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Abstract
Objective lenses are frequently employed in state-of-the-art ultrafast time-resolved microscopy techniques. While it enables tight spatial focusing, its dispersion causes a longer optical pulse duration. Since an objective lens is a combination of different lens materials, finding its correct dispersion can be challenging. In this paper, we propose a dispersion measurement method for an objective lens using white light interferometry. Our proposed method enables the experimental determination of the lens dispersion and improves the temporal resolution of ultrafast time-resolved microscopy techniques.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The dynamics in materials and biological systems span across multiple time and length scales. To understand multiscale interactions in nonequilibrium systems, experimental approaches with high spatial and temporal resolutions are necessary. To achieve this, new microscopic methods that combine femtosecond laser technologies and various imaging techniques are being developed, thereby expanding research frontiers of nonequilibrium dynamics in complex systems [1–3].
In ultrafast time-resolved microscopy, an objective lens is frequently employed to focus femtosecond optical pulses on a target sample. An objective lens comprises multiple lenses of different materials and adds large dispersion to femtosecond pulses. However, in most cases, the specifications of the component lenses are undisclosed, making it difficult to know the dispersion values of the objective lens. Since knowing and compensating dispersion is essential in handling ultrafast optical pulses, accurately evaluating the dispersion added by an objective lens is essential. So far, the dispersion issue of an objective lens has been addressed by measuring the duration of pulses that passes through the target objective lens [4,5]. In this paper, we performed a direct experimental measurement of dispersion values of an objective lens using white light interferometry. By employing a broadband white light source, the dispersion curve including higher-order terms, which is key information for dispersion compensation of few-cycle pulses, of an objective lens was characterized over its entire working spectral range.
A lens has two mechanisms that induce temporal broadening of an ultrashort optical pulse at focus [6]: (i) The difference between the phase and group velocities in the lens material causes the difference in the arrival time between the pulse front propagating through the lens center and the periphery, respectively. (ii) The group velocity dispersion of the lens material causes the pulse to broaden. For singlet lenses, the former effect (i) is dominant. On the other hand, for achromats, including objective lenses, the difference between the pulse and phase fronts are constant over the lens’ cross-section and as such, the former effect is negligible, and the latter effect (ii) is larger than in singlet lenses. However, this effect is constant over the cross-section of the lens, and such spatially homogeneous pulse broadening can be compensated by chirped mirrors, a grating pair, a prism pair, and so on. In summary, the dispersion of an objective lens is compensable, and thus its evaluation is important.
2. Experiment
2.1 Experimental setup
Figure 1 illustrates the experimental setup for measuring the dispersion of an objective lens. We employed the spectral domain method [7–10] of white light interferometry [11–21], but with a programmable delay stage [22]. Since the dispersion of an objective lens is large, an interferometric signal appears only as a part of the whole spectrum, where the group delay difference between the two interferometer paths is close to zero. By changing the relative path length of the interferometer and recording a series of the interferograms, the group delay values over a wide wavelength range can be obtained by our setup. The light source for this setup was a fiber-coupled tungsten halogen lamp (Ocean Optics, HL-2000). The output beam of the lamp was coupled into a multimode fiber with a core diameter of 200 µm. After passing through the fiber, the beam was collimated with an objective lens (Wraymer, GLI-PLACH10X) and sent to the Michelson interferometer. Silver mirrors were placed on both ends of the sample and reference arms of the interferometer. The mirror in the reference arm is on a linear stepping motor stage (SigmaKoki, HPS60-20X-M5), and the length of the reference arm can be controlled both manually and automatically. A pair of objective lenses, whose dispersions are interested in, were set in the sample arm. During the round trip of the sample arm, the light passes through the objective lenses four times forming an erect unmagnified image. Then, the beams from both arms were superimposed by the beam splitter, and their spectral interference was observed by a fiber-coupled optical multichannel analyzer (Ocean Insight, FLAME-S).
Fig. 1. Experimental setup for the dispersion measurement of an objective lens. Lamp: fiber-coupled tungsten halogen lamp (Ocean Optics, HL-2000); MM fiber 1: multimode optical fiber patch cable (Thorlabs, M15L02); OBJ1: objective lens with a magnification of 10x (Wraymer, GLI-PLACH10X); BS: beam splitter (Thorlabs, BS010); Mirror: silver mirror; OBJ2 and OBJ3: objective lenses whose dispersions are interested in; Lens: plano-convex lens with f = 100 mm; MM fiber 2: multimode optical fiber patch cable (Ocean Insight, VIS-NIR low OH fiber with a core size of 400 µm); OMA: fiber-coupled optical multichannel analyzer (Ocean Insight, FLAME-S). The mirror in the reference arm is on a linear stepping motor stage (SigmaKoki, HPS60-20X-M5), and the length of the reference arm can be controlled manually and automatically.
