Abstract

The infrared emission from an external-cavity-tapered diode laser is frequency doubled in a 5-mm β-BaB2O4 crystal that is placed in a resonant-enhancement cavity. For a coupled diode power of 535 mW, 100 mW of continuous-wave, second-harmonic power is generated at λ=392 nm. The ultraviolet (UV) beam is measured to have a nearly ideal beam-quality parameter, M21, in both meridians even though substantial double refraction is present in the nonlinear crystal. Printing of an array of submicron dots on a phase-change film is demonstrated with this UV beam. Also, by printing on photochemical film we demonstrate that improved film sensitivity is possible when a UV source is used.

© 1999 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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1999 (2)

1997 (2)

A. K. Goyal, P. Gavrilovic, and H. Po, “Stable single-frequency operation of a high-power external cavity tapered diode laser at 780 nm,” Appl. Phys. Lett. 71, 1296–1298 (1997).
[CrossRef]

B. Beier, D. Woll, M. Scheidt, K.-J. Boller, and R. Wallenstein, “Second harmonic generation of the output of an AlGaAs diode oscillator amplifier system in critically phase matched LiB3O5 and β-BaB2O4,” Appl. Phys. Lett. 71, 315–317 (1997).
[CrossRef]

1996 (1)

J. N. Walpole, “Semiconductor amplifiers and lasers with tapered gain regions,” Opt. Quantum Electron. 28, 623–645 (1996), and references therein.
[CrossRef]

1994 (2)

L. Goldberg and D. Mehuys, “Blue light generation using a high power tapered amplifier mode-locked laser,” Appl. Phys. Lett. 65, 522–524 (1994).
[CrossRef]

K. Hayasaka, M. Watanabe, H. Imajo, R. Ohmukai, and S. Urabe, “Tunable 397 nm light source for laser cooling of Ca ions based on frequency doubling of diode laser,” Jpn. J. Appl. Phys. 33, 1595–1598 (1994).
[CrossRef]

1993 (1)

S. O’Brien, D. F. Welch, R. A. Parke, D. Mehuys, K. Dzurko, R. J. Lang, R. Waarts, and D. Scifres, “Operating characteristics of a high-power monolithically integrated flared amplifier master oscillator power amplifier,” IEEE J. Quantum Electron. 29, 2052–2057 (1993).
[CrossRef]

1992 (2)

D. A. Roberts, “Simplified characterization of uniaxial and biaxial nonlinear optical crystals: a plea for standardization of nomenclature and conventions,” IEEE J. Quantum Electron. 28, 2057–2074 (1992).
[CrossRef]

C. S. Adams and A. I. Ferguson, “Tunable narrow linewidth ultra-violet light generation by frequency doubling of a ring Ti:sapphire laser using lithium triborate in an external enhancement cavity,” Opt. Commun. 90, 89–94 (1992).
[CrossRef]

1990 (3)

1989 (2)

E. J. Lim, M. M. Fejer, R. L. Byer, and W. J. Kozlovsky, “Blue light generation by frequency doubling in periodically poled lithium niobate channel waveguide,” Electron. Lett. 25, 731–732 (1989).
[CrossRef]

C. T. Chen, Y. Wu, A. Jiang, B. Wu, G. You, R. Li, and S. Lin, “New nonlinear-optical crystal: LiB3O5,” J. Opt. Soc. Am. B 6, 616–621 (1989).
[CrossRef]

1988 (1)

W. J. Kozlovsky, C. D. Nabors, and R. L. Byer, “Efficient second harmonic generation of a diode-laser-pumped CW Nd:YAG laser using monolithic MgO:LiNbO3 external resonant cavities,” IEEE J. Quantum Electron. 24, 913–919 (1988).
[CrossRef]

1987 (1)

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62, 1968–1983 (1987).
[CrossRef]

1985 (1)

C. T. Chen, B. Wu, A. Jiang, and G. You, “A new type ultraviolet SHG crystal β-BaB2O4,” Sci. Sin. Ser. B 28, 235–243 (1985).

