## Comparison of intensity-modulated continuous-wave lasers with a chirped modulation frequency to pulsed lasers for photoacoustic imaging applications |

Biomedical Optics Express, Vol. 1, Issue 4, pp. 1188-1195 (2010)

http://dx.doi.org/10.1364/BOE.1.001188

Acrobat PDF (1446 KB)

### Abstract

Using a Green’s function solution to the photoacoustic wave equation, we compare intensity-modulated continuous-wave (CW) lasers with a chirped modulation frequency to pulsed lasers for photoacoustic imaging applications. Assuming the same transducer is used in both cases, we show that the axial resolution is identical and is determined by the transducer and material properties of the object. We derive a simple formula relating the signal-to-noise ratios (SNRs) of the two imaging systems that only depends on the fluence of each pulse and the time-bandwidth product of the chirp pulse. We also compare the SNR of the two systems assuming the fluence is limited by the American National Standards Institute (ANSI) laser safety guidelines for skin. We find that the SNR is about 20 dB to 30 dB larger for pulsed laser systems for reasonable values of the parameters. However, CW diode lasers have the advantage of being compact and relatively inexpensive, which may outweigh the lower SNR in many applications.

© 2010 OSA

## 1. Introduction

1. C. Li and L. V. Wang, “Photoacoustic tomography and sensing in biomedicine,” Phys. Med. Biol. **54**(19), R59–R97 (2009). [CrossRef] [PubMed]

2. Y. Fan, A. Mandelis, G. Spirou, and I. A. Vitkin, “Development of a laser photothermoacoustic frequency-swept system for subsurface imaging: theory and experiment,” J. Acoust. Soc. Am. **116**(6), 3523–3533 (2004). [CrossRef] [PubMed]

## 2. Pulse compression using matched filters

*I*

_{0}is the average irradiance of the pulse (2

*I*

_{0}is the peak irradiance) with units W/m

^{2},

*b*is the frequency sweep rate with units s

^{−2}, and

*T*is the length of the pulse with units s. The fluence of the pulse is

*I*

_{0}

*T*with units J/m

^{2}. The angular frequency is given by d/d

*t*(

*ω*

_{0}

*t*+

*πbt*

^{2}) =

*ω*

_{0}+ 2

*πbt*, which gives the frequency,

*f*, as

*f*

_{0}−

*bT*/2 ≤

*f*≤

*f*

_{0}+

*bT*/2, the bandwidth as

*bT*, and the time-bandwidth product as

*bT*

^{2}. For example, if the chirp goes from 1 MHz to 5 MHz and

*T*is 1 ms, then

*f*

_{0}is 3 MHz and

*b*is 4×10

^{9}s

^{−2}.

*I*

_{0}and length

*T*) and a high-frequency term (a pulse with amplitude

*I*

_{0}cos(

*ω*

_{0}

*t*+

*πbt*

^{2}) and length

*T*). In most practical situations, the low-frequency term is outside of the bandwidth of the acoustic transducer since

*T*is on the order of 1 ms and the transducer has a bandpass frequency response with a center frequency in the MHz range. The Fourier transform of the high frequency term can be split up into positive and negative frequency components. By completing the square, the positive frequency components (using the unitary Fourier transform) are given by

*f*

_{0}with bandwidth

*bT*for large time-bandwidth products (

*bT*

^{2}> 30) [9]. The spectrum of a chirp signal is shown in Fig. 1(a) for

*f*

_{0}= 3 MHz and

*bT*= 4 MHz. Eq. (4) is then approximately given by

*πbTt*) = sin(

*πbTt*)/(

*πbTt*). This is shown in Fig. 1(b) for

*f*

_{0}= 3 MHz and

*bT*= 4 MHz. The filtered signal has undergone pulse compression by a factor 1/(

*bT*

^{2}) and the axial resolution of the photoacoustic imaging system (ignoring the frequency dependence of photoacoustic generation and the transducer bandwidth, which are discussed later) is approximately

*ν*/(

_{s}*bT*) where

*ν*is the speed of sound (~ 1500 m/s in tissue). The signal is compressed by delaying different frequency components by different times. Since the chirp pulse has an instantaneous frequency that depends on time, the pulse will increase in amplitude and decrease in duration if the delay is done in the same manner as the frequency sweep, which is exactly what the matched filter accomplishes.

