic field. As an extension to the related work in SHG [11] we consider focused Gaussian beams (which is more appropriate for realistic applications of CNM) and also include the inevitable effect of dispersion. We follow the standard theoretical treatment of harmonic generation with focused Gaussian beams [15]. A fundamental beam propagates through an optically nonlinear medium and drives THG. We assume an undepleted pump, i.e., the THG signal intensity shall be very small compared to the driving fundamental intensity. The Gaussian-shaped fundamental and THG beam are represented by: 𝑞 (𝑧)
3𝜔 (3) 𝜒 𝐽 (𝛥𝑘, 𝐿)𝐴31 + 𝐴3𝑏 𝑒𝑖3𝜔∕(2𝑐)𝑛3 𝐿 𝑒𝑖𝜑 . 2𝑛3 𝑐
From this total amplitude we calculate the total stimulated THG power as: √ 𝑃3 = 𝜅𝑃13 + 𝑃3𝑏 + 2 𝜅𝑃13 𝑃3𝑏 sin 𝜗, (7)
2. Theory
𝑞 (𝑟, 𝑧) =
(5)
(4) 2
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Optics Communications 457 (2020) 124660
Fig. 2. Variation of the enhancement in a CaF2 sample vs. temporal delay between fundamental pulse (𝑃1 = 100 mW) and bias harmonic pulse (𝑃3𝑏 = 4 μW). For better visibility we show only the upper half of the data (i.e., for positive values of the y-axis) here. The inset shows an enlarged section around zero delay, with regions of stimulated THG (+) and back conversion (−). To precisely determine the signal power we monitored the standard deviation 𝜎 of power fluctuations in small delay intervals. The standard deviation is directly proportional to the amplitude of a sinusoidal oscillation. Taking four signal periods with 20 samples per period leads to an error in the measured signal power of less than 1%. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 1. Experimental setup (BBO: two stage frequency tripling unit, ND: variable neutral density filter, BP: band pass filter, DS: delay stage, AL: aspheric lens, S: Sample, C: collimator lens, PMT: photo multiplier tube).
and are again overlapped prior to the microscope setup by another dichroic mirror (Layertec, #102854). The group velocity dispersion accumulated in both beam lines stretches the fundamental and third harmonic pulses to roughly 80 fs at the sample position. To provide equal pulse durations, we selected appropriate optics with matching dispersion in the beam lines. In the harmonic microscope the beams are focused with an aspheric objective lens (Edmund Optics, UV fused silica aspheric lenses, EFL = 10 mm) to a beam waist of 𝑤0 ≈ 6 μm (𝑧𝑅 = 140 μm) into the glass sample. The maximum average fundamental power in the sample is 240 mW. The maximal third harmonic beam power is 5 μW. For spatial overlap, we compensate the different focal length in fundamental and THG beam by a second focusing lens in the fundamental beam. The focus can be scanned in propagation direction (z) through the sample. The THG signal generated in the laser focus in the sample is collimated by a lens, separated by two dichroic mirrors and a band pass filter (Laser Components, 280 ± 25 nm BP) and detected on a photo-multiplier tube (Hamamatsu, R4220) Residual background due to SHG from the frequency tripling stage is blocked by a band pass filter (Laser Components, 280 ± 25 nm BP) before the sample. To determine the third harmonic signal generated in the sample, we modulate the fundamental beam with an optical chopper (Scitec Instruments, 310CD) at a modulation frequency of 30 kHz. This enables signal processing in a lock-in amplifier (Scitec Instruments, 450DV2) to separate the THG signal from the large background of the bias harmonic beam at the same wavelength. The experiments require a stable overlap of the fundamental and bias harmonic pulses in time and space. We check the spatial overlap of the laser foci with a CCD camera. The temporal overlap of the pulses is controlled by a delay stage in the interferometric setup (see Fig. 1), permitting a precision in the range of 10 nm, corresponding to a temporal delay of 0.03 fs, and a relative phase difference between the pulses of 0.23 rad with regard to the third harmonic frequency. This is sufficient to vary the relative phase such, that maximal enhancement (or suppression) of THG emission by bias pulses is possible. The phase stability is estimated as 𝜆∕50 over 10–50 ms required for one measurement point and 𝜆∕10 over 10 s required for a full measurement. Thus, the setup is interferometrically sufficiently stable, also without any active phase stabilization, as the data presented in the next section clearly demonstrate.
