Simultaneous self-phase modulation and two-photon absorption measurement by a spectral homodyne Z-scan method

doi:10.1364/OE.16.004192

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Simultaneous self-phase modulation and two-photon absorption measurement by a spectral homodyne Z-scan method

Martin C. Fischer, Henry C. Liu, Ivan R. Piletic, and Warren S. Warren »View Author Affiliations

Optics Express, Vol. 16, Issue 6, pp. 4192-4205 (2008)
http://dx.doi.org/10.1364/OE.16.004192

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Abstract

We developed a technique to simultaneously measure self-phase modulation and two-photon absorption using shaped femtosecond laser pulses. In the conventional Z-scan measurement technique the amount of nonlinearity is determined by measuring the change in shape and intensity of a transmitted laser beam. In contrast, our method sensitively measures nonlinearity-induced changes in the pulse spectrum. In this work we demonstrate the technique in nonlinear absorptive and dispersive samples, quantify the obtained signal, and compare the measurements with traditional Z-scans. This technique is capable of measuring these nonlinearities in highly scattering samples.

1. Introduction

In this work we demonstrate a new Z-scan technique to simultaneously measure two-photon absorption (TPA) and self-phase modulation (SPM). Conventional Z-scan techniques can measure these nonlinear effects in transparent samples, but they are unable to do so if strong scattering is present. Biological tissues are examples of highly scattering environments that preclude conventional TPA and SPM measurements. Recently we have shown that hemoglobin [1

D. Fu, T. Ye, T. E. Matthews, B. J. Chen, G. Yurtserver, and W. S. Warren, “High-resolution in vivo imaging of blood vessels without labeling,” Opt. Lett. 32, 2641–2643 (2007). [CrossRef] [PubMed]

] and melanin [2

D. Fu, T. Ye, T. E. Matthews, G. Yurtsever, and W. S. Warren, “Two-color, two-photon, and excited-state absorption microscopy,” J. Biomed. Opt. 12, 054004 (2007). [CrossRef] [PubMed]

] offer intrinsic TPA contrast of great potential diagnostic value. We have also shown that neuronal activity in brain tissue can result in strong intrinsic SPM signatures [3

M. C. Fischer, H. Liu, I. R. Piletic, Y. Escobedo-Lozoya, R. Yasuda, and W. S. Warren, “High-sensitivity, single-beam n ₂ measurements,” Opt. Lett. 14, 955–957 (1989). [CrossRef]

]. These new contrast mechanisms could provide valuable functional and metabolic information in tissue if they can be extracted efficiently using physiologically acceptable optical power levels.

Traditional Z-scan techniques operate on the underlying principle that a single, transmitted, focused laser beam interacting with a nonlinear medium experiences changes in its spatial phase and amplitude profile. Both phase and amplitude changes result in detectable variations of the intensity pattern in the far field. These changes depend on the position of the nonlinear sample relative to the focal point of the laser beam. The sample is scanned along the beam propagation direction (Z-scan) and the resulting far field intensity pattern is analyzed to yield information on type and magnitude of the nonlinearity. This method was first demonstrated in thin nonlinear samples [4

M. Sheik-Bahae, A. A. Said, and E. W. van Stryland, “High-sensitivity, single-beam n ₂ measurements,” Opt. Lett. 14, 955–957 (1989). [CrossRef] [PubMed]

, 5

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990). [CrossRef]

]. In the case of a two-photon absorbing medium the relative position of the sample with respect to the beam focus determines the attenuation and therefore the total transmitted power, which can be easily measured by collecting the entire beam onto a photodetector. In the case of a medium with self-phase modulation properties the total power remains constant but nonlinearity-induced phase changes in the beam modify the far-field intensity pattern (self-focusing). These beam shape changes can be detected by measuring the transmission through an appropriately placed small aperture (most commonly placed in the center of the beam). For media that exhibit both TPA and SPM, an open-aperture scan can determine the TPA coefficient, whereas the closed-aperture scan contains signatures of both coefficients.

The fundamental difficulty in conventional Z-scan measurements is that for peak intensities that do not damage the sample under study the measured intensity changes can be exceedingly small. The sensitivity to beam shape changes can be enhanced over closed-aperture measurements by eclipse-type measurements [6

T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. van Stryland, “Eclipsing Z-scan measurement of λ/10⁴ wave-front distortion,” Opt. Lett. 19, 317–319 (1994). [CrossRef] [PubMed]

] or by numerically fitting complete beam profiles acquired with an imaging device (e.g. a CCD array) [7

G. Tsigaridas, M. Fakis, I. Polyzos, P. Persephonis, and V. Giannetas, “Z-scan technique through beam radius measurements,” Appl. Phys. B: Lasers and Optics 76, 83–86 (2003). [CrossRef]

]. The detection can also be improved by employing multi-color, pump-probe type schemes [8

H. Ma, A. S. L. Gomes, and C. B. de Araujo, “Measurements of nondegenerate optical nonlinearity using a two-color single beam method,” Appl. Phys. Lett. 59, 2666–2668 (1991). [CrossRef]

, 9

J. Wang, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E. W. Van Stryland, “Time-resolved Z-scan measurements of optical nonlinearities,” J. Opt. Soc. Am. B 11, 1009–1017 (1994). [CrossRef]

], lock-in detection with an intensity-modulated source [10

I. Guedes, L. Misoguti, L. De Boni, and S. C. Zilio, “Heterodyne Z-scan measurements of slow absorbers,” J. Appl. Phys. 101, 063112 (2007). [CrossRef]

] or differential measurements while dithering the sample position [11

J.-M. Menard, M. Betz, I. Sigal, and H. M. Van Driel, “Single-beam differential Z-scan technique,” Appl. Opt. 46, 2119–2122 (2007). [CrossRef] [PubMed]

]. Despite these enhancements the major drawback remains that small changes in the beam profile caused by effects other than the nonlinearity (for example imperfect sample surfaces or scattering within the sample) can complicate if not prevent reliable measurements. Also, a well characterized input beam profile is required for precise measurements (a review of the Z-scan technique including the effects of input beam profiles is given by Chapple, et al. [12

