Determination of the contributions of two simultaneous absorption orders using 2-beam action spectroscopy

Nikolaos Liaros; Samuel R. Cohen; John T. Fourkas

doi:10.1364/OE.26.009492

1. Introduction

Nonlinear absorption plays a key role in many optical applications [1–21]. Although nonlinear absorption was first demonstrated more than 50 years ago, the accurate characterization of this phenomenon remains challenging. For instance, most methods for determining the effective order of a nonlinear absorption process rely on making a logarithmic graph of an observable as a function of the average excitation power [22]. The slope of this type of plot should give the order of the dominant absorption process. However, the accurate determination of this order requires being able to measure the observable over several orders of magnitude of average excitation power. Furthermore, such plots are rarely linear, indicating that processes of two or more orders may contribute to the signal. Even when the plots are linear, the slope is often not integral, again suggesting that multiple processes contribute to the signal.

We recently introduced a technique [23] for the determination of the order of the effective absorptive nonlinearity in multiphoton absorption polymerization [19–21]. In this method, called 2-beam initiation threshold (2-BIT) spectroscopy, two pulse trains that are overlapped in space but are interleaved in time are used to expose a photoresist. The average power of one pulse train required to reach the polymerization threshold is measured as a function of the average power of the other pulse train. The order of the nonlinear absorption process can be determined by plotting one average power versus the other. 2-BIT was employed to demonstrate that various photoinitiators are excited by either 2-photon or 3-photon absorption with ultrafast pulses tuned in the vicinity of 800 nm [23].

2-BIT is the first example of a 2-beam action (2-BA) spectroscopy. The 2-BA concept can, in principle, be applied to any linear or nonlinear absorption process that yields a measureable signal. In the case of the photopolymerization threshold, the observable is single-valued. However, for observables that can take on many values, one can instead make 2-BA spectroscopy measurements for any desired value of the observable. Such a measurement reveals the effective number of photons involved in generating the observable at that particular value. This process can be repeated for different values of the observable.

As a representative example of the generalization of 2-BA spectroscopies, here we study the linear and nonlinear generation of photocurrent or photovoltage in a photodiode. The nonlinear optical production of photocurrent in different types of semiconductors has stimulated extensive theoretical and experimental investigations, and has found a vast number of applications [24,25]. This phenomenon has been used, for instance, to autocorrelate ultrafast laser pulses [16,17,26–30], to explore the electronic properties of organic [31,32] and inorganic [33,34] semiconductors, to control photocurrent generation through coherence [35–37], to generate electrical power [38], to map the structure in composite materials [39], and to “upconvert” mid-infrared light for detection in the near-ultraviolet [40]. Furthermore, the strong multiphoton absorption generally exhibited by semiconductors has been used for applications such as optical limiting, induction of population inversion, processing of signals, and entanglement of photons.

We characterize the linear and nonlinear generation of photocurrent in a GaAsP photodiode with a 2-BA technique that we call 2-beam constant-amplitude photocurrent (2-BCAmP) spectroscopy. We use this example to demonstrate that 2-BA spectroscopy data offer substantial advantages over logarithmic plots in elucidating the nature of such processes. In particular, we introduce a framework for extracting the contributions of two different orders of absorption from 2-BA spectroscopy data. We also show how the dependence of the contributions of the two different absorption orders on the value of the observable can be used to test for self-consistency.

2. Experimental details

A schematic of the 2-BCAmP concept is shown in Fig. 1. The excitation source was a tunable, Ti:sapphire oscillator (Coherent Mira 900-F) that can be operated in either mode-locked (ML) or continuous-wave (CW) mode. The repetition rate of the laser in ML mode was 76 MHz, and the pulse duration was approximately 150 fs. The spatially filtered beam was chopped at 1 KHz and then split in two parts. The power of each beam was adjusted by means of a motorized half-wave plate and a Glan-Taylor polarizer, and the beam powers were measured using a power meter. Each beam was passed through a separate variable beam expander to allow for the adjustment of the beam spot size at the back aperture of the objective. The lengths of the two beam paths were adjusted so that consecutive pulses arrived at the sample with approximately equally spaced timings, giving an effective repetition rate of 152 MHz. The two beams were combined with a polarizing beam cube and made collinear, and then passed through a quarter-wave plate.

