Optoacoustic determination of optical attenuation depth using interferometric detection

. We use a modiﬁed Mach-Zehnder interferometer to measure surface displacement resulting from the thermoelastic response of a target to the absorption of a short laser pulse with axial and temporal resolutions of 0.1 nm and 3 ns, respectively. These measurements are used in conjunction with a solution to the thermoelastic wave equation and a nonlinear optimization algorithm to extract optical attenuation depth. We demonstrate the ability to determine the effective optical attenuation depth of homogeneous targets with either diffuse or specular reﬂecting surfaces with a precision of < 4% for attenuation depths spanning 0.1 to 2 mm. © 2003 Society of Photo-Optical Instrumentation Engineers. [DOI: 10.1117/1.1559731]


Introduction
Tissue optical properties play a central role in the application of light for medical diagnostics and therapeutics. For example, application of laser surgery and photodynamic therapy relies on the achievement of specific spatial distributions of laser energy within the tissue, that is in turn governed by tissue optical properties. Measurement of optical properties also yields diagnostic information regarding the metabolic, physiologic, and structural state of tissue. In many applications, an effective optical attenuation depth is used to characterize light dosimetry within tissues. The effective attenuation depth is the reciprocal of the optical absorption coefficient for nonscattering media, and is a function of the absorption coefficient, scattering coefficient, and single scattering anisotropy for turbid media. 1,2 Several optical methods are available to measure the effective attenuation depth of light within tissue. These methods typically employ optical transmittance and/or reflectance measurements in conjunction with a radiative transport model to estimate optical properties. These techniques can be performed in spatial, temporal, or frequency domains. When using spatially resolved methods, light of constant intensity is launched into tissue and detected at several spatial locations. [3][4][5] When using time-domain or time-resolved methods, a pulse of light is launched into tissue, and changes in the pulse shape resulting from its propagation are used to extract optical absorption and scattering properties. In this case, a single source-detector pair is sufficient to characterize the optical properties. [6][7][8] The third approach uses high-frequency intensity-modulated light to create diffuse photon density waves ͑PDW͒ that propagate in tissue with a wave vector that is a function of optical properties. [9][10][11][12][13][14] Optoacoustic techniques can also be employed to determine optical properties of turbid media. [15][16][17] Optoacoustic techniques typically measure the thermoelastic stresses generated by the absorption of a short laser pulse. These stresses are a function of the spatial distribution of absorbed energy and are therefore dependent on the optical properties of the sample. Current optoacoustic techniques employ transducers placed in contact with the target to measure these stresses. In many cases, the measurement must be made on the same surface as the laser exposure, because access to both sides of the tissue target is not possible. Thus transducers that are placed on the front surface of the target can be impractical, because they block the incident laser light. A variety of techniques have been used to overcome this difficulty, including the use of piezoelectric transducers with separated light and sound fields, transparent transducers that measure pressure-induced changes in optical reflectance, and annular piezoelectric elements and acoustic conductors where the stresses generated by an obliquely incident laser pulse propagates predominantly normal to the tissue surface. 15,16,18 -22 Despite these advances, stress detection using sensors placed in contact with the tissue may be impractical in some medical applications and may not be compatible with minimally invasive procedures.
In this work, a minimally invasive noncontact interferometric technique is used to accurately measure the effective optical attenuation depth of a sample. This technique measures the time-resolved surface displacement resulting from absorption of a short laser pulse. The surface motion is caused by thermoelastic stress relaxation, whose time constant is proportional to the optical attenuation depth of the sample. The magnitude of expansion is a function of the incident radiant exposure and thermophysical properties of the sample. This technique is complementary to interferometric photothermal spectroscopy, a method that determines the effective optical attenuation depth through time-resolved measurement of surface displacement resulting from the thermal diffusion that follows the thermoelastic stress relaxation process. 23,24 The use of an interferometric system to calculate optical attenuation depth by measuring surface motion has been described previously [25][26][27][28] and used to make preliminary attenuation depth measurements of human meniscal cartilage. 29 2 Interferometer System 2. 1 Interferometer Design The interferometer system is shown in Fig. 1 and represents a refined version of a system described previously. 24  cel any inherent laser noise. The subtracted photodiode current is amplified by a commercial 500-MHz broadband lownoise amplifier ͑Mini-Circuits, Branson, Missouri͒ and digitized by a 500-MHz bandwidth oscilloscope ͑Tektronix, Wilsonville, Oregon͒. When the sample surface is stationary, the photodiodes detect interference fringes that oscillate at the modulation frequency of 110 MHz. When the sample surface moves, the path length difference between the two arms changes and the phase of the modulation frequency is altered. The polarity of the phase shift is dependent on the direction of sample movement. The intensity reaching the photodiodes is a function of time t, path length difference ␦, and the sample and reference beam intensities, and is given by: where I 1 and I 2 are the intensities of the sample beam and reference beam. m is the phase of the modulated signal: where k is the propagation vector and ⌬w is the modulation frequency. The surface displacement S(t) is equal to half the path length difference and given by: The surface displacement is computed by subtracting the phase of the AOM drive signal from the phase of the detected signal. The phase of both signals is obtained using the Hilbert transform.

