Temporal and spatial characterization of a pulsed ... - OSA Publishing

Temporal and spatial characterization of a pulsed gas jet by a compact high-speed high-sensitivity second-harmonic interferometer F. Brandi1,∗ , and F. Giammanco2 1 Nanophysics 2 Physics

Division, Italian Institute of Technology, Via Morego 30, Genova, Italy Department, University of Pisa, Largo B. Pontecorvo 3, Pisa, Italy ∗ [email protected]

Abstract: A compact modular high-speed high-sensitivity secondharmonic interferometer is used to characterize a pulsed gas jet. The temporal evolution of the line-integrated gas density is measured with a resolution of 1 μ s revealing detailed information on its dynamics. The actual radial gas density distribution in the jet is obtained applying the Abel’s inversion method. The sensitivity of the interferometer is 1 mrad, and its robustness, compactness and modularity make the instrument suitable for practical application. Possible use of the instrument in monitoring cluster formation, and phase-dispersion microscopy is discussed. © 2011 Optical Society of America OCIS codes: (120.4640) Optical instruments; (120.3180) Interferometry; (120.5050) Phase measurement; (190.1900) Diagnostic applications of nonlinear optics.

References and links 1. W. T. Mohamed, G. Chen, J. Kim, G. X. Tao, J. Ahn, and D. E. Kim, “Controlling the length of a plasma waveguide up to 5mm, produced by femtosecond laser pulses in atomic clusters,” Opt. Express 19, 15919–15928 (2011). 2. M. Krishnan, K. W. Elliott, C. G. R. Geddes, R. A. van Mourik, W. P. Leemans, H. Murphy, and M. Clover, “Electromagnetically driven, fast opening and closing gas jet valve,” Phys. Rev. Spec. Top.- Accel. Beams 14, 033502 (2011) 3. J. Grant-Jacob, B. Mills, T. J. Butcher, R. T. Chapman, W. S Brocklesby, and J.G. Frey, “Gas jet structure influence on high order harmonic generation,” Opt. Express 19, 9801–9806 (2011). 4. C. Altucci, C. Beneduce, R. Bruzzese, C. de Lisio, G. S. Sorrentino, t. Starczewski, and F. Vigilante, “Characterization of pulsed gas sources for intense laser field-atom interaction experiments,” J. Phys. D: Appl. Phys. 29, 68–75 (1996). 5. T. Adachi, K. Kondo, and S. Watanabe, “Gas density measurement of pulsed gas jet using XeF four-photon fluorescence induced by a KrF laser,” Appl. Phys. B 55, 323–326 (1992) 6. H. Lu, G. Ni, R. Li, and Z. Xu, “An experimental investigation on the performance of conical nozzles for argon cluster formation in supersonic jets” J. Chem. Phys. 132 124303 (2010). 7. T. Auguste, M. Bougeard, E. Caprin, P. D’Oliveira, and P. Monot, “Characterization of a high-density large scale pulsed gas jet for laser-gas interaction experiments,” Rev. Sci. Instrum. 70, 2349–2354 (1999) 8. F. Brandi, F. Giammanco, W. S. Harris, T. Roche, E. Trask, and F. J. Wessel, “Electron density measurements of a field-reversed configuration plasma using a novel compact ultrastable second-harmonic interferometer,” Rev. Sci. Instrum. 80, 113501 (2009). 9. C. Yang, A. Wax, I. Georgakoudi, E. B. Hanlon, K. Badizadegan; R. R. Dasari, and M. S. Feld, “Interferometric phase-dispersion microscopy,” Opt. Lett. 25, 1526–1528 (2000). 10. D. Fu, W. Choi, Y. Sung, Z. Yaqoob, R. R. Dasari, and M. Feld, “Quantitative dispersion microscopy,” Biomed. Opt. Express 1, 347–353 (2010).

