Abstract

First order distributed feedback optofluidic dye lasers embedded in a 350 µm thick TOPAS® foil are demonstrated. They are designed in order to give high output pulse energies. Microfluidic channels and first order distributed feedback gratings are fabricated in parallel by thermal nanoimprint into a 100 µm foil. The channels are closed by thermal bonding with a 250 µm thick foil and filled with 5·10−3 mol/l Pyrromethene 597 in benzyl alcohol. The fluid forms a liquid core single mode slab waveguide of 1.6 µm height on a nanostructured grating area of 0.5 × 0.5 mm2. This results in a large gain volume. Two grating periods of 185 nm and 190 nm yield single mode laser light emission at 566 nm and 581 nm respectively. High emitted pulse energies of more than 1 µJ are reported. Stable operation for more than 25 min at 10 Hz pulse repetition rate is achieved.

© 2010 OSA

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References

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  1. S. Balslev and A. Kristensen, “Microfluidic single-mode laser using high-order Bragg grating and antiguiding segments,” Opt. Express 13(1), 344–351 (2005).
    [CrossRef] [PubMed]
  2. D. Psaltis, S. R. Quake, and C. Yang, “Developing optofluidic technology through the fusion of microfluidics and optics,” Nature 442(7101), 381–386 (2006).
    [CrossRef] [PubMed]
  3. C. Monat, P. Domachuk, and B. J. Eggleton, “Integrated optofluidics: A new river of light,” Nat. Photonics 1(2), 106–114 (2007).
    [CrossRef]
  4. M. Gersborg-Hansen and A. Kristensen, “Optofluidic third order distributed feedback dye laser,” Appl. Phys. Lett. 89(10), 103518 (2006).
    [CrossRef]
  5. W. Song, A. E. Vasdekis, Z. Li, and D. Psaltis, “Optofluidic evanescent dye laser based on a distributed feedback circular grating,” Appl. Phys. Lett. 94(16), 161110 (2009).
    [CrossRef]
  6. W. Song, A. E. Vasdekis, Z. Li, and D. Psaltis, “Low-order distributed feedback optofluidic dye laser with reduced threshold,” Appl. Phys. Lett. 94(5), 051117 (2009).
    [CrossRef]
  7. Z. Li, Z. Zhang, T. Emery, A. Scherer, and D. Psaltis, “Single mode optofluidic distributed feedback dye laser,” Opt. Express 14(2), 696–701 (2006).
    [CrossRef] [PubMed]
  8. M. Gersborg-Hansen and A. Kristensen, “Tunability of optofluidic distributed feedback dye lasers,” Opt. Express 15(1), 137–142 (2007).
    [CrossRef] [PubMed]
  9. Z. Li, Z. Zhang, A. Scherer, and D. Psaltis, “Mechanically tunable optofluidic distributed feedback dye laser,” Opt. Express 14(22), 10494–10499 (2006).
    [CrossRef] [PubMed]
  10. W. Song and D. Psaltis, “Pneumatically tunable optofluidic dye laser,” Appl. Phys. Lett. 96(8), 081101 (2010).
    [CrossRef]
  11. TOPAS, 6013 COC, acquired from TOPAS Advanced Polymers, Inc., www.topas.com .
  12. B. Bilenberg, M. Hansen, D. Johansen, V. Özkapici, C. Jeppesen, P. Szabo, I. M. Obieta, O. Arroyo, J. O. Tegenfeldt, and A. Kristensen, “Topas based lab-on-a-chip microsystems fabricated by thermal nanoimprint lithography,” J. Vac. Sci. Technol. B 23(6), 2944–2949 (2005).
    [CrossRef]
  13. R. Francesconi, A. Bigi, and F. Comelli, “Enthalpies of Mixing, Densities, and Refractive Indices for Binary Mixtures of (Anisole or Phenetole) + Three Aryl Alcohols at 308.15 K and at Atmospheric Pressure,” J. Chem. Eng. Data 50(4), 1404–1408 (2005).
    [CrossRef]
  14. Pyrromethene 597, CAS Nr. 137829–79–9, acquired from Exciton Inc., www.exciton.com .
  15. C. Vannahme, S. Klinkhammer, A. Kolew, P.-J. Jakobs, M. Guttmann, S. Dehm, T. Mappes, and U. Lemmer, “Integration of organic semiconductor lasers and single-mode passive waveguides into a PMMA substrate,” Microelectron. Eng. in press.