1980 (1)

T. W. Hänsch and B. Couillaud, “Laser frequency stabilization by polarization spectroscopy of a reflecting reference cavity,” Opt. Commun. 35, 441–444 (1980).
[CrossRef]

1968 (1)

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[CrossRef]

1966 (1)

A. Ashkin, G. D. Boyd, and J. M. Dziedzic, “Resonant optical second harmonic generation and mixing,” IEEE J. Quantum Electron. QE-2, 109–123 (1966).
[CrossRef]

1965 (1)

Adams, C. S.

C. S. Adams and A. I. Ferguson, “Tunable narrow linewidth ultra-violet light generation by frequency doubling of a ring Ti:sapphire laser using lithium triborate in an external enhancement cavity,” Opt. Commun. 90, 89–94 (1992).
[CrossRef]

Ashkin, A.

A. Ashkin, G. D. Boyd, and J. M. Dziedzic, “Resonant optical second harmonic generation and mixing,” IEEE J. Quantum Electron. QE-2, 109–123 (1966).
[CrossRef]

Beier, B.

D. Woll, B. Beier, K.-J. Boller, R. Wallenstein, M. Hagberg, and S. O’Brien, “1 W of blue 465-nm radiation generated by frequency doubling of the output of a high-power diode laser in critically phase-matched LiB3O5,” Opt. Lett. 24, 691–693 (1999).
[CrossRef]

B. Beier, D. Woll, M. Scheidt, K.-J. Boller, and R. Wallenstein, “Second harmonic generation of the output of an AlGaAs diode oscillator amplifier system in critically phase matched LiB3O5 and β-BaB2O4,” Appl. Phys. Lett. 71, 315–317 (1997).
[CrossRef]

Bhawalkar, J. D.

Boller, K.-J.

D. Woll, B. Beier, K.-J. Boller, R. Wallenstein, M. Hagberg, and S. O’Brien, “1 W of blue 465-nm radiation generated by frequency doubling of the output of a high-power diode laser in critically phase-matched LiB3O5,” Opt. Lett. 24, 691–693 (1999).
[CrossRef]

B. Beier, D. Woll, M. Scheidt, K.-J. Boller, and R. Wallenstein, “Second harmonic generation of the output of an AlGaAs diode oscillator amplifier system in critically phase matched LiB3O5 and β-BaB2O4,” Appl. Phys. Lett. 71, 315–317 (1997).
[CrossRef]

Boyd, G. D.

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[CrossRef]

A. Ashkin, G. D. Boyd, and J. M. Dziedzic, “Resonant optical second harmonic generation and mixing,” IEEE J. Quantum Electron. QE-2, 109–123 (1966).
[CrossRef]

Byer, R. L.

E. J. Lim, M. M. Fejer, R. L. Byer, and W. J. Kozlovsky, “Blue light generation by frequency doubling in periodically poled lithium niobate channel waveguide,” Electron. Lett. 25, 731–732 (1989).
[CrossRef]

W. J. Kozlovsky, C. D. Nabors, and R. L. Byer, “Efficient second harmonic generation of a diode-laser-pumped CW Nd:YAG laser using monolithic MgO:LiNbO3 external resonant cavities,” IEEE J. Quantum Electron. 24, 913–919 (1988).
[CrossRef]

Chen, C. T.

C. T. Chen, Y. Wu, A. Jiang, B. Wu, G. You, R. Li, and S. Lin, “New nonlinear-optical crystal: LiB3O5,” J. Opt. Soc. Am. B 6, 616–621 (1989).
[CrossRef]

C. T. Chen, B. Wu, A. Jiang, and G. You, “A new type ultraviolet SHG crystal β-BaB2O4,” Sci. Sin. Ser. B 28, 235–243 (1985).

Conturie, Y.

Couillaud, B.

T. W. Hänsch and B. Couillaud, “Laser frequency stabilization by polarization spectroscopy of a reflecting reference cavity,” Opt. Commun. 35, 441–444 (1980).
[CrossRef]

Davis, L.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62, 1968–1983 (1987).
[CrossRef]

Dziedzic, J. M.

