Abstract

Hollow core waveguides made of TiO2–SiO2 based glasses have been proposed for the IR. A series of TiO2–SiO2 based glasses were prepared and their complex refractive indices measured by reflectometry. Using measured refractive indices, we calculated spectral losses of the TiO2–SiO2 hollow waveguides. With the waveguides, we can expect efficient transmission of mid-IR light including CO2 laser light, since refractive indices of the glasses become less than unity and the absorption coefficients become small in the 8–11-μm wavelength range.

© 1991 Optical Society of America

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References

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  1. E. Garmire, T. McMahon, M. Bass, “Flexible Infrared Waveguides for High-Power Transmission,” IEEE J. Quantum Electron. QE-16, 23–32 (1980).
    [CrossRef]
  2. A. Hongo, K. Morosawa, T. Shiota, Y. Matsuura, M. Miyagi, “Transmission Characteristics of Germanium Thin Film Coated Metallic Hollow Waveguides for High-Power CO2 Laser Light,” IEEE J. Quantum Electron., 26, 1510–1515 (1990).
    [CrossRef]
  3. K. Fujiwara, K. Fuwa, “Liquid Core Optical Fiber Total Reflection Cell as a Colorimetric Detector for Flow Injection Analysis,” Anal. Chem. 57, 1012–1016 (1985).
    [CrossRef]
  4. T. Kano, Y. Saito, A. Nomura, “A Long-Path Differential Monitoring Using a Hollow-Core Fiber in the Infrared Region,” in Proceedings, Symposium on Guided Wave Technology and Its Application for Mid-Infrared, Sendai, Japan (13–14 Mar. 1986) pp. 165–172, in Japanese.
  5. M. E. Marhic, E. Garmire, “Low-Order TE0q Operation of a CO2 Laser for Transmission Through Circular Metallic Waveguides,” Appl. Phys. Lett. 38, 743–745 (1981).
    [CrossRef]
  6. N. Croitoru, J. Dror, E. Goldenberg, D. Mendlovic, S. Ruschin, “Use of Metallic and Dielectric Films for Hollow Fiber,” Fiber Integr. Opt. 6, 347–361 (1987).
    [CrossRef]
  7. Y. Matsuura, M. Miyagi, A. Hongo, “Loss Reduction of Dielectric-Coated Metallic Hollow Waveguides for CO2 Laser Light Transmission,” Opt. Laser Technol. 22, 141–145 (1990).
    [CrossRef]
  8. T. Hidaka, T. Morikawa, J. Shimada, “Hollow-Core Oxide-Glass Cladding Optical Fibers for Middle-Infrared Region,” J. Appl. Phya. 32, 44G7–4471 (1981).
  9. A. Bornstein, N. Croitoru, A. Seidman, “Chalcogenide Hollow Fibers for Infrared Energy Transmission,” Appl. Phys. Lett. 46, 705–707 (1985).
    [CrossRef]
  10. C. A. Worrell, V. Skarda, “CO2 Laser Waveguides from Germania-Based Glasses,” J. Phys. D 22, 535–541 (1989).
    [CrossRef]
  11. J. A. Harrington, C. C. Gregory, R. Nubling, “Hollow Waveguides for CO2 Laser Delivery System,” Proc. Soc. Photo-Opt. Instrum. Eng. 1048, 117–121 (1989).
  12. Y. Matsuura, M. Saito, M. Miyagi, A. Hongo, “Loss Characteristics of Circular Hollow Waveguides for Incoherent Infrared Light,” J. Opt. Soc. Am. A 6, 423–427 (1989).
    [CrossRef]
  13. R. C. Turnbull, W. G. Lawrence, “The Role of Titania in Silica Glass,” J. Am. Ceram. Soc. 35, 48–53 (1952).
    [CrossRef]
  14. B. V. Janakirama Rao, “The Dual Role of Titanium in the System K2O.SiO2.TiO2,” Phys. Chem. Glasses 4, 22–34 (1963).
  15. E. D. Palik, Handbook of Optical Constants of Solids (Academic, Orlando, FL, 1985), Part 2, pp. 749–768.
  16. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1985), pp. 36–66.
  17. W. R. Hunter, “Errors in Using the Reflectance vs Angle of Incidence Method for Measuring Optical Constants,” J. Opt. Soc. Am. 55, 1197–1204 (1965).
    [CrossRef]
  18. M. Miyagi, S. Kawakami, “Design Theory of Dielectric-Coated Circular Metallic Waveguides for Infrared Transmission,” IEEE/OSA J. Lightwave Technol. LT-2, 116–126 (1984).
    [CrossRef]
  19. D. W. Hewak, J. W. Y. Lit, “Fabrication of Tapers and Lenslike Waveguides by a Microcontrolled Dip Coating Procedure,” Appl. Opt. 27, 4562–4564 (1988).
    [CrossRef] [PubMed]

