## Abstract

Nonlinear optical microscopy is a new and rapidly growing technique within which ultrafast laser technology finds a wide range of applications. Pulse widening, due to the microscope optics, is an issue of major concern for nonlinear excitation efficiency. We herewith describe a novel, simple and inexpensive autocorrelation technique to characterize the laser temporal behavior at the microscope focal plane. The method is based on a wavefront-division lateral shearing interferometer which is inserted into the microscope optical path like an ordinary filter, while a spatially uniform fluorescent specimen is observed. The two-photon excited fluorescent image provides the second-order autocorrelation curve.

©2004 Optical Society of America

## 1. Introduction

Nonlinear optical microscopy [1] is one of the most promising and widespread new field of application for ultrafast laser technology. Second-order autocorrelation techniques aimed at evaluating the widening of the laser pulse duration caused by the microscope optics, have been widely developed in recent years.

In the present paper it will be described a novel and very simple setup in which the microscope tools and capabilities are exploited at best. The technique is based on a lateral shearing interferometer (LSI) which will be inserted into the microscope light path, like an ordinary optical filter, and a spatially uniform fluorescent specimen. The second order autocorrelation curve can then be easily obtained from the two-photon excited fluorescent image.

A typical autocorrelator [2, 3] is made up of three functional subsystems: an interferometer I, a nonlinear detector D, and a control electronics and display unit E (Fig. 1).

The interferometer I creates two replicas of an input laser pulse with a variable temporal delay ∆t among them. The device has to be of the white-light compensated type in order to allow delays ranging down to zero. Though many interferometric configurations have been exploited [4–11], the Michelson one is the most commonly used. The displacement of a corner reflector R at the end of one of the arms can easily accomplish the time delay mechanism.

The autocorrelator detection system D is made up of a nonlinear medium N, which is usually a second harmonic crystal, an emission filter F, to prevent the fundamental laser frequency from reaching the detector, and a photomultiplier tube PMT. In some recent papers, simpler and inexpensive non-linear detection devices have been described [6–14]. Two-photon conductivity and two-photon absorption [15–17] are among the nonlinear phenomena exploited.

The two beam replicas, besides a temporal delay ∆t, can also undergo a lateral spatial shift s caused by a transverse displacement of the corner reflector R. Once they are focused onto the nonlinear medium N by an objective O, they produce a fringe modulated diffraction pattern. The spatially integrated nonlinear output signal is finally detected by PMT.

Although second-order autocorrelation curves produced by spatially shifted beams have been reported in some recent papers [9–10], and numerical considerations have also been presented [10], we herewith propose a more general analytical treatment.

#### 1.1. Second-order autocorrelation functions

The total complex electric field E_{T} at the objective back focal plane (z = z_{o}), given by the contributions of the two beam replicas, is:

Let E be a separable function of the spatial and temporal coordinates:

where u(x, y, z) is the complex field spatial function, ε(t) the temporal envelope and ω the optical angular frequency. The Fraunhofer diffraction field at the objective focal plane (z = z_{f}) is proportional to the Fourier transform of Eq. (1) with respect to the spatial coordinates:

where the coordinates at the focal plane are kept expressed in terms of their respective spatial frequencies: f_{x}=x/λf and f_{y}=y/λf.

The detected nonlinear output, integrated both in space and time, is proportional to the second order autocorrelation function:

After some trivial computations we have:

$$\times \underset{-\infty}{\overset{+\infty}{\int}}\mathrm{dt}\left[{\mid \epsilon \left(t\right)\mid}^{4}+{\mid \epsilon \left(t-\mathrm{\Delta t}\right)\mid}^{4}+4{\mid \epsilon \left(t\right)\mid}^{2}{\mid \epsilon \left(t-\mathrm{\Delta t}\right)\mid}^{2}\right]+$$

$$+\{2\phantom{\rule{.2em}{0ex}}\mathrm{exp}\left(-\mathrm{i\omega \Delta t}\right)\underset{-\infty}{\overset{+\infty}{\int}}\underset{-\infty}{\overset{+\infty}{\int}}{\mathrm{df}}_{x}{\mathrm{df}}_{y}{\mid u({f}_{x},{f}_{y},{z}_{f})\mid}^{4}\mathrm{exp}\left(i2\mathrm{\pi s}{f}_{x}\right)\times \text{}$$

