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Virtual Journal for Biomedical Optics

Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editors: Andrew Dunn and Anthony Durkin
  • Vol. 6, Iss. 8 — Aug. 26, 2011

Optics InfoBase > Virtual Journal for Biomedical Optics > Volume 6 > Issue 8 > Page 13365

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Fine depth resolution of two-photon absorption-induced photoacoustic microscopy using low-frequency bandpass filtering

Yoshihisa Yamaoka, Mika Nambu, and Tetsuro Takamatsu  »View Author Affiliations


Optics Express, Vol. 19, Issue 14, pp. 13365-13377 (2011)
http://dx.doi.org/10.1364/OE.19.013365


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Abstract

Photoacoustic microscopy usually uses high-frequency photoacoustic waves, which provide not only high spatial resolution but also limitation of the penetration depth. In this study, we developed two-photon absorption-induced photoacoustic microscopy (TP-PAM) to improve the depth resolution without use of high-frequency photoacoustic waves. The spatial resolution in TP-PAM is determined by two-photon absorption. TP-PAM with a 20X objective lens (numerical aperture: 0.4) provides an optically-determined depth resolution of 44.9 ± 2.0 μm, which is estimated by the full width at half maximum of the photoacoustic signal from an infinitely small target, using low-frequency bandpass filtering of photoacoustic waves. The combination of TP-PAM and frequency filtering provides high spatial resolution.

© 2011 OSA

1. Introduction

Optical microscopy such as confocal laser-scanning microscopy is commonly used to visualize various structures with high spatial resolution [1

A. Diaspro, Confocal and Two-Photon Microscopy (Wiley-Liss, 2002).

]. However, it is difficult to observe the inner structures in media where light is strongly absorbed and scattered, such as polymers, semiconductors, metals, ceramics, and biological materials, by optical microscopy. In order to visualize the inner structures, photoacoustic microscopy (PAM) was recently developed [2

H. F. Zhang, K. Maslov, G. Stoica, and L. V. Wang, “Functional photoacoustic microscopy for high-resolution and noninvasive in vivo imaging,” Nat. Biotechnol. 24(7), 848–851 (2006). [CrossRef] [PubMed]

–9

C. Zhang, K. Maslov, and L. V. Wang, “Subwavelength-resolution label-free photoacoustic microscopy of optical absorption in vivo ,” Opt. Lett. 35(19), 3195–3197 (2010). [CrossRef] [PubMed]

]. It provides high-contrast images based on the distribution of optical absorption along with the use of photoacoustic waves propagating through a long distance, even in opaque media. In conventional PAM, high-frequency transducers (e. g., >50 MHz) provide improved spatial resolution compared to low-frequency transducers (e. g., 10 MHz). However, the penetration depth is decreased when the spatial resolution is improved, as shown in Fig. 1(a) , because high-frequency ultrasonic waves can generally propagate through only a short distance [10

C. Guittet, F. Ossant, L. Vaillant, and M. Berson, “ In vivo high-frequency ultrasonic characterization of human dermis,” IEEE Trans. Biomed. Eng. 46(6), 740–746 (1999). [CrossRef] [PubMed]

–13

C. R. Hill, J. C. Bamber, and G. R. t. Haar, eds., Physical principles of medical ultrasonics (John Weily & Sons, Chichester, 2004).

]. In biological tissues, the attenuation depends on the kind of tissues. For example, the attenuation coefficient for epidermis and dermis is 3.5 dB/MHz/cm [12

L. V. Wang, ed., Photoacoustic Imaging and Spectroscopy (CRC press, Boca Raton, 2009).

, 13

C. R. Hill, J. C. Bamber, and G. R. t. Haar, eds., Physical principles of medical ultrasonics (John Weily & Sons, Chichester, 2004).

]. When 100-MHz ultrasonic waves are used to visualize 2-mm deep layers, 70-dB attenuation of ultrasonic waves occurs. Thus, avoiding use of high-frequency transducer is suitable for the investigation of inner structures.

Fig. 1 Comparison between the depth resolutions and the detection frequencies of photoacoustic waves for (a) one-photon photoacoustic microscopy (PAM) and (b) two-photon absorption-induced photoacoustic microscopy (TP-PAM).

In this study, we have developed two-photon absorption-induced photoacoustic microscopy (TP-PAM) in order to improve the depth resolution together with low-frequency bandpass filtering of photoacoustic waves. Our method utilizes photoacoustic signals generated by highly localized two-photon absorption (Fig. 1(b)). This is the first report of imaging based on two-photon absorption-induced photoacoustic generation in nonlinear photoacoustic spectroscopy [14

Y. Bae, J. J. Song, and Y. B. Kim, “Photoacoustic study of two-photon absorption in hexagonal ZnS,” J. Appl. Phys. 53(1), 615–619 (1982). [CrossRef]

–16

P. Sathy, R. Philip, V. P. N. Nampoori, and C. P. G. Vallabhan, “Observation of two-photon absorption in rhodamine 6G using photoacoustic technique,” Opt. Commun. 74(5), 313–317 (1990). [CrossRef]

]. Because the high depth resolution is achieved by two-photon absorption, high-frequency transducers are not needed, as shown in Fig. 1(b). In the present study, the objective was to clarify whether TP-PAM with low-frequency bandpass filtering can visualize structures with optical resolution determined by the two-photon absorption.

2. Materials and methods

2.1 Experimental setup for TP-PAM

The experimental apparatus of the TP-PAM system was assembled as shown schematically in Fig. 2 . As a laser source, we employed a passively Q-switched LD-pumped microchip sub-nanosecond pulse laser (SNP-13E; Teem Photonics) (wavelength, 1064 nm; pulse duration, 0.6 ns; pulse energy, 14.1 μJ; repetition rate, 7.3 KHz) for inducing nonlinear absorption. The laser beam was collimated by the combination of two lenses (f = 25.4 mm and f = 100 mm). The distance between the two collimator lenses was adjusted by a collimation checker (SPV-25; Sigma Koki Co., Ltd.). Sub-nanosecond light pulses were focused on target samples by a 20X objective lens (LMH-20X-1064; Thorlabs Japan, Inc.) of which the numerical aperture (NA) is 0.4. Generated photoacoustic signals were detected by a 10-MHz flat transducer (10K6.4I; Japan Probe, Inc.) which was submerged in the water bath shown in Fig. 2. The transducer was set along the direction of light propagation at a slight angle to avoid direct striking of the pulses onto the transducer. The bandwidth, diameter, and thickness of the transducer were about 8 MHz at full width at half maximum (FWHM), 6.4 mm, and 0.2 mm, respectively. Sample position was controlled by an xyz mechanical stage (KS701-30LMS; Suruga Seiki, Ltd.). The distance between the sample and transducer was about 2 cm. A set of neutral density filters (Thorlabs Japan, Inc.) was used for decreasing the pulse energy. Photoacoustic signals from the transducer were amplified by a low-noise wide-band preamplifier (9913; NF Corp.) of which polarity was inverting. Photoacoustic signals as a function of time were measured by a high-speed PCI digitizer (U1082A-AVG; Agilent Technologies, Inc.). Obtained signals in one position were 104-time averaged by on-board real-time FPGA (Field-Programmable Gate Array) control. The movement of the xyz mechanical stage was automatically controlled by a computer. The step size of the xyz controller was set to 2 μm to measure the depth and transverse profiles and to 5 μm to obtain photoacoustic images. Power spectra of photoacoustic signals were calculated by Fourier transform of the product of the time-dependent photoacoustic signal and Hanning function [17