Fig. 1 Schematic depiction of a 2-beam constant-amplitude photocurrent experiment. Two trains of pulses whose amplitudes can be adjusted independently are interleaved and focused onto a photodiode. Multiple sets of average powers for the two pulse trains that generate the same photocurrent (or photovoltage) are determined, allowing for the measurement of the effective order of the absorption process in the photodiode.

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The two beams were sent through the reflected-light illumination port of an inverted microscope and filled the back aperture of a 0.30 NA, 10 × , infinity-corrected, microscope objective (Zeiss, Plan-NEOFLUAR). The objective focused the beams onto a GaAsP photodiode (Hamamatsu G1117, with the resin coating removed). Average power values cited here were measured at the back aperture of the objective; the loss from the objective was ~30%. The beam diameter at the focal plane was approximately 3.25 μm. To eliminate any contribution from dark current, the photodiode output was sent to a lock-in amplifier (Stanford Research Systems, SR810) that was referenced to the chopping frequency. To maximize the signal-to-noise ratio, data were collected in current mode at low excitation powers and in voltage mode at high excitation powers. The detection mode did not influence the 2-BCAmP exponent.

The GaAsP photodiode was mounted on a motor-driven stage that allowed for sample positioning in the plane transverse to the laser beam. A separate motor drive controlled the distance between the objective and the photodiode. The movements of the stages were controlled using LabVIEW programs. To control the position of each of the two excitation beams focused on the sample, the reflection of each laser beam from the photodiode surface was observed in real time using a CCD camera and a monitor. For measurements made in ML mode the beams were overlapped on the photodiode, whereas for measurements made in CW mode the beams were focused to neighboring positions but were not overlapped.

3. Results and discussion

Figure 2 shows photocurrent excitation data collected for a GaAsP photodiode at an excitation wavelength of 800 nm. For one set of data the laser was mode locked and the photodiode was placed at the focal plane of the objective. In this case the slope of the logarithmic photocurrent excitation (PE) plot is 2.02 ± 0.03, which is indicative of 2-photon absorption. The magnitude of the photovoltage is consistent with that reported previously [28]. For the other set of data the laser was operated in CW mode and the photodiode was moved 2.8 mm out of the focal plane of the objective. In this case the slope of the logarithmic PE plot is 1.07 ± 0.03, indicating that the signal is dominated by linear absorption. Although the red edge of the specified detection range of this photodiode is at 680 nm, there is still a small linear signal even at wavelengths of 800 nm and longer due to the Urbach-tail absorption of the semiconductor. At the highest average excitation powers the plot begins to diverge slightly from the fit line. This divergence is even clearer if the slope of the fit is constrained to unity, as seen in the dashed line in Fig. 2(a). The average ML powers here range from 0.39 mW to 2.3 mW, corresponding to irradiances of 0.82 to 4.8 GW/cm².

Fig. 2 (a) Logarithmic photocurrent excitation plots for a GaAsP photodiode for 800 nm excitation with a ML laser (at the focal plane) and a CW laser (out of the focal plane). The solid lines are free fits and the dashed line is a fit with the slope constrained to 1. (b) 2-BCAmP data collected under the same conditions. The error bars are smaller than the symbols in all cases.

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Figure 2(b) shows 2-BCAmP plots for ML excitation in the focal plane and CW excitation away from the focal plane, for a specific chosen value of the photocurrent. In both cases the average laser power for each pulse train is normalized by the average power at which that pulse train alone yields the target value of the photocurrent. Using these normalized average powers, the 2-BCAmP plot follows the equation

{\bar{P}}_{1}^{n} + {\bar{P}}_{2}^{n} = 1

where n is the order of the absorption process [23]. For instance, a linear absorption process yields a linear 2-BCAmP plot, a 2-photon absorption process yields a 2-BCAmP plot that is a quarter of a circle, and so on. The data were fit using a nonlinear least squares routine [41], yielding exponents of 1.99 ± 0.03 for the ML case and 0.99 ± 0.03 for the CW case. Thus, when the observed photocurrent arises through a single absorption order, the 2-BCAmP results are consistent with the exponent derived from a conventional logarithmic PE plot.