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. McKay, and R. G. McDuff, “Single-beam Z-scan: measurement techniques and analysis,” J. Nonlinear Opt. Phys. Mater. 6, 253–293 (1997). [CrossRef]

]). These stringent requirements on the detected beam profile can only partially be alleviated by performing measurements relative to a well characterized standard sample [13

R. E. Bridges, G. L. Fischer, and R. W. Boyd, “Z-scan measurement technique for non-Gaussian beams and arbitrary sample thicknesses,” Opt. Lett. 20, 1821–1823 (1995). [CrossRef] [PubMed]

We believe that none of the existing Z-Scan techniques is suitable to highly scattering samples, especially when scattering is as large as to completely destroy the incoming beam profile or when no transmitted beam is available at all (as is the case for microscopy in thick tissue samples). To overcome this limitation, we developed a technique that does not encode the signature of the nonlinear interaction in the spatial beam profile but in the pulse spectrum. As a consequence we then do not detect small beam shape changes but small spectral changes. Analyzing the pulse spectrum has the advantage that, unlike the beam shape, it is generally not affected by confounding linear effects such as scattering. However, as would be expected, these spectral changes can be just as minuscule as the beam shape changes. To overcome the challenge of measuring these small changes we have developed the “hole-refilling” technique that uses femtosecond laser pulse shaping to substantially improve sensitivity [14

M. C. Fischer, T. Ye, G. Yurtsever, A. Miller, M. Ciocca, W. Wagner, and W. S. Warren, “Two-photon absorption and self-phase modulation measurements with shaped femtosecond laser pulses,” Opt. Lett. 30, 1551–1553 (2005). [CrossRef] [PubMed]

]. Here we describe the implementation of a “hole refilling”-based Z-Scan technique, verify its nonlinear scaling behavior, provide a quantitative analysis, and compare the technique to traditional Z-scans in nonlinear absorptive and dispersive samples.

2. Hole refilling

2.1. Principle

The idea of using femtosecond pulse shaping to enhance the sensitivity of nonlinear measurements is described in [14

] and is only briefly summarized here. The key concept of this new measurement technique is to sensitively monitor spectral changes in a laser pulse undergoing nonlinear interactions in a medium. Even when these changes are small, we can efficiently detect them by appropriately pre-shaping the spectrum such that these changes show against a small background. For this purpose we introduce a “hole” in the center of the pulse spectrum. Two-photon absorption and self-phase modulation both change the laser pulse spectrum in a way as to refill the hole. TPA attenuates the propagating laser pulse; therefore the added field in the spectral hole is negative with respect to the incoming pulse. In contrast, SPM does not absorb energy from the laser beam but merely modulates the phase of the pulse, leading to an added field in the center of the spectral hole that is 90° out of phase with the remainder of the pulse. In both cases we can find the weak nonlinear signal by analyzing the frequency component in the center of the spectral hole—in principle a background free measurement task. The trick in separating the two contributions is the phase difference of the electric field generated by the nonlinear polarization in the spectral hole (180° for the absorptive TPA, 90° for the dis-sipationless SPM). The phase can be detected by a homodyne technique. If the spectral hole is not completely empty but a few percent of the peak intensity remain we can use the remaining portion as a local oscillator (LO). The LO interferes with the nonlinear signal so that the signal amplitude in the spectral hole depends on the relative phase of the LO and the nonlinear signal. Rotating the phase of the LO allows the reconstruction of the phase of the nonlinear signal from the measured spectral intensity in the hole.

2.2. Experimental implementation

Figure 1 shows a simplified schematic of our experimental setup. The laser source was a regenerative amplifier (RegA, Coherent) seeded by a mode-locked oscillator (Vitesse, Coherent). The output pulse length was about 50 fs at a repetition rate of 20 kHz. The pulses were spectrally shaped in a 4 - f pulse shaper arrangement [15

C. W. Hillegas, J. X. Tull, D. Goswami, D. Strickland, and W. S. Warren, “Femtosecond laser pulse shaping by use of microsecond radio-frequency pulses,” Opt. Lett. 19, 737–739 (1994). [CrossRef] [PubMed]

]. The central portion of the radio-frequency (RF) waveform driving the shaper’s acousto-optic modulator (AOM) was attenuated, leading to a dip (the local oscillator) in the laser spectrum. In addition, this central portion was frequency shifted by 5 kHz, leading to a LO phase rotation at the same frequency. The optical power in the 3 nm wide LO portion of the spectrum was adjusted to 2.5% of its original value. The beam was then focused with a 10× objective lens into the sample, which was mounted on a translation stage for scanning along the beam axis. The transmitted light was re-collimated and part of it was diverted by a beam splitter (BS) onto a photodiode (D1) for the measurement of sample transmission. The remaining beam passed through a band-pass filter (BPF, the filter was either a monochromator or an angle-tuned interference filter with no notable difference in performance). The central 1 nm portion of the spectral hole passing through the filter was collected onto an amplified photodiode (D2). The signal from D2 was time gated with a boxcar integrator and analyzed in a lock-in amplifier whose reference was a signal at the frequency difference between the two sections of the RF waveform driving the pulse shaper AOM.

Fig. 1. Schematic of the experimental setup.

2.3. Numerical simulations

To support our experiments we performed numerical simulations of the pulse propagation in a nonlinear medium. The propagation equation in the slowly varying amplitude approximation can be expressed (neglecting third- and higher-order dispersion) [16

R. W. Boyd, Nonlinear Optics , (Academic Press, 2003).

] as:

[\frac{1}{2 k_{0}} \nabla_{⊥}^{2} + i \partial_{z} - \frac{β_{2}}{2} \partial_{τ}^{2} + i \frac{α_{0}}{2} + i (α_{2} - i η_{2}) {∣ A (z, τ) ∣}^{2}] A (z, τ) = 0 .