W. H. Press, Numerical recipes in C: the art of scientific computing (Cambridge University Press, 2002).

]. To construct the TP-PAM image of samples, brightness at each xyz coordinate was determined by the integral of the power spectrum in the selected frequency range of the photoacoustic signal.

Fig. 2 TP-PAM system.

2.2 Frequency filtering in TP-PAM

In the investigation of target structures, the extraction of TP-PAM signals is important because one-photon photoacoustic signals are mixed into TP-PAM signals under actual circumstances. In this study, frequency filtering was used for the extraction. Frequency filtering in TP-PAM is based on the fact that materials with different absorptions generate acoustic waves with different frequency contents [18

P. C. Li, C. W. Wei, and Y. L. Sheu, “Subband photoacoustic imaging for contrast improvement,” Opt. Express 16(25), 20215–20226 (2008). [CrossRef] [PubMed]

, 19

Z. Guo, S. Hu, and L. V. Wang, “Calibration-free absolute quantification of optical absorption coefficients using acoustic spectra in 3D photoacoustic microscopy of biological tissue,” Opt. Lett. 35(12), 2067–2069 (2010). [CrossRef] [PubMed]

]. In one-photon absorption, absorbed energy gradually decreases depending on the depth position. On the other hand, two-photon absorption only occurs near the focus point. This difference between one-photon and two-photon absorptions results in the difference between the spectra of one-photon and two-photon photoacoustic waves. If we select the frequency region where the difference of their spectra is large, TP-PAM signals can be effectively extracted. Frequency filtering is extremely important to improve the signal-to-noise ratio of TP-PAM.

2.3 Imaging targets

As target structures, we employed transparent blocks (height: 24 mm, width: 19 mm, depth: 5 mm) made of silicone elastomer (SYLGARD 184; Dow Corning Toray Co., Ltd.) with a depth-centrally-located cylindrical-shaped hollow, as shown in Fig. 3(a) . The diameter of the cylindrical-shaped hollow was about 300 μm as estimated by using a confocal laser-scanning microscope (IX81 with FV1000; Olympus Corp.). Reservoirs in the silicone block were attached to both ends of the cylindrical-shaped hollow. The reservoirs, which were used in order not to alter the concentration of the solution in the hollow during the measurement, and the cylindrical-shaped hollow were filled with solution containing two-photon absorption dye. We employed Rhodamine B (R6626; Sigma-Aldrich, Inc., molar extinction coefficient; 1.05 X 105 M−1cm−1 at 550 nm), which has an absorption peak near 540 nm and is transparent in the infrared region, as the two-photon absorption dye. Here we confirmed that the photoacoustic signal of Rhodamine B/ethanol is generated by two-photon absorption because the signal intensity is nearly proportional to the square of the pulse energy. The transmittances of Rhodamine B/ethanol (58 mM) and ethanol in a 1-cm glass cell were measured when 3.2-μJ pulses were focused with 20X objective lens inside the glass cell. The results showed that there was no difference between their transmittances. Based on the reported value of the two-photon cross-section of Rhodamine B at 1064 nm (14.3 X 10−50 cm4 s photon−1) [20

J. P. Hermann and J. Ducuing, “Dispersion of the two-photon cross section in rhodamine dyes,” Opt. Commun. 6(2), 101–105 (1972). [CrossRef]

], extremely low transmittance was estimated. This discrepancy is caused by the saturation of the two-photon absorption. The saturation of the two-photon absorption of Rhodamine B occurs over 2 X 1025 photon cm−2 s−1 at 690 nm [20

J. P. Hermann and J. Ducuing, “Dispersion of the two-photon cross section in rhodamine dyes,” Opt. Commun. 6(2), 101–105 (1972). [CrossRef]

]. Our photon density per unit area at focus is about 3 X 1029 photon cm−2 s−1 for 4.3 μJ (pulse width: 0.6 ns) at 1.9-μm radius focus spot (the same experimental conditions as described in section 3). Therefore, two-photon absorption may be saturated under these conditions, though the concentration and wavelength in ref. [20

J. P. Hermann and J. Ducuing, “Dispersion of the two-photon cross section in rhodamine dyes,” Opt. Commun. 6(2), 101–105 (1972). [CrossRef]

] are different from those in our experiment. Thus, we calculated the spatial resolution of TP-PAM based on the assumption of non-depletion of the laser beam. The feasibility of TP-PAM with frequency filtering was evaluated by using the cross-sectional images of hollows in silicone blocks, as shown in Fig. 3(a). The expected cross-sectional image of the hollow is shown in Fig. 3(b).

Fig. 3 (a) Target sample for evaluating spatial resolution and (b) expected cross sectional image of the solution-filled hollow.

2.4 Evaluation of depth and lateral resolutions

The spatial resolution is one of the important parameters in evaluating the performance of the TP-PAM system. Ideally TP-PAM can achieve the same depth and transverse resolutions as two-photon microscopy (2PM) [21

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21(11), 1369–1377 (2003). [CrossRef] [PubMed]

, 22

P. Theer and W. Denk, “On the fundamental imaging-depth limit in two-photon microscopy,” J. Opt. Soc. Am. A 23(12), 3139–3149 (2006). [CrossRef] [PubMed]

]. In order to estimate the actual resolution of our TP-PAM system, we measured the cross sections of the hollows in silicone blocks filled with Rhodamine B/ethanol solution, as shown in Fig. 3. By using the depth and transverse profiles at the center of the hollow, we fit them to the theoretical curves to derive the spatial resolution.