Interferometer Performance
The interferometer shown here was inspired by an original design presented by Albagli coworkers. 27 In this original design, a moving mirror placed in the reference arm was used to accomplish the frequency shift in the reference beam, a single photomultiplier tube was used for detection, and no polarization-sensitive optics were used. The standard deviation of the baseline traces obtained from this interferometer design was approximately 4 nm. A few years later, Yablon and coworkers demonstrated a modified interferometer design that reduced the baseline standard deviation by a factor of four. Their instrument employment an AOM instead of a moving mirror to accomplish the frequency shift in the reference arm, polarization-sensitive optics to improve the strength of the optical signal, and an avalanche photodiode for detection. 24 The performance of the present interferometer is best illustrated by comparing a surface displacement trace with one obtained using this earlier interferometer that did not employ dual PIN diode detection. 24 Surface displacement traces of colored glass after Q-switched Nd:YAG laser irradiation at 355 nm are shown in Fig. 2. Both traces were obtained under similar conditions. The axial resolution as defined by the standard deviation ͑STD͒ of the displacement signal has been improved by an order of magnitude from 0.93 to 0.09 nm. The maximum displacement in both cases was approximately 30 nm.
In general, there is a tradeoff between the temporal and spatial resolution provided by the interferometric system. Although subnanosecond temporal resolution is possible, we ap-plied an 80-MHz bandpass filter centered at 110 MHz to all displacement traces to eliminate high-frequency noise and limit our signal to record velocities that are physically possible. This resulted in displacement traces with a 4-ns temporal resolution and STD that varied between 0.08 nm for targets with specularly reflecting surfaces to 0.16 nm for diffusely reflecting targets.

Theoretical Modeling
The one-dimensional time-resolved surface expansion resulting from the absorption of an instantaneous laser pulse is derived in Sec. 8 and given by: where ␤ is the coefficient of thermal expansion, K is the bulk modulus, 0 is the relative surface temperature, D is the effective optical attenuation depth, is the density, c L is the longitudinal sound speed, is the Poisson's ratio, 0 is the incident radiant exposure, R s is the specular reflectance, and c v is the specific heat capacity at constant volume. When the pump laser pulse has a finite duration, the impulse solution must be convolved with the temporal profile of the incident radiation. The temporal profile of the laser pulse used in all experiments resembled a Gaussian distribution. The one-dimensional surface displacement caused by the absorption of such a laser pulse is derived in Sec. 9 and given by: where FWHM is the full width at half maximum of the laser temporal profile and dϵ(c L /D).

Measurement of Colored Glass
To validate this technique, a set of well-characterized absorbing neutral density glass samples ͑Schott Glass Technologies Incorporated, Duryea, Pennsylvania͒ were measured. Optical and thermal properties were obtained from the manufacturer and are listed in Table 1. A Q-switched Nd:YAG laser was frequency doubled to operate at 532 nm and used as the pump laser. The laser pulse possessed a Gaussian temporal profile with a FWHM duration of 7 ns. The spatial profile, analyzed using a CCD camera ͑Cohu 4800, Cohu Incorporated͒, resembled a flat top with a FWHM diameter of 6 mm. The angle of incidence of the pump laser at the sample was 15 deg. The refractive index of glass is 1.5 and results in the propagation of the pump laser pulse within the glass at an angle of 10 deg. The measured attenuation depth was corrected by dividing the measured value by cos͑10 deg͒. A diagram of the experimental setup is shown in Figure 3. The interferometer probe beam was centered within the incident pump beam on the target surface. The incident radiant exposures used were 10 to 550 mJ/cm 2 .