(1)

Here, A(z,τ) is the electric field envelope (E(z,τ)=A(z,τ)exp[ik ₀ z-iω ₀τ]+c.c.) as a function of time, τ=t-z/v_g , measured in the reference frame moving with group velocity v_g . We defined the TPA coefficient α ₂ as α=α ₀+2α ₂|A|², the SPM coefficient η ₂ as n=n ₀+2n ₂|A|² with η ₂=2n ₂ ω ₀/c. The numerical integration of the beam propagation equationwas performed with a split-step Fourier transform method [17

G. P. Agrawal, Nonlinear Fiber Optics , (Elsevier / Academic Press, 2007).

]. The temporal dispersion term was included as a simple multiplication in the spectral domain. The remaining terms (diffraction, attenuation, TPA, and SPM) were accounted for in the time domain by a recursive solution method for cylindrically symmetric geometries [18

S. T. Hendow and S. A. Shakir, “Recursive numerical solution for nonlinear wave propagation in fibers and cylindrically symmetrical systems,” Appl. Opt. 25, 1759–1764 (1986). [CrossRef] [PubMed]

]. In order to avoid having to perform multiple propagations of pulses with different phases of the local oscillator we assumed that the LO portion of the beam is too small to contribute to nonlinear effects. Using this approximation we only needed to propagate one pulse without LO and interfere the resulting field with a known LO of varying phase. To obtain the electric field distribution in the far field we calculated the diffraction integral of the field yielded by the nonlinear simulations (for our cylindrically symmetric geometry this has the form of a Hankel transform).

3. Scaling

To verify the nonlinear origin of the acquired signal we investigated the scaling with the input power. To illustrate the expected scaling we can assume a plane wave input and neglect temporal dispersion and linear absorption. For small nonlinearities we can approximate the refilling of the spectral hole by

Δ \tilde{A} (ω) = z (i η_{2} - α_{2}) \tilde{{∣ A_{h} ∣}^{2} A_{h}},

(2)

where

\tilde{{∣ A_{h} ∣}^{2} A_{h}}

is the Fourier transform of the product |A_h (τ)|² A_h (τ) and Ã_h (ω) is the incoming spectrum with a clean hole (no LO). A typical pulse shape is illustrated in Fig. 2. We can then write the homodyne signal as the interference of the local oscillator with the hole refill contribution in the spectral domain as

Fig. 2. Sketch of an incoming pulse with a spectral hole in the frequency domain (a) and the time domain (b). Part (c) illustrates

\tilde{{∣ A_{h} ∣}^{2} A_{h}}

in Eq. 2 (the nonlinear polarization term). Note the non-zero component at the hole location, which in principle allows for a background-free detection. Part (d) indicates a pulse spectrum with a square hole (as used in the experiments) with a local oscillator component.

{\tilde{A}}_{hom} = {\tilde{A}}_{LO} \exp (i ϕ) + Δ \tilde{A},

(3)

where ϕ is the phase of the local oscillator relative to the remainder of the pulse. With our pulse shaping technique we can create arbitrary values of ϕ for each laser pulse. The spectral intensity in the hole is

{∣ {\tilde{A}}_{hom} ∣}^{2} = {∣ {\tilde{A}}_{LO} ∣}^{2} + {∣ Δ \tilde{A} ∣}^{2} + 2 z {\tilde{A}}_{LO} \tilde{{∣ A_{h} ∣}^{2} A_{h}} (η_{2} \sin ϕ - α_{2} \cos ϕ),

(4)

where for simplicity the incoming spectra Ã _LO and Ã_h were assumed real. In the experiment we chose a linearly increasing ϕ for successive pulses, leading to a modulation of the measured spectral intensity. Comparing this signal to a reference of known phase in a lock-in amplifier separates the SPM from the TPA contribution. The phase calibration of the lock-in amplifier was performed with a glass sample showing only SPM but not TPA.

If no local oscillator is present, the power in the spectral hole scales as |ΔÃ|²∝P ³ _in, where P _in is the input power into the sample. In the presence of a local oscillator the phase-dependent signal scales as

{\tilde{A}}_{LO} \tilde{{∣ A_{h} ∣}^{2} A_{h}} \propto P_{in}^{2}

if both A_h and A _LO are a constant fraction of the input power. Besides the ability to distinguish between TPA and SPM, Eq. (4) also points out another advantage of the homodyne technique: the magnitude of the oscillatory part of the homodyne signal is larger than the refilling itself by a factor of 2Ã _LO/ΔÃ.

Fig. 3. Scaling of the hole refilling and homodyne signal as a function of input power. The symbols are experimental data, the lines are linear fit on a double-log scale. The fit results for the slope is indicated in the graph. For the R6G solution, only the first 6 points (with lower power) were used for fitting. The background data was measured when the focus was outside the sample and was subtracted from the sample data at each power level.

Figure 3 shows experimental measurements of the hole refill signal as a function of input power in a glass cuvette filled with a 30mM solution of Rhodamine 6G (R6G) in methanol. The focus was either positioned outside the cuvette (background), in the glass, or in the R6G solution. The Rayleigh length of the beam was substantially smaller than the sample and glass thicknesses. Panel (a) shows the intensity in the spectral hole without local oscillator. The scaling of the hole refilling in glass – which shows only SPM but no TPA – follows a cubic scaling over the entire measured range (slope: 2.98±0.02, the error denotes the error of the fit). The refilling in R6G scales approximately cubic for lower power levels (slope: 2.84±0.06) but drops noticeably at higher power. This saturation-like behavior was to be expected since the pulse attenuates while passing through the nonlinear absorber – an effect that is not taken into account in the first-order approximation for the hole refilling in Eq. (2). Figure 3(b) shows the two quadrature components of the homodyne signal as a function of power. For the acquisition, the phase reference of the lock-in amplifier was chosen to yield only an SPM component when the focus was located in glass. The scaling of the SPM signal in glass is quadratic with the input power (slope: 2.03±0.01) for the entire measured power range, whereas in R6G it is quadratic only for lower power levels (SPM slope: 2.08±0.02, TPA slope: 2.00±0.05). This scaling changes at higher powers due to pulse attenuation. The interaction of temporal pulse reshaping (dispersion) and spatial pulse re-shaping (e.g. self-focusing) during passage through the nonlinear media can also alter the power scaling of the hole refilling at high power levels. In Fig. 3 it is apparent that the homodyne signal can be measured at lower input power than the hole refill signal owing to the amplification effect of the local oscillator.