In order to derive the theoretical curves of TP-PAM signals as functions of depth and transverse positions, we assumed that the sub-nanosecond pulse laser emits a beam with Gaussian profile [23

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart and Winston, Inc., New York, 1985).

], as shown by
I(x',y',z')= I0 z02 z '2+ z02exp [ − 2 z02(x '2+y '2) w02(z '2+ z02)],
(1)
where I0, z0 and, w0 denote the beam intensity at focus position, the Rayleigh length, and the beam radius at focus position, respectively. Here we assumed the non-depletion of the laser beam. Because the beam was collimated by two lenses, the objective lens was assumed to be located at the beam waist position of the Gaussian beam. The back aperture of the objective lens was not overfilled by the beam in order not to reduce the pulse energy. Because of this, we assumed that the propagating beam after passing through the objective lens was also a Gaussian beam (not a truncated Gaussian beam [24

H. Urey, “Spot size, depth-of-focus, and diffraction ring intensity formulas for truncated Gaussian beams,” Appl. Opt. 43(3), 620–625 (2004). [CrossRef] [PubMed]

, 25

V. N. Mahajan, “Uniform versus Gaussian beams: a comparison of the effects of diffraction, obscuration, and aberrations,” J. Opt. Soc. Am. 3(4), 470–485 (1986). [CrossRef]

]). The truncation ratio [24

H. Urey, “Spot size, depth-of-focus, and diffraction ring intensity formulas for truncated Gaussian beams,” Appl. Opt. 43(3), 620–625 (2004). [CrossRef] [PubMed]

] was about 0.5. Here we assumed that the photoacoustic signal f TPPA(x,y,z,a) was generated by two-photon absorption in the solution-filled hollow (radius: a) when the center of the Gaussian beam is placed at relative coordinates (x,y,z) with its origin at the center of the hollow. The signal f TPPA(x,y,z,a) is proportional to the integral of the square of the beam intensity I( x′, y′, z′) in the integral range for the volume occupied by the two-photon absorption solution:
f TPPA(x,y,z,a)∝ ∬ (x− x′)2+ (z− z′)2≤ a2 d x′d z′ ∫ −∞∞ d y′ I2( x′, y′, z′).
(2)
Here we defined the y axis as the long axis of the hollow. Because the length of the hollow in the direction of the long axis is considerably greater than the beam radius at the focus point, the square of the beam intensity was integrated in the range from −∞ to ∞.

For the estimation of depth resolution of TP-PAM, we measured the depth profile where x=0 and y=0. We assumed that the beam diameter was considerably smaller than that of the target hollow ( a>> w0). Consequently, substituting Eq. (1) into Eq. (2), we integrated the square of the intensity in the range from −∞ to ∞ in the transverse direction (x) instead of the finite range for the volume occupied by the two-photon absorption solution. The photoacoustic signal at (0, 0, z) is therefore shown by
f TPPA(0,0,z,a)= I02 ∫ z−a z+a d z′ ∫ −∞∞ d x′ ∫ −∞∞ d y′ z04 ( z′2+ z02)2exp [ − 4 z02 ( x′2+ y′2) w02 ( z′2+ z02)]
(3)
∝ tan −1 ( z+a z0)− tan −1 ( z−a z0).
(4)
By fitting the experimentally-obtained depth profile at the center of the solution-filled hollow to Eq. (4), we determined the Rayleigh length z0. The depth resolution in TP-PAM was estimated to be 2 z0 based on FWHM of TP-PAM signal generated by the sample of which thickness is infinitely small.

For the estimation of transverse resolution of TP-PAM, we measured the transverse profile where y=0 and z=0. We assumed that the square of the beam intensity has a certain value only when z′ is near zero, because the volume where the two-photon absorption occurred was considerably smaller than that of the target hollow ( a>> w0).
f TPPA(x,0,0,a)= I02 ∫ −∞∞ d z′ ∫ x−a x+a d x′ ∫ −∞∞ d y′ z04 ( z′2+ z02)2exp [ − 4 z02 ( x′2+ y′2) w02 ( z′2+ z02)]
(5)
∝erf ( 2(x−a) w0, 2(x+a) w0),
(6)
where
erf( x1, x2)≡ ∫ x1 x2 d x′exp [ − x′2].
(7)
The transverse resolution in TP-PAM was estimated to be ( ln2) 1/2 w0 based on the FWHM of the TP-PAM signal generated by the sample of which thickness is infinitely small.

2.5 Evaluation of beam focusing

To compare the measured depth and transverse resolutions in TP-PAM with those optically expected in 2PM, we also measured the beam radius near the focus depending on the depth position by a knife-edge method [26

J. M. Khosrofian and B. A. Garetz, “Measurement of a Gaussian laser beam diameter through the direct inversion of knife-edge data,” Appl. Opt. 22(21), 3406–3410 (1983). [CrossRef] [PubMed]

]. For precise estimation of the beam focusing, the knife-edge experiment was done using the same optical apparatus as for the TP-PAM experiment. To estimate precisely the defocusing by the silicone between the surface of the silicone block and the target hollow, silicone of the same thickness was attached to the knife. Because the target samples in the TP-PAM experiments were placed in water, the knife-edge experiment was also performed in water. From the obtained beam radii at all depth positions ( w(z)), we calculated the expected TP-PAM signal from infinity small target as a function of the depth position using the equation
  f TPPA(0,0,z,a≃0)∝ 1 w (z)2= 1 w02 [ 1+ ( z z0)2],
(8)
where

z0= π w02nλ.
(9)

Here, n and λ denote the refractive index of water (1.33) and the excitation wavelength (1064 nm), respectively. We determined the beam radius ( w0) at the focus point and the Rayleigh length ( z0) [23] by fitting the expected TP-PAM signal to Eq. (8). Comparing the depth and transverse resolutions in TP-PAM from the experimental and theoretical points of view.

3. Experimental and numerical results

3.1 Temporal and spectral differences of photoacoustic signals between one-photon and two-photon absorptions

In order to determine the appropriate frequency region for effective detection of TP-PAM signals, we investigated the temporal and spectral differences between one-photon and two-photon absorptions. We measured the photoacoustic signals generated by Rhodamine B/ethanol in the hollow depending on the depth position. When the beam focus is inside the hollow filled with Rhodamine B/ethanol, two-photon photoacoustic signals are dominant. On the other hand, one-photon photoacoustic signals generated by solvent (ethanol) are dominant when the beam focus is outside the hollow. In order to compare the photoacoustic signals generated by one-photon and two-photon absorptions, we compared the photoacoustic signals when the beam focus was inside and outside the hollow. The concentration of Rhodamine B/ethanol was 25.4 mM. The pulse energy was 4.3 μJ after passing through the objective lens.