Measurement of Gelatin Phantoms
To demonstrate the use of this technique on diffusely reflecting targets, the surface displacement of gelatin phantoms doped with potassium chromate (K 2 CrO 4 , Aldrich Chemical Company, Milwaukee, Wisconsin͒ was measured. Potassium chromate was used to increase optical absorption of the phan-  tom. This was done to provide a geometry wherein the effective optical attenuation depth of the pump laser radiation was much smaller than the diameter of the pump laser beam, and to ensure validity of the one-dimensional modeling treatment given in §3. Potassium chromate was selected, as its absorption properties are known to be photochemically stable. 16,21 The experimental setup was similar to the setup used in the colored glass experiments described in § 4.1. The Q-switched Nd:YAG pump laser was operated at its third harmonic with a wavelength of 354.7 nm. The spatial profile resembled a top hat with FWHM diameter of 6 mm. The temporal profile was Gaussian in shape with FWHM duration of 8 ns. When measuring surface displacements on diffusely reflecting samples, a smaller interferometer spot size is required to ensure that sufficient coherent light is reflected. The incident radiant exposure was 70 mJ/cm 2 .
Three different solutions of potassium chromate were prepared with molar concentrations of 0.0188, 0.0169, and 0.0154 M, yielding estimated attenuation depths of 95, 106, and 117 m, respectively. All solutions were prepared with distilled water and filtered. Tissue phantoms were prepared by mixing 10% gelatin ͑G-1890 Type A gelatin, Sigma, Saint Louis, Missouri͒ by weight with a 0.0169-M potassium chromate solution.

Measurement of Colored Glass
A sample displacement trace for a glass sample with the bestfit solution is shown in Fig. 4. A single pulse from the Nd:YAG pump laser was absorbed by the glass sample beginning at 0 ns, generating thermoelastic stresses within the glass sample. These stresses propagated through the glass, reflected off the free surface, and produced a surface displacement that is a function of effective optical attenuation depth D and longitudinal sound speed c L of the target. The glass sample was stationary prior to the delivery of the laser pulse. At times greater than 300 ns, shear waves influenced the surface profile, making a one-dimensional treatment of the thermoelastic stress dynamics invalid. The maximum surface displacement was 45 nm. The best-fit trace was obtained by performing a Gauss-Newton least-squared optimization of Eq. ͑6͒, with the maximum displacement and the effective attenuation depth D as floating parameters. The best-fit trace corresponds to an optical attenuation depth of 101 m and is in good agreement with the known properties of the target. The measurements were repeated for incident radiant exposures ranging from 12 to 440 mJ/cm 2 . The displacement traces and best-fit curves are shown in Fig. 5. Figure 6 shows surface displacement traces with best-fit solutions for all Schott glass samples tested. The incident radiant exposure was 550 mJ/cm 2 . For ease of comparison, the maximum displacements have been normalized. Table 2 summarizes the measured optical penetration depths ͑meanϮone standard de-viation͒ for all glass samples, compared to values computed from transmission curves given by the manufacturer. To evaluate the reproducibility of the results, the attenuation depth of the NG 9 glass was determined from 350 separate surface displacement traces. The traces were obtained over a Fig. 3 Schematic of experimental setup to determine optical attenuation depth using interferometric detection. Fig. 4 Sample displacement trace and best-fit solution for Schott glass type NG1. The symbols represent the interferometer data taken at a sampling rate of 2 GS/s, of which only 1 of every 6 points is shown for clarity.