A. Ashkin, G. D. Boyd, and J. M. Dziedzic, “Resonant optical second harmonic generation and mixing,” IEEE J. Quantum Electron. QE-2, 109–123 (1966).
[CrossRef]

Dzurko, K.

S. O’Brien, D. F. Welch, R. A. Parke, D. Mehuys, K. Dzurko, R. J. Lang, R. Waarts, and D. Scifres, “Operating characteristics of a high-power monolithically integrated flared amplifier master oscillator power amplifier,” IEEE J. Quantum Electron. 29, 2052–2057 (1993).
[CrossRef]

Eimerl, D.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62, 1968–1983 (1987).
[CrossRef]

Fejer, M. M.

E. J. Lim, M. M. Fejer, R. L. Byer, and W. J. Kozlovsky, “Blue light generation by frequency doubling in periodically poled lithium niobate channel waveguide,” Electron. Lett. 25, 731–732 (1989).
[CrossRef]

Ferguson, A. I.

C. S. Adams and A. I. Ferguson, “Tunable narrow linewidth ultra-violet light generation by frequency doubling of a ring Ti:sapphire laser using lithium triborate in an external enhancement cavity,” Opt. Commun. 90, 89–94 (1992).
[CrossRef]

Gavrilovic, P.

J. D. Bhawalkar, Y. Mao, H. Po, A. K. Goyal, P. Gavrilovic, Y. Conturie, and S. Singh, “High-power 390-nm laser source based on efficient frequency doubling of a tapered diode laser in an external resonant cavity,” Opt. Lett. 24, 823–825 (1999).
[CrossRef]

A. K. Goyal, P. Gavrilovic, and H. Po, “Stable single-frequency operation of a high-power external cavity tapered diode laser at 780 nm,” Appl. Phys. Lett. 71, 1296–1298 (1997).
[CrossRef]

Goldberg, L.

L. Goldberg and D. Mehuys, “Blue light generation using a high power tapered amplifier mode-locked laser,” Appl. Phys. Lett. 65, 522–524 (1994).
[CrossRef]

Goyal, A. K.

J. D. Bhawalkar, Y. Mao, H. Po, A. K. Goyal, P. Gavrilovic, Y. Conturie, and S. Singh, “High-power 390-nm laser source based on efficient frequency doubling of a tapered diode laser in an external resonant cavity,” Opt. Lett. 24, 823–825 (1999).
[CrossRef]

A. K. Goyal, P. Gavrilovic, and H. Po, “Stable single-frequency operation of a high-power external cavity tapered diode laser at 780 nm,” Appl. Phys. Lett. 71, 1296–1298 (1997).
[CrossRef]

Graham, E. K.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62, 1968–1983 (1987).
[CrossRef]

Guerra, J. M.

Hagberg, M.

Hänsch, T. W.

A. Hemmerich, D. H. McIntyre, C. Zimmermann, and T. W. Hänsch, “Second-harmonic generation and optical stabilization of a diode laser in an external ring resonator,” Opt. Lett. 15, 372–374 (1990).
[CrossRef] [PubMed]

T. W. Hänsch and B. Couillaud, “Laser frequency stabilization by polarization spectroscopy of a reflecting reference cavity,” Opt. Commun. 35, 441–444 (1980).
[CrossRef]

Hayasaka, K.

K. Hayasaka, M. Watanabe, H. Imajo, R. Ohmukai, and S. Urabe, “Tunable 397 nm light source for laser cooling of Ca ions based on frequency doubling of diode laser,” Jpn. J. Appl. Phys. 33, 1595–1598 (1994).
[CrossRef]

Hemmerich, A.

Imajo, H.

K. Hayasaka, M. Watanabe, H. Imajo, R. Ohmukai, and S. Urabe, “Tunable 397 nm light source for laser cooling of Ca ions based on frequency doubling of diode laser,” Jpn. J. Appl. Phys. 33, 1595–1598 (1994).
[CrossRef]

Jiang, A.