1990 (2)

A. Hongo, K. Morosawa, T. Shiota, Y. Matsuura, M. Miyagi, “Transmission Characteristics of Germanium Thin Film Coated Metallic Hollow Waveguides for High-Power CO2 Laser Light,” IEEE J. Quantum Electron., 26, 1510–1515 (1990).
[CrossRef]

Y. Matsuura, M. Miyagi, A. Hongo, “Loss Reduction of Dielectric-Coated Metallic Hollow Waveguides for CO2 Laser Light Transmission,” Opt. Laser Technol. 22, 141–145 (1990).
[CrossRef]

1989 (3)

C. A. Worrell, V. Skarda, “CO2 Laser Waveguides from Germania-Based Glasses,” J. Phys. D 22, 535–541 (1989).
[CrossRef]

J. A. Harrington, C. C. Gregory, R. Nubling, “Hollow Waveguides for CO2 Laser Delivery System,” Proc. Soc. Photo-Opt. Instrum. Eng. 1048, 117–121 (1989).

Y. Matsuura, M. Saito, M. Miyagi, A. Hongo, “Loss Characteristics of Circular Hollow Waveguides for Incoherent Infrared Light,” J. Opt. Soc. Am. A 6, 423–427 (1989).
[CrossRef]

1988 (1)

1987 (1)

N. Croitoru, J. Dror, E. Goldenberg, D. Mendlovic, S. Ruschin, “Use of Metallic and Dielectric Films for Hollow Fiber,” Fiber Integr. Opt. 6, 347–361 (1987).
[CrossRef]

1985 (2)

K. Fujiwara, K. Fuwa, “Liquid Core Optical Fiber Total Reflection Cell as a Colorimetric Detector for Flow Injection Analysis,” Anal. Chem. 57, 1012–1016 (1985).
[CrossRef]

A. Bornstein, N. Croitoru, A. Seidman, “Chalcogenide Hollow Fibers for Infrared Energy Transmission,” Appl. Phys. Lett. 46, 705–707 (1985).
[CrossRef]

1984 (1)

M. Miyagi, S. Kawakami, “Design Theory of Dielectric-Coated Circular Metallic Waveguides for Infrared Transmission,” IEEE/OSA J. Lightwave Technol. LT-2, 116–126 (1984).
[CrossRef]

1981 (2)

M. E. Marhic, E. Garmire, “Low-Order TE0q Operation of a CO2 Laser for Transmission Through Circular Metallic Waveguides,” Appl. Phys. Lett. 38, 743–745 (1981).
[CrossRef]

T. Hidaka, T. Morikawa, J. Shimada, “Hollow-Core Oxide-Glass Cladding Optical Fibers for Middle-Infrared Region,” J. Appl. Phya. 32, 44G7–4471 (1981).

1980 (1)

E. Garmire, T. McMahon, M. Bass, “Flexible Infrared Waveguides for High-Power Transmission,” IEEE J. Quantum Electron. QE-16, 23–32 (1980).
[CrossRef]

1965 (1)

1963 (1)

B. V. Janakirama Rao, “The Dual Role of Titanium in the System K2O.SiO2.TiO2,” Phys. Chem. Glasses 4, 22–34 (1963).

1952 (1)

R. C. Turnbull, W. G. Lawrence, “The Role of Titania in Silica Glass,” J. Am. Ceram. Soc. 35, 48–53 (1952).
[CrossRef]

Bass, M.