Fig. 5
Sample displacement traces and best-fit solutions for Schott glass type NG1. The incident radiant exposure ranged from 12 to 440 mJ/cm 2 . The symbols represent the interferometer data taken at a sampling rate of 2 GS/s, of which only 1 of every 6 points is shown for clarity.
period of two months under different operating and alignment conditions. The resulting values for optical attenuation depth varied by no more than 0.6%.

Measurement of Gelatin Phantoms
A sample displacement trace and best-fit solution for a 0.0169-M potassium chromate solution is shown in Fig. 7. The best-fit solution matched the measured displacement trace and had an attenuation depth of 91 m compared with the estimated value of 106 m. A sample displacement trace and best-fit solution for a gelatin phantom doped with a 0.0169-M potassium chromate solution is shown in Fig. 8. The best-fit solution matched the measured displacement trace with an effective attenuation depth of 136 m. This attenuation depth was higher than the attenuation depth of the 0.0169-M potassium chromate solution alone ͑91 m͒. The effective attenuation depth of gelatindoped tissue phantoms appeared to decrease with repeated laser pulses. Although this decrease was small, it was clearly detectable. Figure 9 compares the calculated effective attenuation depth for a 0.0169-M potassium chromate solution with a gelatin phantom prepared with the same solution.

Instrumentation
We have designed and built a high-resolution interferometric system capable of measuring surface deformation with angstrom displacement and subnanosecond temporal resolution. The displacement resolution of this system was 10 to 50 times better than those obtained by previous investigators. 24 -29 The current system could be improved further by increasing the power of the He-Ne probe laser. This would provide shot noise limited detection for targets that provide low-reflectance or possess diffusely reflecting surfaces. For highly reflecting targets, noise calculations indicated that the system was already limited by shot noise. Note that the system could be configured using fiber optic components. This would simplify alignment, increase modulation depth, and enable the system to be used in clinical applications, including minimally invasive catheter-based procedures.

Interferometric Determination of Optical Properties
The accuracy of the proposed interferometric technique was assessed via measurement of the effective attenuation depth of well-characterized absorbing glass samples. The variation in the measured values was generally more than an order of magnitude lower than the range given by the glass manufacturer. In addition, the technique provided consistent results when performed under different alignment and laser operating conditions. The consistently accurate results are primarily due to the high axial and temporal resolution of the interferometer system. The effective attenuation depth is determined by fitting the time-resolved surface displacement measurement to that predicted by a one-dimensional treatment of the thermoelastic wave equation. For purely absorbing samples, this simplified treatment is sufficient so long as the pump laser beam diameter is much larger than the optical penetration depth. This is an attractive feature that eliminates the need for a full three-dimensional treatment of thermoelastic wave propagation to extract the attenuation depth of weakly absorbing samples.
Determination of the effective attenuation depth in optically turbid samples can also be achieved using this approach. In this case, one must employ a pump laser beam whose diameter is much larger than the optical transport mean free path of the target. If this additional condition is not satisfied, the light distribution within the sample will have a threedimensional character and will result in an underestimation of the effective attenuation depth. In the future, it may be possible to extract both absorption a and reduced scattering s Ј coefficients or accommodate spatially heterogeneous media by incorporating a different formulation for the spatial distribution of the absorbed radiation or by explicit consideration of the inverse problem in the form of an integral equation. 30 In situations where the optical properties of the sample are known, the maximum surface displacement could be used to extract other properties of the target, such as the thermal expansion coefficient, Poisson ratio, density, and/or specific heat capacity.
This technique was also used to investigate laser-induced surface deformation of gelatin tissue phantoms. The best displacement resolution achieved on these diffusely reflecting surfaces was 0.16 nm. The slight deterioration in displacement resolution was a consequence of the reduced amount of light reflected from the gelatin surfaces in comparison to the glass samples. The effective optical attenuation depth was found to decrease with the number of additional laser pulses. The attenuation depth and its rate of decrease appeared to be a function of gelatin type, potassium chromate concentration, and incident radiant exposure. Explanations for these effects may include photochemical processes, dehydration, or laserinduced changes in the optical properties of the phantom. Although these effects resulted in a small change in the attenuation depth of the gelatin phantom, they were clearly detectable. These effects may be important when preparing tissue phantoms for the testing and validation of new techniques or applications.
The use of radiant exposures as large as 550 mJ/cm 2 to obtain the measured displacement traces may attract attention, as these exposures are 5.5 times greater than those allowed by the ANSI standard. 31 However, these measurements were made in the colored glass samples, a material that expands approximately six times less than water or tissue for an identical radiant exposure. In fact, for the gelatin-based phantoms, the largest radiant exposure used was 70 mJ/cm 2 . These exposures resulted in maximum displacements of approximately 35 nm and were more than ample to obtain accurate determinations of the effective optical attenuation depth.
The challenge of diagnostic spectroscopy is to measure changes in optical properties, such as effective attenuation depth, and correlate these changes with specific changes in tissue state. Often, changes in the optical properties are subtle, and must therefore be measured with a high-resolution technique. The interferometric technique shown here provides high resolution and is capable of differentiating small changes in attenuation depth ͑ϳ5 m͒. In addition, the system could be fabricated using fiber optic compatible instrumentation and  used as a clinically valuable minimally invasive diagnostic tool.