C. T. Chen, Y. Wu, A. Jiang, B. Wu, G. You, R. Li, and S. Lin, “New nonlinear-optical crystal: LiB3O5,” J. Opt. Soc. Am. B 6, 616–621 (1989).
[CrossRef]

C. T. Chen, B. Wu, A. Jiang, and G. You, “A new type ultraviolet SHG crystal β-BaB2O4,” Sci. Sin. Ser. B 28, 235–243 (1985).

Kleinman, D. A.

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[CrossRef]

Kozlovsky, W. J.

E. J. Lim, M. M. Fejer, R. L. Byer, and W. J. Kozlovsky, “Blue light generation by frequency doubling in periodically poled lithium niobate channel waveguide,” Electron. Lett. 25, 731–732 (1989).
[CrossRef]

W. J. Kozlovsky, C. D. Nabors, and R. L. Byer, “Efficient second harmonic generation of a diode-laser-pumped CW Nd:YAG laser using monolithic MgO:LiNbO3 external resonant cavities,” IEEE J. Quantum Electron. 24, 913–919 (1988).
[CrossRef]

Lang, R. J.

S. O’Brien, D. F. Welch, R. A. Parke, D. Mehuys, K. Dzurko, R. J. Lang, R. Waarts, and D. Scifres, “Operating characteristics of a high-power monolithically integrated flared amplifier master oscillator power amplifier,” IEEE J. Quantum Electron. 29, 2052–2057 (1993).
[CrossRef]

Li, R.

Lim, E. J.

E. J. Lim, M. M. Fejer, R. L. Byer, and W. J. Kozlovsky, “Blue light generation by frequency doubling in periodically poled lithium niobate channel waveguide,” Electron. Lett. 25, 731–732 (1989).
[CrossRef]

Lin, S.

Malitson, I. H.

Mao, Y.

McIntyre, D. H.

Mehuys, D.

L. Goldberg and D. Mehuys, “Blue light generation using a high power tapered amplifier mode-locked laser,” Appl. Phys. Lett. 65, 522–524 (1994).
[CrossRef]

S. O’Brien, D. F. Welch, R. A. Parke, D. Mehuys, K. Dzurko, R. J. Lang, R. Waarts, and D. Scifres, “Operating characteristics of a high-power monolithically integrated flared amplifier master oscillator power amplifier,” IEEE J. Quantum Electron. 29, 2052–2057 (1993).
[CrossRef]

Nabors, C. D.

W. J. Kozlovsky, C. D. Nabors, and R. L. Byer, “Efficient second harmonic generation of a diode-laser-pumped CW Nd:YAG laser using monolithic MgO:LiNbO3 external resonant cavities,” IEEE J. Quantum Electron. 24, 913–919 (1988).
[CrossRef]

O’Brien, S.

D. Woll, B. Beier, K.-J. Boller, R. Wallenstein, M. Hagberg, and S. O’Brien, “1 W of blue 465-nm radiation generated by frequency doubling of the output of a high-power diode laser in critically phase-matched LiB3O5,” Opt. Lett. 24, 691–693 (1999).
[CrossRef]

S. O’Brien, D. F. Welch, R. A. Parke, D. Mehuys, K. Dzurko, R. J. Lang, R. Waarts, and D. Scifres, “Operating characteristics of a high-power monolithically integrated flared amplifier master oscillator power amplifier,” IEEE J. Quantum Electron. 29, 2052–2057 (1993).
[CrossRef]

Ohmukai, R.

K. Hayasaka, M. Watanabe, H. Imajo, R. Ohmukai, and S. Urabe, “Tunable 397 nm light source for laser cooling of Ca ions based on frequency doubling of diode laser,” Jpn. J. Appl. Phys. 33, 1595–1598 (1994).
[CrossRef]

Parke, R. A.

S. O’Brien, D. F. Welch, R. A. Parke, D. Mehuys, K. Dzurko, R. J. Lang, R. Waarts, and D. Scifres, “Operating characteristics of a high-power monolithically integrated flared amplifier master oscillator power amplifier,” IEEE J. Quantum Electron. 29, 2052–2057 (1993).
[CrossRef]

Po, H.