4. Comparison to the traditional Z-scan method

In this section we describe the implementation of the homodyne hole refill technique in a Z-scan setup and point out similarities and differences to the traditional Z-Scan method. To illustrate these points we provide data of SPM and TPA measurements in a quartz cuvette filled with an R6G solution (Fig. 4). In these measurements the cuvette was scanned along the beam propagation direction through the focal point.

Figure 4(a) shows the transmission measured with the monitor photodiode (D1) as a function of cuvette position, similar to a conventional open-aperture Z-scan measurement. Without an aperture, traditional Z-scans are not sensitive to SPM and only TPA can be detected. In comparison, Fig. 4(b) shows the homodyne hole refill signal, which we acquired for the same sample. In contrast to the Z-scan data, the homodyne signal clearly distinguishes the three regions in the cuvette (SPM in the glass walls and TPA/SPM in the solution). It is important to note that SPM and TPA signals are acquired simultaneously as two quadrature components of the photodiode signal from photodiode D2; no aperture was required for the SPM measurement. If an aperture is introduced in the homodyne hole refilling setup, features reminiscent of closed-aperture Z-scan can be observed. Figure 4(c) shows a scan where an aperture was placed after the sample that blocked 75% of the transmitted light (corresponding to the commonly used aperture transmission value S=0.25) with otherwise identical parameters. As can be seen in this graph, transitions in the SPM coefficients caused a TPA signature and vice versa. To confirm this finding we performed numerical simulations for the case of pure SPM and pure TPA media. The results are displayed in Fig. 5(a) and are in qualitative agreement with the experiment.

To take into account focusing analytically, the diffraction term in Eq.(1) needs to be retained. The solution of the resulting propagation equation has been explored in the Z-scan literature. Since in the present case the Rayleigh length is appreciably shorter than the sample length, we need to consider the “thick-sample” limit. Hermann and McDuff [19

J. A. Hermann and R. G. McDuff, “Analysis of spatial scanning with thick optically nonlinear media,” J. Opt. Soc. Am. B 10, 2056–2064 (1993). [CrossRef]

] have derived an on-axis expression for the far field in the case of absorptive and dispersive media; Hermann [20

J. A. Hermann, “Nonlinear optical absorption in thick media,” J. Opt. Soc. Am. B 14, 814–823 (1997). [CrossRef]

] has provided an off-axis generalization. The field envelope A normalized to the field in the absence of any nonlinearity A _lin is given as

\frac{A}{A_{lin}} = 1 - \frac{1}{8} B Q with Q = E_{i} (λ_{a} x) - E_{i} (λ_{b} x) .

(5)

In our notation, B=2(η ₂+iα ₂)|A ₀|² z_R , where z_R =nπw ² ₀/λ is the Rayleigh length in the material. E_i is the exponential integral function and the quantities x (a scaled radial variable) and λ_a,b are defined in Hermann [20

J. A. Hermann, “Nonlinear optical absorption in thick media,” J. Opt. Soc. Am. B 14, 814–823 (1997). [CrossRef]

]. For a Z-scan experiment, the closed-aperture (on-axis) transmission is determined by the on-axis limiting value Q ₀=Q(x→0), the open-aperture transmission by the intensity-weighted radial average (to first order in B):

T_{closed} \approx 1 - \frac{1}{4} Re (B Q_{0})

(6)

T_{open} \approx 1 - \frac{2 π}{4} Re \frac{(B \int Q I_{lin} ρ d ρ)}{P_{lin}},

(7)

Fig. 4. SPM and TPA measurements in a quartz cuvette filled with 30mM R6G in methanol. The shaded areas indicate different materials (glass thickness 1.25 mm, gap 1 mm; both thicknesses are scaled by their respective index of refraction). The input power in this experiment was 60µW. Part (a) shows the transmission through the sample measured in the far field without aperture (conventional open-aperture Z-scan). The boxcar averager was set to average 300 consecutive laser pulses. The lower parts show the homodyne signal measured without aperture (b) and with aperture (c). The lock-in time constant was set to 30 ms and the glass region was used to set the reference phase. The portion of transmitted power onto the photo detector in (a) was attenuated to reach a similar electrical signal level as the level from the homodyne photo detector in (b) for comparison.

where P _lin is the power and I _lin is the intensity at the detector plane of the field in the absence of nonlinearities. Considering only the spatial dependence, our homodyne signal S _hom scales as

Fig. 5. (a) Simulations of SPM and TPA measurements in a medium with SPM (left) and TPA (right). The sample thickness was taken as 200λ and the Rayleigh length as 12λ. The solid lines indicate spectral analysis in the center of the far field distribution (on-axis), the dashed lines are averages over the entire radial distribution. (b) Real and imaginary parts of the integral expression Q as defined in the text. The ratio of z_R to the sample length L was taken as 1/20. The solid lines are on-axis values, the dashed lines are radial averages.

S_{hom} \propto \int A_{lin}^{*} e^{- i Φ} Δ A ρ d ρ \propto e^{- i Φ} B \int Q I_{lin} ρ d ρ .

(8)

Figure 5(b) shows the real and imaginary values of the integral expression Q on-axis (solid lines) and as an intensity-weighted radial average (dashed lines). The real part has a dispersion-like shape on-axis, which vanishes in the weighted radial average. This behavior reflects the fundamental difference between an open- and closed-aperture Z-scan. Equation (6) shows that in the transmission measurement of a conventional Z-Scan only the real part of B×Q contributes to the irradiance onto the detector. For SPM, the real part of Q contributes on-axis but vanishes when integrated over the entire beam (this is obvious on physical grounds, since the pulse energy needs to be conserved). For TPA, however, the imaginary part of Q contributes and integration over the beam profile does not appreciably change the shape of the z-dependence. Our homodyne measurement is fundamentally different from a traditional Z-scan in that the local oscillator’s phase is adjustable, allowing us to measure the real and imaginary parts of B×Q. Since the real part of Q is zero in an open aperture measurement we can adjust the LO to be sensitive to only TPA (as in the Z-scan case) or to only SPM without the requirement for an aperture.