Figure 4(a) shows a representative photoacoustic signal acquired when the beam focus was outside the hollow (one-photon absorption). The beam focus was located 400 μm apart from the center of the hollow ( z=−400 μm). Figure 4(b) shows the photoacoustic signal acquired when the beam focus was in the center of the hollow (two-photon absorption, z=0 μm). Here we note that, because the preamplifier has inverse polarity, the signals are inverted. Both signals have fluctuation with frequency of about 22 MHz. This is the noise received and generated by electrical components such as the transducer and the preamplifier, because the fluctuation was still present when the excitation beam was blocked. Moreover, the fluctuation was reduced when the transducer was taken out of the water bath and when the preamplifier was turned off. In the case of the two-photon absorption indicated in Fig. 4(b), the signal was generated from the narrower area than in the case of the one-photon absorption. According to the difference of temporal profiles, the spectra calculated by measured photoacoustic signals for the one-photon (black line) and the two-photon (red line) absorptions are shown in Fig. 4(c). The spectrum for the one-photon absorption was obtained as the average spectrum in the ranges of the depth positions between −400 μm and −300 μm and between 300 μm and 400 μm. The spectrum for the two-photon absorption was obtained as the average spectrum in the range of the depth positions between −100 μm and 100 μm. Figure 4(c) makes it clear that the photoacoustic signals generated by two-photon absorption ( z=0 μm) and those generated by one-photon absorption ( z=−400 μm) are different.

Fig. 4 Experimental photoacoustic signals as a function of time acquired when the beam focus is outside the Rhodamine B/ethanol-filled hollow ((a), one-photon, black line) and when it is inside the hollow ((b), two-photon, red lines) and their power spectra (c). Calculated photoacoustic signals for one-photon ((d), black line) and two-photon ((e), red line) absorbers and their power spectra (f).

In order to validate the frequency analysis shown in Fig. 4(c), we calculated the photoacoustic signals generated by one-photon and two-photon absorptions [8

M. Xu and L. V. Wang, “Photoacoustic imaging in biomedicine,” Rev. Sci. Instrum. 77(4), 041101 (2006). [CrossRef]

, 12

L. V. Wang, ed., Photoacoustic Imaging and Spectroscopy (CRC press, Boca Raton, 2009).

, 19

Z. Guo, S. Hu, and L. V. Wang, “Calibration-free absolute quantification of optical absorption coefficients using acoustic spectra in 3D photoacoustic microscopy of biological tissue,” Opt. Lett. 35(12), 2067–2069 (2010). [CrossRef] [PubMed]

, 27

L. H. Wang, and H.-I. Wu, Biomedical Optics (John Wiley & Sons, Hoboken, 2007). [PubMed]

]. We assumed that the thermal expansion occurs only on the optical (z) axis without any x-y spatial extent and is induced instantaneously in accordance with the waveform of the sub-nanosecond optical pulses. Based on these assumptions, the temporal waveforms of generated photoacoustic signals are then proportional to the derivatives of the absorption profiles in the target objects. Calculated photoacoustic waves for one-photon and two-photon absorbers filled in 300-μm-diameter silicone hollows under the same experimental conditions (Fig. 4(a) and (b)) are shown by a black line in Fig. 4(d) and a red line in Fig. 4(e), respectively. The photoacoustic signals are inverted in order to compare them with those obtained in the experiment. In the theoretical calculation, we made the background noise zero. The absorption is assumed to occur uniformly independent of the depth position inside the hollow in the case of one-photon absorption. To calculate the photoacoustic signal generated by two-photon absorption shown in Fig. 4(e), we used Eq. (8) with w0 and z0 estimated by the knife-edge experiment using 20X objective lens, as explained in subsection 2.5. From the knife-edge experiment, we found that estimated w0 and z0 were 2.3 μm and 20.1 μm, respectively. Depth position is converted into time by assuming that the velocity of ultrasonic waves in water is 1500 m/s. As shown in Fig. 4(d), the one-photon signal has two peaks that are delta functions. In the comparison with the experimental result as shown in Fig. 4(a), two peaks shown in Fig. 4(a) are broader than those shown in Fig. 4(d). There are several reasons for this discrepancy. First is the finite response of the acoustic transducer. Even if the photoacoustic signal is generated from one point, the response of the acoustic signal has duration. Because of this, the photoacoustic signal becomes broader. Second is the x-y extent of the absorption object. The x-y extent also broadens the photoacoustic signal. Third is the tilting of the transducer to the laser direction. These factors broaden and deform the photoacoustic signals. The measured photoacoustic signal generated by two-photon absorption is also broader than that calculated by measured beam parameters, as shown in Fig. 4(b) and (e). This broadening is also explained by the same reasons that apply to the case of one-photon photoacoustic signals.

The power spectra shown in Fig. 4(f) were calculated by performing Fourier transform of the photoacoustic signals. The spectral difference between one-photon (black line) and two-photon (red line) absorptions qualitatively agrees with the experimental results. Theoretical calculation shown in Fig. 4(f) indicates that the contribution of the two-photon photoacoustic signal increases with increasing frequency of the acoustic waves. However, experimental results show that the difference between one-photon and two-photon disappears above 10 MHz. This is attributed to the fact that the experimentally-obtained spectrum over 10 MHz is buried by the foot of the electronic noise near 22 MHz. In addition, in the frequency range outside of the bandwidth of the acoustic transducer, the difference disappears. Humpy structure in the one-photon case (black line in Fig. 4(c)) is also reproduced by the calculation (black line in Fig. 4(f)). The frequency at the spectral first dip (5 MHz in Fig. 4(f)) corresponds to the speed of acoustic waves divided by the diameter of the target hollow. The photoacoustic detection using frequency components above the frequency where the power begins to drop to the first dip is important to extract the two-photon photoacoustic signals effectively. When the pulse reached the opposite side of the target hollow, the calculation indicated that the dip frequency was independent of the amount of the one-photon absorption. It can be inferred that the drop frequency related to the dip frequency is largely determined by the hollow diameter independent of the one-photon absorption. In this experiment for a 300-μm-diameter hollow filled with 25.4-mM Rhodamine B/ethanol, we chose 1 MHz as the drop frequency based on the experimental results. As the high-frequency limit, we used 10 MHz, where the difference between the one-photon and two-photon photoacoustic signals disappears due to the electronic noise near 22 MHz.