_{s}## 3. Green’s function solution to the photoacoustic equation

*T*(

**r**,

*t*) is the temperature with units K,

*D*is the thermal diffusivity (~ 1×10

_{T}^{−7}m

^{2}/s for tissue),

*H*(

**r**,

*t*) is the heating function with units W/m

^{3},

*ρ*is the mass density (~ 1000 kg/m

^{3}for tissue), and

*C*is the specific heat at constant volume (~4000 J/(kg·K) for tissue). The first two terms in Eq. (10) go like

_{V}*ωT*and

*D*/

_{T}T*l*

^{2}, respectively, where

*l*is the characteristic length of the absorber, which means we can ignore the second term in most practical situations [10

10. R. A. Kruger, P. Liu, Y. R. Fang, and C. R. Appledorn, “Photoacoustic ultrasound (PAUS)--reconstruction tomography,” Med. Phys. **22**(10), 1605–1609 (1995). [CrossRef] [PubMed]

11. L. V. Wang, “Tutorial on photoacoustic microscopy and computed tomography,” IEEE J. Sel. Top. Quantum Electron. **14**(1), 171–179 (2008). [CrossRef]

*p*(

**r**,

*t*) is the pressure with units Pa,

*β*is the thermal coefficient of volume expansion (~ 2×10

^{−4}K

^{−1}for tissue), and

*κ*is the isothermal compressibility (~ 5×10

^{−10}Pa

^{−1}for tissue). We use

*κ*=

*C*/(

_{P}*ρC*

_{V}ν^{2}

*), where*

_{s}*C*is the specific heat at constant pressure (~ 4000 J/(kg·K) for tissue), Eq. (11), and Eq. (12) to get

_{P}*k*= −

*ω*/

*ν*. The Green’s function solution is

_{s}*r*≫

*r*

_{0}we have |

**r**−

**r**| ≈

_{0}*r*−

**r̂**·

**r**. We keep both terms for the phase but only the first term for the amplitude. Eq. (15) becomes

_{0}*x*= 0,

*y*= 0,

*z*≫

*a*). The heating function for a cubic absorber has the form

*χ*is the dimensionless efficiency of absorbed energy converted to heat,

*µ*is the absorption coefficient with units m

^{−1},

*z*axis,

*I*(

*t*) is the irradiance of the plane wave light source, and a is the length of one side of the absorber with units m. This is shown in Fig. 2. Using

**r̂**=

**ẑ**=

**ẑ**and

_{0}*k*= −

*ω*/

*ν*, Eq. (16) becomes

_{s}*p̃*(

*ω*) is shorthand for

*p̃*(

*x*= 0,

*y*=0,

*z*≫

*a*,

*ω*). Even though photoacoustic generation is less efficient at high frequencies [7

7. S. A. Telenkov and A. Mandelis, “Photothermoacoustic imaging of biological tissues: maximum depth characterization comparison of time and frequency-domain measurements,” J. Biomed. Opt. **14**(4), 044025 (2009). [CrossRef] [PubMed]

## 4. Far-field pressure for a pulse excitation

*F*

_{0}is the fluence per pulse and

*τ*is the pulse length with units s, we have

*p*

_{pulse}(

*t*) is shorthand for

*p*

_{pulse}(

*x*= 0,

*y*= 0,

*z*≫

*a*,

*t*),

*u*(

*t*) is the unit step function,

*t*

_{1}=

*t*+

*τ*/2 − (2

*z*−

*a*)/(2

*ν*),

_{s}*t*

_{2}=

*t*−

*τ*/2 − (2

*z*−

*a*)/(2

*ν*),

_{s}*t*

_{3}=

*t*+

*τ*/2 − (2

*z*+

*a*)/(2

*ν*), and

_{s}*t*

_{4}=

*t*−

*τ*/2 − (2

*z*+

*a*)/(2

*ν*). This is shown in Fig. 3 for

_{s}*χ*= 0.25,

*µ*= 1 cm

^{−1},

*a*= 100

*µ*m,

*F*

_{0}= 20 mJ/cm

^{2},

*z*= 3 cm, and

*τ*= 10 ns.