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Received 8 Sep 2011; accepted 4 Oct 2011; published 29 Nov 2011 5 December 2011 / Vol. 19, No. 25 / OPTICS EXPRESS 25479

11. M. Cui, M. G. Zeitouny, N. Bhattacharya, S. A. van den Berg, and H. P. Urbach, “Long distance measurement with femtosecond pulses using a dispersive interferometer,” Opt. Express 19, 6549–6562 (2011). 12. F. Hopf, A. Tomita, and G. Al-Jumaily, “Second-harmonic interferometers,” Opt. Lett. 5, 386–388 (1980). 13. K. Alum, Y. Koval’chuk, and G. Ostrovskaya, “Nonlinear dispersive interferometer,” Sov. Tech. Phys. Lett. 7, 581–582 (1981). 14. M. J. Buie, J. T. P. Pender, J. P. Holloway, T. Vincent, P. L. G. Ventzek, and M. L. Brake, “Abel’s inversion applied to exprerimental spectroscopic data with off axis peaks,” J. Quant. Spectrosc. Radiat. Transfer 55, 231– 243 (1996). 15. V. Licht and H. Bluhm, “A sensitive dispersion interferometer with high temporal resolution for electron density measurements,” Rev. Sci. Instrum. 71, 2710–2715 (2000). 16. V. Drachev, Y. Krasnikov, and P. Bagryansky, “Dispersion interferometer for controlled fusion devices,” Rev. Sci. Instrum. 64, 1010–1013 (1993). 17. S. Velasko and D. Eimerl, “Precise measurements of optical dispersion using a new interferometric technique,” Appl. Opt. 25, 1344–1349 (1986). 18. F. Brandi, and F. Giammanco, “Harmonic interferometry in the visible and UV, based on second- and thirdharmonic generation of a 25 ps mode-locked Nd:YAG laser,” Opt. Lett. 33, 2071–2073 (2008) 19. P. Bagryansky, A. Khilchenko, A. Kvashnin, A. Solomakhin, H. Koslowsky, and T. team, “Dispersion interferometer based on a CO2 laser for TEXTOR and burning plasma experiments,” Rev. Sci. Instrum. 77, 053501 (2006). 20. P. Burdack, M. Tr¨obs, M. Hunnekuhl, C. Fallnich, and I. Freitag, “Modulation-free sub-Doppler laser frequency stabilization to molecular iodine with a common-path, two-color interferometer,” Opt. Express 12, 644–650 (2004). 21. F. Brandi, and F. Giammanco, “Versatile second-harmonic interferometer with high temporal resolution and high sensitivity based on a continuous-wave Nd:YAG laser,” Opt. Lett. 32, 2327–2329 (2009). 22. V. Drachev, “Nonlinear regime of a dispersion interferometer,” Opt. Spectrosc. 75, 278–281 (1993). 23. F. Brandi, P. Marsili, and F. Giammanco, “ Compact high-speed high-sensitivity second-harmonic interferometer for electron density measurement,” in AIP conference proceedings: Burning plasma diagniostics, 988, 132–135 (2008) 24. A. Behjat, G. J. Tallents, and D. Neely, “The characterization of a high-density gas jet,” J. Phys. D: Appl. Phys. 30, 2872–2879 (1997). 25. G. Chen, B. Kim, B. Ahn, and D. E. Kim, “Experimental investigation on argon cluster sizes for conical nozzles with different opening angles,” J. Appl. Phys. 108, 064329 (2010). 26. K. Y. Kim, V. Kumarappan, and H. M. Milchberg, “Measurements of the average size and density of clusters in a gas jet,” Appl. Phys. Lett. 83, 3210–3212 (2003). 27. A. Ahn, C. Yang, A. Wax, G. Popescu, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Harmonic phase-dispersion microscope with a MachZehnder interferometer,” Appl. Opt. 44, 1188–1190 (2005).

1.

Introduction

The accurate knowledge of the molecular density in pulsed gas jet is of great importance for laser generated plasma experiments [1, 2], and high-order harmonic generation studies [3]. Temporal evolution of pulsed gas jet have been investigated by means of third-harmonic generation [4], fluorescence [5], Rayleigh scattering [6], and Mach-Zender interferometry [7]. These techniques relay on pulsed laser source and repeated measurements varying the time delay between laser pulses and valve opening, and cannot be implemented for on-line monitoring of the gas density temporal evolution. The optical dispersion, i.e, the wavelength dependent refractive index n(λ ), is a basic property of matter whose accurate measure is used in many applications, e.g., to monitor free electron density in plasma [8], to image biological tissues with phase-contrast measurements [9, 10], or to accurately measure distance [11]. By measuring optical dispersion, also the molecular density of gases can be determined provided that the dispersion is known for a defined density value, e.g., standard condition for temperature and pressure. An attractive and accurate method to precisely measure the optical dispersion is based on the so-called second harmonic interferometer (SHI) [12, 13] that is a special common-path, twocolor interferometer. The SHI principles of operation are the following: i) the second harmonic of a laser beam is generated in a single pass frequency converter; ii) the fundamental and second harmonic beams propagate collinearly through the sample; iii) the fundamental beam is #154330 - $15.00 USD (C) 2011 OSA