3.2 Comparison between TP-PAM images with and without frequency filtering for measuring cross sections of silicone hollow filled with solutions of Rhodamine B/ethanol

In the previous subsection, we showed the spectral difference between one-photon and two-photon absorptions and determined the appropriate frequency region to extract the two-photon photoacoustic signals. We next measured the cross sections of silicone hollows filled with Rhodamine B/ethanol solution by TP-PAM without (a) and with (b) frequency filtering (1-10 MHz), as shown in Fig. 5 . The concentration of Rhodamine B/ethanol was 25.4 mM. The pulse energy we used was 4.3 μJ after passing through the objective lens. The time range of measured photoacoustic signals was 10 μs. Arrows in Fig. 5 denote the traveling directions of the excitation beam. Without frequency filtering, the image of the cross section of the target hollow is unclear. On the other hand, the cross section of the hollow is clearly observed with frequency filtering. Photoacoustic signals in TP-PAM are usually mixed with one-photon photoacoustic signals originating from linear absorptions. Because frequency filtering reduced the contribution of linear absorption of the solvents and silicone, the contrast of TP-PAM was improved. The frequency filtering also reduces low-frequency noise which is easily generated in acoustic measurements. These observations lead us to infer that TP-PAM with frequency filtering can provide improved contrast and spatial resolution.

Fig. 5 Comparison between TP-PAM images acquired without (a) and with (b) frequency filtering of 1-10 MHz of a 300-μm-diameter silicone hollow filled with Rhodamine B/ethanol.

3.3 Estimation of depth and transverse resolutions in TP-PAM with frequency filtering by using silicone hollow filled with Rhodamine B/ethanol

To estimate the depth and transverse resolutions, we investigated the intensity profiles along with the depth and transverse axes passing through the center of silicone hollow (diameter 2a: 300 μm) filled with Rhodamine B/ethanol at concentration of 25.4 mM. We measured the depth and transverse profiles with the frequency filtering of 1-10 MHz using a 20X objective lens (NA: 0.4). Representative depth (Fig. 6(a) ) and transverse (Fig. 6(b)) intensity profiles at the center of the hollow are plotted by black open dots. By fitting the depth and transverse profiles respectively to Eqs. (4) and (6), we obtained depth and transverse resolutions of TP-PAM using the 20X objective lens as 44.9 ± 2.0 μm and 6.5 ± 1.5 μm, respectively. Small increments were observed at the leading and trailing edges in the depth and transverse profiles. Signal deviation at the edges is larger than that in the middle of the hollow. Though the representative depth profile shows the increment of the leading edge, the signal at the trailing edge is sometimes increased. One of the reasons for these increments is an edge effect [28

R. A. McFarlane and L. D. Hess, “Photoacoustic measurements of ion-implanted and laser-annealed GaAs,” Appl. Phys. Lett. 36(2), 137–139 (1980). [CrossRef]

]. Limited thermal evacuation at the edge makes the thermal elevation high, and strong photoacoustic waves are then generated. We investigated the depth resolution of TP-PAM at other concentrations (43.1, 36.9, 18.5, 12.7, and 7.4 mM) in addition to 25.4 mM. The results showed no clear concentration dependence of determined depth resolution, although the error bar did increase with decrease of the concentration. In addition, we note that the width of the obtained depth profile was narrower than the diameter of the solution-filled hollow. This is because of the refractive-index differences between water, silicone, and target solution (solvent: ethanol). The refractive indices of water, silicone, and ethanol for sodium D line are 1.333, 1.4103, and 1.3618, respectively [29

National Astronomical Observatory, Rika Nenpyo (Chronological Scientific Tables 2008) (Maruzen Co., Ltd., 2008).

]. Slight differences of refractive indices of the materials through which the beam passes distort the TP-PAM image. Actually, the cross-sectional image shown in Fig. 5(b) looks slightly ellipsoidal (not true circular). When we treated the diameter of the solution-filled hollow as one of the fitting parameters, we obtained the values for the distorted diameter of the hollow and the depth resolution of TP-PAM with the 20X objective lens of 271.8 ± 0.8 μm and 44.9 ± 2.0 μm, respectively. We verified that the evaluated depth resolution does not change, even if the diameter of the solution-filled hollow is fitted smaller than the actual diameter.

Fig. 6 Depth (a) and transverse (b) intensity profiles of TP-PAM image of 300-μm Rhodamine B/ethanol-filled hollow with frequency filtering (1-10 MHz) obtained using a 20X objective lens (NA: 0.4). Black open dots and red solid lines denote the experimental data and fitting results obtained using Eq. (4) for depth profiles and Eq. (6) for transverse profiles.

From the values of w0 (2.3 μm) and z0 (20.1 μm) obtained by the knife-edge method, we estimated the depth and transverse resolutions of TP-PAM as 40.2 μm and 1.9 μm, respectively. These resolutions almost entirely agree with those experimentally obtained by using TP-PAM. These results lead us to infer that TP-PAM has the capacity to visualize the structures with optical resolution. The obtained value of the depth resolution of TP-PAM is comparable to that experimentally estimated using FWHM in a recent study with detailed investigation of the depth resolution of PAM with a high-frequency transducer (75 MHz) [4

G. Ku, K. Maslov, L. Li, and L. V. Wang, “Photoacoustic microscopy with 2-μm transverse resolution,” J. Biomed. Opt. 15(2), 021302 (2010). [CrossRef] [PubMed]

]. Our method provides the optical depth resolution determined by two-photon absorption with low-frequency transducer (10 MHz) as a means to improve the spatial resolution.

4. Discussion

In the present study, we found that TP-PAM with appropriate frequency filtering provides optical resolution. The depth and transverse resolutions obtained almost entirely agree with those optically estimated by the knife-edge experiment.

There are several factors that limit the spatial resolution and penetration depth in TP-PAM. One of them is the beam focusing related to the quality of the beam source (collimation, distortion, and spatial mode) and the optical properties of the targets (scattering and absorption). The beam focusing in TP-PAM is more important than that in the one-photon PAM because the efficiency of the two-photon absorption strongly depends on the beam focusing. Despite the agreement between the expected and experimentally obtained resolutions, slight mismatch still occurs even in the transparent and reduced-scattering target samples. The reason for the mismatch is the estimation error of the beam focusing induced by the deviation from the Gaussian beam theory we assumed for the beam source. In the knife-edge experiment, the transmitted intensity at each depth position as a function of the knife position is erf function [26

J. M. Khosrofian and B. A. Garetz, “Measurement of a Gaussian laser beam diameter through the direct inversion of knife-edge data,” Appl. Opt. 22(21), 3406–3410 (1983). [CrossRef] [PubMed]

]. Though the plateau and steep-slope parts of the transmitted intensity were well fitted by the erf function, the leading and falling edges were not especially near the focus position. The deviation from the erf function is provoked by the loose beam focusing, which makes the spatial resolution poor.