2010 (1)

W. Song and D. Psaltis, “Pneumatically tunable optofluidic dye laser,” Appl. Phys. Lett. 96(8), 081101 (2010).
[CrossRef]

2009 (2)

W. Song, A. E. Vasdekis, Z. Li, and D. Psaltis, “Optofluidic evanescent dye laser based on a distributed feedback circular grating,” Appl. Phys. Lett. 94(16), 161110 (2009).
[CrossRef]

W. Song, A. E. Vasdekis, Z. Li, and D. Psaltis, “Low-order distributed feedback optofluidic dye laser with reduced threshold,” Appl. Phys. Lett. 94(5), 051117 (2009).
[CrossRef]

2007 (2)

C. Monat, P. Domachuk, and B. J. Eggleton, “Integrated optofluidics: A new river of light,” Nat. Photonics 1(2), 106–114 (2007).
[CrossRef]

M. Gersborg-Hansen and A. Kristensen, “Tunability of optofluidic distributed feedback dye lasers,” Opt. Express 15(1), 137–142 (2007).
[CrossRef] [PubMed]

2006 (4)

D. Psaltis, S. R. Quake, and C. Yang, “Developing optofluidic technology through the fusion of microfluidics and optics,” Nature 442(7101), 381–386 (2006).
[CrossRef] [PubMed]

Z. Li, Z. Zhang, T. Emery, A. Scherer, and D. Psaltis, “Single mode optofluidic distributed feedback dye laser,” Opt. Express 14(2), 696–701 (2006).
[CrossRef] [PubMed]

Z. Li, Z. Zhang, A. Scherer, and D. Psaltis, “Mechanically tunable optofluidic distributed feedback dye laser,” Opt. Express 14(22), 10494–10499 (2006).
[CrossRef] [PubMed]

M. Gersborg-Hansen and A. Kristensen, “Optofluidic third order distributed feedback dye laser,” Appl. Phys. Lett. 89(10), 103518 (2006).
[CrossRef]

2005 (3)

B. Bilenberg, M. Hansen, D. Johansen, V. Özkapici, C. Jeppesen, P. Szabo, I. M. Obieta, O. Arroyo, J. O. Tegenfeldt, and A. Kristensen, “Topas based lab-on-a-chip microsystems fabricated by thermal nanoimprint lithography,” J. Vac. Sci. Technol. B 23(6), 2944–2949 (2005).
[CrossRef]

R. Francesconi, A. Bigi, and F. Comelli, “Enthalpies of Mixing, Densities, and Refractive Indices for Binary Mixtures of (Anisole or Phenetole) + Three Aryl Alcohols at 308.15 K and at Atmospheric Pressure,” J. Chem. Eng. Data 50(4), 1404–1408 (2005).
[CrossRef]

S. Balslev and A. Kristensen, “Microfluidic single-mode laser using high-order Bragg grating and antiguiding segments,” Opt. Express 13(1), 344–351 (2005).
[CrossRef] [PubMed]

Arroyo, O.

B. Bilenberg, M. Hansen, D. Johansen, V. Özkapici, C. Jeppesen, P. Szabo, I. M. Obieta, O. Arroyo, J. O. Tegenfeldt, and A. Kristensen, “Topas based lab-on-a-chip microsystems fabricated by thermal nanoimprint lithography,” J. Vac. Sci. Technol. B 23(6), 2944–2949 (2005).
[CrossRef]

Balslev, S.

Bigi, A.

R. Francesconi, A. Bigi, and F. Comelli, “Enthalpies of Mixing, Densities, and Refractive Indices for Binary Mixtures of (Anisole or Phenetole) + Three Aryl Alcohols at 308.15 K and at Atmospheric Pressure,” J. Chem. Eng. Data 50(4), 1404–1408 (2005).
[CrossRef]

Bilenberg, B.

B. Bilenberg, M. Hansen, D. Johansen, V. Özkapici, C. Jeppesen, P. Szabo, I. M. Obieta, O. Arroyo, J. O. Tegenfeldt, and A. Kristensen, “Topas based lab-on-a-chip microsystems fabricated by thermal nanoimprint lithography,” J. Vac. Sci. Technol. B 23(6), 2944–2949 (2005).
[CrossRef]

Comelli, F.