J. D. Bhawalkar, Y. Mao, H. Po, A. K. Goyal, P. Gavrilovic, Y. Conturie, and S. Singh, “High-power 390-nm laser source based on efficient frequency doubling of a tapered diode laser in an external resonant cavity,” Opt. Lett. 24, 823–825 (1999).
[CrossRef]

A. K. Goyal, P. Gavrilovic, and H. Po, “Stable single-frequency operation of a high-power external cavity tapered diode laser at 780 nm,” Appl. Phys. Lett. 71, 1296–1298 (1997).
[CrossRef]

Roberts, D. A.

D. A. Roberts, “Simplified characterization of uniaxial and biaxial nonlinear optical crystals: a plea for standardization of nomenclature and conventions,” IEEE J. Quantum Electron. 28, 2057–2074 (1992).
[CrossRef]

Scheidt, M.

B. Beier, D. Woll, M. Scheidt, K.-J. Boller, and R. Wallenstein, “Second harmonic generation of the output of an AlGaAs diode oscillator amplifier system in critically phase matched LiB3O5 and β-BaB2O4,” Appl. Phys. Lett. 71, 315–317 (1997).
[CrossRef]

Scifres, D.

S. O’Brien, D. F. Welch, R. A. Parke, D. Mehuys, K. Dzurko, R. J. Lang, R. Waarts, and D. Scifres, “Operating characteristics of a high-power monolithically integrated flared amplifier master oscillator power amplifier,” IEEE J. Quantum Electron. 29, 2052–2057 (1993).
[CrossRef]

Siegman, A. E.

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).
[CrossRef]

Singh, S.

Urabe, S.

K. Hayasaka, M. Watanabe, H. Imajo, R. Ohmukai, and S. Urabe, “Tunable 397 nm light source for laser cooling of Ca ions based on frequency doubling of diode laser,” Jpn. J. Appl. Phys. 33, 1595–1598 (1994).
[CrossRef]

Velsko, S.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, “Optical, mechanical, and thermal properties of barium borate,” J. Appl. Phys. 62, 1968–1983 (1987).
[CrossRef]

Waarts, R.

S. O’Brien, D. F. Welch, R. A. Parke, D. Mehuys, K. Dzurko, R. J. Lang, R. Waarts, and D. Scifres, “Operating characteristics of a high-power monolithically integrated flared amplifier master oscillator power amplifier,” IEEE J. Quantum Electron. 29, 2052–2057 (1993).
[CrossRef]

Wallenstein, R.

D. Woll, B. Beier, K.-J. Boller, R. Wallenstein, M. Hagberg, and S. O’Brien, “1 W of blue 465-nm radiation generated by frequency doubling of the output of a high-power diode laser in critically phase-matched LiB3O5,” Opt. Lett. 24, 691–693 (1999).
[CrossRef]

B. Beier, D. Woll, M. Scheidt, K.-J. Boller, and R. Wallenstein, “Second harmonic generation of the output of an AlGaAs diode oscillator amplifier system in critically phase matched LiB3O5 and β-BaB2O4,” Appl. Phys. Lett. 71, 315–317 (1997).
[CrossRef]

Walpole, J. N.

J. N. Walpole, “Semiconductor amplifiers and lasers with tapered gain regions,” Opt. Quantum Electron. 28, 623–645 (1996), and references therein.
[CrossRef]

Watanabe, M.

K. Hayasaka, M. Watanabe, H. Imajo, R. Ohmukai, and S. Urabe, “Tunable 397 nm light source for laser cooling of Ca ions based on frequency doubling of diode laser,” Jpn. J. Appl. Phys. 33, 1595–1598 (1994).
[CrossRef]

Welch, D. F.

S. O’Brien, D. F. Welch, R. A. Parke, D. Mehuys, K. Dzurko, R. J. Lang, R. Waarts, and D. Scifres, “Operating characteristics of a high-power monolithically integrated flared amplifier master oscillator power amplifier,” IEEE J. Quantum Electron. 29, 2052–2057 (1993).
[CrossRef]

Woll, D.