E. Garmire, T. McMahon, M. Bass, “Flexible Infrared Waveguides for High-Power Transmission,” IEEE J. Quantum Electron. QE-16, 23–32 (1980).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1985), pp. 36–66.

Bornstein, A.

A. Bornstein, N. Croitoru, A. Seidman, “Chalcogenide Hollow Fibers for Infrared Energy Transmission,” Appl. Phys. Lett. 46, 705–707 (1985).
[CrossRef]

Croitoru, N.

N. Croitoru, J. Dror, E. Goldenberg, D. Mendlovic, S. Ruschin, “Use of Metallic and Dielectric Films for Hollow Fiber,” Fiber Integr. Opt. 6, 347–361 (1987).
[CrossRef]

A. Bornstein, N. Croitoru, A. Seidman, “Chalcogenide Hollow Fibers for Infrared Energy Transmission,” Appl. Phys. Lett. 46, 705–707 (1985).
[CrossRef]

Dror, J.

N. Croitoru, J. Dror, E. Goldenberg, D. Mendlovic, S. Ruschin, “Use of Metallic and Dielectric Films for Hollow Fiber,” Fiber Integr. Opt. 6, 347–361 (1987).
[CrossRef]

Fujiwara, K.

K. Fujiwara, K. Fuwa, “Liquid Core Optical Fiber Total Reflection Cell as a Colorimetric Detector for Flow Injection Analysis,” Anal. Chem. 57, 1012–1016 (1985).
[CrossRef]

Fuwa, K.

K. Fujiwara, K. Fuwa, “Liquid Core Optical Fiber Total Reflection Cell as a Colorimetric Detector for Flow Injection Analysis,” Anal. Chem. 57, 1012–1016 (1985).
[CrossRef]

Garmire, E.

M. E. Marhic, E. Garmire, “Low-Order TE0q Operation of a CO2 Laser for Transmission Through Circular Metallic Waveguides,” Appl. Phys. Lett. 38, 743–745 (1981).
[CrossRef]

E. Garmire, T. McMahon, M. Bass, “Flexible Infrared Waveguides for High-Power Transmission,” IEEE J. Quantum Electron. QE-16, 23–32 (1980).
[CrossRef]

Goldenberg, E.

N. Croitoru, J. Dror, E. Goldenberg, D. Mendlovic, S. Ruschin, “Use of Metallic and Dielectric Films for Hollow Fiber,” Fiber Integr. Opt. 6, 347–361 (1987).
[CrossRef]

Gregory, C. C.

J. A. Harrington, C. C. Gregory, R. Nubling, “Hollow Waveguides for CO2 Laser Delivery System,” Proc. Soc. Photo-Opt. Instrum. Eng. 1048, 117–121 (1989).

Harrington, J. A.

J. A. Harrington, C. C. Gregory, R. Nubling, “Hollow Waveguides for CO2 Laser Delivery System,” Proc. Soc. Photo-Opt. Instrum. Eng. 1048, 117–121 (1989).

Hewak, D. W.

Hidaka, T.

T. Hidaka, T. Morikawa, J. Shimada, “Hollow-Core Oxide-Glass Cladding Optical Fibers for Middle-Infrared Region,” J. Appl. Phya. 32, 44G7–4471 (1981).

Hongo, A.

A. Hongo, K. Morosawa, T. Shiota, Y. Matsuura, M. Miyagi, “Transmission Characteristics of Germanium Thin Film Coated Metallic Hollow Waveguides for High-Power CO2 Laser Light,” IEEE J. Quantum Electron., 26, 1510–1515 (1990).
[CrossRef]

Y. Matsuura, M. Miyagi, A. Hongo, “Loss Reduction of Dielectric-Coated Metallic Hollow Waveguides for CO2 Laser Light Transmission,” Opt. Laser Technol. 22, 141–145 (1990).
[CrossRef]

Y. Matsuura, M. Saito, M. Miyagi, A. Hongo, “Loss Characteristics of Circular Hollow Waveguides for Incoherent Infrared Light,” J. Opt. Soc. Am. A 6, 423–427 (1989).
[CrossRef]

Hunter, W. R.