R. Francesconi, A. Bigi, and F. Comelli, “Enthalpies of Mixing, Densities, and Refractive Indices for Binary Mixtures of (Anisole or Phenetole) + Three Aryl Alcohols at 308.15 K and at Atmospheric Pressure,” J. Chem. Eng. Data 50(4), 1404–1408 (2005).
[CrossRef]

Dehm, S.

C. Vannahme, S. Klinkhammer, A. Kolew, P.-J. Jakobs, M. Guttmann, S. Dehm, T. Mappes, and U. Lemmer, “Integration of organic semiconductor lasers and single-mode passive waveguides into a PMMA substrate,” Microelectron. Eng. in press.

Domachuk, P.

C. Monat, P. Domachuk, and B. J. Eggleton, “Integrated optofluidics: A new river of light,” Nat. Photonics 1(2), 106–114 (2007).
[CrossRef]

Eggleton, B. J.

C. Monat, P. Domachuk, and B. J. Eggleton, “Integrated optofluidics: A new river of light,” Nat. Photonics 1(2), 106–114 (2007).
[CrossRef]

Emery, T.

Francesconi, R.

R. Francesconi, A. Bigi, and F. Comelli, “Enthalpies of Mixing, Densities, and Refractive Indices for Binary Mixtures of (Anisole or Phenetole) + Three Aryl Alcohols at 308.15 K and at Atmospheric Pressure,” J. Chem. Eng. Data 50(4), 1404–1408 (2005).
[CrossRef]

Gersborg-Hansen, M.

M. Gersborg-Hansen and A. Kristensen, “Tunability of optofluidic distributed feedback dye lasers,” Opt. Express 15(1), 137–142 (2007).
[CrossRef] [PubMed]

M. Gersborg-Hansen and A. Kristensen, “Optofluidic third order distributed feedback dye laser,” Appl. Phys. Lett. 89(10), 103518 (2006).
[CrossRef]

Guttmann, M.

C. Vannahme, S. Klinkhammer, A. Kolew, P.-J. Jakobs, M. Guttmann, S. Dehm, T. Mappes, and U. Lemmer, “Integration of organic semiconductor lasers and single-mode passive waveguides into a PMMA substrate,” Microelectron. Eng. in press.

Hansen, M.

B. Bilenberg, M. Hansen, D. Johansen, V. Özkapici, C. Jeppesen, P. Szabo, I. M. Obieta, O. Arroyo, J. O. Tegenfeldt, and A. Kristensen, “Topas based lab-on-a-chip microsystems fabricated by thermal nanoimprint lithography,” J. Vac. Sci. Technol. B 23(6), 2944–2949 (2005).
[CrossRef]

Jakobs, P.-J.

C. Vannahme, S. Klinkhammer, A. Kolew, P.-J. Jakobs, M. Guttmann, S. Dehm, T. Mappes, and U. Lemmer, “Integration of organic semiconductor lasers and single-mode passive waveguides into a PMMA substrate,” Microelectron. Eng. in press.

Jeppesen, C.

B. Bilenberg, M. Hansen, D. Johansen, V. Özkapici, C. Jeppesen, P. Szabo, I. M. Obieta, O. Arroyo, J. O. Tegenfeldt, and A. Kristensen, “Topas based lab-on-a-chip microsystems fabricated by thermal nanoimprint lithography,” J. Vac. Sci. Technol. B 23(6), 2944–2949 (2005).
[CrossRef]

Johansen, D.

B. Bilenberg, M. Hansen, D. Johansen, V. Özkapici, C. Jeppesen, P. Szabo, I. M. Obieta, O. Arroyo, J. O. Tegenfeldt, and A. Kristensen, “Topas based lab-on-a-chip microsystems fabricated by thermal nanoimprint lithography,” J. Vac. Sci. Technol. B 23(6), 2944–2949 (2005).
[CrossRef]

Klinkhammer, S.

C. Vannahme, S. Klinkhammer, A. Kolew, P.-J. Jakobs, M. Guttmann, S. Dehm, T. Mappes, and U. Lemmer, “Integration of organic semiconductor lasers and single-mode passive waveguides into a PMMA substrate,” Microelectron. Eng. in press.

Kolew, A.