As for the strongly-scattering and -absorbing targets such as biological tissues, scattering and absorption also impair the beam focusing. The degradation of the spatial resolution and the penetration depth in TP-PAM is based on the principle of 2PM [22

P. Theer and W. Denk, “On the fundamental imaging-depth limit in two-photon microscopy,” J. Opt. Soc. Am. A 23(12), 3139–3149 (2006). [CrossRef] [PubMed]

] except that there is acoustic detection instead of optical detection. The fact that overfilling the back aperture of the objective lens improves the spatial resolution is also valid for TP-PAM. With increasing NA of the objective lens, the spatial resolution is improved in TP-PAM. On the other hand, the imaging depth in 2PM is fundamentally limited by the generation of out-of-focus fluorescence which cannot be distinguished from the fluorescence near the focus in 2PM [22

P. Theer and W. Denk, “On the fundamental imaging-depth limit in two-photon microscopy,” J. Opt. Soc. Am. A 23(12), 3139–3149 (2006). [CrossRef] [PubMed]

]. Contrary to 2PM, TP-PAM has the potential to distinguish the signals from the out-of-focal and focal planes, because the speed of acoustic waves is slower than that of light. This improves the penetration depth compared with 2PM.

The second limiting factor is the photon number at the focus point. The mechanism of the generation of photoacoustic waves varies depending on the photon number per unit volume of interacted materials. Large photon number causes photodamage involving material alterations such as dielectric breakdown, vaporization, and material ablation [30

S. Boonsang, “Photoacoustic generation mechanisms and measurement systems for biomedical applications,” Int. J. Appl. Biomed. Eng. 2(1), 17–23 (2009).

]. The high-NA objective lens enables TP-PAM to provide improved spatial resolution. However, the high NA increases the photon number per unit volume and thus causes photodamage. The photodamage threshold of the photon number, therefore, limits the spatial resolution of TP-PAM. Because the improvement of the detection efficiency of photoacoustic signals decreases the required photon number, the detection improvement surmounts the limitation of the spatial resolution. The optimization of the pulse width to improve the efficiency of two-photon absorption also improves the spatial resolution in TP-PAM. In 2PM, photobleaching due to large photon number is one of the important causes of decrease of the fluorescence signals and subsequent impairment of the spatial resolution. Unlike fluorescence microscopy, photoacoustic microscopy utilizes nonradiative process instead of fluorescence process [31

H. Vargas and L. C. M. Miranda, “Photoacoustic and related photothermal techniques,” Phys. Rep. 161(2), 43–101 (1988). [CrossRef]

]. Photobleaching occurs when nonradiative processes become dominant [32

C. Eggeling, A. Volkmer, and C. A. Seidel, Molecular photobleaching kinetics of Rhodamine 6G by one- and two-photon induced confocal fluorescence microscopy, ChemPhysChem 6(5), 791–804 (2005). [CrossRef] [PubMed]

]. Nonradiative processes generate thermal expansion and then increase photoacoustic signals. Though the photobleaching is a severe problem in 2PM, it works to advantage in TP-PAM.

The third limiting factor is the frequency filtering. We have shown the effectiveness of the frequency filtering to extract TP-PAM signals. The frequency filtering is based on the absorption profile in the target. When the one-photon absorption is more extensive compared with the two-photon absorption, it becomes difficult to extract TP-PAM signals using frequency filtering. If the target structures are very small, the extraction also becomes difficult. Especially for in vivo applications, the frequency filtering range should be appropriately determined by focusing on specific target structures, because there are various sized and tiny structures in biological tissues. Such structures limit the spatial resolution and worsen the contrast of TP-PAM. However, even when the two-photon and one-photon photoacoustic signals generated from the small area, the shapes of their signals are slightly different. This means that the included spectral components are different. Detailed study of spectral analysis of generated photoacoustic signals may lead to better extraction of TP-PAM and the improvement of the spatial resolution and contrast of TP-PAM.

The fourth limiting factor is the noise of TP-PAM. There are several origins of this noise. First is the electronic noise received by the acoustic transducer. Because the electrical current generated in acoustic transducers is very small, electronic noise from nearby equipment and public airwaves easily generates relatively large output current in acoustic transducers, and the necessary signals are then buried. This is improved by appropriate shielding of the electronic equipment and the electronic frequency filtering of the output of the acoustic transducers. Second is the fluctuation of the energy and pulse width of sub-nanosecond laser pulses. Because the signal of TP-PAM is proportional to the square of the peak intensity of the laser pulses, the fluctuation of the energy and pulse width is critical compared with conventional one-photon PAM. This problem is solved by the laser stabilization techniques using feedback circuits. Third is the thermal elevation effect, such as edge effect. To generate the two-photon photoacoustic signals, we need relatively large power compared with the one-photon PAM. Because of this, the edge effect occurs relatively more often than in one-photon PAM. The improvement of the detection efficiency of photoacoustic waves and the fast scanning, to reduce the energy to generate photoacoustic waves, may reduce the signal enhancement due to the edge effect.

5. Conclusion

We have demonstrated high depth resolution and high contrast by using TP-PAM with low-frequency bandpass filtering of photoacoustic waves. The depth resolution of TP-PAM is determined by two-photon absorption. The combination of TP-PAM and frequency filtering will provide new insights helpful in detailed investigation of inner structures of various targets.

Acknowledgments

The helpful discussion with Prof. Ryuji Morita (Hokkaido University) is gratefully acknowledged. This work was partially supported by grants-in-aid for scientific research (B) from the Japan Society for the Promotion of Science (JSPS), Japan (18300153), and grants-in-aid for young scientists (B) from The Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan (19700384 and 21700469).

References and links

1.

A. Diaspro, Confocal and Two-Photon Microscopy (Wiley-Liss, 2002).

2.

H. F. Zhang, K. Maslov, G. Stoica, and L. V. Wang, “Functional photoacoustic microscopy for high-resolution and noninvasive in vivo imaging,” Nat. Biotechnol. 24(7), 848–851 (2006). [CrossRef] [PubMed]

3.

L. V. Wang, “Multiscale photoacoustic microscopy and computed tomography,” Nat. Photonics 3(9), 503–509 (2009). [CrossRef] [PubMed]

4.

G. Ku, K. Maslov, L. Li, and L. V. Wang, “Photoacoustic microscopy with 2-μm transverse resolution,” J. Biomed. Opt. 15(2), 021302 (2010). [CrossRef] [PubMed]

5.