Janakirama Rao, B. V.

B. V. Janakirama Rao, “The Dual Role of Titanium in the System K2O.SiO2.TiO2,” Phys. Chem. Glasses 4, 22–34 (1963).

Kano, T.

T. Kano, Y. Saito, A. Nomura, “A Long-Path Differential Monitoring Using a Hollow-Core Fiber in the Infrared Region,” in Proceedings, Symposium on Guided Wave Technology and Its Application for Mid-Infrared, Sendai, Japan (13–14 Mar. 1986) pp. 165–172, in Japanese.

Kawakami, S.

M. Miyagi, S. Kawakami, “Design Theory of Dielectric-Coated Circular Metallic Waveguides for Infrared Transmission,” IEEE/OSA J. Lightwave Technol. LT-2, 116–126 (1984).
[CrossRef]

Lawrence, W. G.

R. C. Turnbull, W. G. Lawrence, “The Role of Titania in Silica Glass,” J. Am. Ceram. Soc. 35, 48–53 (1952).
[CrossRef]

Lit, J. W. Y.

Marhic, M. E.

M. E. Marhic, E. Garmire, “Low-Order TE0q Operation of a CO2 Laser for Transmission Through Circular Metallic Waveguides,” Appl. Phys. Lett. 38, 743–745 (1981).
[CrossRef]

Matsuura, Y.

Y. Matsuura, M. Miyagi, A. Hongo, “Loss Reduction of Dielectric-Coated Metallic Hollow Waveguides for CO2 Laser Light Transmission,” Opt. Laser Technol. 22, 141–145 (1990).
[CrossRef]

A. Hongo, K. Morosawa, T. Shiota, Y. Matsuura, M. Miyagi, “Transmission Characteristics of Germanium Thin Film Coated Metallic Hollow Waveguides for High-Power CO2 Laser Light,” IEEE J. Quantum Electron., 26, 1510–1515 (1990).
[CrossRef]

Y. Matsuura, M. Saito, M. Miyagi, A. Hongo, “Loss Characteristics of Circular Hollow Waveguides for Incoherent Infrared Light,” J. Opt. Soc. Am. A 6, 423–427 (1989).
[CrossRef]

McMahon, T.

E. Garmire, T. McMahon, M. Bass, “Flexible Infrared Waveguides for High-Power Transmission,” IEEE J. Quantum Electron. QE-16, 23–32 (1980).
[CrossRef]

Mendlovic, D.

N. Croitoru, J. Dror, E. Goldenberg, D. Mendlovic, S. Ruschin, “Use of Metallic and Dielectric Films for Hollow Fiber,” Fiber Integr. Opt. 6, 347–361 (1987).
[CrossRef]

Miyagi, M.

A. Hongo, K. Morosawa, T. Shiota, Y. Matsuura, M. Miyagi, “Transmission Characteristics of Germanium Thin Film Coated Metallic Hollow Waveguides for High-Power CO2 Laser Light,” IEEE J. Quantum Electron., 26, 1510–1515 (1990).
[CrossRef]

Y. Matsuura, M. Miyagi, A. Hongo, “Loss Reduction of Dielectric-Coated Metallic Hollow Waveguides for CO2 Laser Light Transmission,” Opt. Laser Technol. 22, 141–145 (1990).
[CrossRef]

Y. Matsuura, M. Saito, M. Miyagi, A. Hongo, “Loss Characteristics of Circular Hollow Waveguides for Incoherent Infrared Light,” J. Opt. Soc. Am. A 6, 423–427 (1989).
[CrossRef]

M. Miyagi, S. Kawakami, “Design Theory of Dielectric-Coated Circular Metallic Waveguides for Infrared Transmission,” IEEE/OSA J. Lightwave Technol. LT-2, 116–126 (1984).
[CrossRef]

Morikawa, T.

T. Hidaka, T. Morikawa, J. Shimada, “Hollow-Core Oxide-Glass Cladding Optical Fibers for Middle-Infrared Region,” J. Appl. Phya. 32, 44G7–4471 (1981).

Morosawa, K.