C. Vannahme, S. Klinkhammer, A. Kolew, P.-J. Jakobs, M. Guttmann, S. Dehm, T. Mappes, and U. Lemmer, “Integration of organic semiconductor lasers and single-mode passive waveguides into a PMMA substrate,” Microelectron. Eng. in press.

Kristensen, A.

M. Gersborg-Hansen and A. Kristensen, “Tunability of optofluidic distributed feedback dye lasers,” Opt. Express 15(1), 137–142 (2007).
[CrossRef] [PubMed]

M. Gersborg-Hansen and A. Kristensen, “Optofluidic third order distributed feedback dye laser,” Appl. Phys. Lett. 89(10), 103518 (2006).
[CrossRef]

S. Balslev and A. Kristensen, “Microfluidic single-mode laser using high-order Bragg grating and antiguiding segments,” Opt. Express 13(1), 344–351 (2005).
[CrossRef] [PubMed]

B. Bilenberg, M. Hansen, D. Johansen, V. Özkapici, C. Jeppesen, P. Szabo, I. M. Obieta, O. Arroyo, J. O. Tegenfeldt, and A. Kristensen, “Topas based lab-on-a-chip microsystems fabricated by thermal nanoimprint lithography,” J. Vac. Sci. Technol. B 23(6), 2944–2949 (2005).
[CrossRef]

Lemmer, U.

C. Vannahme, S. Klinkhammer, A. Kolew, P.-J. Jakobs, M. Guttmann, S. Dehm, T. Mappes, and U. Lemmer, “Integration of organic semiconductor lasers and single-mode passive waveguides into a PMMA substrate,” Microelectron. Eng. in press.

Li, Z.

W. Song, A. E. Vasdekis, Z. Li, and D. Psaltis, “Optofluidic evanescent dye laser based on a distributed feedback circular grating,” Appl. Phys. Lett. 94(16), 161110 (2009).
[CrossRef]

W. Song, A. E. Vasdekis, Z. Li, and D. Psaltis, “Low-order distributed feedback optofluidic dye laser with reduced threshold,” Appl. Phys. Lett. 94(5), 051117 (2009).
[CrossRef]

Z. Li, Z. Zhang, T. Emery, A. Scherer, and D. Psaltis, “Single mode optofluidic distributed feedback dye laser,” Opt. Express 14(2), 696–701 (2006).
[CrossRef] [PubMed]

Z. Li, Z. Zhang, A. Scherer, and D. Psaltis, “Mechanically tunable optofluidic distributed feedback dye laser,” Opt. Express 14(22), 10494–10499 (2006).
[CrossRef] [PubMed]

Mappes, T.

C. Vannahme, S. Klinkhammer, A. Kolew, P.-J. Jakobs, M. Guttmann, S. Dehm, T. Mappes, and U. Lemmer, “Integration of organic semiconductor lasers and single-mode passive waveguides into a PMMA substrate,” Microelectron. Eng. in press.

Monat, C.

C. Monat, P. Domachuk, and B. J. Eggleton, “Integrated optofluidics: A new river of light,” Nat. Photonics 1(2), 106–114 (2007).
[CrossRef]

Obieta, I. M.

B. Bilenberg, M. Hansen, D. Johansen, V. Özkapici, C. Jeppesen, P. Szabo, I. M. Obieta, O. Arroyo, J. O. Tegenfeldt, and A. Kristensen, “Topas based lab-on-a-chip microsystems fabricated by thermal nanoimprint lithography,” J. Vac. Sci. Technol. B 23(6), 2944–2949 (2005).
[CrossRef]

Özkapici, V.

B. Bilenberg, M. Hansen, D. Johansen, V. Özkapici, C. Jeppesen, P. Szabo, I. M. Obieta, O. Arroyo, J. O. Tegenfeldt, and A. Kristensen, “Topas based lab-on-a-chip microsystems fabricated by thermal nanoimprint lithography,” J. Vac. Sci. Technol. B 23(6), 2944–2949 (2005).
[CrossRef]

Psaltis, D.

W. Song and D. Psaltis, “Pneumatically tunable optofluidic dye laser,” Appl. Phys. Lett. 96(8), 081101 (2010).
[CrossRef]

W. Song, A. E. Vasdekis, Z. Li, and D. Psaltis, “Low-order distributed feedback optofluidic dye laser with reduced threshold,” Appl. Phys. Lett. 94(5), 051117 (2009).
[CrossRef]

W. Song, A. E. Vasdekis, Z. Li, and D. Psaltis, “Optofluidic evanescent dye laser based on a distributed feedback circular grating,” Appl. Phys. Lett. 94(16), 161110 (2009).
[CrossRef]

D. Psaltis, S. R. Quake, and C. Yang, “Developing optofluidic technology through the fusion of microfluidics and optics,” Nature 442(7101), 381–386 (2006).
[CrossRef] [PubMed]

Z. Li, Z. Zhang, A. Scherer, and D. Psaltis, “Mechanically tunable optofluidic distributed feedback dye laser,” Opt. Express 14(22), 10494–10499 (2006).
[CrossRef] [PubMed]

Z. Li, Z. Zhang, T. Emery, A. Scherer, and D. Psaltis, “Single mode optofluidic distributed feedback dye laser,” Opt. Express 14(2), 696–701 (2006).
[CrossRef] [PubMed]

Quake, S. R.

D. Psaltis, S. R. Quake, and C. Yang, “Developing optofluidic technology through the fusion of microfluidics and optics,” Nature 442(7101), 381–386 (2006).
[CrossRef] [PubMed]

Scherer, A.

Song, W.

W. Song and D. Psaltis, “Pneumatically tunable optofluidic dye laser,” Appl. Phys. Lett. 96(8), 081101 (2010).
[CrossRef]

W. Song, A. E. Vasdekis, Z. Li, and D. Psaltis, “Low-order distributed feedback optofluidic dye laser with reduced threshold,” Appl. Phys. Lett. 94(5), 051117 (2009).
[CrossRef]

W. Song, A. E. Vasdekis, Z. Li, and D. Psaltis, “Optofluidic evanescent dye laser based on a distributed feedback circular grating,” Appl. Phys. Lett. 94(16), 161110 (2009).
[CrossRef]

Szabo, P.

B. Bilenberg, M. Hansen, D. Johansen, V. Özkapici, C. Jeppesen, P. Szabo, I. M. Obieta, O. Arroyo, J. O. Tegenfeldt, and A. Kristensen, “Topas based lab-on-a-chip microsystems fabricated by thermal nanoimprint lithography,” J. Vac. Sci. Technol. B 23(6), 2944–2949 (2005).
[CrossRef]

Tegenfeldt, J. O.

B. Bilenberg, M. Hansen, D. Johansen, V. Özkapici, C. Jeppesen, P. Szabo, I. M. Obieta, O. Arroyo, J. O. Tegenfeldt, and A. Kristensen, “Topas based lab-on-a-chip microsystems fabricated by thermal nanoimprint lithography,” J. Vac. Sci. Technol. B 23(6), 2944–2949 (2005).
[CrossRef]

Vannahme, C.

C. Vannahme, S. Klinkhammer, A. Kolew, P.-J. Jakobs, M. Guttmann, S. Dehm, T. Mappes, and U. Lemmer, “Integration of organic semiconductor lasers and single-mode passive waveguides into a PMMA substrate,” Microelectron. Eng. in press.

Vasdekis, A. E.

W. Song, A. E. Vasdekis, Z. Li, and D. Psaltis, “Optofluidic evanescent dye laser based on a distributed feedback circular grating,” Appl. Phys. Lett. 94(16), 161110 (2009).
[CrossRef]

W. Song, A. E. Vasdekis, Z. Li, and D. Psaltis, “Low-order distributed feedback optofluidic dye laser with reduced threshold,” Appl. Phys. Lett. 94(5), 051117 (2009).
[CrossRef]

Yang, C.

D. Psaltis, S. R. Quake, and C. Yang, “Developing optofluidic technology through the fusion of microfluidics and optics,” Nature 442(7101), 381–386 (2006).
[CrossRef] [PubMed]

Zhang, Z.