Z. Xie, S. Jiao, H. F. Zhang, and C. A. Puliafito, “Laser-scanning optical-resolution photoacoustic microscopy,” Opt. Lett. 34(12), 1771–1773 (2009). [CrossRef] [PubMed]

6.

R. Bitton, R. Zemp, J. Yen, L. V. Wang, and K. K. Shung, “A 3-D high-frequency array based 16 channel photoacoustic microscopy system for in vivo micro-vascular imaging,” IEEE Trans. Med. Imaging 28(8), 1190–1197 (2009). [CrossRef] [PubMed]

7.

S. Hu, K. Maslov, and L. V. Wang, “Noninvasive label-free imaging of microhemodynamics by optical-resolution photoacoustic microscopy,” Opt. Express 17(9), 7688–7693 (2009). [CrossRef] [PubMed]

8.

M. Xu and L. V. Wang, “Photoacoustic imaging in biomedicine,” Rev. Sci. Instrum. 77(4), 041101 (2006). [CrossRef]

9.

C. Zhang, K. Maslov, and L. V. Wang, “Subwavelength-resolution label-free photoacoustic microscopy of optical absorption in vivo ,” Opt. Lett. 35(19), 3195–3197 (2010). [CrossRef] [PubMed]

10.

C. Guittet, F. Ossant, L. Vaillant, and M. Berson, “ In vivo high-frequency ultrasonic characterization of human dermis,” IEEE Trans. Biomed. Eng. 46(6), 740–746 (1999). [CrossRef] [PubMed]

11.

K. Maslov, H. F. Zhang, S. Hu, and L. V. Wang, “Optical-resolution photoacoustic microscopy for in vivo imaging of single capillaries,” Opt. Lett. 33(9), 929–931 (2008). [CrossRef] [PubMed]

12.

L. V. Wang, ed., Photoacoustic Imaging and Spectroscopy (CRC press, Boca Raton, 2009).

13.

C. R. Hill, J. C. Bamber, and G. R. t. Haar, eds., Physical principles of medical ultrasonics (John Weily & Sons, Chichester, 2004).

14.

Y. Bae, J. J. Song, and Y. B. Kim, “Photoacoustic study of two-photon absorption in hexagonal ZnS,” J. Appl. Phys. 53(1), 615–619 (1982). [CrossRef]

15.

J. J. Barrett and M. J. Berry, “Photoacoustic Raman spectroscopy (PARS) using cw laser sources,” Appl. Phys. Lett. 34(2), 144–146 (1979). [CrossRef]

16.

P. Sathy, R. Philip, V. P. N. Nampoori, and C. P. G. Vallabhan, “Observation of two-photon absorption in rhodamine 6G using photoacoustic technique,” Opt. Commun. 74(5), 313–317 (1990). [CrossRef]

17.

W. H. Press, Numerical recipes in C: the art of scientific computing (Cambridge University Press, 2002).

18.

P. C. Li, C. W. Wei, and Y. L. Sheu, “Subband photoacoustic imaging for contrast improvement,” Opt. Express 16(25), 20215–20226 (2008). [CrossRef] [PubMed]

19.

Z. Guo, S. Hu, and L. V. Wang, “Calibration-free absolute quantification of optical absorption coefficients using acoustic spectra in 3D photoacoustic microscopy of biological tissue,” Opt. Lett. 35(12), 2067–2069 (2010). [CrossRef] [PubMed]

20.

J. P. Hermann and J. Ducuing, “Dispersion of the two-photon cross section in rhodamine dyes,” Opt. Commun. 6(2), 101–105 (1972). [CrossRef]

21.

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21(11), 1369–1377 (2003). [CrossRef] [PubMed]

22.

P. Theer and W. Denk, “On the fundamental imaging-depth limit in two-photon microscopy,” J. Opt. Soc. Am. A 23(12), 3139–3149 (2006). [CrossRef] [PubMed]

23.

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart and Winston, Inc., New York, 1985).

24.

H. Urey, “Spot size, depth-of-focus, and diffraction ring intensity formulas for truncated Gaussian beams,” Appl. Opt. 43(3), 620–625 (2004). [CrossRef] [PubMed]

25.

V. N. Mahajan, “Uniform versus Gaussian beams: a comparison of the effects of diffraction, obscuration, and aberrations,” J. Opt. Soc. Am. 3(4), 470–485 (1986). [CrossRef]

26.

J. M. Khosrofian and B. A. Garetz, “Measurement of a Gaussian laser beam diameter through the direct inversion of knife-edge data,” Appl. Opt. 22(21), 3406–3410 (1983). [CrossRef] [PubMed]

27.

L. H. Wang, and H.-I. Wu, Biomedical Optics (John Wiley & Sons, Hoboken, 2007). [PubMed]

28.

R. A. McFarlane and L. D. Hess, “Photoacoustic measurements of ion-implanted and laser-annealed GaAs,” Appl. Phys. Lett. 36(2), 137–139 (1980). [CrossRef]

29.

National Astronomical Observatory, Rika Nenpyo (Chronological Scientific Tables 2008) (Maruzen Co., Ltd., 2008).

30.

S. Boonsang, “Photoacoustic generation mechanisms and measurement systems for biomedical applications,” Int. J. Appl. Biomed. Eng. 2(1), 17–23 (2009).

31.

H. Vargas and L. C. M. Miranda, “Photoacoustic and related photothermal techniques,” Phys. Rep. 161(2), 43–101 (1988). [CrossRef]

32.

C. Eggeling, A. Volkmer, and C. A. Seidel, Molecular photobleaching kinetics of Rhodamine 6G by one- and two-photon induced confocal fluorescence microscopy, ChemPhysChem 6(5), 791–804 (2005). [CrossRef] [PubMed]

OCIS Codes
(170.5120) Medical optics and biotechnology : Photoacoustic imaging
(070.2615) Fourier optics and signal processing : Frequency filtering
(180.4315) Microscopy : Nonlinear microscopy

ToC Category:
Microscopy

History
Original Manuscript: December 20, 2010
Revised Manuscript: March 14, 2011
Manuscript Accepted: March 18, 2011
Published: June 27, 2011

Virtual Issues
Vol. 6, Iss. 8 Virtual Journal for Biomedical Optics

Citation
Yoshihisa Yamaoka, Mika Nambu, and Tetsuro Takamatsu, "Fine depth resolution of two-photon absorption-induced photoacoustic microscopy using low-frequency bandpass filtering," Opt. Express 19, 13365-13377 (2011)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-19-14-13365