D. Woll, B. Beier, K.-J. Boller, R. Wallenstein, M. Hagberg, and S. O’Brien, “1 W of blue 465-nm radiation generated by frequency doubling of the output of a high-power diode laser in critically phase-matched LiB3O5,” Opt. Lett. 24, 691–693 (1999).
[CrossRef]

B. Beier, D. Woll, M. Scheidt, K.-J. Boller, and R. Wallenstein, “Second harmonic generation of the output of an AlGaAs diode oscillator amplifier system in critically phase matched LiB3O5 and β-BaB2O4,” Appl. Phys. Lett. 71, 315–317 (1997).
[CrossRef]

Wu, B.

C. T. Chen, Y. Wu, A. Jiang, B. Wu, G. You, R. Li, and S. Lin, “New nonlinear-optical crystal: LiB3O5,” J. Opt. Soc. Am. B 6, 616–621 (1989).
[CrossRef]

C. T. Chen, B. Wu, A. Jiang, and G. You, “A new type ultraviolet SHG crystal β-BaB2O4,” Sci. Sin. Ser. B 28, 235–243 (1985).

Wu, Y.

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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Other (7)

Y. Kitaoka, K. Yamamoto, K. Mizuuchi, K. Narumi, and M. Kato, “Blue second-harmonic generation waveguide device and its application to high-density optical disk,” in Conference on Lasers and Electro-Optics, Vol. 6 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), paper CWG1.

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Replacement tapered gain element, part SDL-8630E, SDL, Inc., San Jose, Calif. 95134.

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986). For a discussion of the delta notation in characterizing resonant-cavity losses, see Sections 11.4 and 11.5; for ABCD matrices and their application to paraxial beams, see Chapters 15 and 20; and for resonator design using ABCD matrices, see Chapter 21.

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BeamScan Model 1180 single-axis slit profiler with 2.5 μm slit width and silicon detector. Photon Inc., 1115 Space Park Drive, Santa Clara, Calif. 95054.

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Figures (7)

Fig. 1
Fig. 1

Experimental setup.

Fig. 2
Fig. 2

UV power generated as a function of coupled pump power, Pi=ηcPtot. 100 mW of UV is generated for a coupled power of 535 mW. The theoretical curve is calculated from Eq. (6) with δx=1.0% and γshg=6.0×10-5/W.

Fig. 3
Fig. 3

Comparison of UV power generated in swept and locked modes. The inset shows locking at a UV power level of ∼70 mW for a period of >3 h.

Fig. 4
Fig. 4

Beam size, ω, as a function of distance from the focusing lens for the IR and the UV beams in both meridians.

Fig. 5
Fig. 5

Evolution of the UV beam profile in the walk-off meridian through focus. The centerline of each profile coincides with its axial position from the focusing lens. The relative width of each spectrum is maintained.

Fig. 6
Fig. 6

Schematic diagram of printing setup.

Fig. 7
Fig. 7

Array of submicron dots printed with the UV beam as resolved by a photon-tunneling microscope.

Tables (1)

Tables Icon

Table 1 Gaussian Beam Parameters for the IR and UV Beams Measured Outside the Cavity and Inferred Inside the Cavity

Equations (17)

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ω=2σ.
σx2(z)=(x-x¯)2I(x, y, z)dxdyI(x, y, z)dxdy.
ω2(z)=ωo21+(z-zo)2zR2,
zR=πωo2M2λ
δc=δ1+δshg+δx.
Pi=Puvγshg δc24δ1,δc1,
PtottotalIRpowerincidentuponthecavity,
PontotalreflectedIRpowerwhenthecavityison-resonance,
PirIRpowertransmittedbytheoutputcoupler,
Puv,collUVpowertransmittedbytheoutputcoupler.
PcPi4δ1δc2,δc1,
PrPiδc-2δ1δc2,δc1.
PonPtot=(1-ηc)Ptot+PrPtot.
η˜c=1-PonPtot,
η˜cPtotPc=δc-δ1=δx+δshg.
ηc=η˜c1-δc-2δ1δc2.
F=2πδc

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