the blue line shows a Gaussian fit of the envelope with a constant offset. The FWHM of the Gaussian fit is 100(5) fs, i.e., in good agreements with the fundamental pulse duration. When the pulses do not overlap, we get conventional THG, driven by the fundamental beam alone. When the pulses overlap, the THG signal either strongly increases or decreases, depending on the exact choice of the delay (i.e., the relative phase). The inset in Fig. 2, showing an enlarged section around zero delay, clearly reveals the sinusoidal interferometric oscillation pattern. We observe either THG enhancements of 𝜂 = 100 by stimulated THG, or THG suppression by back conversion to the fundamental wave. The measured oscillation period (0.79 fs) is smaller than the expected value without dispersion (a wavelength of 270 nm corresponds to an oscillation period of 0.9 fs), but also larger than the value of 0.69 fs calculated from the phase 𝛷(𝑧). This indicates, that our simple model overestimates the effect of dispersion, but gives the correct tendency. For the following measurements we set now the pulse delay to zero, i.e., such, that THG emission is enhanced by the bias harmonic pulses. We systematically vary the powers of fundamental beam and bias harmonic beam, and monitor the possible enhancement 𝜂. 4.1. Power dependencies of stimulated THG Fig. 3 shows the dependence of the THG signal power (and enhancement) vs. the power of the fundamental beam, while the power of the bias harmonic beam is kept fixed. As expected, conventional THG (red data points in Fig. 3) increases with third order in the fundamental power. As also expected from theory (see Eq. (7) stimulated THG increases in a lower order with the fundamental power. Hence, the enhancement is larger for lower fundamental powers. The simulations fit very well with the experimental data (compare solid lines√and dots
in Fig. 3), confirming dependencies 𝑃𝑐 ∝ 𝑃13 and 𝑃𝑠 ∝ 𝑃13 . We note, that in the calculation we had to reduce the effective bias power 𝑃3𝑏 by a factor of 1200 compared to the experimental value. This is mainly due to non-perfect overlap of the foci and additional averaging effects: As the different frequency components in the laser pulses experience different phase velocities, we get averaging over areas of THG enhancement and back conversion, which reduces the enhancement. Also residual chirps of different strengths in the two beamlines could cause the observed reduction of the signal enhancement In the range of
4. Results Fig. 2 shows the THG signal from the first surface of a 1 mm thick calcium fluoride slide, when we vary the pulse delay of the fundamental beam in the delay line. Gray dots show the experimental data points, 3
C. Stock, K. Zlatanov and T. Halfmann
Optics Communications 457 (2020) 124660
4.2. Improving spatial resolution by stimulated THG THG signal enhancements are of obvious relevance for harmonic microscopy, e.g., to increase the signal yield and image quality (e.g., as defined by the achievable image contrast or signal-to-noise-ratio). We will focus now on another potential application with relevance to nonlinear imaging. The approach uses inevitable dispersion in the sample to improve the spatial resolution in propagation direction (i.e., along the optical axis). We will discuss the concept in an instructive demonstration experiment, when we scan the laser focus in our THG microscope through a fused silica slide with a thickness of 1 mm. In order to demonstrate the increase in spatial resolution, we work with a large Rayleigh length of the microscope objective. Hence, we replace the aspheric lens in the objective by two plano-convex lenses (EFL = 150 mm, one for each beam). The beam waists increase now to (𝑤0 ≈ 20 μm, 𝑧𝑅 ≈ 1.5 mm) for the fundamental beam and (𝑤0 ≈ 8 μm, 𝑧𝑅 ≈ 240 μm) for the TH beam. We compare now the signal yield for stimulated THG and conventional THG, when we scan the laser focus across the sample (or, in our setup, the sample across the laser focus). We remember, that THG is emitted at the interfaces only. Hence, we expect to see two peaks (for the two interfaces at the front and back of the sample) in the plot of the THG signal vs. sample position. Fig. 5 shows the results of this measurement. In the case of conventional THG (see data points set as gray dots in Fig. 5(b)) we observe only a single peak. The conventional THG microscope cannot resolve the separation of the two interfaces, as the Rayleigh length of the focus is too large. For stimulated THG we must align now the pulse delay such, that the fundamental pulse and the bias harmonic pulse temporally perfectly overlap, i.e., with interferometric precision required for constructive coherent interaction. If we align for zero delay at the front interface, due to dispersion in the sample the pulses will move apart from each other. Hence, at the back interface the pulses will no more overlap. Therefore, we will observe large stimulated THG from the front surface only, but none from the back surface (see data points set as pink dots in Fig. 5(b)). If we align for zero delay between the pulses at the back surface, the situation is vice versa, i.e., large stimulated THG from the back surface and none from the front surface (see data points set as light blue dots in Fig. 5(b)). Hence, due to dispersion, the bias pulse enhances THG at only one interface at the time. As a consequence, stimulated THG