frequency doubled again in a second frequency converter; iv) interference then takes place between the two collinear second harmonic beams. The measured phase shift between the second harmonic beams is Δφ = 4λπ L Δn(λ )dl, where Δn(λ ) = n(λ ) − n(λ /2), and L is the optical path between the two harmonic converters. Being a common-path interferometer the SHI is insensitive to vibration, and compared with typical two-color interferometers it has the advantage of using a single laser source. For non-homogeneous samples with cylindrical symmetry, the actual radial distribution of the refractive index dispersion can be obtained by measuring the phase shift along several chords and then applying the well known Abel’s inversion technique[14]. Pulsed laser sources are used in SHI to enhance the efficiency of single-pass second-harmonic generation, however this limits the temporal resolution and measurable time window range, depending on the laser repetition rate and the pulse time duration. The fastest SHI developed up to date has a time resolution in the ns range [15], and it comprises the pulsed amplification of a CW laser source limiting the measurable interval to tens of μ s. With a 100kHz Nd:YAG modelocked laser a time resolution of 10 μ s and sensitivity of about 1 mrad have been achieved [16]. In [17] an accurate measurement of the optical dispersion of several gases between 1064 nm and 532 nm has been performed in a static cell using a pulsed laser source, resulting in the determination of optical dispersion at standard temperature and pressure. Using a Nd:YAG modelocked ps laser third-harmonic interferometry has also been demonstrated [18], thus bringing dispersion interferometry in the UV wavelength range. The use of a continuous-wave (CW) laser source has in principle no limit in terms of time resolution, the only constrain being given by the detection system. A SHI based on a CW CO2 laser was developed [19] resulting in a sensitivity of 10 mrad with time resolution of 1 ms. A CW SHI was used to demonstrate modulation-free sub-Doppler laser frequency stabilization on a molecular iodine transition [20]. Recently, we have demonstrated a high-speed high-sensitivity SHI employing a CW Nd:YAG laser [21], and a remotely controlled version of such device has been used on a large plasma machine to measure free electron density [8]. Here we present a complete spatiotemporal characterization of the gas density in a pulsed jet using an improved compact and modular high-speed CW SHI with phase sensitivity of 1 mrad, and spatial resolution of 400 μ m. The time resolution of 1 μ s allows for real-time monitoring of the gas dynamics. To our knowledge, on-line high-resolution measurement of temporal evolution of a gas density in a pulsed jet have not been reported to date. Possible implementation of the technique in monitoring cluster formation, and in phasedispersion microscopy is also discussed. 2.

The experimental setup and procedure

The schematic of the experimental setup is shown in Fig. 1(a). The SHI is composed by a fiber coupled CW Nd:YAG laser (λ = 1064 nm, output power 500 mW) and two compact optical units (OUs) with dimensions of about 10x10x20 cm3 . The optical elements inside the OUs are aligned and define the optical axes of each OU. The OUs are supported by tilt stages used to align the optical axes. The beam from the laser is brought to the first OU by a singlemode polarization-maintaining optical fiber. The first OU comprises an adjustable collimator, a temperature stabilized second harmonic generatio (SHG) unit, and a dual-wavelength waveplate, half-wave at 532 nm and full-wave at 1064 nm. The adjustable collimator at the fiber out-put is set such that: i) the polarization axis of the fundamental beam matches the optical axis of the SHG crystal; ii) the fundamental beam waist radius is 400 μ m, while the beam diameter on the SHG crystals is about 1mm. The second OU comprises a tilted BK7 window AR coated at both 1064 nm and 532 nm (C), a second temperature stabilized SHG unit, a harmonics separator (HS) with high reflectivity at 1064 nm and high transmission at 532nm, a beam dump (BD) for the 1064 nm reflected beam, an interferential filter centered at 532 nm

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Laser

OI Coll

DW λ

MMF

Vaccum Chamber

OU1 Coll

D1

(a)

SMPMF

SHG1

Valve

λ λ/2

OU2 C

λ

BD Coll F λ/2

SHG2 Gas Jet

HS

PC D2

Coll

V− V+

MMF

λ/2

PBS

(b) Armature machanical stop

x z

Fig. 1. (a) Experimental apparatus: OI-optical isolator; SMPMF-single mode polarization maintaining fiber; OU-optical unit; Coll-collimator; SHG-second harmonic generation unit; DW-dual-wavelength wave-plate; C-tilted glass window; HS-harmonic separator; BDbeam dump; F-interferential filter; PBS-polarizing beam splitter; MMF-multi-mode fiber; D-detector; V± -detector signals. (b) Schematic of the solenoid valve: the central axis of the valve is denoted as z, while the optical axis of the interferometer is parallel to the x-axis.