## 5. Comparison of SNRs

*T̃*(

*ω*), and a unity-gain matched filter,

*G̃*(

*ω*), in Eq. (18), we have

*S̃*(

*ω*) is the detected signal spectrum and

*T̃*(

*ω*). If the pulse is short enough,

*Ĩ*

_{pulse}(

*ω*) in Eq. (20) will be approximately constant with amplitude

*ω*

_{0}is the transducer center frequency, Δ

*f*is the transducer bandwidth, and we have used

*T̃*(

*ω*); however,

*G̃*

_{pulse}(

*ω*) is not matched exactly unless the transducer spectrum is a rect and

*Ã*(

*ω*) is constant over the transducer bandwidth.

*f*as in the short pulse case), in which case we have

*bT*= Δ

*f*. Using Eq. (8), where

*Ĩ*

_{fc}(

*ω*) =

*Ĩ*

_{chirp}(

*ω*)

*G̃*

_{chirp}(

*ω*), we can write Eq. (22) as

*G̃*

_{chirp}(

*ω*) is not matched exactly unless the transducer spectrum is a rect and

*Ã*(

*ω*) is constant over the transducer bandwidth. We see that for both the short pulse and the chirp signals, the axial resolution of the imaging system is the same and is determined by the transducer and

*Ã*(

*ω*), which is dependent on the material properties of the object. Note that for actual objects,

*Ã*(

*ω*) will be more complicated than what is given in Eq. (23). If we ignore

*Ã*(

*ω*), then the resolution is approximately

*ν*/Δ

_{s}*f*.

*Ñ*(

*ω*), is filtered by |

*G̃*(

*ω*)|. This means the filtered noise amplitudes are the same in both cases and are given by

## 6. SNR in the context of the ANSI limits

*τ*≤ 100 ns), as used in pulsed systems, are given by

*T*≤ 10 s), as used in chirped systems, are given by

^{2}, where

*t*is the total exposure time in seconds,

*N*is the total number of pulses (

*N*=

_{τ}*N*

_{pulse}=

*R*and

_{τ}t*N*=

_{T}*N*

_{chirp}=

*R*, where

_{T}t*R*is the short pulse repetition rate and

_{τ}*R*is the chirp pulse repetition rate), and

_{T}*C*is a wavelength correction factor defined in [12]. The maximum

_{A}*F*

_{0}for a short pulse is 20

*C*mJ/cm

_{A}^{2}when

*R*is set to the optimum rate given by 55/

_{τ}*t*

^{3/4}in units of Hz for

*t*≤ 10 s and 10 Hz for

*t* 10 s. The maximum

*I*

_{0}for a chirp pulse is 1100

*C*/

_{A}*T*

^{3/4}mW/cm

^{2}when

*R*is set to the optimum rate given by 1/(

_{T}*T*

^{1/4}

*t*

^{3/4}) in units of Hz for

*t*≤ 10 s and 10/(55

*T*

^{1/4}) in units of Hz for

*t* 10 s. The optimum rates are plotted in Fig. 4 for a short pulse and a 1 ms chirp pulse. With multiple pulses, we can perform averaging, which improves the SNR by √

*N*. Note that the SNR is also maximized by choosing the same repetition rates that optimize the fluence and irradiance. Including averaging and assuming the optimum repetition rates are used, Eq. (28) becomes

*T*in such a way as to create that dependence. In the following discussion, we assume white noise for ease of explanation. Since white noise depends on

*T*

^{1/4}. However, for applications where the fluence is not determined by the ANSI limits, the maximum repetition rate is 1/

*T*and so SNR

_{chirp}∝

*I*

_{0}√

*t*and does not depend on

*T*. In both cases, SNR

_{chirp}does not depend on the bandwidth. The

## 7. Conclusion

## Acknowledgments

## References and links

1. | C. Li and L. V. Wang, “Photoacoustic tomography and sensing in biomedicine,” Phys. Med. Biol. |

2. | Y. Fan, A. Mandelis, G. Spirou, and I. A. Vitkin, “Development of a laser photothermoacoustic frequency-swept system for subsurface imaging: theory and experiment,” J. Acoust. Soc. Am. |

3. | Y. Fan, A. Mandelis, G. Spirou, I. A. Vitkin, and W. M. Whelan, “Laser photothermoacoustic heterodyned lock-in depth profilometry in turbid tissue phantoms,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

4. | S. A. Telenkov and A. Mandelis, “Fourier-domain biophotoacoustic subsurface depth selective amplitude and phase imaging of turbid phantoms and biological tissue,” J. Biomed. Opt. |

5. | S. A. Telenkov and A. Mandelis, “Fourier-domain methodology for depth-selective photothermoacoustic imaging of tissue chromophores,” Eur. Phys. J. Spec. Top. |