Appl. Phys. Lett. (4)

M. Gersborg-Hansen and A. Kristensen, “Optofluidic third order distributed feedback dye laser,” Appl. Phys. Lett. 89(10), 103518 (2006).
[CrossRef]

W. Song, A. E. Vasdekis, Z. Li, and D. Psaltis, “Optofluidic evanescent dye laser based on a distributed feedback circular grating,” Appl. Phys. Lett. 94(16), 161110 (2009).
[CrossRef]

W. Song, A. E. Vasdekis, Z. Li, and D. Psaltis, “Low-order distributed feedback optofluidic dye laser with reduced threshold,” Appl. Phys. Lett. 94(5), 051117 (2009).
[CrossRef]

W. Song and D. Psaltis, “Pneumatically tunable optofluidic dye laser,” Appl. Phys. Lett. 96(8), 081101 (2010).
[CrossRef]

J. Chem. Eng. Data (1)

R. Francesconi, A. Bigi, and F. Comelli, “Enthalpies of Mixing, Densities, and Refractive Indices for Binary Mixtures of (Anisole or Phenetole) + Three Aryl Alcohols at 308.15 K and at Atmospheric Pressure,” J. Chem. Eng. Data 50(4), 1404–1408 (2005).
[CrossRef]

J. Vac. Sci. Technol. B (1)

B. Bilenberg, M. Hansen, D. Johansen, V. Özkapici, C. Jeppesen, P. Szabo, I. M. Obieta, O. Arroyo, J. O. Tegenfeldt, and A. Kristensen, “Topas based lab-on-a-chip microsystems fabricated by thermal nanoimprint lithography,” J. Vac. Sci. Technol. B 23(6), 2944–2949 (2005).
[CrossRef]

Microelectron. Eng. (1)

C. Vannahme, S. Klinkhammer, A. Kolew, P.-J. Jakobs, M. Guttmann, S. Dehm, T. Mappes, and U. Lemmer, “Integration of organic semiconductor lasers and single-mode passive waveguides into a PMMA substrate,” Microelectron. Eng. in press.

Nat. Photonics (1)

C. Monat, P. Domachuk, and B. J. Eggleton, “Integrated optofluidics: A new river of light,” Nat. Photonics 1(2), 106–114 (2007).
[CrossRef]

Nature (1)

D. Psaltis, S. R. Quake, and C. Yang, “Developing optofluidic technology through the fusion of microfluidics and optics,” Nature 442(7101), 381–386 (2006).
[CrossRef] [PubMed]

Opt. Express (4)

Other (2)

TOPAS, 6013 COC, acquired from TOPAS Advanced Polymers, Inc., www.topas.com .

Pyrromethene 597, CAS Nr. 137829–79–9, acquired from Exciton Inc., www.exciton.com .

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Figures (5)

Fig. 1
Fig. 1

(a) Illustration of an imprinted microchannel in a TOPAS® foil with support pillars and a phase shifted DFB grating in the center. (b) Scheme of the finished laser: A liquid core of Pyrromethene 597 dissolved in benzyl alcohol in the microchannel structured with the phase shifted DFB grating of period Λ. Arrows denote the direction of the pump light and the emission.

Fig. 2
Fig. 2

(a) Atomic force micrograph illustrating the shape of imprinted grating lines. Here, the tip reached only 80 nm deep because of the narrow trenches but a depth of 140 nm has been measured on broader trenches on the same foil (b) Scanning electron micrograph taken under an angle of 30° of an imprinted grating with Λ/2 phase shift on a 100 µm thick TOPAS® foil.

Fig. 3
Fig. 3

(a) Photograph of a foil (size 18 × 18 × 0.35 mm3) with two microchannels and 1 ml syringe filled with Pyrromethene 597 in benzyl alcohol in the background. Blue light is diffracted from the gratings on the right in the direction of the camera. (b) Microscope image of microchannel with pillars and distributed feedback grating in the center.

Fig. 4
Fig. 4

Optical characteristics. (a, b) Output spectra of two lasers emitting at 565.8 nm and 581.2 nm. The full width at half maximum of the spectral peaks is 0.20 nm and 0.14 nm respectively. (c) Curve showing the output intensity as function of pump fluence for a laser emitting at 565.8 nm. The threshold is ~20 µJ/mm2. A linear response is obtained for pump fluences above threshold. (d) Curve illustrating that the threshold at 581.2 nm is ~30 µJ/mm2.

Fig. 5
Fig. 5

Total output pulse energy of a laser emitting at 565.8 nm measured over a time of more than 25 min.

Equations (2)

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

d c = λ 2 n c 2 n s 2 ,
λ i = 2 n eff Λ i .

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