$$\times \underset{-\infty}{\overset{+\infty}{\int}}\mathrm{dt}{\phantom{\rule{.2em}{0ex}}\epsilon}^{*}\left(t\right)\epsilon \left(t-\mathrm{\Delta t}\right)\left[{\mid \epsilon \left(t\right)\mid}^{2}+{\mid \epsilon \left(t-\mathrm{\Delta t}\right)\mid}^{2}\right]+c.c\}+$$

$$+\{\mathrm{exp}(-i2\mathrm{\omega \Delta t})\underset{-\infty}{\overset{+\infty}{\int}}\underset{-\infty}{\overset{+\infty}{\int}}{\mathrm{df}}_{x}{\mathrm{df}}_{y}{\mid u({f}_{x},{f}_{y},{z}_{f})\mid}^{4}\mathrm{exp}\left(i2\pi 2\text{s}{f}_{x}\right)\times $$

$$\times \underset{-\infty}{\overset{+\infty}{\int}}\mathrm{dt}\phantom{\rule{.2em}{0ex}}{\epsilon}^{*2}\left(t\right){\epsilon}^{2}\left(t-\mathrm{\Delta t}\right)+c.c\}$$

The spatial integrations in Eq. (5) can be rewritten using the Fourier formalism:

where ℑ^{-1} denotes the inverse Fourier transform. Eq. (6) can be simplified using the relationship:

$$\phantom{\rule{8em}{0ex}}=u\left(x,y,{z}_{o}\right)\otimes {u}^{*}\left(-x,-y,{z}_{o}\right)\otimes u\left(x,y,{z}_{o}\right)\otimes {u}^{*}\left(-x,-y,{z}_{o}\right)=$$

$$\phantom{\rule{8em}{0ex}}\equiv C\left(x,y\right)$$

where ⊗ represents a convolution. The final fourfold convolution integral C(x, y) is then replaced into Eq. (5) after being evaluated at the coordinates (0, 0), (s, 0), (2s, 0):

$$+\left\{C\left(s,0\right)\mathrm{exp}\left(-\mathrm{i\omega \Delta t}\right)2\underset{-\infty}{\overset{+\infty}{\int}}\mathrm{dt}{\phantom{\rule{.2em}{0ex}}\epsilon}^{*}\left(t\right)\epsilon \left(t-\mathrm{\Delta t}\right)\left[{\mid \epsilon \left(t\right)\mid}^{2}+{\mid \epsilon \left(t-\mathrm{\Delta t}\right)\mid}^{2}\right]+c.c\right\}+$$

$$+\left\{C\left(2s,0\right)\mathrm{exp}\left(-i2\mathrm{\omega \Delta t}\right)\underset{-\infty}{\overset{+\infty}{\int}}\mathrm{dt}\phantom{\rule{.2em}{0ex}}{\epsilon}^{*2}\left(t\right){\epsilon}^{2}\left(t-\mathrm{\Delta t}\right)+c.c\right\}$$

For s = 0, the constant factor C(0, 0) can be factorized in Eq. (8) and the well known expression for the second order interferometric autocorrelaton function results [3].

Figure 2(a) shows the second order interferometric autocorrelation curve computed for an unchirped gaussian pulse. This situation corresponds to the collinear, fringe resolved autocorrelation method.

The two light beams have a finite width w along the x coordinate: u(x, y, z_{o}) = 0 for ∣x∣ ≥ w/2 (Fig. 1). It follows that C(x, y) is zero, for ∣x∣ ≥ 2w, because of the lack of overlap among the four convolving functions in Eq. (7). C(s, 0) and C(2s, 0) are then both zero for ∣s∣ ≥ 2w, and Eq. (8) simplifies into the intensity autocorrelation with background formula, plotted in Fig. 2(b) [3].

The intermediate situation: ∣s∣ = w is shown in Fig. 2(c) [9, 10]. This time C(2w, 0) = 0 while C(w, 0) ≠ 0. In order to evaluate C(w, 0), the particular case of a separable function with a rectangular shape along x will be considered:

where f(y) is any arbitrary complex function. The analytical expression for C(s, 0) results:

From Eq. (10): C(w, 0)/C(0, 0) = 0.25, which is in good agreement with the experimental value of 0.26 in Ref. [10].

Equation (8) can be rewritten:

$$+0.25\left\{2\phantom{\rule{.2em}{0ex}}\mathrm{exp}\left(-\mathrm{i\omega \Delta t}\right)\underset{-\infty}{\overset{+\infty}{\int}}\mathrm{dt}\phantom{\rule{.2em}{0ex}}{\epsilon}^{*}\left(t\right)\epsilon \left(t-\mathrm{\Delta t}\right)\left[{\mid \epsilon \left(t\right)\mid}^{2}+{\mid \epsilon \left(t-\mathrm{\Delta t}\right)\mid}^{2}\right]+c.c\right\}$$

## 2. Microscope autocorrelation measurements

Several experimental setups have been devised to perform second order autocorrelation at the objective focal plane of a nonlinear optical microscope [15–22]. Following the standard measuring method, the ultrafast laser light is firstly sent into the autocorrelator interferometer I, where the two temporally delayed beam replicas are generated. The two beams are then sent into the scanning head, through all the microscope optics, and are eventually focused onto the autocorrelator detection system D, located at the objective focus.

Two-photon conductivity and two-photon absorption are the most suitable autocorrelation techniques for pulse temporal characterization in nonlinear optical microscopy. Second harmonic generation imposes, in fact, stringent phase-matching conditions which are not fulfilled by the wide focusing angles of high-NA objectives.

In the present paper, a novel and far more simple setup is proposed, where the microscope tools and capabilities are exploited at their best. No additional instrumentation but a few optical components are needed.

A lateral shearing interferometer LSI is inserted into the microscope, like an ordinary optical filter, and a spatially uniform fluorescent specimen N is observed. The second order interferometric autocorrelation function is then easily recovered from the two-photon excited fluorescent image.

#### 2.1. Lateral shearing interferometer

In a LSI an input beam is split into two output replicas which are laterally spaced by a distance s (Fig. 3) [23]. In a white-light compensated setup, an incoming plane wave with a propagation vector perpendicular to the shear direction, yields two output wavefronts with zero optical path difference OPD between them. Otherwise, any angular deviation α from perpendicularity gives rise to a finite OPD and a corresponding temporal delay:

where c is the velocity of light.

Unlike the Michelson interferometer, the temporal delay in a LSI is generated by angularly scanning the input beam. The use of a LSI to perform autocorrelation measurements in scanning optical microscopy is thus obvious.

Figure 4 shows the principle of operation [24, 25]. The LSI was introduced into the microscope parallel optical path at the rear of the infinity corrected objective O, a space usually allocated to optional optical devices such as, for example, epifluorescence filter blocks and polarization components. Moreover, a spatially uniform fluorescent material N was placed onto the microscope specimen stage. These are the only optical elements which are needed to perform autocorrelation measurements at the microscope specimen plane.

During image acquisition, the collimated laser excitation beam scans the object field along the x (and y) direction and a variable temporal delay among the two LSI outputs arises due to the angular scanning according to Eq. (12): ∆t = s sinα /c ≅ sα/c ≅ x s/f c, where the x axis was chosen along the LSI shear direction, and f is the objective focal length.

Once the two beams are finally focused by the objective O at the point P(x ≅ fα, y), a two-photon excited fluorescence is emitted. The signal goes back along the light path passes through the short-pass dichroic SD and a band-pass interference filter F and it is finally detected by the microscope photomultiplier tube PMT. A 2D image is then built up point by point by x, y scanning. Being N an homogeneously fluorescent material, the image is a function of the excitation electric field distribution only and it is given by Eq. (8).

#### 2.2. Wavefront-division LSI

White-light compensated LSIs working in collimated light [23] are all suited for Lateral Shearing Autocorrelation [24]. Figure 5 shows the extremely simple configuration adopted here. Two glass plane parallel plates are rotated by an angle 2φ and joined at one edge.

An input collimated beam reaches the device with a mean propagation direction parallel to the angle bisector. The refraction from each plate displaces the beam by s/2 giving rise to a total shear of [4, 9]:

where d is the plate thickness and n the refractive index.

This LSI belongs to the wavefront-division type. The two output beams (Fig. 5(b)) are spatially separated, laying side by side, so that Eq. (11) applies.

## 3. Experimental setup

The instrument is a commercial confocal laser scanning microscope CLSM (PCM2000, Nikon Corp.) modified for nonlinear operation (Fig. 6(a)) [21, 26].

A short-pass dichroic SD with a wavelength cutoff at 650 nm and a 3 mm thick BG39 emission filter F were chosen to assure total extinction of the excitation light while maximizing the fluorescence collection.