6. | S. Telenkov, A. Mandelis, B. Lashkari, and M. Forcht, “Frequency-domain photothermoacoustics: Alternative imaging modality of biological tissues,” J. Appl. Phys. |

7. | S. A. Telenkov and A. Mandelis, “Photothermoacoustic imaging of biological tissues: maximum depth characterization comparison of time and frequency-domain measurements,” J. Biomed. Opt. |

8. | H. H. Barrett, and K. J. Myers, |

9. | C. E. Cook, and M. Bernfeld, |

10. | R. A. Kruger, P. Liu, Y. R. Fang, and C. R. Appledorn, “Photoacoustic ultrasound (PAUS)--reconstruction tomography,” Med. Phys. |

11. | L. V. Wang, “Tutorial on photoacoustic microscopy and computed tomography,” IEEE J. Sel. Top. Quantum Electron. |

12. | Laser Institute of America, |

**OCIS Codes**

(140.2020) Lasers and laser optics : Diode lasers

(170.3880) Medical optics and biotechnology : Medical and biological imaging

(170.5120) Medical optics and biotechnology : Photoacoustic imaging

**ToC Category:**

Photoacoustic Imaging and Spectroscopy

**History**

Original Manuscript: July 30, 2010

Revised Manuscript: September 16, 2010

Manuscript Accepted: October 15, 2010

Published: October 20, 2010

**Citation**

Adam Petschke and Patrick J. La Rivière, "Comparison of intensity-modulated continuous-wave lasers with a chirped modulation frequency to pulsed lasers for photoacoustic imaging applications," Biomed. Opt. Express **1**, 1188-1195 (2010)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-1-4-1188

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### References

- C. Li and L. V. Wang, “Photoacoustic tomography and sensing in biomedicine,” Phys. Med. Biol. 54(19), R59–R97 (2009). [CrossRef] [PubMed]
- Y. Fan, A. Mandelis, G. Spirou, and I. A. Vitkin, “Development of a laser photothermoacoustic frequency-swept system for subsurface imaging: theory and experiment,” J. Acoust. Soc. Am. 116(6), 3523–3533 (2004). [CrossRef] [PubMed]
- Y. Fan, A. Mandelis, G. Spirou, I. A. Vitkin, and W. M. Whelan, “Laser photothermoacoustic heterodyned lock-in depth profilometry in turbid tissue phantoms,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(5), 051908 (2005). [CrossRef] [PubMed]
- S. A. Telenkov and A. Mandelis, “Fourier-domain biophotoacoustic subsurface depth selective amplitude and phase imaging of turbid phantoms and biological tissue,” J. Biomed. Opt. 11(4), 044006 (2006). [CrossRef] [PubMed]
- S. A. Telenkov and A. Mandelis, “Fourier-domain methodology for depth-selective photothermoacoustic imaging of tissue chromophores,” Eur. Phys. J. Spec. Top. 153(1), 443–448 (2008). [CrossRef]
- S. Telenkov, A. Mandelis, B. Lashkari, and M. Forcht, “Frequency-domain photothermoacoustics: Alternative imaging modality of biological tissues,” J. Appl. Phys. 105(10), 102029 (2009). [CrossRef]
- S. A. Telenkov and A. Mandelis, “Photothermoacoustic imaging of biological tissues: maximum depth characterization comparison of time and frequency-domain measurements,” J. Biomed. Opt. 14(4), 044025 (2009). [CrossRef] [PubMed]
- H. H. Barrett, and K. J. Myers, Foundations of Image Science (Wiley, Hoboken, NJ, 2004).
- C. E. Cook, and M. Bernfeld, Radar Signals: An Introduction to Theory and Application (Academic, New York, NY, 1967).
- R. A. Kruger, P. Liu, Y. R. Fang, and C. R. Appledorn, “Photoacoustic ultrasound (PAUS)--reconstruction tomography,” Med. Phys. 22(10), 1605–1609 (1995). [CrossRef] [PubMed]
- L. V. Wang, “Tutorial on photoacoustic microscopy and computed tomography,” IEEE J. Sel. Top. Quantum Electron. 14(1), 171–179 (2008). [CrossRef]
- Laser Institute of America, American National Standard for Safe Use of Lasers ANSI Z136.1–2007 (American National Standards Institute, Orlando, FL, 2007).

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