(F), a polarizing beam splitter (PBS), and two collimators. The nonlinear crystals are BIBO cut for Type I SHG at 1064 nm, with dimension 4x4x10 mm3 , and anti-reflection coated for both fundamental and second harmonic radiation. The axes of the two BIBO crystals are set parallel and the DW is used to rotate by 90◦ the polarization of the first second harmonic beam (SHB) while leaving unaffected the polarization of the 1064 nm beam. In this way the second SHG process is independent on the first SHG process, in order to obtain the proper interferometric signal [22]. The window C is held on a motorized mirror mount which allows remote control of the tilt angle in order to obtain very good compensation of the spurious phase signal due to the dispersion in BK7 glass. The tilted window also induces a transversal displacement between the fundamental beam and SHB, however its magnitude is small and does not affect the interference signal significantly. The HS and the filter F are used to completely suppress the fundamental radiation. The actual interference between the two SHBs takes place at the PBS which is set with transmission axis at 45◦ with respect to the polarization of both SHBs. The transmitted and reflected beams show interference signals with a π phase shift off-set one with respect to the other, and are sent to the detector by means of multi-mode fibers. The detector is a Si-PIN photodiode with a transimpedance gain of 107 Ohm and a response time of 0.9 μ s. The signals are acquired with a digital card (NI PCI 6120, 1 MS/s, 16 bit) and saved in a personal computer for on-line analysis and data storage. The sample to be tested is placed in between the two OUs. In the present experiments the sample is a pulsed xenon gas jet free expanding in a vacuum chamber. The light beams pass the chamber through two AR coated windows, and the residual pressure in the chamber is kept

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at about 10−2 mbar by means of a rotative pump. The pulsed gas jet is produced with a fast solenoid valve (Parker-Pulse Valve series 9) operated at 10 Hz, whose pulse width can be set from few hundred microsecond upwards using the valve driver (Parker-IOTA ONE). The valve has a 0.79 mm cone shaped orifice with 45◦ nominal half-angle. In Fig 1(b) a schematic of the valve is presented. A poppet, held by a magnetic armature, is pushed forward by the main spring and closes the orifice. A second spring is used as buffer for the armature. When a current pulse flows in the solenoid coil the generated magnetic field pushes backwords the armature and the poppet, opening the orifice. The armature has a mechanical stop that set the full opening. The valve is mounted on a translation stages that allows movement in all three spatial directions, and it is set on the beam waist position along the optical axis direction x. It is worth mentioning that the apparatus, comprising the SHI and the vacuum chamber, is held on an optical table without vibration dumping systems, and no realignment was needed during the experimental campaign, which demonstrate the stability and insensitivity to vibration of the device. The photodiode signals are V± = α±

P02 [β1 + β2 ± 2(β1 β2 )1/2 cos(Δφ + φ0 )], 2

(1)

where P0 is the fundamental beam power, βi with i = 1, 2 are the SHG efficiency in the two crystals, α± are the detector responsivities, Δφ is the phase to be measured, and φ0 = π 2 (2m + 1) + Δφ0 , with m = 0, ±1, ±2..., is the off-set phase due to spurious dispersions plus the compensator phase. The recorded quantity is α ±V sin(Δφ + Δφ0 ) V+ −V− , (2) = V+ +V− 1 ± α V sin(Δφ + Δφ0 ) α− 1/2 /(β + β ) ≤ 1 is the where the actual ± sign value depends on m, α = αα++ − 1 2 +α− , V = 2(β1 β2 ) fringe visibility. The value of α is adjusted to less than 1%, while Δφ0 can be controlled and set below the mrad level by tilting the compensator window. For small values of Δφ , to an accuracy up to the first order in α , Eq. (2) can be approximated by (V+ −V− )/(V+ +V− ) = V ×Δφ +Δφ0 . The value of V is directly obtained by scanning the tilt angle of the compensator window over half fringe, as reported in [8, 23], and for the present experiments is V = 0.95.