Conclusions
We have demonstrated the use of an interferometric system to determine the effective optical attenuation depth of homogenous samples by measuring the optoacoustic response to the absorption of a short laser pulse. The system measures the time-resolved surface displacement that is a function of the thermal, mechanical, and optical properties of the target. We have successfully used this technique to measure effective optical attenuation depths ranging from 0.1 to 2 mm in both diffusely and specularly reflecting targets with high precision. We expect that the application of this technique will be extended to measure optical absorption and scattering coefficients in homogeneous and heterogeneous turbid media.

Appendix A
We derive the relationship between surface displacement and optical attenuation depth of a sample after absorption of a laser impulse. The following derivation is based on the linear theory of elasticity. 32,33 The stress tensor, the equation of motion, and the constitutive relation for the material govern the stress and strain relationship of any material. The effects of thermal diffusion were neglected because the time scale for stress relaxation is much more rapid compared to the time scale for thermal diffusion. The strain tensor ( i j ) is given by Eq. ͑8͒ and is the small strain approximation of the Lagrangian strain tensor ( i j ). The subscripts represent standard indicial notation.
where u i is the displacement vector. The equation of motion is derived from the conservation of momentum and given by: where i j is the stress tensor, b i is the body force vector, is the density, and V i is the acceleration vector. The constitutive relation for a linear isotropic material is given by: where is Poisson's ratio, E is the elastic modulus, ␤ is the coefficient of thermal expansion, is the relative temperature, and ␦ i j is the Kronecker delta. Assuming the laser energy is absorbed exponentially, the temperature distribution will be: where 0 is the relative surface temperature, and D is the effective optical attenuation depth, which for purely absorbing samples, is equal to the inverse of the absorption coefficient. Substituting Eqs. ͑11͒ and ͑10͒ into Eq. ͑9͒, we obtain the partial differential equation for displacement as a function of depth z and time t.
where c L is the longitudinal speed of sound and K is the bulk modulus given by: and The initial conditions and boundary conditions are: The general solution is given by: Solving for the constants of integration C 1 and C 2 by applying the boundary conditions of Eq. ͑15͒ yields the final solution in the Laplace domain: The final solution is transformed back into the time domain using the following relationships: The one-dimensional surface displacement is thus given by: At the surface (zϭ0), the solution simplifies to:

͑26͒
where 0 is the incident radiant exposure, R s is the specular reflectance, and c v is the specific heat at constant volume.

Appendix B
Here we derive the relationship between surface displacement and attenuation depth of a sample after absorption of a laser pulse of finite duration. To do so the impulse solution must be convolved with the temporal profile of the laser pulse: Ϫdx͔͒g͑xϪt ͒dx, ͑27͒ where dϵc L /D and g(x) is the temporal profile of the laser pulse. The temporal profile of the laser pulse used in this study resembled a Gaussian function and is given by: g͑x ͒ϭͱ a exp͑Ϫax 2 ͒, ͑28͒ where aϵ 2.8 FWHM 2 . ͑29͒ FWHM is the full width half max of the laser temporal profile. Solving Eq. ͑27͒ yields the final one-dimensional surface displacement caused by the finite laser pulse: