Abstract

We present the optical design of a 9.6-mm diameter fiber-coupled probe for combined femtosecond laser microsurgery and nonlinear optical imaging. Towards enabling clinical use, we successfully reduced the dimensions of our earlier 18-mm microsurgery probe by half, while improving optical performance. We use analytical and computational models to optimize the miniaturized lens system for off-axis scanning aberrations. The optimization reveals that the optical system can be aberration-corrected using simple aspheric relay lenses to achieve diffraction-limited imaging resolution over a large field of view. Before moving forward with custom lenses, we have constructed the 9.6-mm probe using off-the-shelf spherical relay lenses and a 0.55 NA aspheric objective lens. In addition to reducing the diameter by nearly 50% and the total volume by 5 times, we also demonstrate improved lateral and axial resolutions of 1.27 µm and 13.5 µm, respectively, compared to 1.64 µm and 16.4 µm in our previous work. Using this probe, we can successfully image various tissue samples, such as rat tail tendon that required 2-3 × lower laser power than the current state-of-the-art. With further development, image-guided, femtosecond laser microsurgical probes such as this one can enable physicians to achieve the highest level of surgical precision anywhere inside the body.

©2011 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Design of a fiber-optic multiphoton microscopy handheld probe

Yuan Zhao, Mingyu Sheng, Lin Huang, and Shuo Tang
Biomed. Opt. Express 7(9) 3425-3437 (2016)

Miniaturized probe for femtosecond laser microsurgery and two-photon imaging

Christopher L. Hoy, Nicholas J. Durr, Pengyuan Chen, Wibool Piyawattanametha, Hyejun Ra, Olav Solgaard, and Adela Ben-Yakar
Opt. Express 16(13) 9996-10005 (2008)

In vivo imaging of unstained tissues using long gradient index lens multiphoton endoscopic systems

David M. Huland, Christopher M. Brown, Scott S. Howard, Dimitre G. Ouzounov, Ina Pavlova, Ke Wang, David R. Rivera, Watt W. Webb, and Chris Xu
Biomed. Opt. Express 3(5) 1077-1085 (2012)

More Recommended Articles

References

  • View by:
  • Article Order
  • |
  • Year
  • |
  • Author
  • |
  • Publication
  1. A. Vogel, J. Noack, G. Hüttman, and G. Paltauf, “Mechanisms of femtosecond laser nanosurgery of cells and tissues,” Appl. Phys. B 81(8), 1015–1047 (2005).
    [Crossref]
  2. H. K. Soong and J. B. Malta, “Femtosecond lasers in ophthalmology,” Am. J. Ophthalmol. 147(2), 189–197, e2 (2009).
    [Crossref]
  3. A. V. Rode, E. G. Gamaly, B. Luther-Davies, B. T. Taylor, J. Dawes, A. Chan, R. M. Lowe, and P. Hannaford, “Subpicosecond laser ablation of dental enamel,” J. Appl. Phys. 92(4), 2153–2158 (2002).
    [Crossref]
  4. M. H. Niemz, A. Kasenbacher, M. Strassl, A. Bäcker, A. Beyertt, D. Nickel, and A. Giesen, “Tooth ablation using a CPA-free thin disk femtosecond laser system,” Appl. Phys. B 79, 269–271 (2004).
    [Crossref]
  5. W. B. Armstrong, J. A. Neev, L. B. Da Silva, A. M. Rubenchik, and B. C. Stuart, “Ultrashort pulse laser ossicular ablation and stapedotomy in cadaveric bone,” Lasers Surg. Med. 30(3), 216–220 (2002).
    [Crossref] [PubMed]
  6. J. Ilgner, M. Wehner, J. Lorenzen, M. Bovi, and M. Westhofen, “Morphological effects of nanosecond- and femtosecond-pulsed laser ablation on human middle ear ossicles,” J. Biomed. Opt. 11(1), 014004 (2006).
    [Crossref] [PubMed]
  7. R. G. McCaughey, H. Sun, V. S. Rothholtz, T. Juhasz, and B. J. F. Wong, “Femtosecond laser ablation of the stapes,” J. Biomed. Opt. 14(2), 024040–024046 (2009).
    [Crossref] [PubMed]
  8. H. Wisweh, U. Merkel, A. K. Huller, K. Lurben, and H. Lubatschowski, “Optical coherence tomography monitoring of vocal fold femtosecond laser microsurgery ” in Therapeutic Laser Applications and Laser-Tissue Interaction III, A. Vogel, ed. (2007), p. 63207.
  9. C. L. Hoy, W. N. Everett, J. Kobler, and A. Ben-Yakar, “Toward endoscopic ultrafast laser microsurgery of vocal folds,” Proc. SPIE 7548, 754831 (2010).
  10. C. L. Hoy, N. J. Durr, P. Chen, W. Piyawattanametha, H. Ra, O. Solgaard, and A. Ben-Yakar, “Miniaturized probe for femtosecond laser microsurgery and two-photon imaging,” Opt. Express 16(13), 9996–10005 (2008).
    [Crossref] [PubMed]
  11. N. J. Durr, T. Larson, D. K. Smith, B. A. Korgel, K. Sokolov, and A. Ben-Yakar, “Two-photon luminescence imaging of cancer cells using molecularly targeted gold nanorods,” Nano Lett. 7(4), 941–945 (2007).
    [Crossref] [PubMed]
  12. H. Ürey, “Spot size, depth-of-focus, and diffraction ring intensity formulas for truncated Gaussian beams,” Appl. Opt. 43(3), 620–625 (2004).
    [Crossref] [PubMed]
  13. R. Le Harzic, M. Weinigel, I. Riemann, K. König, and B. Messerschmidt, “Nonlinear optical endoscope based on a compact two axes piezo scanner and a miniature objective lens,” Opt. Express 16(25), 20588–20596 (2008).
    [Crossref] [PubMed]
  14. K. Carlsson, “The influence of specimen refractive index, detector signal integration, and non-uniform scan speed on the imaging properties in confocal microscopy,” J. Microsc. 163, 167–178 (1991).
    [Crossref]
  15. T. D. Visser, J. L. Oud, and G. J. Brakenhoff, “Refractive index and axial distance measurements in 3-D microscopy,” Optik (Stuttg.) 90, 17–19 (1992).
  16. D. Lee, and O. Solgaard, “Two-axis gimbaled microscanner in double SOI layers actuated by self-aligned vertical electrostatic combdrive,” in Proceedings of the Solid-State Sensors, Actuators and Microsystems Workshop, Hilton Head Island, (Hilton Head Island, South Carolina, 2004), pp. 352–355.
  17. G. F. Marshall, Handbook of optical and laser scanning (Marcel Dekker, 2004).
  18. R. R. Shannon, The Art and Science of Optical Design (Cambridge University Press, 1997).
  19. E. Salomatina, B. Jiang, J. Novak, and A. N. Yaroslavsky, “Optical properties of normal and cancerous human skin in the visible and near-infrared spectral range,” J. Biomed. Opt. 11(6), 064026 (2006).
    [Crossref]
  20. R. Drezek, K. Sokolov, U. Utzinger, I. Boiko, A. Malpica, M. Follen, and R. Richards-Kortum, “Understanding the contributions of NADH and collagen to cervical tissue fluorescence spectra: modeling, measurements, and implications,” J. Biomed. Opt. 6(4), 385–396 (2001).
    [Crossref] [PubMed]
  21. C. L. Hoy, N. J. Durr, and A. Ben-Yakar, “Fast-updating and non-repeating Lissajous image reconstruction method for capturing increased dynamic information,” Appl. Opt.in press.
    [PubMed]
  22. W. Göbel, J. N. D. Kerr, A. Nimmerjahn, and F. Helmchen, “Miniaturized two-photon microscope based on a flexible coherent fiber bundle and a gradient-index lens objective,” Opt. Lett. 29(21), 2521–2523 (2004).
    [Crossref] [PubMed]
  23. W. Piyawattanametha, E. D. Cocker, R. P. J. Barretto, J. C. Jung, B. A. Flusberg, H. Ra, O. Solgaard, and M. J. Schnitzer, “A portable two-photon fluorescence microendoscope based on a two-dimensional scanning mirror,” in IEEE/LEOS International Conference on Optical MEMS and Nanophotonics, (Hualien, Taiwan, 2007).
  24. G. Liu, T. Xie, I. V. Tomov, J. Su, L. Yu, J. Zhang, B. J. Tromberg, and Z. Chen, “Rotational multiphoton endoscopy with a 1 microm fiber laser system,” Opt. Lett. 34(15), 2249–2251 (2009).
    [Crossref] [PubMed]
  25. Y. Wu, Y. Leng, J. Xi, and X. Li, “Scanning all-fiber-optic endomicroscopy system for 3D nonlinear optical imaging of biological tissues,” Opt. Express 17(10), 7907–7915 (2009).
    [Crossref] [PubMed]
  26. G. Liu, K. Kieu, F. W. Wise, and Z. Chen, “Multiphoton microscopy system with a compact fiber-based femtosecond-pulse laser and handheld probe,” J Biophotonics 4(1-2), 34–39 (2011).
    [Crossref]
  27. C. L. Hoy, N. Durr, P. Chen, D. K. Smith, T. Larson, W. Piyawattanametha, H. Ra, B. Korgel, K. Sokolov, O. Solgaard, and A. Ben-Yakar, “Two-Photon Luminescence Imaging Using a MEMS-Based Miniaturized Probe,” in Conference on Lasers and Electro-Optics (CLEO), (Optical Society of America, 2008), paper CThG5.
  28. L. Fu, A. Jain, C. Cranfield, H. Xie, and M. Gu, “Three-dimensional nonlinear optical endoscopy,” J. Biomed. Opt. 12(4), 040501 (2007).
    [Crossref] [PubMed]

2011 (1)

G. Liu, K. Kieu, F. W. Wise, and Z. Chen, “Multiphoton microscopy system with a compact fiber-based femtosecond-pulse laser and handheld probe,” J Biophotonics 4(1-2), 34–39 (2011).
[Crossref]

2010 (1)

C. L. Hoy, W. N. Everett, J. Kobler, and A. Ben-Yakar, “Toward endoscopic ultrafast laser microsurgery of vocal folds,” Proc. SPIE 7548, 754831 (2010).

2009 (4)

2008 (2)

2007 (2)

N. J. Durr, T. Larson, D. K. Smith, B. A. Korgel, K. Sokolov, and A. Ben-Yakar, “Two-photon luminescence imaging of cancer cells using molecularly targeted gold nanorods,” Nano Lett. 7(4), 941–945 (2007).
[Crossref] [PubMed]

L. Fu, A. Jain, C. Cranfield, H. Xie, and M. Gu, “Three-dimensional nonlinear optical endoscopy,” J. Biomed. Opt. 12(4), 040501 (2007).
[Crossref] [PubMed]

2006 (2)

J. Ilgner, M. Wehner, J. Lorenzen, M. Bovi, and M. Westhofen, “Morphological effects of nanosecond- and femtosecond-pulsed laser ablation on human middle ear ossicles,” J. Biomed. Opt. 11(1), 014004 (2006).
[Crossref] [PubMed]

E. Salomatina, B. Jiang, J. Novak, and A. N. Yaroslavsky, “Optical properties of normal and cancerous human skin in the visible and near-infrared spectral range,” J. Biomed. Opt. 11(6), 064026 (2006).
[Crossref]

2005 (1)

A. Vogel, J. Noack, G. Hüttman, and G. Paltauf, “Mechanisms of femtosecond laser nanosurgery of cells and tissues,” Appl. Phys. B 81(8), 1015–1047 (2005).
[Crossref]

2004 (3)

2002 (2)

W. B. Armstrong, J. A. Neev, L. B. Da Silva, A. M. Rubenchik, and B. C. Stuart, “Ultrashort pulse laser ossicular ablation and stapedotomy in cadaveric bone,” Lasers Surg. Med. 30(3), 216–220 (2002).
[Crossref] [PubMed]

A. V. Rode, E. G. Gamaly, B. Luther-Davies, B. T. Taylor, J. Dawes, A. Chan, R. M. Lowe, and P. Hannaford, “Subpicosecond laser ablation of dental enamel,” J. Appl. Phys. 92(4), 2153–2158 (2002).
[Crossref]

2001 (1)

R. Drezek, K. Sokolov, U. Utzinger, I. Boiko, A. Malpica, M. Follen, and R. Richards-Kortum, “Understanding the contributions of NADH and collagen to cervical tissue fluorescence spectra: modeling, measurements, and implications,” J. Biomed. Opt. 6(4), 385–396 (2001).
[Crossref] [PubMed]

1992 (1)

T. D. Visser, J. L. Oud, and G. J. Brakenhoff, “Refractive index and axial distance measurements in 3-D microscopy,” Optik (Stuttg.) 90, 17–19 (1992).

1991 (1)

K. Carlsson, “The influence of specimen refractive index, detector signal integration, and non-uniform scan speed on the imaging properties in confocal microscopy,” J. Microsc. 163, 167–178 (1991).
[Crossref]

Armstrong, W. B.

W. B. Armstrong, J. A. Neev, L. B. Da Silva, A. M. Rubenchik, and B. C. Stuart, “Ultrashort pulse laser ossicular ablation and stapedotomy in cadaveric bone,” Lasers Surg. Med. 30(3), 216–220 (2002).
[Crossref] [PubMed]

Bäcker, A.

M. H. Niemz, A. Kasenbacher, M. Strassl, A. Bäcker, A. Beyertt, D. Nickel, and A. Giesen, “Tooth ablation using a CPA-free thin disk femtosecond laser system,” Appl. Phys. B 79, 269–271 (2004).
[Crossref]

Ben-Yakar, A.

C. L. Hoy, W. N. Everett, J. Kobler, and A. Ben-Yakar, “Toward endoscopic ultrafast laser microsurgery of vocal folds,” Proc. SPIE 7548, 754831 (2010).

C. L. Hoy, N. J. Durr, P. Chen, W. Piyawattanametha, H. Ra, O. Solgaard, and A. Ben-Yakar, “Miniaturized probe for femtosecond laser microsurgery and two-photon imaging,” Opt. Express 16(13), 9996–10005 (2008).
[Crossref] [PubMed]

N. J. Durr, T. Larson, D. K. Smith, B. A. Korgel, K. Sokolov, and A. Ben-Yakar, “Two-photon luminescence imaging of cancer cells using molecularly targeted gold nanorods,” Nano Lett. 7(4), 941–945 (2007).
[Crossref] [PubMed]

C. L. Hoy, N. J. Durr, and A. Ben-Yakar, “Fast-updating and non-repeating Lissajous image reconstruction method for capturing increased dynamic information,” Appl. Opt.in press.
[PubMed]

Beyertt, A.

M. H. Niemz, A. Kasenbacher, M. Strassl, A. Bäcker, A. Beyertt, D. Nickel, and A. Giesen, “Tooth ablation using a CPA-free thin disk femtosecond laser system,” Appl. Phys. B 79, 269–271 (2004).
[Crossref]

Boiko, I.

R. Drezek, K. Sokolov, U. Utzinger, I. Boiko, A. Malpica, M. Follen, and R. Richards-Kortum, “Understanding the contributions of NADH and collagen to cervical tissue fluorescence spectra: modeling, measurements, and implications,” J. Biomed. Opt. 6(4), 385–396 (2001).
[Crossref] [PubMed]

Bovi, M.

J. Ilgner, M. Wehner, J. Lorenzen, M. Bovi, and M. Westhofen, “Morphological effects of nanosecond- and femtosecond-pulsed laser ablation on human middle ear ossicles,” J. Biomed. Opt. 11(1), 014004 (2006).
[Crossref] [PubMed]

Brakenhoff, G. J.

T. D. Visser, J. L. Oud, and G. J. Brakenhoff, “Refractive index and axial distance measurements in 3-D microscopy,” Optik (Stuttg.) 90, 17–19 (1992).

Carlsson, K.

K. Carlsson, “The influence of specimen refractive index, detector signal integration, and non-uniform scan speed on the imaging properties in confocal microscopy,” J. Microsc. 163, 167–178 (1991).
[Crossref]

Chan, A.

A. V. Rode, E. G. Gamaly, B. Luther-Davies, B. T. Taylor, J. Dawes, A. Chan, R. M. Lowe, and P. Hannaford, “Subpicosecond laser ablation of dental enamel,” J. Appl. Phys. 92(4), 2153–2158 (2002).
[Crossref]

Chen, P.

Chen, Z.

G. Liu, K. Kieu, F. W. Wise, and Z. Chen, “Multiphoton microscopy system with a compact fiber-based femtosecond-pulse laser and handheld probe,” J Biophotonics 4(1-2), 34–39 (2011).
[Crossref]

G. Liu, T. Xie, I. V. Tomov, J. Su, L. Yu, J. Zhang, B. J. Tromberg, and Z. Chen, “Rotational multiphoton endoscopy with a 1 microm fiber laser system,” Opt. Lett. 34(15), 2249–2251 (2009).
[Crossref] [PubMed]

Cranfield, C.

L. Fu, A. Jain, C. Cranfield, H. Xie, and M. Gu, “Three-dimensional nonlinear optical endoscopy,” J. Biomed. Opt. 12(4), 040501 (2007).
[Crossref] [PubMed]

Da Silva, L. B.

W. B. Armstrong, J. A. Neev, L. B. Da Silva, A. M. Rubenchik, and B. C. Stuart, “Ultrashort pulse laser ossicular ablation and stapedotomy in cadaveric bone,” Lasers Surg. Med. 30(3), 216–220 (2002).
[Crossref] [PubMed]

Dawes, J.

A. V. Rode, E. G. Gamaly, B. Luther-Davies, B. T. Taylor, J. Dawes, A. Chan, R. M. Lowe, and P. Hannaford, “Subpicosecond laser ablation of dental enamel,” J. Appl. Phys. 92(4), 2153–2158 (2002).
[Crossref]

Drezek, R.

R. Drezek, K. Sokolov, U. Utzinger, I. Boiko, A. Malpica, M. Follen, and R. Richards-Kortum, “Understanding the contributions of NADH and collagen to cervical tissue fluorescence spectra: modeling, measurements, and implications,” J. Biomed. Opt. 6(4), 385–396 (2001).
[Crossref] [PubMed]

Durr, N. J.

C. L. Hoy, N. J. Durr, P. Chen, W. Piyawattanametha, H. Ra, O. Solgaard, and A. Ben-Yakar, “Miniaturized probe for femtosecond laser microsurgery and two-photon imaging,” Opt. Express 16(13), 9996–10005 (2008).
[Crossref] [PubMed]

N. J. Durr, T. Larson, D. K. Smith, B. A. Korgel, K. Sokolov, and A. Ben-Yakar, “Two-photon luminescence imaging of cancer cells using molecularly targeted gold nanorods,” Nano Lett. 7(4), 941–945 (2007).
[Crossref] [PubMed]

C. L. Hoy, N. J. Durr, and A. Ben-Yakar, “Fast-updating and non-repeating Lissajous image reconstruction method for capturing increased dynamic information,” Appl. Opt.in press.
[PubMed]

Everett, W. N.

C. L. Hoy, W. N. Everett, J. Kobler, and A. Ben-Yakar, “Toward endoscopic ultrafast laser microsurgery of vocal folds,” Proc. SPIE 7548, 754831 (2010).

Follen, M.

R. Drezek, K. Sokolov, U. Utzinger, I. Boiko, A. Malpica, M. Follen, and R. Richards-Kortum, “Understanding the contributions of NADH and collagen to cervical tissue fluorescence spectra: modeling, measurements, and implications,” J. Biomed. Opt. 6(4), 385–396 (2001).
[Crossref] [PubMed]

Fu, L.

L. Fu, A. Jain, C. Cranfield, H. Xie, and M. Gu, “Three-dimensional nonlinear optical endoscopy,” J. Biomed. Opt. 12(4), 040501 (2007).
[Crossref] [PubMed]

Gamaly, E. G.

A. V. Rode, E. G. Gamaly, B. Luther-Davies, B. T. Taylor, J. Dawes, A. Chan, R. M. Lowe, and P. Hannaford, “Subpicosecond laser ablation of dental enamel,” J. Appl. Phys. 92(4), 2153–2158 (2002).
[Crossref]

Giesen, A.

M. H. Niemz, A. Kasenbacher, M. Strassl, A. Bäcker, A. Beyertt, D. Nickel, and A. Giesen, “Tooth ablation using a CPA-free thin disk femtosecond laser system,” Appl. Phys. B 79, 269–271 (2004).
[Crossref]

Göbel, W.

Gu, M.

L. Fu, A. Jain, C. Cranfield, H. Xie, and M. Gu, “Three-dimensional nonlinear optical endoscopy,” J. Biomed. Opt. 12(4), 040501 (2007).
[Crossref] [PubMed]

Hannaford, P.

A. V. Rode, E. G. Gamaly, B. Luther-Davies, B. T. Taylor, J. Dawes, A. Chan, R. M. Lowe, and P. Hannaford, “Subpicosecond laser ablation of dental enamel,” J. Appl. Phys. 92(4), 2153–2158 (2002).
[Crossref]

Helmchen, F.

Hoy, C. L.

C. L. Hoy, W. N. Everett, J. Kobler, and A. Ben-Yakar, “Toward endoscopic ultrafast laser microsurgery of vocal folds,” Proc. SPIE 7548, 754831 (2010).

C. L. Hoy, N. J. Durr, P. Chen, W. Piyawattanametha, H. Ra, O. Solgaard, and A. Ben-Yakar, “Miniaturized probe for femtosecond laser microsurgery and two-photon imaging,” Opt. Express 16(13), 9996–10005 (2008).
[Crossref] [PubMed]

C. L. Hoy, N. J. Durr, and A. Ben-Yakar, “Fast-updating and non-repeating Lissajous image reconstruction method for capturing increased dynamic information,” Appl. Opt.in press.
[PubMed]

Hüttman, G.

A. Vogel, J. Noack, G. Hüttman, and G. Paltauf, “Mechanisms of femtosecond laser nanosurgery of cells and tissues,” Appl. Phys. B 81(8), 1015–1047 (2005).
[Crossref]

Ilgner, J.

J. Ilgner, M. Wehner, J. Lorenzen, M. Bovi, and M. Westhofen, “Morphological effects of nanosecond- and femtosecond-pulsed laser ablation on human middle ear ossicles,” J. Biomed. Opt. 11(1), 014004 (2006).
[Crossref] [PubMed]

Jain, A.

L. Fu, A. Jain, C. Cranfield, H. Xie, and M. Gu, “Three-dimensional nonlinear optical endoscopy,” J. Biomed. Opt. 12(4), 040501 (2007).
[Crossref] [PubMed]

Jiang, B.

E. Salomatina, B. Jiang, J. Novak, and A. N. Yaroslavsky, “Optical properties of normal and cancerous human skin in the visible and near-infrared spectral range,” J. Biomed. Opt. 11(6), 064026 (2006).
[Crossref]

Juhasz, T.

R. G. McCaughey, H. Sun, V. S. Rothholtz, T. Juhasz, and B. J. F. Wong, “Femtosecond laser ablation of the stapes,” J. Biomed. Opt. 14(2), 024040–024046 (2009).
[Crossref] [PubMed]

Kasenbacher, A.

M. H. Niemz, A. Kasenbacher, M. Strassl, A. Bäcker, A. Beyertt, D. Nickel, and A. Giesen, “Tooth ablation using a CPA-free thin disk femtosecond laser system,” Appl. Phys. B 79, 269–271 (2004).
[Crossref]

Kerr, J. N. D.

Kieu, K.

G. Liu, K. Kieu, F. W. Wise, and Z. Chen, “Multiphoton microscopy system with a compact fiber-based femtosecond-pulse laser and handheld probe,” J Biophotonics 4(1-2), 34–39 (2011).
[Crossref]

Kobler, J.

C. L. Hoy, W. N. Everett, J. Kobler, and A. Ben-Yakar, “Toward endoscopic ultrafast laser microsurgery of vocal folds,” Proc. SPIE 7548, 754831 (2010).

König, K.

Korgel, B. A.

N. J. Durr, T. Larson, D. K. Smith, B. A. Korgel, K. Sokolov, and A. Ben-Yakar, “Two-photon luminescence imaging of cancer cells using molecularly targeted gold nanorods,” Nano Lett. 7(4), 941–945 (2007).
[Crossref] [PubMed]

Larson, T.

N. J. Durr, T. Larson, D. K. Smith, B. A. Korgel, K. Sokolov, and A. Ben-Yakar, “Two-photon luminescence imaging of cancer cells using molecularly targeted gold nanorods,” Nano Lett. 7(4), 941–945 (2007).
[Crossref] [PubMed]

Le Harzic, R.

Leng, Y.

Li, X.

Liu, G.

G. Liu, K. Kieu, F. W. Wise, and Z. Chen, “Multiphoton microscopy system with a compact fiber-based femtosecond-pulse laser and handheld probe,” J Biophotonics 4(1-2), 34–39 (2011).
[Crossref]

G. Liu, T. Xie, I. V. Tomov, J. Su, L. Yu, J. Zhang, B. J. Tromberg, and Z. Chen, “Rotational multiphoton endoscopy with a 1 microm fiber laser system,” Opt. Lett. 34(15), 2249–2251 (2009).
[Crossref] [PubMed]

Lorenzen, J.

J. Ilgner, M. Wehner, J. Lorenzen, M. Bovi, and M. Westhofen, “Morphological effects of nanosecond- and femtosecond-pulsed laser ablation on human middle ear ossicles,” J. Biomed. Opt. 11(1), 014004 (2006).
[Crossref] [PubMed]

Lowe, R. M.

A. V. Rode, E. G. Gamaly, B. Luther-Davies, B. T. Taylor, J. Dawes, A. Chan, R. M. Lowe, and P. Hannaford, “Subpicosecond laser ablation of dental enamel,” J. Appl. Phys. 92(4), 2153–2158 (2002).
[Crossref]

Luther-Davies, B.

A. V. Rode, E. G. Gamaly, B. Luther-Davies, B. T. Taylor, J. Dawes, A. Chan, R. M. Lowe, and P. Hannaford, “Subpicosecond laser ablation of dental enamel,” J. Appl. Phys. 92(4), 2153–2158 (2002).
[Crossref]

Malpica, A.

R. Drezek, K. Sokolov, U. Utzinger, I. Boiko, A. Malpica, M. Follen, and R. Richards-Kortum, “Understanding the contributions of NADH and collagen to cervical tissue fluorescence spectra: modeling, measurements, and implications,” J. Biomed. Opt. 6(4), 385–396 (2001).
[Crossref] [PubMed]

Malta, J. B.

H. K. Soong and J. B. Malta, “Femtosecond lasers in ophthalmology,” Am. J. Ophthalmol. 147(2), 189–197, e2 (2009).
[Crossref]

McCaughey, R. G.

R. G. McCaughey, H. Sun, V. S. Rothholtz, T. Juhasz, and B. J. F. Wong, “Femtosecond laser ablation of the stapes,” J. Biomed. Opt. 14(2), 024040–024046 (2009).
[Crossref] [PubMed]

Messerschmidt, B.

Neev, J. A.

W. B. Armstrong, J. A. Neev, L. B. Da Silva, A. M. Rubenchik, and B. C. Stuart, “Ultrashort pulse laser ossicular ablation and stapedotomy in cadaveric bone,” Lasers Surg. Med. 30(3), 216–220 (2002).
[Crossref] [PubMed]

Nickel, D.

M. H. Niemz, A. Kasenbacher, M. Strassl, A. Bäcker, A. Beyertt, D. Nickel, and A. Giesen, “Tooth ablation using a CPA-free thin disk femtosecond laser system,” Appl. Phys. B 79, 269–271 (2004).
[Crossref]

Niemz, M. H.

M. H. Niemz, A. Kasenbacher, M. Strassl, A. Bäcker, A. Beyertt, D. Nickel, and A. Giesen, “Tooth ablation using a CPA-free thin disk femtosecond laser system,” Appl. Phys. B 79, 269–271 (2004).
[Crossref]

Nimmerjahn, A.

Noack, J.

A. Vogel, J. Noack, G. Hüttman, and G. Paltauf, “Mechanisms of femtosecond laser nanosurgery of cells and tissues,” Appl. Phys. B 81(8), 1015–1047 (2005).
[Crossref]

Novak, J.

E. Salomatina, B. Jiang, J. Novak, and A. N. Yaroslavsky, “Optical properties of normal and cancerous human skin in the visible and near-infrared spectral range,” J. Biomed. Opt. 11(6), 064026 (2006).
[Crossref]

Oud, J. L.

T. D. Visser, J. L. Oud, and G. J. Brakenhoff, “Refractive index and axial distance measurements in 3-D microscopy,” Optik (Stuttg.) 90, 17–19 (1992).

Paltauf, G.

A. Vogel, J. Noack, G. Hüttman, and G. Paltauf, “Mechanisms of femtosecond laser nanosurgery of cells and tissues,” Appl. Phys. B 81(8), 1015–1047 (2005).
[Crossref]

Piyawattanametha, W.

Ra, H.

Richards-Kortum, R.

R. Drezek, K. Sokolov, U. Utzinger, I. Boiko, A. Malpica, M. Follen, and R. Richards-Kortum, “Understanding the contributions of NADH and collagen to cervical tissue fluorescence spectra: modeling, measurements, and implications,” J. Biomed. Opt. 6(4), 385–396 (2001).
[Crossref] [PubMed]

Riemann, I.

Rode, A. V.

A. V. Rode, E. G. Gamaly, B. Luther-Davies, B. T. Taylor, J. Dawes, A. Chan, R. M. Lowe, and P. Hannaford, “Subpicosecond laser ablation of dental enamel,” J. Appl. Phys. 92(4), 2153–2158 (2002).
[Crossref]

Rothholtz, V. S.

R. G. McCaughey, H. Sun, V. S. Rothholtz, T. Juhasz, and B. J. F. Wong, “Femtosecond laser ablation of the stapes,” J. Biomed. Opt. 14(2), 024040–024046 (2009).
[Crossref] [PubMed]

Rubenchik, A. M.

W. B. Armstrong, J. A. Neev, L. B. Da Silva, A. M. Rubenchik, and B. C. Stuart, “Ultrashort pulse laser ossicular ablation and stapedotomy in cadaveric bone,” Lasers Surg. Med. 30(3), 216–220 (2002).
[Crossref] [PubMed]

Salomatina, E.

E. Salomatina, B. Jiang, J. Novak, and A. N. Yaroslavsky, “Optical properties of normal and cancerous human skin in the visible and near-infrared spectral range,” J. Biomed. Opt. 11(6), 064026 (2006).
[Crossref]

Smith, D. K.

N. J. Durr, T. Larson, D. K. Smith, B. A. Korgel, K. Sokolov, and A. Ben-Yakar, “Two-photon luminescence imaging of cancer cells using molecularly targeted gold nanorods,” Nano Lett. 7(4), 941–945 (2007).
[Crossref] [PubMed]

Sokolov, K.

N. J. Durr, T. Larson, D. K. Smith, B. A. Korgel, K. Sokolov, and A. Ben-Yakar, “Two-photon luminescence imaging of cancer cells using molecularly targeted gold nanorods,” Nano Lett. 7(4), 941–945 (2007).
[Crossref] [PubMed]

R. Drezek, K. Sokolov, U. Utzinger, I. Boiko, A. Malpica, M. Follen, and R. Richards-Kortum, “Understanding the contributions of NADH and collagen to cervical tissue fluorescence spectra: modeling, measurements, and implications,” J. Biomed. Opt. 6(4), 385–396 (2001).
[Crossref] [PubMed]

Solgaard, O.

Soong, H. K.

H. K. Soong and J. B. Malta, “Femtosecond lasers in ophthalmology,” Am. J. Ophthalmol. 147(2), 189–197, e2 (2009).
[Crossref]

Strassl, M.

M. H. Niemz, A. Kasenbacher, M. Strassl, A. Bäcker, A. Beyertt, D. Nickel, and A. Giesen, “Tooth ablation using a CPA-free thin disk femtosecond laser system,” Appl. Phys. B 79, 269–271 (2004).
[Crossref]

Stuart, B. C.

W. B. Armstrong, J. A. Neev, L. B. Da Silva, A. M. Rubenchik, and B. C. Stuart, “Ultrashort pulse laser ossicular ablation and stapedotomy in cadaveric bone,” Lasers Surg. Med. 30(3), 216–220 (2002).
[Crossref] [PubMed]

Su, J.

Sun, H.

R. G. McCaughey, H. Sun, V. S. Rothholtz, T. Juhasz, and B. J. F. Wong, “Femtosecond laser ablation of the stapes,” J. Biomed. Opt. 14(2), 024040–024046 (2009).
[Crossref] [PubMed]

Taylor, B. T.

A. V. Rode, E. G. Gamaly, B. Luther-Davies, B. T. Taylor, J. Dawes, A. Chan, R. M. Lowe, and P. Hannaford, “Subpicosecond laser ablation of dental enamel,” J. Appl. Phys. 92(4), 2153–2158 (2002).
[Crossref]

Tomov, I. V.

Tromberg, B. J.

Ürey, H.

Utzinger, U.

R. Drezek, K. Sokolov, U. Utzinger, I. Boiko, A. Malpica, M. Follen, and R. Richards-Kortum, “Understanding the contributions of NADH and collagen to cervical tissue fluorescence spectra: modeling, measurements, and implications,” J. Biomed. Opt. 6(4), 385–396 (2001).
[Crossref] [PubMed]

Visser, T. D.

T. D. Visser, J. L. Oud, and G. J. Brakenhoff, “Refractive index and axial distance measurements in 3-D microscopy,” Optik (Stuttg.) 90, 17–19 (1992).

Vogel, A.

A. Vogel, J. Noack, G. Hüttman, and G. Paltauf, “Mechanisms of femtosecond laser nanosurgery of cells and tissues,” Appl. Phys. B 81(8), 1015–1047 (2005).
[Crossref]

Wehner, M.

J. Ilgner, M. Wehner, J. Lorenzen, M. Bovi, and M. Westhofen, “Morphological effects of nanosecond- and femtosecond-pulsed laser ablation on human middle ear ossicles,” J. Biomed. Opt. 11(1), 014004 (2006).
[Crossref] [PubMed]

Weinigel, M.

Westhofen, M.

J. Ilgner, M. Wehner, J. Lorenzen, M. Bovi, and M. Westhofen, “Morphological effects of nanosecond- and femtosecond-pulsed laser ablation on human middle ear ossicles,” J. Biomed. Opt. 11(1), 014004 (2006).
[Crossref] [PubMed]

Wise, F. W.

G. Liu, K. Kieu, F. W. Wise, and Z. Chen, “Multiphoton microscopy system with a compact fiber-based femtosecond-pulse laser and handheld probe,” J Biophotonics 4(1-2), 34–39 (2011).
[Crossref]

Wong, B. J. F.

R. G. McCaughey, H. Sun, V. S. Rothholtz, T. Juhasz, and B. J. F. Wong, “Femtosecond laser ablation of the stapes,” J. Biomed. Opt. 14(2), 024040–024046 (2009).
[Crossref] [PubMed]

Wu, Y.

Xi, J.

Xie, H.

L. Fu, A. Jain, C. Cranfield, H. Xie, and M. Gu, “Three-dimensional nonlinear optical endoscopy,” J. Biomed. Opt. 12(4), 040501 (2007).
[Crossref] [PubMed]

Xie, T.

Yaroslavsky, A. N.

E. Salomatina, B. Jiang, J. Novak, and A. N. Yaroslavsky, “Optical properties of normal and cancerous human skin in the visible and near-infrared spectral range,” J. Biomed. Opt. 11(6), 064026 (2006).
[Crossref]

Yu, L.

Zhang, J.

Am. J. Ophthalmol. (1)

H. K. Soong and J. B. Malta, “Femtosecond lasers in ophthalmology,” Am. J. Ophthalmol. 147(2), 189–197, e2 (2009).
[Crossref]

Appl. Opt. (2)

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

C. L. Hoy, N. J. Durr, and A. Ben-Yakar, “Fast-updating and non-repeating Lissajous image reconstruction method for capturing increased dynamic information,” Appl. Opt.in press.
[PubMed]

Appl. Phys. B (2)

M. H. Niemz, A. Kasenbacher, M. Strassl, A. Bäcker, A. Beyertt, D. Nickel, and A. Giesen, “Tooth ablation using a CPA-free thin disk femtosecond laser system,” Appl. Phys. B 79, 269–271 (2004).
[Crossref]

A. Vogel, J. Noack, G. Hüttman, and G. Paltauf, “Mechanisms of femtosecond laser nanosurgery of cells and tissues,” Appl. Phys. B 81(8), 1015–1047 (2005).
[Crossref]

J Biophotonics (1)

G. Liu, K. Kieu, F. W. Wise, and Z. Chen, “Multiphoton microscopy system with a compact fiber-based femtosecond-pulse laser and handheld probe,” J Biophotonics 4(1-2), 34–39 (2011).
[Crossref]