We next consider what happens when more than one absorption order contributes to the photocurrent. In Fig. 3(a) we show logarithmic PE plots for 800 nm ML excitation with the photodiode at the focal plane of the objective and at two different distances away from the focal plane. As the distance from the focal plane increases, so does the spot size, reducing the peak irradiance and, concomitantly, the contribution of 2-photon absorption. Thus, the slope of the PE plot decreases from a value of about 2 in the focal plane to a value of about 1.5 when the photodiode is 2.8 mm from the focal plane. None of the PE traces is completely linear. The in-plane trace shows signs of saturation at high average excitation powers, and the out-of-plane traces show signs of the slope increasing at higher average excitation powers.

Fig. 3 (a) Logarithmic PE plots for ML 800 nm excitation, with the GaAsP photodiode different distances from the focal plane. (b) 2-BCAmP data collected under the same conditions.

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Figure 3(A) illustrates some of the challenges inherent in using logarithmic PE plots to determine absorptive nonlinearities. Typically, the slopes of such plots are not integers. Furthermore, the accurate determination of the order of an absorption process generally requires data that span several orders of magnitude of average excitation power, a problem that is related to the accurate fitting of power-law functions [42]. However, logarithmic PE plots are rarely linear over many orders of magnitude of average excitation power.

Logarithmic PE plots that are not linear or that have non-integer slopes can indicate that two or more orders of absorption contribute to the observed signal over the range of average excitation powers used. However, even if the two exponents are known, accurate fitting to such a function is also challenging, and is rarely performed for PE data. Consider the case of two contributions to the absorption that differ by an order of one, such that the signal S as a function of irradiance I is given by

S (I) = A I^{n} + B I^{n + 1} .

Here, the coefficients A and B encompass the multiphoton absorption cross section, the quantum yield for photocurrent generation, and pulse shape factors relating to the absorption probability for the respective orders n and n + 1 [43]. At the irradiance for which 90% of the absorption probability is of order n we have

\frac{B I^{n + 1}}{A I^{n}} = \frac{B I}{A} = \frac{1}{9} .

Conversely, at the irradiance for which 90% of the absorption is of order n + 1 we have

\frac{B I^{n + 1}}{A I^{n}} = \frac{B I}{A} = 9 .

Thus, going from 90% order n to 90% order n + 1 requires changing the irradiance, and therefore the average excitation power, by a factor of 81. Similarly, if the orders of absorption vary by 2, going from 90% order n to 90% order n + 2 requires a factor of 9 increase in irradiance. Reliable fitting of two different orders of absorption therefore generally requires data that span at least two orders of magnitude in average excitation power. Even in this scenario, the average excitation powers must lie in the range over which the dominant order of absorption changes. If these conditions are not met, as is the case for the data in Fig. 3(a), then it is not possible to make an accurate determination of the relative contributions of different absorption orders at any given average excitation power for which the slope is not an integer.

Figure 3(b) shows 2-BCAmP data taken at the same three positions as the PE data in Fig. 3(a), all at the same average excitation power of 15.6 mW for a single pulse train (this convention is used throughout this paper when reporting powers). As is the case for PE data, the 2-BCAmP data that were not collected at the focal plane have best-fit exponents that are not integers. If the exponents between 1 and 2 arise from a combination of linear and 2-photon absorption, then we have

a ({\bar{P}}_{1} + {\bar{P}}_{2}) + b ({\bar{P}}_{1}^{2} + {\bar{P}}_{2}^{2}) = 1.

We impose the constraint that a + b = 1, such that a is the fractional contribution of linear absorption and b is the fractional contribution of 2-photon absorption in the conditions under which the curve was obtained. There is then a unique value of a for a given value of n under these conditions. To determine the value of a for a 2-BCAmP data set, we make the substitution b = a – 1 in Eq. (5) and rearrange to obtain

1 - {\bar{P}}_{1}^{2} - {\bar{P}}_{2}^{2} = a ({\bar{P}}_{1} + {\bar{P}}_{2} - {\bar{P}}_{1}^{2} - {\bar{P}}_{2}^{2}) .

Thus, the slope of a plot of

1 - {\bar{P}}_{1}^{2} - {\bar{P}}_{2}^{2}

as a function of

{\bar{P}}_{1} + {\bar{P}}_{2} - {\bar{P}}_{1}^{2} - {\bar{P}}_{2}^{2}

will be a. An analogous approach can be used to extract the fractional contribution of any pair of orders of absorption from a 2-BA spectroscopy plot.