permits resolution of the two interfaces now, i.e., the peaks of the two stimulated THG signals in Fig. 5(b) are clearly separated by 1.04(1) mm. We investigated now the effect of dispersion (or the pulse delay, rp.) on the stimulated THG signal in an extended measurement, when we monitor the stimulated THG power both vs. pulse delay as well as the sample position (see Fig. 5(c)). The optimal pulse delay for stimulated THG varies by 0.48 ps between the front and back surface of the sample (see green line in Fig. 5(c)). On the upper side of the dashed green line the bias harmonic pulse follows the fundamental pulse, on the lower side of the dashed green line the fundamental pulse follows the bias pulse. As the strongly elliptical regions of maximal THG signal indicate, the variation of stimulated THG signal with the pulse delay is much stronger than with the sample position (under conditions of weak focusing (NA = 0.02) and for ultra-short laser pulses). Thus, variation of the pulse delay enables a strong variation of the THG signal and enables improved resolution. Finally, we performed a measurement of the stimulated THG signal vs. optimal pulse delay, i.e., a scan along the green line in Fig. 5(c). Thus, we moved the sample across the laser focus and, for each position z, choose the optimal pulse delay 𝛿𝜏(𝑧) to maintain perfect pulse overlap and maximal THG signal. Fig. 5(d) shows the experimental results. From the optimal pulse delay and using values for indices of refraction and dispersion from literature [16], we calculated an equivalent propagation distance, which is given as an additional upper 𝑥-axis in Fig. 5(d). We fit the experimental data with the sum of two Gaussian functions. The calculated thickness of the sample is 1.015(1) mm.
Fig. 3. Dependence of the average THG signal power vs. average fundamental power, while keeping the bias harmonic power fixed at 𝑃3𝑏 = 4.8μ W. Note the double logarithmic scale. Red dots indicate the contribution from conventional THG. Blue dots represent the contribution of stimulated THG. Solid lines show simulations based on ′ Eq. (7), with 𝜒 (3) = 3⋅10−23 𝑉 2 ∕𝑚2 and 𝑃3𝑏 = 4 nW. Green dots indicate the enhancement factor calculated from the THG data (with the dashed green line based on Eq. (9)). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 4. Dependence of the average THG signal power vs. average bias harmonic power ′ (𝑃3𝑏 = 𝑃3𝑏 ∕1200), while keeping the average fundamental power fixed at 𝑃1 = 53 mW. Note the double logarithmic scale. The color code is the same as in Fig. 3.
our laser powers, the maximal enhancement by bias harmonic pulses is around 𝜂 ≈ 3000, i.e., a signal gain of more than three orders of magnitude. Even for the largest possible fundamental power in our setup (i.e., with already strong conventional THG, hence, the least advantageous case for stimulated THG), the enhancement still is 𝜂 ≈ 20. Our above conclusions are confirmed by a second measurement, where we measure the dependence of the THG signal power (and enhancement) vs. the power of the bias harmonic beam, while the power of the fundamental beam is kept fixed (see Fig. 4). As the fundamental power does not vary now, conventional THG remains constant (see red data points in Fig. 4). Stimulated THG increases with the power of the bias harmonic beam (see blue data points in Fig. 4). 0 (trivial case) and The simulation confirms the dependencies 𝑃𝑐 ∝ 𝑃3𝑏 √ 𝑃𝑠 ∝ 𝑃3𝑏 (compare solid lines and dots in Fig. 4). We note, that the observable enhancements in our setup are limited by the maximal available bias harmonic power in the range of 5 μW, the maximal fundamental power around 240 mW, and the noise level of our detection setup, which requires a minimum fundamental power of 10 mW. As discussed in the theory section above, large THG enhancements are expected for weak fundamental beam and sufficiently intense bias harmonic beam. The optimal power ratio of 𝑃1 ∕𝑃3𝑏 = 3 would require a bias harmonic power of 3 mW, which we cannot achieve in our setup, or a fundamental power of only 15 μW, which is not enough to provide THG signal above the detection limit in our setup. Hence, in the measurement of Fig. 3 we applied the maximal available bias power. Nevertheless, already at the resulting far-from-optimal power ratio 𝑃1 ∕𝑃3𝑏 > 2000 we obtain a large enhancement of 3000. When we extrapolate this finding on the basis of Eq. (9), THG enhancements by six orders of magnitude should be possible for 𝑃1 ∕𝑃3𝑏 = 3. 4
C. Stock, K. Zlatanov and T. Halfmann
Optics Communications 457 (2020) 124660
Fig. 5. (a) Geometry of laser foci and sample. (b) Average THG signal power vs. sample position relative to the laser focus. To precisely determine the signal power we apply the same approach as described in the caption of Fig. 2. Gray dots show conventional THG, magenta and cyan dots show stimulated THG with optimized delay at the two interfaces. (c) Average THG signal power vs. sample position and pulse delay. Red color indicates large THG signal, blue color indicates low THG signal. The pink and light blue lines indicate the data already discussed above in Fig. 5. The green line depicts the variation of the optimal (relative) pulse delay across the sample. (d) Average THG signal power vs. optimal pulse delay (or equivalent propagation distance in the sample). Experimental data (green dots) and fit with a double-Gaussian function (black dashed line).