5. Quantification

To quantify the hole refill signal for a focused beam, we would need to solve Eq. (1) given a specified input pulse. For the case of a Gaussian spatial input profile we can obtain an approximate expression for the hole refill signal by reducing the 3-dimensional equation to an “effective” 1-dimensional equation. Disregarding the effect of the nonlinearities in the medium on the spatial beam profile, we can insert the linear solution for the field amplitude

A (ρ, z, τ) = A_{0} (z, τ) \frac{w_{0}}{w (z)} \exp (\frac{- ρ^{2}}{{w (z)}^{2}} + i Φ (ρ, z))

(9)

into Eq. (1) and replace the remaining radial dependence exp(-2ρ ²/w ²) in the nonlinear polarization term by its intensity-weighted average of 1/2 to arrive at an effective propagation equation for A ₀:

\partial_{z} A_{0} (z, τ) = \frac{1}{2} (i η_{2} - α_{2}) \frac{w_{0}^{2}}{{w (z)}^{2}} {∣ A_{0} (z, τ) ∣}^{2} A_{0} (z, τ) .

(10)

Assuming a uniform nonlinearity we can obtain a first order expression for the amplitude change as

Δ A_{out} (τ) \approx (i η_{2} - α_{2}) \frac{π}{2} z_{R} {∣ A_{in} (τ) ∣}^{2} A_{in} (τ) .

(11)

This expression indicates that we can approximate the case of a Gaussian beam focused well inside a uniform nonlinear medium as a plane wave traversing a similar medium of length

\frac{π}{2} z_{R}

. When we add a local oscillator term and use a notation similar to Eq. (2) we obtain

A_{out} (τ) = A_{LO} (τ) e^{iϕ} + (i η_{2} - α_{2}) \frac{π}{2} z_{R} {∣ A_{h} (τ) ∣}^{2} A_{h} (τ) .

(12)

Ultimately, the quantity that is measured by the photodiode is the integrated optical power passing through the band-pass filter

P_{\det} = \frac{1}{2 π} \frac{π w_{0}^{2}}{2} \frac{n_{0} c}{2 π} \int_{slit} {∣ {\tilde{A}}_{out} (ω) ∣}^{2} d ω .

(13)

We can further assume Ã _out(ω) to be constant across the width Δω _slit of the of the band-pass filter. We denote

P_{ref} = \frac{1}{2 π} \frac{π w_{0}^{2}}{2} \frac{n_{0} c}{2 π} Δ ω_{slit} {∣ {\tilde{A}}_{0} (ω = 0) ∣}^{2}

to be the optical power passing through the slit when the nonlinear medium is removed and no spectral hole is present (that is for an input pulse A(τ)=A ₀ exp(-τ²/2σ ² _τ)). We then can write the ϕ -dependent terms in Eq. (13) as two quadrature components (rotating with cos ϕ and sin ϕ respectively) with amplitudes

\frac{P_{\det, ϕ}}{P_{ref}} = {\begin{matrix} - \sqrt{\frac{π}{2}} α_{2} z_{R} r_{LO} {∣ A_{0} ∣}^{2} g (\cos ϕ) \\ \sqrt{\frac{π}{2}} η_{2} z_{R} r_{LO} {∣ A_{0} ∣}^{2} g (\sin ϕ), \end{matrix}

(14)

where r _LO is the fraction of the original spectral amplitude that is used as a local oscillator. The dimensionless scaling factor is

g = \int {∣ a_{h} (τ) ∣}^{2} a_{h} (τ) d τ,

(15)

with a_h (τ)=A_h (τ)/Ã ₀(ω=0) being the normalized pulse shape corresponding to a pulse with a spectral hole of width Δω _hole. If we assume a transform limited input pulse the factor g depends only on the ratio Δω _hole/Δω _FWHM between the hole width and the bandwidth of the input pulse. Figure 6 shows numerical values for g as a function of this ratio.

Fig. 6. Numerical values of the scaling factor g as a function of the ratio between the hole width and the bandwidth of the input pulse. A square hole in a transform-limited pulse spectrum was assumed.

It is useful to compare expression (14) to the transmission changes for a traditional Z-scan using a Gaussian input pulse [19

J. A. Hermann and R. G. McDuff, “Analysis of spatial scanning with thick optically nonlinear media,” J. Opt. Soc. Am. B 10, 2056–2064 (1993). [CrossRef]

Δ T = {\begin{matrix} \frac{1}{\sqrt{2}} \frac{π}{2} α_{I} z_{R} I_{0} (TPA) \\ \frac{1}{\sqrt{2}} \ln 3 \frac{2 π}{λ} n_{I} z_{R} I_{0} (SPM), \end{matrix}

(16)

with α_II ₀=2α ₂|A ₀|² and n_II ₀=2n ₂|A ₀|². Using these expressions we obtain

\frac{P_{\det, ϕ}}{P_{ref}} = {\begin{matrix} \frac{1}{\sqrt{π}} r_{LO} g Δ T (TPA) \\ \frac{\sqrt{π}}{\ln 3} r_{LO} g Δ T (SPM), \end{matrix}

(17)

where it is important to note that ΔT in these expressions denotes the Z-scan transmission change for a Gaussian input pulse, not for a pulse with a spectral hole.

For quantitative measurements of SPM and TPA with the hole refilling technique it is required to carefully characterize the setup. The calibrations that are required to evaluate Eq. (14) are as follows. The reference signal corresponding to P _ref can be obtained by removing the hole from the pulse spectrum and the sample from the beam path (or moving the sample to a position far from the focal point) and measuring the lock-in signal while toggling (or modulating) the power of the input beam. When a hole with a local oscillator is added, r _LO can be determined in a similar way. The temporal pulse shape needs to be known to determine the scaling factor g; if a transform limited pulse is used, knowledge of the ratio of the hole width to the pulse bandwidth is sufficient. Finally, the pulse energy incident onto the sample needs to be known. Note that the signal is independent of focusing conditions if the sample is much thicker than the Rayleigh length.