2.2 Evaluation of the instrument response
Before measuring the dispersions of objective lenses, we first evaluated the instrument response due to the dispersion unbalance between the interferometer arms with no sample inserted. Then, we measured the dispersion of a 19.824-mm-thick N-BK7 substrate, whose dispersion can be calculated from the Sellmeier equation [23], to analyze the accuracy of the measurement method.
Since a cubic beam splitter and two identical silver mirrors were employed as the beam splitter and as the end mirrors of both arms of the interferometer (Fig. 1), the dispersions in both arms should be balanced with no sample inserted. To confirm this, an interferogram obtained with no sample inserted, as shown in Fig. 2(a), was analyzed. Note that, since the dispersions of the arms were nearly balanced and interference was observed in the whole part of the spectrum, the scanning method, i.e., scanning the relative delay and tracing the ridgeline of the interferometric component, was not applicable. Hence, we employed spectral interferometry, where the information of the dispersion difference between the arms of the interferometer can be extracted from an interferogram using the Fourier transform [10]. Figure 2(b) shows the result of the analysis, i.e., the GDD difference between the arms. A quarter of the GDD difference is considerably small compared to the GDD of the objective lenses (Fig. 6(a)), and hence, in the analysis, we ignored the GDD difference between the arms.
Fig. 2. (a) Interferogram measured with no sample inserted, (b) Group delay dispersion calculated from the interferogram.
To determine the accuracy of the scanning method, we measured the GDD of a 19.824-mm-thick N-BK7 substrate by inserting the substrate in the sample arm of the interferometer, and compared the measured GDD (GDDE) with the GDD calculated using the Sellmeier equation for N-BK7 (GDDS) [23]. The Figs. 3(a) and 3(b) show the measured spectrogram and the interferogram at delay = 0 clipped out from the spectrogram, respectively. The ridgeline of the spectrogram was extracted and fitted by a 6th-order polynomial. Figure 4(a) compares the experimentally obtained GDD (GDDE) and the GDD calculated from the Sellmeier equation (GDDS), and Fig. 4(b) represents their difference normalized by GDDS. The difference was less than 1.5% for a wavelength range of 430–840 nm. The positive deviation of the error values indicates that the increased effective thickness of the sample due to oblique incidence.
Fig. 3. (a) Spectrogram of the interferometric component obtained by scanning the relative delay between the sample and the reference electric fields with a 19.824-mm-thick N-BK7 substrate inserted in the sample arm. The non-interferometric component obtained with the large relative delay was subtracted as the background. (b) Spectral interferogram at delay = 0 ps.
Fig. 4. Group delay dispersion (GDD) of a 39.648-mm-long N-BK7. The solid and dotted curves in (a) show the experimentally obtained values (GDDE) and the values calculated from the Sellmeier equation (GDDS) [23], respectively, and (b) shows their difference normalized by the calculated values, (GDDE-GDDS)/GDDS.
2.3 Measurement of the dispersion of an objective lens
For the demonstration of the dispersion measurement of objective lenses, we prepared two types of objective lenses, a high numerical aperture (NA) lens (Wraymer GLI-PLACH80XLM 80X, NA = 0.80) and a low-NA lens (Mitutoyo M-PLAN NIR 10X, NA = 0.26). Their dispersions were measured with three experimental setups: (1) High-NA lenses consisting of OBJ2 and OBJ3 as seen in Fig. 1, (2) low-NA lenses consisting of OBJ2 and OBJ3, (3) high-NA lens consisting of OBJ2 and low-NA lens consisting of OBJ3. Setups 1 and 2 require a pair of identical lenses. On the other hand, Setup 3 assumes known and unknown dispersions of each objective lens.