We show a representative plot of Eq. (6), for the 2-BCAmP data obtained 1.26 mm from the focal plane, in Fig. 4(a). This approach was used in Fig. 4(b) to determine the fractional contribution of 2-photon absorption at a fixed average power as a function of distance from the focal plane, based on the three 2-BCAmP data sets in Fig. 3(b). In the case of a combination of linear and 2-photon absorption, the corresponding local slope of a logarithmic plot at a given excitation intensity is one plus the fraction of 2-photon absorption. For the data obtained 2.80 mm from the focal plane, the corresponding local slope calculated in this manner is 1.51, and for the data obtained 1.26 mm from the focal plane the corresponding slope is 1.80. Note that these values differ somewhat from the 2-BCAmP exponents, but the 2-BCAmP data allow the local slope to be calculated directly.

Fig. 4 (a) A representative linearized plot used to extract the fractional contribution of linear absorption from a 2-BCAmP data set. The line is a linear least-squares fit constrained to pass through the origin. (b) Fraction of photocurrent arising from 2-photon absorption at a fixed average laser power as a function of the distance of the photodiode from the focal plane.

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These data show that it is straightforward to use 2-BA spectroscopy to determine the relative contributions of two different known orders of absorption to an observable at a particular value. In contrast, the conventional logarithmic plot method requires measuring the observable over orders of magnitude in its value to determine the contributions of different orders of nonlinearity.

The relative contributions of linear and 2-photon absorption should depend on the average excitation power in a predictable manner, so we next consider 2-BCAmP data obtained with the photodiode 1.28 mm from the focal plane at three different average excitation powers: 15.6 mW, 19.2 mW, and 23.2 mW. As can be seen from the data in Fig. 5(a), the measured exponent changes from 1.72 to 1.80 over this range of average excitation powers, with an uncertainty of approximately ± 0.03 in each value. It is not possible to determine this small of a change in slope in a logarithmic PE plot over such a limited range of average excitation powers.

Fig. 5 (a) 2-BCAmP data for a GaAsP photodiode 1.28 mm from the focal plane of the objective for mode-locked 800 nm excitation at three different average powers. (b) Ratio of 2-photon to linear absorption as a function of average power from the three 2-BCAmP data sets (red symbols). The solid line is a linear least-squares fit, constrained to pass through the origin. The blue symbols are the values of b/a for fits to a linear absorption process and a 3-photon absorption process, in which case the ratio should depend on the square of the average power.

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Although in this case we can safely assume that the photocurrent should arise from linear absorption and 2-photon absorption, the 2-BCAmP data in Fig. 5(a) can in principle be fit to the sum of a linear absorption process and any higher-order absorption process, giving different values of a and b. As shown above, if the orders of the two contributing absorption processes differ by one, then the ratio of the contribution of the higher-order process to that of the lower-order process is BI/A. In the 2-BCAmP fits we instead determine the normalized contributions a and b. Together, these results imply that b/a should increase linearly with the average power of the laser if the photocurrent arises from linear and 2-photon absorption. As can be seen in Fig. 5(b), our results are in good agreement with this prediction. If we instead assume that the signal comes from a combination of linear and 3-photon absorption, we can extract different values of a and b from the 2-BCAmP plots. If the signal arises from those two orders of absorption, then b/a should scale with the square of I. As can be seen from blue symbols in Fig. 5(b), a plot of b/a as a function of I in this case is not a parabola that passes through the origin. Whenever two orders of absorption dominate in the generation of an observable, the ability to perform this type of consistency check makes it possible for 2-BA spectroscopy to be used to determine these orders, even if the orders are not known a priori.

As was shown in Fig. 2, 2-BCAmP measurements can be made not just with ML beams, but also with CW beams. Indeed, Ranka et al. observed a deviation in linearity in a logarithmic PE plot of data for CW irradiation of a GaAsP photodiode [28]. Their focusing was considerably gentler than what we have used here, so the increase in slope that they observed at high power was modest. They attributed this phenomenon to thermal effects rather than to the onset of 2-photon absorption [28]. To investigate this effect, we performed additional 2-BCAmP measurements with CW beams.