5. Conclusion
With the Rayleigh criterion we determine a resolution limit of 240 μm. Comparing this number to the resolution limit of 3 mm for conventional THG (as deduced from Fig. 5(b)), we find that stimulated THG yields an improvement of the spatial resolution along the optical axis by more than an order of magnitude. We note, that the improvement depends upon the pulse duration and the group velocity dispersion in the sample. To get a resolution of 2 μm, which is easily obtainable with tighter focusing, we would need a pulse duration around 1 fs (under the same experimental conditions). If possible, tighter focusing remains the simplest way to improve the spatial resolution for ultra-short pulses. Hence, the discussed approach with a bias field offers an advantage only in situations, when tighter focusing is not possible, e.g., if the sample geometry does not permit lenses close to the sample surface. As an interesting perspective, it would be possible to implement the approach also without moving the laser focus across the sample, as long as the Rayleigh length is not much shorter than the sample thickness. Application of a fast piezo-electric actuator in the delay line, with a scan range larger than the delay in the sample, would permit to shift the overlap through the sample in milliseconds (or less). This enables a fast analysis of a sample. Because of the negligible dispersion in air, the optimal delay is independent of the sample position. Therefore this method is also tolerant to small movements of the sample (in air), i.e. in the range of the Rayleigh length. Finally we note, that during the experiments on stimulated THG we discovered an additional phaseindependent THG signal, which also varied with the temporal overlap between fundamental and bias pulse in the sample. The signal is generated also inside homogeneous media and permitted large enhancements up to 105 . The effect is not yet understood and still under investigation in our laboratory.
We investigated THG enhanced by a bias harmonic field in microscopic demonstration samples, driven by ultra-short laser pulses at the fundamental wavelength and a small amount of bias radiation at the third harmonic wavelength. When the relative phase between the driving laser pulses is varied, the third harmonic yield from the sample either experiences strong suppression or enhancement. The enhancements are largest for small nonlinear susceptibility or low fundamental pulse power, i.e., the quite typical situation of low signal yield in nonlinear microscopy. At an appropriate relative phase and with a bias harmonic power of less then 0.1% compared to the fundamental beam, we boost the THG yield by a factor of 3000 compared to conventional THG imaging with a fundamental beam only. From the data we extrapolate potential signal enhancements of six orders of magnitude at an optimized power ratio of 1/3 between bias harmonic beam and fundamental beam. We performed numerical simulations, based on a simple model for harmonic generation of focused Gaussian beams, also taking dispersion of the microscopic sample into account. The simulations fitted very well to the experimental data and confirmed, e.g., the power dependencies of the THG signal and the effect of dispersion. Finally, we exploited the inevitable dispersion in a sample to improve the spatial resolution along the optical axis by stimulated THG imaging. In the approach an appropriate delay between fundamental pulse and bias harmonic pulse is used to compensate dispersion and temporally overlap the two pulses at an interface. Such, we demonstrated an resolution improvement by more than an order of magnitude. The effect is most evident when the sample geometry allows only moderate focusing. The demonstrations shall pave the way for stimulated THG 5
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microscopy, enabling larger signals and image contrast, as well as less energy deposition in samples of small nonlinear susceptibilities.
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Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments The authors thank J. F. Kinder, F. Cipura and P. Ackermann (Technische Universität Darmstadt) for valuable discussions. This project received funding by the Deutsche Forschungsgemeinschaft (3791/131), and the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 641272. References [1] S. Yue, M.N. Slipchenko, J.-X. Cheng, Laser Photonics Rev. (ISSN: 18638880) 5 (2011) 496.
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