The Ultrafast Laser system UL is a mode-locked Ti:Sapphire oscillator (Mira 900 F, Coherent Inc.) pumped by a 5W frequency-doubled Nd:YVO_{4} laser (Verdi V5). The laser output is directly coupled to the scanning head SH.

The epifluorescence filter holder was removed from the microscope and the LSI was placed in its place (Fig. 6(b)).

The LSI (Fig. 6(c)) was made up of two BK7 glass plates, with refractive index n = 1.51 and a thickness d = 6 mm, placed at an angle φ =π/4. From Eq. (13), a shear value s = 4.0 mm results. The two parts were cut from the same plane parallel plate to avoid differences in their thickness greater than a few wavelengths, so as to ensure white-light compensation.

A uniformly fluorescent specimen N, with single-photon absorption and emission peaks at 458 nm and 516 nm respectively, was then taken an image of (Fig. 6(d)).

A PlanApo 60x, 1.4 NA oil immersion objective was used to perform the measurement and the excitation laser wavelength was set at 780 nm.

The total scan angle range of our CLSM is 56.7 mrad, giving rise (Eq. (12)) to a maximum time delay range of 756 fs across the whole field of view.

As expected, the two-photon excited fluorescence intensity profile in the y direction is constant while, along the x direction (the LSI shear direction) it is proportional to the second-order autocorrelation function given by Eq. (11) (Fig. 7).

A typical experimental result is shown in Fig. 8(a). A best fit was calculated for comparison with the theoretical curve and the absolute value of the relative error is plotted in Fig. 8(b). From the fit, a pulse temporal width of 246 fs results. This value is in good agreement with other experimental data obtained on a quite similar microscope setup [22].

## 4. Conclusion

In conclusion, we have shown here a novel autocorrelator design which is at our knowledge the most suitable and simplest setup for pulse temporal characterization at the objective focal plane of a nonlinear optical microscope as it takes advantage of the instrument tools and capabilities. An LSI inserted into the microscope optical path like an ordinary filter, and a homogeneous fluorescent specimen are the only external components needed to perform two-photon absorption autocorrelation measurements.

## References and links

**1. **W. Denk, J.H. Strickler, and W.W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science **248**, 73–76 (1990). [CrossRef] [PubMed]

**2. **J.C. Diels, J. J. Fontaine, I. C. McMichael, and F. Simoni, “Control and measurement of ultrashort pulse shapes (in amplitude and phase) with femtosecond accuracy,” Appl. Opt. **24**, 1270–1282 (1985). [CrossRef] [PubMed]

**3. **J.C. Diels and W. Rudolph, *Ultrashort Laser Pulse Phenomena: Fundamentals, Techniques, & Applications on a Femtosecond Time Scale* (Academic, San Diego, 1996), Chap. 8.

**4. **D. Welford and B.C. Johnson, “Real time monitoring of CW mode-locked dye laser pulses using a rapid-scanning autocorrelator,” Opt. Commun. **45**, 101–104 (1983). [CrossRef]

**5. **P. Yeh, “Autocorrelation of ultrashort optical pulses using polarization interferometry,” Opt. Lett. **8**, 330–332 (1983). [CrossRef] [PubMed]

**6. **D. T. Reid, M. Padgett, C. McGowan, W. E. Sleat, and W. Sibbett, “Light-emitting diodes as measurement devices for femtosecond laser pulses,” Opt. Lett. **22**, 233–235 (1997). [CrossRef] [PubMed]

**7. **S. M. Kobtsev, S.V. Smirnov, S.V. Kukarin, and V. B. Sorokin, “Femtosecond autocorrelator based on swinging birifrangent plate,” Quantum Electron . **31**, 829–833 (2001). [CrossRef]

**8. **P. Wasylczyk, “Ultracompact autocorrelator for femtosecond laser pulses,” Rev. Sci. Instrum. **72**, 2221–2223 (2001). [CrossRef]

**9. **K. Wada, H. Fukuta, R. Kawashima, N. Kurahashi, T. Matsuyama, H. Horinaka, M. Okuno, A. Watanabe, and Y. Cho, “Simple real-time fringe-resolved autocorrelator for measuring picosecond optical pulses,” Opt. Commun. **214**, 343–351 (2002). [CrossRef]