J. Appl. Phys. (1)

A. V. Rode, E. G. Gamaly, B. Luther-Davies, B. T. Taylor, J. Dawes, A. Chan, R. M. Lowe, and P. Hannaford, “Subpicosecond laser ablation of dental enamel,” J. Appl. Phys. 92(4), 2153–2158 (2002).
[Crossref]

J. Biomed. Opt. (5)

J. Ilgner, M. Wehner, J. Lorenzen, M. Bovi, and M. Westhofen, “Morphological effects of nanosecond- and femtosecond-pulsed laser ablation on human middle ear ossicles,” J. Biomed. Opt. 11(1), 014004 (2006).
[Crossref] [PubMed]

R. G. McCaughey, H. Sun, V. S. Rothholtz, T. Juhasz, and B. J. F. Wong, “Femtosecond laser ablation of the stapes,” J. Biomed. Opt. 14(2), 024040–024046 (2009).
[Crossref] [PubMed]

E. Salomatina, B. Jiang, J. Novak, and A. N. Yaroslavsky, “Optical properties of normal and cancerous human skin in the visible and near-infrared spectral range,” J. Biomed. Opt. 11(6), 064026 (2006).
[Crossref]

R. Drezek, K. Sokolov, U. Utzinger, I. Boiko, A. Malpica, M. Follen, and R. Richards-Kortum, “Understanding the contributions of NADH and collagen to cervical tissue fluorescence spectra: modeling, measurements, and implications,” J. Biomed. Opt. 6(4), 385–396 (2001).
[Crossref] [PubMed]

L. Fu, A. Jain, C. Cranfield, H. Xie, and M. Gu, “Three-dimensional nonlinear optical endoscopy,” J. Biomed. Opt. 12(4), 040501 (2007).
[Crossref] [PubMed]

J. Microsc. (1)

K. Carlsson, “The influence of specimen refractive index, detector signal integration, and non-uniform scan speed on the imaging properties in confocal microscopy,” J. Microsc. 163, 167–178 (1991).
[Crossref]

Lasers Surg. Med. (1)

W. B. Armstrong, J. A. Neev, L. B. Da Silva, A. M. Rubenchik, and B. C. Stuart, “Ultrashort pulse laser ossicular ablation and stapedotomy in cadaveric bone,” Lasers Surg. Med. 30(3), 216–220 (2002).
[Crossref] [PubMed]

Nano Lett. (1)

N. J. Durr, T. Larson, D. K. Smith, B. A. Korgel, K. Sokolov, and A. Ben-Yakar, “Two-photon luminescence imaging of cancer cells using molecularly targeted gold nanorods,” Nano Lett. 7(4), 941–945 (2007).
[Crossref] [PubMed]

Opt. Express (3)

Opt. Lett. (2)

Optik (Stuttg.) (1)

T. D. Visser, J. L. Oud, and G. J. Brakenhoff, “Refractive index and axial distance measurements in 3-D microscopy,” Optik (Stuttg.) 90, 17–19 (1992).

Proc. SPIE (1)

C. L. Hoy, W. N. Everett, J. Kobler, and A. Ben-Yakar, “Toward endoscopic ultrafast laser microsurgery of vocal folds,” Proc. SPIE 7548, 754831 (2010).

Other (6)

H. Wisweh, U. Merkel, A. K. Huller, K. Lurben, and H. Lubatschowski, “Optical coherence tomography monitoring of vocal fold femtosecond laser microsurgery ” in Therapeutic Laser Applications and Laser-Tissue Interaction III, A. Vogel, ed. (2007), p. 63207.

D. Lee, and O. Solgaard, “Two-axis gimbaled microscanner in double SOI layers actuated by self-aligned vertical electrostatic combdrive,” in Proceedings of the Solid-State Sensors, Actuators and Microsystems Workshop, Hilton Head Island, (Hilton Head Island, South Carolina, 2004), pp. 352–355.

G. F. Marshall, Handbook of optical and laser scanning (Marcel Dekker, 2004).

R. R. Shannon, The Art and Science of Optical Design (Cambridge University Press, 1997).

W. Piyawattanametha, E. D. Cocker, R. P. J. Barretto, J. C. Jung, B. A. Flusberg, H. Ra, O. Solgaard, and M. J. Schnitzer, “A portable two-photon fluorescence microendoscope based on a two-dimensional scanning mirror,” in IEEE/LEOS International Conference on Optical MEMS and Nanophotonics, (Hualien, Taiwan, 2007).

C. L. Hoy, N. Durr, P. Chen, D. K. Smith, T. Larson, W. Piyawattanametha, H. Ra, B. Korgel, K. Sokolov, O. Solgaard, and A. Ben-Yakar, “Two-Photon Luminescence Imaging Using a MEMS-Based Miniaturized Probe,” in Conference on Lasers and Electro-Optics (CLEO), (Optical Society of America, 2008), paper CThG5.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Optical system of the new 9.6-mm probe. a) A photograph of the 9.6-mm probe housing (right) next to the housing of the 18-mm probe (left) showing the reduction in packaged probe size. A US penny is shown for scale. b) The components include: (1a) three meters of PBF; (1b) the fiber collimation assembly; (2) a two-axis MEMS scanning mirror; (3a) a pair of spherical relay lenses with 2.32 × magnification; (3b) a right angle TIR prism; (4) a 0.55 NA aspheric objective lens; (5a) a dichroic mirror; and (5b) one meter of 2 mm core plastic optical fiber. c) The packaged endoscope is shown transparently overlaid with the optical system.

Download Full Size | PPT Slide | PDF

Fig. 2
Fig. 2

Experimental setup for measuring the PSF of miniature objective lenses. The system consists of a home-built inverted two-photon microscope in which a miniature lens is used in place of a traditional objective lens. An optional beam path can deliver the laser pulses through one meter of PBF. Care was taken to maintain a beam diameter at the miniature objective lens back aperture of approximately 4 mm (1/e2) for both optical paths. λ/2: half wave plate; PBS: polarizing beam splitter; FM: flipping mirror; Obj.: 0.25-NA objective lens; PBF: one meter of photonic bandgap fiber; BE: beam expander; SM: galvanometric scanning mirrors; DM: dichroic mirror; PZT: piezoelectric translation stage; MOL: miniature objective lens; F: laser blocking filter; PMT: photomultiplier tube; I→V: current-to-voltage preamplifier; PS: scanning mirror power supply.