3. 3.1.

Experiments, results and discussion Gas jet temporal evolution

The phase shift induced by a xenon pulsed gas jet measured for a backing pressure of 5 bar at different values of the valve opening time setting τ is shown in Fig. 2, where the baseline of each curve is vertically shifted for clarity. The valve orifice is centered on the laser beam and set at 1 mm distance from the beam waist in the z direction. The phase shift reflects the line-integrated gas density temporal evolution which has distinctive profiles depending on the opening time setting. For τ < 500 μ s the gas pulse has a quasi-symmetric bell-shaped temporal evolution. For 1 ms < τ < 2 ms the gas pulse has a fast rising edge of few hundreds μ s followed by oscillations. At longer opening time settings, τ > 2 ms, a steady gas flow is achieved. Since the valve orifice is set close to the laser beam the measured temporal evolution of the gas jet follows the motion of the poppet. Specifically, the data are interpreted as follows: i) for τ in the range of the pulses valve response time the poppet is pressed back by the main spring in the closed position before reaching the mechanical stop, and so a smooth temporal profile of the gas jet is obtained; ii) for intermediate opening time settings the armature reaches the mechanical stop, then it bounces back several time inducing the sharp peaks in the gas density, before the #154330 - $15.00 USD (C) 2011 OSA


4.5 ms 3.5 ms 2.0 ms 1.0 ms 0.5 ms 0.2 ms

0,06

0,05

Δφ, Rad

0,04

0,03

0,02

0,01

0

0

2

1

3

4

5

6

Time, ms Fig. 2. Temporal evolution of the phase shift for different opening time settings. Note that each curve is vertically shifted for clarity.

solenoid get discharged and the poppet is press back to the closed position; iii) for τ >2 ms the armature reach stably the complete opening after the bouncing on the mechanical stop is damped, and the steady gas flow is archived. Thanks to the high-speed of the interferometer it is possible to investigate in detail the transition between the different temporal profiles. In Fig. 3 a the phase shift temporal evolution between 400 μ s and 1.4 ms from the valve opening is reported for three opening time settings, clearly showing the different temporal evolutions. It is noticeable that the first bounce of the armature on the mechanical stop has a duration shorter than the temporal resolution of the SHI, i.e. 1 μ s, due to the high velocity, i.e., momentum, acquired by the armature during the fast opening. The subsequent bounce lasts about ten μ s and is resolved by the SHI. The effect of the gas backing pressure is also investigated. In Fig. 4 the measured temporal

0,04 3.5 ms 1.0 ms 0.5 ms

Δφ, Rad

0,03

0,02

0,01

400

500

600

700

800

900

Time, μs

1000

1100

1200

1300

1400

Fig. 3. Detailed view of the temporal evolution of the phase shift for three selected opening time settings.

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Δφmax, Rad

0,02

Experimental data Linear least square fit 0,01

0

0

1

2

3

4

Backing pressure, bar

6

5

Fig. 4. Measured phase shift for different backing pressures. In the inset the peak value of the phase shift is plotted ad function of the backing pressure along with the least-square fit.

evolution of the phase shift as function of the backing pressure is shown for an opening time setting of 400 μ s. In the inset of Fig. 4 the maximum value of the phase shift is plotted clearly revealing a linear increase on the gas density as function of the backing pressure in agreement with previous findings reported in the literature [4, 5, 7]. 3.2.

Gas jet spatial distribution

The actual spatial distribution of the gas density in the jet is investigated by measuring the phase shift for several chords, i.e., y values, along a x − y plane at z = 3 mm from the valve orifice. An opening time setting of 3.5 ms, and a backing pressure of 5 bar are used. The results are reported in Fig. 5(a), where the measured phase shift Δφ (y) shows a symmetric profile falling to zero at 4.2 mm from the central axis. A cylindrical symmetry of the gas jet around the central axis of the valve is assumed, thus 4π λ

Δφ (y) =

Δφ, Rad

0,015

+(r2 −y2 )1/2 0 −(r02 −y2 )1/2

Δn(r)dx =

4π Δn0 λ N0

+(r2 −y2 )1/2 0 −(r02 −y2 )1/2

N(r)dx,

(3)

(a)

0,01

0,005

Gas density, 10

17

cm

-3

0

2,5

(b)

2 1,5 1 0,5 0

-4

-3

-2

0 -1 1 Radial position, mm

2

3

4

Fig. 5. (a) Phase shift measured as function of the lateral position of the valve orifice (black dots), and sixth order polynomial fit on the experimental data (dashed line). (b) Gas density radial distribution obtained by Abel’s inversion.