Figure 5(a) shows the spectrogram of the interferometric component obtained with Setup 1, where the non-interferometric component obtained with the large relative delay between the interferometer arms was subtracted as a background. The ridgeline of the spectrogram directly shows a four-fold quantity of the group delay (GD) given by the high-NA objective lens. We extracted the ridgeline by a Gaussian peak fitting of the interferograms of each spectral component extracted along the delay axis, and the results were further fitted by a 6th-order polynomial in ω [rad/fs], as shown in Fig. 5(b). Since GDD is obtained by differentiating GD with respect to angular frequency, GDD is obtained as a 5th-order polynomial. Similar measurements were performed with Setup 2 and Setup 3. Their results are summarized in Table 1, where the experimentally obtained values were divided by four to represent the GDD values of a single objective lens for Setup 1 and Setup 2. If we assume an optical pulse whose central wavelength is 600 nm and duration is 10 fs in full-width half-maximum (FWHM) of its intensity, the pulse duration becomes ∼970 fs and ∼610 fs (FWHM) after passing through the single high-NA and low-NA lenses, respectively (Fig. 6). This result shows that the dispersions of the objective lenses are large, and hence their measurement and compensation are essential for ultrafast spectroscopy.
Fig. 5. (a) Spectrogram of the interferometric component obtained by scanning the relative delay between the sample and the reference electric fields where the non-interferometric component obtained with the large relative delay was subtracted as background. The inset shows the spectral interferogram at delay = 0 ps. (b) Group delay obtained from the spectrogram by the Gaussian peak fitting to trace the ridgeline of the spectrogram (black dots), and their polynomial fitting result (red curve).
Fig. 6. Pulse broadening of a transform-limited 10-fs pulse with its center wavelength of 600 nm after passing through the single high-NA (blue dash-dot-dot curve) and low-NA (red dash-dot curve) objectives lenses. The pulse duration becomes ∼970 fs and ∼610 fs and the peak intensity decreases to 1.0% and 1.6% of that of the transform-limited pulse, for the high-NA and low-NA lenses, respectively.
Table 1. Experimentally obtained GDDs in the form of 5th order polynomials of $\omega $ [rad/fs], ${C_0} + {C_1}\omega + {C_2}{\omega ^2} + \cdots $. GDD1 corresponds to a quoter of the GDD values measured with Setup 1, i.e., GDD1 is the GDD of the single high-NA objective lens (Wraymer GLI-PLACH80XLM 80X, NA = 0.80). Similarly, GDD2 is the GDD of the single low-NA objective lens (Mitutoyo M-PLAN NIR 10X, NA = 0.26). GDD3 corresponds to the average of the GDDs of the high- and low-NA lenses
Figure 7(a) compares the experimentally obtained GDD values of the high-NA lens (GDD1), that of the low-NA lens (GDD2), and their average (GDD3). Figure 7(b) represents the difference between GDD3 and the average of GDD1 and GDD2, i.e., GDD3 − (GDD1 + GDD2)/2. The differences were a few percent of their original values within the wavelength range of 470–800 nm. This proves the consistency of the three measurements and the feasibility of characterizing an objective lens with unknown dispersion by pairing it with an objective lens with known dispersion.
Fig. 7. (a) Comparison of the quarters of the experimentally obtained GDD values with Setup 1(GDD1, blue dash-dot-dot curve), Setup 2 (GDD2, red dash-dot curve), and Setup 3 (GDD3, green solid curve). (b) Difference between GDD3 and the average of GDD1 and GDD2, i.e., GDD3 - (GDD1 + GDD2)/2.
3. Summary
In this paper, we demonstrated the measurement of the dispersion of an objective lens using white light interferometry. Due to the large dispersion of the objective lens, extracting its phase information over its entire working spectral range from an interferogram at a certain relative delay is difficult. Hence, we scanned the relative delay between the electric fields passing through the sample and the reference arms of the interferometer. Then, we obtained the dispersion information of the objective lens by tracing the ridgeline of the spectrogram. The obtained dispersions of the objective lenses were large enough to stretch a 10-fs-duration pulse at 600 nm to ∼1 ps by a single pass through an objective lens. As mentioned earlier, this large dispersion given by achromatic lenses can be compensated, and therefore, it is important to evaluate its value for correct compensation. Furthermore, we confirmed that the method is applicable with a pair of different lenses if the dispersion of either one is known. We hope that the method demonstrated here contributes to improving the time resolution of ultrafast time-resolved microscopy techniques.
Funding
Japan Society for the Promotion of Science (19H04397, 19K15460, 20H05662, 22K14620).
Acknowledgments
This work was supported by JSPS KAKENHI Grant Numbers 19H04397, 19K15460, 20H05662, 22K14620.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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