Download Full Size | PPT Slide | PDF

Fig. 3
Fig. 3

Lateral (a) and axial (b) resolution measurements for the commercially available miniature lenses summarized in Table 1. The data in blue were measured with the free-space propagated beam. The data in red were measured using a beam collimated by a 0.49 NA GRIN lens after propagation through one meter of PBF. The dashed bars denote the diffraction limit for the specified NA when focusing in media with index of refraction n = 1.34 [12]. Error bars correspond to the standard error of the mean (n = 12).

Download Full Size | PPT Slide | PDF

Fig. 4
Fig. 4

Calculated diffraction-limited lateral and axial resolutions versus FOV for the specific collimated beam size and the MEMS mirror used in the new probe and the miniature objective lens candidates given in Table 1. The plots correspond to the outer and inner axes of the MEMS mirrors, (a) and (b) respectively. The optical deflection angles and effective apertures of the outer and inner mirror axes are ± 15.3° and ± 7.1°, and 530 µm and 750 µm, respectively. The collimated beam size of 425 µm corresponds to the 0.5-mm GRIN collimation lens used in the probe. Data points represent ZEMAX simulation results, with solid markers indicating optimized custom lenses and the hollow circle indicating a system using commercially available spherical relay lenses (only the diffraction-limited portion of the maximum FOV shown). Magnifications for the models are (1) 1.75 × , (2) 2.38 × , (3) 1.98 × , (4) 1.82 × , and (5) 2.32 × .

Download Full Size | PPT Slide | PDF

Fig. 5
Fig. 5

Optical layout used in ZEMAX modeling of the 9.6-mm probe. The blue, green, and red lines represent the scanned laser beam (1) and correspond to field angles of 0°, 7.1°, and 15.3°, respectively. This model shows a custom-designed relay lens pair (2 & 3) with an effective relay lens magnification of 2.38 × , a right angle prism (4), dichroic mirror (5), and the 0.55 NA aspheric objective lens (6). The sample (7) is modeled as seawater, covered by a glass coverslip.

Download Full Size | PPT Slide | PDF

Fig. 6
Fig. 6

RMS wavefront error variation with scan angle for the ZEMAX-optimized relay lens pair with 2.38 × magnification (red line) and commercially-available spherical lens pair with 2.32 × magnification (blue line). The calculated PSFs at scan angles of 0°, 7.1°, and 15.3°, are shown above for the commercially-available lens model. The corresponding Strehl ratios are 0.99, 0.36, and 0.04, respectively. The calculated PSFs for the optimized lens pair case at all angles are indistinguishable from the diffraction-limited PSF shown at 0° for the commercial lens pair.

Download Full Size | PPT Slide | PDF

Fig. 7
Fig. 7

Addition of a 45° bevel to the collection fiber to improve collection of scattered light. a) The model utilizes (1) a scattering tissue block containing a point source as the sample, (2) the 0.55 NA aspheric objective lens with a long 0.88 mm working distance, (3) the dichroic mirror, and (4) the 2 mm plastic optical collection fiber. b) The addition of a 45° bevel (red arrow) enables the collection fiber to sit flush with the dichroic mirror, enhancing collection of scattered rays. c) The impact of the beveled fiber tip on the number of rays detected from the PMT for varying imaging depths, where depth has been normalized by scattering length, ls , of 29 µm.

Download Full Size | PPT Slide | PDF

Fig. 8
Fig. 8

Experimental setup for testing the 9.6-mm probe mounted above a piezoelectric sample holder. λ/2: half wave plate; PBS: polarizing beam splitter; Obj.: 0.25-NA objective lens; PBF: three meters of photonic bandgap fiber; PZT: piezoelectric translation stage; CF: one meter of plastic optical collection fiber; F: laser blocking filter; PMT: photomultiplier tube; I→V: current-to-voltage preamplifier; Amp: 200 V MEMS driving amplifier.

Download Full Size | PPT Slide | PDF

Fig. 9
Fig. 9

Example lateral (a) and axial (b) point spread function measurements for the 9.6-mm probe. The raw data from a representative bead is shown in black and the Gaussian fit to the data is provided in red while the diffraction-limited PSF is shown as a dashed gray line. The average FWHM of the lateral and axial PSFs are 1.27 µm and 13.5 µm, respectively. The inset images show x-y (a) and x-z (b) plane reconstructions of a single 500 nm bead used for PSF measurement, with 1 µm scale bar. The average power of the laser at the sample was 27 mW.

Download Full Size | PPT Slide | PDF

Fig. 10
Fig. 10

Representative two-photon fluorescence and second harmonic generation images taken with the 9.6-mm probe. a) A series of images taken at axial locations separated by 4.2 µm through a single pollen grain. The distances from the central focal plane are provided below the images. b) A high-magnification second harmonic generation image of rat tail tendon, showing highly aligned collagen fiber bundles. c) A maximum intensity projection of a ~70 µm thick image stack of freshly excised porcine vocal fold, stained with Hoechst 3342, showing nuclear detail. Average laser power at the sample is 9.8 mW, 20 mW, and 35 mW, for a) – c), respectively. Scale bars are 5 µm in a) and b), and 10 µm in c). All images are 14 frame (2 second) averages.

Download Full Size | PPT Slide | PDF

Tables (3)

Tables Icon

Table 1 Summary of Commercially Available Lenses Considered for Use as the Miniature Objective Lens

View Table | View all tables in this article
Tables Icon

Table 2 Summary of Modeled Optical System Performance Using Optimized Relay Lens Designs

View Table | View all tables in this article
Tables Icon

Table 3 Manufacturing Tolerances Sufficient for Maintaining Diffraction-limited Performance

View Table | View all tables in this article

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

Δ z f o c u s = tan ( sin − 1 ( N A n a i r ) ) tan ( sin − 1 ( N A n a g a r ) ) ⋅ Δ z s t a g e ,
FOV ≈ f θ o b j = f θ m i r r o r M ,
δ x y ≈ K 1 λ 2 2 NA ,
δ z ≈ K 2 λ n 2 2 NA 2 ,
K 1 = 1.036 − 0.058 T + 0.156 T 2 ,
K 2 = 3.5 + 0.33 T − 0.73 T 2 + 0.52 T 3 ,
NA ≈ D 2 f ,

Metrics