**10. **H. Mashiko, A. Suda, and K. Midorikawa, “All-reflective interferometric autocorrelator for the measurement of ultra-short optical pulses,” Appl. Phys. B **76**, 525–530 (2003). [CrossRef]

**11. **D. Panasenko and Y. Fainman, “Interferometric correlation of infrared femtosecond pulses with two-photon conductivity in a silicon CCD,” Appl. Opt. **41**, 3748–3752 (2002). [CrossRef] [PubMed]

**12. **J. K. Ranka, A. L. Gaeta, A. Baltuska, M. S. Pshenichnikov, and D. A. Wiersma, “Autocorrelation measurement of 6-fs pulses based on the two-photon-induced photocurrent in a GaAsP photodiode,” Opt. Lett. **22**, 1344–1346 (1997). [CrossRef]

**13. **J. M. Roth, T. E. Murphy, and C. Xu, “Ultrasensitive and high-dynamic-range two-photon absorption in a GaAs photomultiplier tube,” Opt. Lett. **27**, 2076–2078 (2002). [CrossRef]

**14. **Y. Takagi, T. Kobayashi, K. Yoshihara, and S. Imamura, “Multiple- and single-shot autocorrelator based on two-photon conductivity in semiconductors,” Opt. Lett. **17**, 658–660 (1992). [CrossRef] [PubMed]

**15. **M. Müller, J. Squier, and G.J. Brakenhoff, “Measurement of femtosecond pulses in the focal point of a high-numerical-aperture lens by two-photon absorption,” Opt. Lett. **20**, 1038–1040 (1995). [CrossRef] [PubMed]

**16. **M. Müller, J. Squier, R. Wolleschensky, U. Simon, and G.J. Brakenhoff, “Dispersion precompensation of 15 femtosecond optical pulses for high numerical aperture objectives,” J. Microsc. **191**, 141–150 (1998). [CrossRef] [PubMed]

**17. **R. Wolleschensky, T. Feurer, R. Sauerbrey, and U. Simon, “Characterization and optimization of a laser-scanning microscope in the femtosecond regime,” Appl. Phys. B **67**, 87–94 (1998). [CrossRef]

**18. **C. Millard, D. N. Fittinghoff, J. A. Squier, M. Müller, and A. L. Gaeta, “Using GaAsP photodiodes to characterize ultrashort pulses under high numerical aperture focusing in microscopy,” J. Microsc. **193**, 179–181 (1999). [CrossRef]

**19. **G.J. Brakenhoff, M. Müller, and J. Squier, “Femtosecond pulse width control in microscopy by Two-photon absorption autocorrelation,” J. Microsc. **179**, 253–260 (1995). [CrossRef]

**20. **J.B. Guild, C. Xu, and W.W. Webb, “Measurement of group delay dispersion of high numerical aperture lenses using two-photon excited fluorescence,” Appl. Opt. **36**, 397–401 (1997). [CrossRef] [PubMed]

**21. **F. Quercioli, A. Ghirelli, B. Tiribilli, and M. Vassalli, “Ultracompact autocorrelator for multiphoton microscopy,” Microsc. Res. Tech. **63**, 27–33 (2004). [CrossRef]

**22. **F. Cannone, G. Chirico, G. Baldini, and A. Diaspro, “Measurement of the laser pulse width on the microscope objective plane by modulated autocorrelation method,” J. Microsc. **210**, 149–157 (2003). [CrossRef] [PubMed]

**23. **M. V. Mantravadi, “Lateral Shearing Interferometers” in *Optical Shop Testing* , D. Malacara ed. (Wiley, New York, 1992).

**24. **F. Quercioli, B. Tiribilli, M. Vassalli, and A. Ghirelli, “Microscopio confocale multifotone con autocorrelatore basato su un interferometro a spostamento laterale e relativo metodo di caratterizzazione temporale,” Italian patent FI2003A000261 (2003).

**25. **F. Quercioli, A. Ghirelli, B. Tiribilli, and M. Vassalli, “Autocorrelator for multiphoton microscopy," in *Optical Sensing* , B. Culshaw, A. G. Mignani, and R. Riesenbergeds., Proc. SPIE5459 (to be published).

**26. **A. Diaspro, M. Corosu, P. Ramoino, and M. Robello, “Adapting a compact confocal microscope system to a two-photon excitation fluorescence imaging architecture,” Microsc. Res. Tech. **47**, 196–205 (1999). [CrossRef] [PubMed]