Intensity fluctuations and non-ideal spatial and temporal pulse shapes in our setup precluded us from performing a precise calibration. With only a rough calibration we obtained estimates for the nonlinear coefficients as a proof of principle. For the TPA coefficient of R6G we measured a value of δ _R6G≈15GM, consistent with literature values [21

P. Tian and W. S. Warren, “Ultrafast measurement of two-photon absorption by loss modulation,” Opt. Lett. 27, 1634–1636 (2002). [CrossRef]

] obtained for similar laser parameters. The value we obtained for the SPM coefficient in glass n _I,glass≈6×10^-17cm²/W is substantially lower than a common literature value (2×10^-16cm²/W) for fused silica [22

W. Liu, O. Kosareva, I. S. Golubtsov, A. Iwasaki, A. Becker, V. P. Kandidov, and S. L. Chin, “Femtosecond laser pulse filamentation versus optical breakdown in H₂O,” Appl Phys B 76, 215–229 (2003). [CrossRef]

]. However, it needs to be noted that measured SPM values can vary substantially with the laser parameters used for the measurements (e.g. pulse length or repetition rate) due to effects like thermal lensing or time-dependent Kerr effects. Because of this discrepancy we compared our hole refill measurements against results from traditional Z-scans using the same laser system. As a reference value we compared the ratio between TPA of R6G in methanol and SPM in the walls of the glass cuvette. To extract these values we performed closed-aperture scans on the empty cuvette (for the measurement of SPM in glass) and open-aperture scans on the R6G-filled cuvette (for the measurement of TPA). The ratio of TPA_R6G to SPM_glass that we obtained from traditional Z-scans was α _2,R6G/η _2,glass=74×C[M], where C[M] is the molar R6G concentration. From our hole refilling experiments with similar laser parameters (Fig. 7) we extracted a ratio of α _2,R6G/η _2,glass=65×C[M], which is in reasonable agreement with the traditional Z-scan data.

Fig. 7. TPA and SPM measurements of R6G solutions of various concentrations in a glass cuvette. The solid lines connect the first 2 points (and the origin in the case of TPA). Saturation effects due to pulse reshaping are visible for higher concentrations.

5.1. Noise considerations

In this section we compare the noise sources in conventional Z-scan and hole refilling measurements. The measured power in a Z-scan can be viewed as an interference of the propagating fundamental beam with nonlinear polarization it created in the medium. Since the nonlinear polarization is generally very much smaller than the propagating fundamental, the interference signal is overwhelmed by the much larger “local oscillator” signal (the fundamental beam). In comparison, the hole refilling method offers several distinct advantages. First, the local oscillator is adjustable in strength and therefore can be chosen to produce an interference pattern without offset (complete constructive or destructive interference). Second, the interference signal naturally occurs at the rate of rotation of the local oscillator phase (in contrast to a “dcmeasurement” in the Z-scan). Since the output of most lasers are typically 1/f -noise limited, a measurement at a non-zero frequency can lead to a dramatic increase in the signal-to-noise ratio (SNR). Third, as discussed previously, the phase between local oscillator and nonlinear polarization is adjustable in the hole refilling setup while it is fixed in a Z-scan. The adjustability is the key to simultaneous measurements of TPA and SPM in a single scan.

The optimum choice of operation parameters (e.g. local oscillator strength, slit width, etc.) generally depends on the type of dominant noise. A simple estimate can be obtained in the case of shot noise limited operation. We can estimate the signal-to-noise ratios of the Z-scan (Z) and the hole refill measurement (H) as:

{SNR}_{Z} = \frac{{S Δ T P}_{in} 𝓡}{{(2 e Δ f 𝓡 {S P}_{in})}^{\frac{1}{2}}} = {(\frac{𝓡}{2 e Δ f})}^{\frac{1}{2}} Δ T \sqrt{{S P}_{in}} and

(18)

{SNR}_{H} = \frac{P_{sig} 𝓡}{{(2 e Δ f 𝓡 P_{LO})}^{\frac{1}{2}}} = {(\frac{𝓡}{2 e Δ f})}^{\frac{1}{2}} \frac{P_{sig}}{{r_{LO} (R_{slit} P_{in})}^{\frac{1}{2}}},

(19)

where 𝓡 is the detector responsivity, e the electron charge, Δf the detection bandwidth, P _in the input power, P _LO=r ² _LO P _ref the power in the local oscillator, and R _slit=P _ref/P _in the fraction of the power passing through the optical band-pass filter. Using expressions (17) we obtain

\frac{{SNR}_{H}}{{SNR}_{Z}} = {\begin{matrix} \frac{1}{\sqrt{π}} g \sqrt{R_{slit}} (TPA) \\ \frac{\sqrt{π}}{\ln 3} g \sqrt{\frac{R_{slit}}{S}} (SPM) . \end{matrix}

(20)

For an optimum ratio of Δω _hole/Δω _FWHM≈0.2 we obtain the ratio SNR_H/SNRZ of about 1:4 for a TPA measurement, whereas assuming S=0.01 we get a ratio of about 8:1 for an SPM measurement. However, unless considerable effort is spent, typical measurements are not shot-noise limited. For a laser system that is 1/f -noise limited the ratio is much more favorable for the hole refilling because of the lower noise at the LO rotation frequency.

5.2. Effect of scattering

In the conventional Z-Scan the information about the measured nonlinearity is encoded in the spatial beam profile and is completely lost if the beam profile is destroyed or inaccessible. In contrast, in the hole refill technique this information is encoded in the frequency spectrum. The advantage of frequency encoding is that linear effects (such as absorption or elastic scattering) do not create new frequency components in the spectrum. This means that once the pulse spectrum has been modified – which in microscopy applications occurs only at the focal point – this information is preserved and can be analyzed even when only a small fraction of the incoming light is accessible.