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where, r = (x2 + y2 )1/2 is the radial coordinate, r0 is the radius of the gas jet, Δn0 = 192 × 10−7 is the dispersion of xenon between 1064 nm and 532 nm at 0 ◦ C and 1 atm [17], N0 = 2.69 × 1019 cm−3 is the Loschmidt’s number, i.e., the density of an ideal gas at 0 ◦ C and 1 atm, and N(r) is the xenon atomic density in the gas jet. To retreive the radial gas density distribution an analytical method to performe the Abel’s inversion is used [24]. The measured phase shift is fitted with a polynomial curve to the power sixth, Δφ (y) = a + cy2 + ey4 + gy6 , as shown by the dashed line in Fig. 4(a). Note that only even powers are used due to symmetry reasons. The polynomial curve very well follows the experimental data points. The radial gas density distribution, shown in Fig. 5(b), is obtained from the formula N(r) = −(λ N0 )/(4π 2 Δn0 )[2cu + 4e(ur2 +u3 /3)+6g(ur4 +2u3 r2 /3+u5 /5)] [24], where u = (r02 −r2 )1/2 , with r0 = 4.2 mm. The full-width half-maximum of the gas density radial distribution is D(3mm)=6.7 mm. Assuming a conical expansion of the gas jet, D(z) = D0 + 2 tan(β )z where D0 = 0.79 mm is the orifice diameter, a cone half-angle β = 44.6◦ is obtained in very good agreement with the nominal value of the valve orifice cone angle. 4.

Conclusions and outlook

A compact second-harmonic interferometer capable of fully characterize a pulsed gas jet is presented. Temporal evolution of the gas density is measured with a resolution of 1 μ s. The observed gas density dynamics is explained in terms of the pulsed valve poppet motion. Precise reconstruction of the actual radial gas density distribution is obtained by applying Abel’s inversion. Detail knowledge of pulsed gas jet dynamics can aid research in high-intensity lasermatter intercation, like optimization of high-order harmonic generation, and laser generated plasma. In the present experimental conditions (backing pressure ≤ 5 bar) cluster formation in the expanding pulsed gas jet is limited, and therefore neglected. In general, cluster formation in an expanding gas jet with high backing pressure is monitored by means of Rayleigh scattering measurements [25]. However, Rayleigh scattering alone cannot be used to determine cluster average size and density since both quantities are unknown. In [26], an elegant all-optical method to measure both the cluster average size and density has been presented based on the combination of Rayleigh scattering and classical Mach-Zender interferometry. In principle also second-harmonic interferometry can be implemented, in combination with Rayleigh scattering, to characterize a gas jet containing a significant amount of clusters, provided that the dielectric function of the bulk material internal to the cluster is known at the fundamental and second harmonic wavelength, and that the contribution of residual monomers, i.e., isolated atoms, on the optical dispersion of the gas jet can be neglected [26]. Due to its high-sensitivity the presented SHI is an ideal instrument to monitor small variation of dispersion at high-speed. Also, the compactness and stability of the instrument enable its operation in any laboratory and field conditions, as well as its integration into existing apparatus. As outlook, the implementation of the SHI in microscopy is forseen. The spatial resolution can be reduced to the μ m level by using diffraction limited microscope objectives, while the common-path geometry allows for an easy integration into standard transmission microscopic devices. As application example we consider the imaging of biological tissues. In [27] an harmonic phase-dispersion microscope at λ = 1064 nm comprising a Mach-Zender interferometer and two acusto-optic modulators has been presented. Biological samples have been imaged with a phase resolution of 7 × 10−2 rad. However, the acquisition time of 50 ms/pixel hampers practical application of such system. The present SHI has a simple optical setup, with a sensitivity of 10−3 rad for a time resolution as small as 1 μ s, reducing the acquisition time by 4 orders of magnitude and thus enabling practical implementation of harmonic phase-dispersion microscopy. Moreover, the sensitivity can be increased to the 10−4 rad level [21], for an ac-

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quisition time of 10 μ s, further improving the potential capability of the SHI for challenging microscopic measurements.

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