The CW data in Fig. 2 were obtained far from the focal plane of the objective. Figure 6(a) is a logarithmic PE plot for CW irradiation at the focal plane. It is clear from this plot that the nonlinear absorption in the photodiode is so strong that it can readily be driven by tightly focused CW light. The line in Fig. 6(a) is a fit to a linear term plus a quadratic term, which is consistent with a combination of linear and 2-photon absorption. Because the data encompass only about an order of magnitude in excitation power, the data can also be fit nearly as well to the sum of a linear term and a cubic term.

Fig. 6 (a) A logarithmic PE plot for a GaAsP photodiode for 800 nm CW excitation at the focal plane of the objective. The solid line is a fit to a linear term and a quadratic term. (b) 2-BCAmP data for different CW laser powers. (c) Ratio of 2-photon to linear absorption as a function of power from the three 2-BCAmP data sets. The solid line in the plot is a linear least-squares fit, constrained to pass through the origin.

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Figure 6(b) shows corresponding 2-BCAmP data for three different values of the excitation power. The exponent for these three plots ranges from 1.31 to 1.56. Values of b/a extracted using Eq. (6) by assuming the combination of a linear and a 2-photon absorption process are shown in Fig. 6(c). This ratio is once again linear in the laser power, confirming the orders of absorption that contribute to the photocurrent. These data are consistent with the nonlinear response at high average excitation powers arising from nonlinear absorption.

We can compare the data in Fig. 2(a) and Fig. 6(a) to test whether nonlinear absorption could be of a sufficient magnitude to account for the nonlinearity in Fig. 6(a). We find that CW excitation appears to be more than 34 times as efficient at generating a quadratic nonlinearity than is ML excitation. Although it is known that a correction must be made for non-square pulse shapes [43], this factor can account for only about half of the observed difference in efficiencies.

The rise in temperature in GaAs under CW laser irradiation is known to be quadratic in power [44], and we can assume that the same holds for GaAsP. Furthermore, the bandgap of GaAs is known to decrease linearly as a function of temperature [45]. If we assume that the Urbach tail of the GaAsP absorption is a linear (or nearly linear) function of frequency at 800 nm, then we would expect that thermal effects would exhibit a quadratic power dependence in this system. Thus, either 2-photon absorption or thermal effects could be consistent with the observed quadratic contribution at higher CW intensities. However, the small temperature rise expected in the Urbach tail [44] in conjunction with the weak dependence of the bandgap on temperature [45] suggest that thermal effects should be minimal in this system. This conundrum is a subject of ongoing research.

4. Conclusions

We have characterized linear and nonlinear absorption in a photodiode using 2-beam constant-amplitude photocurrent spectroscopy. This work demonstrates the generalizability of the 2-beam action spectroscopy concept. Here we report, for the first time, 2-BA exponents that, within experimental uncertainty, are not integers. The non-integer exponents were shown to arise from a combination of linear and 2-photon absorption. We presented a framework for extracting the relative contributions of different orders of absorption from 2-BA spectroscopy data. We also used the ratio of the contributions to validate that the absorption does arise from the two expected orders, and that any contribution of thermal effects is negligible.

The 2-BA spectroscopy approach offers significant advantages over traditional logarithmic plots of an observable as a function of average excitation power, particularly when multiple orders of absorption are involved. Rather than requiring data obtained over several orders of magnitude in the average excitation power (and, consequently, even more orders of magnitude in the observable), 2-BA spectroscopies can be used to determine the effective exponent at each chosen value of the observable. A 2-BA plot does involve a combination of different average excitation powers that may span an order of magnitude, but offers the ability to determine a well-defined exponent at each chosen value of the observable, as well as the ability to use these exponents to test models for the absorption processes that contribute to the signal.

The 2-BA spectroscopy approach should be applicable to virtually any observable that arises from linear absorption, nonlinear absorption, or any combination thereof. The ability to determine unambiguously the orders of absorption that contribute to the observable can further provide invaluable mechanistic information that can be used to optimize materials. Although the treatment presented here allows for the determination of two orders of absorption that contribute to an observable, this approach should be able to be extended to a larger number of orders. Such an extension will require obtaining 2-BA spectroscopy data over a greater range of values of the observable.

Funding

National Science Foundation (NSF) (CMMI-1449309).

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Determination of the contributions of two simultaneous absorption orders using 2-beam action spectroscopy

Abstract

1. Introduction

2. Experimental details

3. Results and discussion

4. Conclusions

Funding

References and links

Cited By

Figures (6)

Equations (6)

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