To demonstrate this effect we performed both types of measurements (traditional Z-scan and hole refill Z-scan) on a cuvette with an R6G solution, where we simulated scattering by adding a stack of diffusers after the sample. The diffusers completely suppressed the ballistic (unscattered) component of the transmitted light. Even though this procedure is not equivalent to a measurement in a scattering sample, where the scattering occurs before and after the focal point, it demonstrates the qualitative difference between the two types of measurements. When we inserted the diffusers, the power onto the detection photodiode decreased according to the reduced collection efficiency in both types of measurements. For the traditional open-aperture Z-scan, the TPA informationwas preserved when the diffusers were inserted in the light path. In a traditional closed-aperture scan, the SPM information was completely lost and only the TPA component remained. In contrast, in the hole refilling Z-scan both contributions – TPA and SPM – remained unaffected by the insertion of the diffusers, and could be extracted without difficulty.

6. Conclusion

We have demonstrated a Z-scan technique based on femtosecond laser pulse shaping that can simultaneously measure two-photon absorption and self-phase modulation with moderate power and without the requirement of placing an aperture in a transmitted beam. We compared measurements in absorptive and dispersive samples with results of conventional Z-scans and obtained reasonable agreement. We also demonstrated that this technique is applicable for measurements of nonlinearities in scattering samples.

One of the most interesting applications of our technique is its use in nonlinear microscopy of highly scattering biological tissue. As a nonlinear measurement technique our method shares the well known advantages of two-photon fluorescence microscopy over linear microscopy: high axial and lateral resolution, the ability for optical sectioning, and larger imaging depth and lower susceptibility for photodamage. However, our technique is not restricted to fluorescent targets. With a different TPA microscopy technique we have previously shown that we can image melanin and hemoglobin – two important endogenous tissue constituents – with low power well below the surface [1

, 2

D. Fu, T. Ye, T. E. Matthews, G. Yurtsever, and W. S. Warren, “Two-color, two-photon, and excited-state absorption microscopy,” J. Biomed. Opt. 12, 054004 (2007). [CrossRef] [PubMed]

]. With the hole refilling microscope described here we can now access SPM as an additional structural and functional contrast mechanism. With this technique we have observed strong SPM signatures of neuronal activity in brain slices [3

M. C. Fischer, H. Liu, I. R. Piletic, Y. Escobedo-Lozoya, R. Yasuda, and W. S. Warren, “High-sensitivity, single-beam n ₂ measurements,” Opt. Lett. 14, 955–957 (1989). [CrossRef]

], promising a fundamentally new approach to neuroimaging.

References and links

1.	D. Fu, T. Ye, T. E. Matthews, B. J. Chen, G. Yurtserver, and W. S. Warren, “High-resolution in vivo imaging of blood vessels without labeling,” Opt. Lett. 32, 2641–2643 (2007). [CrossRef] [PubMed]
2.	D. Fu, T. Ye, T. E. Matthews, G. Yurtsever, and W. S. Warren, “Two-color, two-photon, and excited-state absorption microscopy,” J. Biomed. Opt. 12, 054004 (2007). [CrossRef] [PubMed]
3.	M. C. Fischer, H. Liu, I. R. Piletic, Y. Escobedo-Lozoya, R. Yasuda, and W. S. Warren, “High-sensitivity, single-beam n ₂ measurements,” Opt. Lett. 14, 955–957 (1989). [CrossRef]
4.	M. Sheik-Bahae, A. A. Said, and E. W. van Stryland, “High-sensitivity, single-beam n ₂ measurements,” Opt. Lett. 14, 955–957 (1989). [CrossRef] [PubMed]
5.	M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990). [CrossRef]
6.	T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. van Stryland, “Eclipsing Z-scan measurement of λ/10⁴ wave-front distortion,” Opt. Lett. 19, 317–319 (1994). [CrossRef] [PubMed]
7.	G. Tsigaridas, M. Fakis, I. Polyzos, P. Persephonis, and V. Giannetas, “Z-scan technique through beam radius measurements,” Appl. Phys. B: Lasers and Optics 76, 83–86 (2003). [CrossRef]
8.	H. Ma, A. S. L. Gomes, and C. B. de Araujo, “Measurements of nondegenerate optical nonlinearity using a two-color single beam method,” Appl. Phys. Lett. 59, 2666–2668 (1991). [CrossRef]
9.	J. Wang, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E. W. Van Stryland, “Time-resolved Z-scan measurements of optical nonlinearities,” J. Opt. Soc. Am. B 11, 1009–1017 (1994). [CrossRef]
10.	I. Guedes, L. Misoguti, L. De Boni, and S. C. Zilio, “Heterodyne Z-scan measurements of slow absorbers,” J. Appl. Phys. 101, 063112 (2007). [CrossRef]
11.	J.-M. Menard, M. Betz, I. Sigal, and H. M. Van Driel, “Single-beam differential Z-scan technique,” Appl. Opt. 46, 2119–2122 (2007). [CrossRef] [PubMed]
12.	P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. McKay, and R. G. McDuff, “Single-beam Z-scan: measurement techniques and analysis,” J. Nonlinear Opt. Phys. Mater. 6, 253–293 (1997). [CrossRef]
13.	R. E. Bridges, G. L. Fischer, and R. W. Boyd, “Z-scan measurement technique for non-Gaussian beams and arbitrary sample thicknesses,” Opt. Lett. 20, 1821–1823 (1995). [CrossRef] [PubMed]
14.	M. C. Fischer, T. Ye, G. Yurtsever, A. Miller, M. Ciocca, W. Wagner, and W. S. Warren, “Two-photon absorption and self-phase modulation measurements with shaped femtosecond laser pulses,” Opt. Lett. 30, 1551–1553 (2005). [CrossRef] [PubMed]
15.	C. W. Hillegas, J. X. Tull, D. Goswami, D. Strickland, and W. S. Warren, “Femtosecond laser pulse shaping by use of microsecond radio-frequency pulses,” Opt. Lett. 19, 737–739 (1994). [CrossRef] [PubMed]
16.	R. W. Boyd, Nonlinear Optics , (Academic Press, 2003).
17.	G. P. Agrawal, Nonlinear Fiber Optics , (Elsevier / Academic Press, 2007).
18.	S. T. Hendow and S. A. Shakir, “Recursive numerical solution for nonlinear wave propagation in fibers and cylindrically symmetrical systems,” Appl. Opt. 25, 1759–1764 (1986). [CrossRef] [PubMed]
19.	J. A. Hermann and R. G. McDuff, “Analysis of spatial scanning with thick optically nonlinear media,” J. Opt. Soc. Am. B 10, 2056–2064 (1993). [CrossRef]
20.	J. A. Hermann, “Nonlinear optical absorption in thick media,” J. Opt. Soc. Am. B 14, 814–823 (1997). [CrossRef]
21.	P. Tian and W. S. Warren, “Ultrafast measurement of two-photon absorption by loss modulation,” Opt. Lett. 27, 1634–1636 (2002). [CrossRef]
22.	W. Liu, O. Kosareva, I. S. Golubtsov, A. Iwasaki, A. Becker, V. P. Kandidov, and S. L. Chin, “Femtosecond laser pulse filamentation versus optical breakdown in H₂O,” Appl Phys B 76, 215–229 (2003). [CrossRef]