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References

  1. A. Diaspro, Confocal and Two-Photon Microscopy (Wiley-Liss, 2002).
  2. H. F. Zhang, K. Maslov, G. Stoica, and L. V. Wang, “Functional photoacoustic microscopy for high-resolution and noninvasive in vivo imaging,” Nat. Biotechnol. 24(7), 848–851 (2006). [CrossRef] [PubMed]
  3. L. V. Wang, “Multiscale photoacoustic microscopy and computed tomography,” Nat. Photonics 3(9), 503–509 (2009). [CrossRef] [PubMed]
  4. G. Ku, K. Maslov, L. Li, and L. V. Wang, “Photoacoustic microscopy with 2-μm transverse resolution,” J. Biomed. Opt. 15(2), 021302 (2010). [CrossRef] [PubMed]
  5. Z. Xie, S. Jiao, H. F. Zhang, and C. A. Puliafito, “Laser-scanning optical-resolution photoacoustic microscopy,” Opt. Lett. 34(12), 1771–1773 (2009). [CrossRef] [PubMed]
  6. R. Bitton, R. Zemp, J. Yen, L. V. Wang, and K. K. Shung, “A 3-D high-frequency array based 16 channel photoacoustic microscopy system for in vivo micro-vascular imaging,” IEEE Trans. Med. Imaging 28(8), 1190–1197 (2009). [CrossRef] [PubMed]
  7. S. Hu, K. Maslov, and L. V. Wang, “Noninvasive label-free imaging of microhemodynamics by optical-resolution photoacoustic microscopy,” Opt. Express 17(9), 7688–7693 (2009). [CrossRef] [PubMed]
  8. M. Xu and L. V. Wang, “Photoacoustic imaging in biomedicine,” Rev. Sci. Instrum. 77(4), 041101 (2006). [CrossRef]
  9. C. Zhang, K. Maslov, and L. V. Wang, “Subwavelength-resolution label-free photoacoustic microscopy of optical absorption in vivo,” Opt. Lett. 35(19), 3195–3197 (2010). [CrossRef] [PubMed]
  10. C. Guittet, F. Ossant, L. Vaillant, and M. Berson, “In vivo high-frequency ultrasonic characterization of human dermis,” IEEE Trans. Biomed. Eng. 46(6), 740–746 (1999). [CrossRef] [PubMed]
  11. K. Maslov, H. F. Zhang, S. Hu, and L. V. Wang, “Optical-resolution photoacoustic microscopy for in vivo imaging of single capillaries,” Opt. Lett. 33(9), 929–931 (2008). [CrossRef] [PubMed]
  12. L. V. Wang, ed., Photoacoustic Imaging and Spectroscopy (CRC press, Boca Raton, 2009).
  13. C. R. Hill, J. C. Bamber, and G. R. t. Haar, eds., Physical principles of medical ultrasonics (John Weily & Sons, Chichester, 2004).
  14. Y. Bae, J. J. Song, and Y. B. Kim, “Photoacoustic study of two-photon absorption in hexagonal ZnS,” J. Appl. Phys. 53(1), 615–619 (1982). [CrossRef]
  15. J. J. Barrett and M. J. Berry, “Photoacoustic Raman spectroscopy (PARS) using cw laser sources,” Appl. Phys. Lett. 34(2), 144–146 (1979). [CrossRef]
  16. P. Sathy, R. Philip, V. P. N. Nampoori, and C. P. G. Vallabhan, “Observation of two-photon absorption in rhodamine 6G using photoacoustic technique,” Opt. Commun. 74(5), 313–317 (1990). [CrossRef]
  17. W. H. Press, Numerical recipes in C: the art of scientific computing (Cambridge University Press, 2002).
  18. P. C. Li, C. W. Wei, and Y. L. Sheu, “Subband photoacoustic imaging for contrast improvement,” Opt. Express 16(25), 20215–20226 (2008). [CrossRef] [PubMed]
  19. Z. Guo, S. Hu, and L. V. Wang, “Calibration-free absolute quantification of optical absorption coefficients using acoustic spectra in 3D photoacoustic microscopy of biological tissue,” Opt. Lett. 35(12), 2067–2069 (2010). [CrossRef] [PubMed]
  20. J. P. Hermann and J. Ducuing, “Dispersion of the two-photon cross section in rhodamine dyes,” Opt. Commun. 6(2), 101–105 (1972). [CrossRef]
  21. W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21(11), 1369–1377 (2003). [CrossRef] [PubMed]
  22. P. Theer and W. Denk, “On the fundamental imaging-depth limit in two-photon microscopy,” J. Opt. Soc. Am. A 23(12), 3139–3149 (2006). [CrossRef] [PubMed]
  23. A. Yariv, Introduction to Optical Electronics (Holt, Rinehart and Winston, Inc., New York, 1985).
  24. H. Urey, “Spot size, depth-of-focus, and diffraction ring intensity formulas for truncated Gaussian beams,” Appl. Opt. 43(3), 620–625 (2004). [CrossRef] [PubMed]
  25. V. N. Mahajan, “Uniform versus Gaussian beams: a comparison of the effects of diffraction, obscuration, and aberrations,” J. Opt. Soc. Am. 3(4), 470–485 (1986). [CrossRef]
  26. J. M. Khosrofian and B. A. Garetz, “Measurement of a Gaussian laser beam diameter through the direct inversion of knife-edge data,” Appl. Opt. 22(21), 3406–3410 (1983). [CrossRef] [PubMed]
  27. L. H. Wang, and H.-I. Wu, Biomedical Optics (John Wiley & Sons, Hoboken, 2007). [PubMed]
  28. R. A. McFarlane and L. D. Hess, “Photoacoustic measurements of ion-implanted and laser-annealed GaAs,” Appl. Phys. Lett. 36(2), 137–139 (1980). [CrossRef]
  29. National Astronomical Observatory, Rika Nenpyo (Chronological Scientific Tables 2008) (Maruzen Co., Ltd., 2008).
  30. S. Boonsang, “Photoacoustic generation mechanisms and measurement systems for biomedical applications,” Int. J. Appl. Biomed. Eng. 2(1), 17–23 (2009).
  31. H. Vargas and L. C. M. Miranda, “Photoacoustic and related photothermal techniques,” Phys. Rep. 161(2), 43–101 (1988). [CrossRef]
  32. C. Eggeling, A. Volkmer, and C. A. Seidel, "Molecular photobleaching kinetics of Rhodamine 6G by one- and two-photon induced confocal fluorescence microscopy," Chemphyschem 6(5), 791-804 (2005). [CrossRef] [PubMed]

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