OCIS Codes
(190.0190) Nonlinear optics : Nonlinear optics
(190.7110) Nonlinear optics : Ultrafast nonlinear optics
(320.5540) Ultrafast optics : Pulse shaping

ToC Category:
Nonlinear Optics

History
Original Manuscript: February 5, 2008
Revised Manuscript: March 10, 2008
Manuscript Accepted: March 10, 2008
Published: March 12, 2008

Virtual Issues
Vol. 3, Iss. 4 Virtual Journal for Biomedical Optics

Citation
Martin C. Fischer, Henry C. Liu, Ivan R. Piletic, and Warren S. Warren, "Simultaneous self-phase modulation and two-photon absorption measurement by a spectral homodyne Z-scan method," Opt. Express 16, 4192-4205 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-6-4192

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References

D. Fu, T. Ye, T. E. Matthews, B. J. Chen, G. Yurtserver, and W. S. Warren, "High-resolution in vivo imaging of blood vessels without labeling," Opt. Lett. 32, 2641-2643 (2007). [CrossRef] [PubMed]
D. Fu, T. Ye, T. E. Matthews, G. Yurtsever, and W. S. Warren, "Two-color, two-photon, and excited-state absorption microscopy," J. Biomed. Opt. 12, 054004 (2007). [CrossRef] [PubMed]
M. C. Fischer, H. Liu, I. R. Piletic, Y. Escobedo-Lozoya, R. Yasuda, and W. S. Warren, "Self-phase modulation signatures of neuronal activity," Opt. Lett. 33219-221 (2008). [CrossRef]
M. Sheik-Bahae, A. A. Said, and E. W. van Stryland, "High-sensitivity, single-beam n2 measurements," Opt. Lett. 14, 955-957 (1989). [CrossRef] [PubMed]
M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E.W. van Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990). [CrossRef]
T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. van Stryland, "Eclipsing Z-scan measurement of ?/104 wavefront distortion," Opt. Lett. 19, 317-319 (1994). [CrossRef] [PubMed]
G. Tsigaridas, M. Fakis, I. Polyzos, P. Persephonis, and V. Giannetas, "Z-scan technique through beam radius measurements," Appl. Phys. B: Lasers and Optics 76, 83-86 (2003). [CrossRef]
H. Ma, A. S. L. Gomes, and C. B. de Araujo, "Measurements of nondegenerate optical nonlinearity using a two-color single beam method," Appl. Phys. Lett. 59, 2666-2668 (1991). [CrossRef]
J. Wang, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E.W. Van Stryland, "Time-resolved Z-scan measurements of optical nonlinearities," J. Opt. Soc. Am. B 11, 1009-1017 (1994). [CrossRef]
I. Guedes, L. Misoguti, L. De Boni, and S. C. Zilio, "Heterodyne Z-scan measurements of slow absorbers," J. Appl. Phys. 101, 063112 (2007). [CrossRef]
J.-M. Menard, M. Betz, I. Sigal, and H. M. Van Driel, "Single-beam differential Z-scan technique," Appl. Opt. 46, 2119-2122 (2007). [CrossRef] [PubMed]
P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. McKay, and R. G. McDuff, "Single-beam Z-scan: measurement techniques and analysis," J. Nonlinear Opt. Phys. Mater. 6, 253-293 (1997). [CrossRef]
R. E. Bridges, G. L. Fischer, and R. W. Boyd, "Z-scan measurement technique for non-Gaussian beams and arbitrary sample thicknesses," Opt. Lett. 20, 1821-1823 (1995). [CrossRef] [PubMed]
M. C. Fischer, T. Ye, G. Yurtsever, A. Miller, M. Ciocca, W. Wagner, and W. S. Warren, "Two-photon absorption and self-phase modulation measurements with shaped femtosecond laser pulses," Opt. Lett. 30, 1551-1553 (2005). [CrossRef] [PubMed]
C. W. Hillegas, J. X. Tull, D. Goswami, D. Strickland, and W. S. Warren, "Femtosecond laser pulse shaping by use of microsecond radio-frequency pulses," Opt. Lett. 19, 737-739 (1994). [CrossRef] [PubMed]
R. W. Boyd, Nonlinear Optics, (Academic Press, 2003).
G. P. Agrawal, Nonlinear Fiber Optics, (Elsevier / Academic Press, 2007).
S. T. Hendow and S. A. Shakir, "Recursive numerical solution for nonlinear wave propagation in fibers and cylindrically symmetrical systems," Appl. Opt. 25, 1759-1764 (1986). [CrossRef] [PubMed]
J. A. Hermann and R. G. McDuff, "Analysis of spatial scanning with thick optically nonlinear media," J. Opt. Soc. Am. B 10, 2056-2064 (1993). [CrossRef]
J. A. Hermann, "Nonlinear optical absorption in thick media," J. Opt. Soc. Am. B 14, 814-823 (1997). [CrossRef]
P. Tian and W. S. Warren, "Ultrafast measurement of two-photon absorption by loss modulation," Opt. Lett. 27, 1634-1636 (2002). [CrossRef]
W. Liu, O. Kosareva, I. S. Golubtsov, A. Iwasaki, A. Becker, V. P. Kandidov, and S. L. Chin, "Femtosecond laser pulse filamentation versus optical breakdown in H2O," Appl Phys B 76, 215-229 (2003). [CrossRef]

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