A. Hongo, K. Morosawa, T. Shiota, Y. Matsuura, M. Miyagi, “Transmission Characteristics of Germanium Thin Film Coated Metallic Hollow Waveguides for High-Power CO2 Laser Light,” IEEE J. Quantum Electron., 26, 1510–1515 (1990).
[CrossRef]

Nomura, A.

T. Kano, Y. Saito, A. Nomura, “A Long-Path Differential Monitoring Using a Hollow-Core Fiber in the Infrared Region,” in Proceedings, Symposium on Guided Wave Technology and Its Application for Mid-Infrared, Sendai, Japan (13–14 Mar. 1986) pp. 165–172, in Japanese.

Nubling, R.

J. A. Harrington, C. C. Gregory, R. Nubling, “Hollow Waveguides for CO2 Laser Delivery System,” Proc. Soc. Photo-Opt. Instrum. Eng. 1048, 117–121 (1989).

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, Orlando, FL, 1985), Part 2, pp. 749–768.

Ruschin, S.

N. Croitoru, J. Dror, E. Goldenberg, D. Mendlovic, S. Ruschin, “Use of Metallic and Dielectric Films for Hollow Fiber,” Fiber Integr. Opt. 6, 347–361 (1987).
[CrossRef]

Saito, M.

Saito, Y.

T. Kano, Y. Saito, A. Nomura, “A Long-Path Differential Monitoring Using a Hollow-Core Fiber in the Infrared Region,” in Proceedings, Symposium on Guided Wave Technology and Its Application for Mid-Infrared, Sendai, Japan (13–14 Mar. 1986) pp. 165–172, in Japanese.

Seidman, A.

A. Bornstein, N. Croitoru, A. Seidman, “Chalcogenide Hollow Fibers for Infrared Energy Transmission,” Appl. Phys. Lett. 46, 705–707 (1985).
[CrossRef]

Shimada, J.

T. Hidaka, T. Morikawa, J. Shimada, “Hollow-Core Oxide-Glass Cladding Optical Fibers for Middle-Infrared Region,” J. Appl. Phya. 32, 44G7–4471 (1981).

Shiota, T.

A. Hongo, K. Morosawa, T. Shiota, Y. Matsuura, M. Miyagi, “Transmission Characteristics of Germanium Thin Film Coated Metallic Hollow Waveguides for High-Power CO2 Laser Light,” IEEE J. Quantum Electron., 26, 1510–1515 (1990).
[CrossRef]

Skarda, V.

C. A. Worrell, V. Skarda, “CO2 Laser Waveguides from Germania-Based Glasses,” J. Phys. D 22, 535–541 (1989).
[CrossRef]

Turnbull, R. C.

R. C. Turnbull, W. G. Lawrence, “The Role of Titania in Silica Glass,” J. Am. Ceram. Soc. 35, 48–53 (1952).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1985), pp. 36–66.

Worrell, C. A.

C. A. Worrell, V. Skarda, “CO2 Laser Waveguides from Germania-Based Glasses,” J. Phys. D 22, 535–541 (1989).
[CrossRef]

Anal. Chem. (1)

K. Fujiwara, K. Fuwa, “Liquid Core Optical Fiber Total Reflection Cell as a Colorimetric Detector for Flow Injection Analysis,” Anal. Chem. 57, 1012–1016 (1985).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

M. E. Marhic, E. Garmire, “Low-Order TE0q Operation of a CO2 Laser for Transmission Through Circular Metallic Waveguides,” Appl. Phys. Lett. 38, 743–745 (1981).
[CrossRef]

A. Bornstein, N. Croitoru, A. Seidman, “Chalcogenide Hollow Fibers for Infrared Energy Transmission,” Appl. Phys. Lett. 46, 705–707 (1985).
[CrossRef]

Fiber Integr. Opt. (1)

N. Croitoru, J. Dror, E. Goldenberg, D. Mendlovic, S. Ruschin, “Use of Metallic and Dielectric Films for Hollow Fiber,” Fiber Integr. Opt. 6, 347–361 (1987).
[CrossRef]

IEEE J. Quantum Electron. (2)

E. Garmire, T. McMahon, M. Bass, “Flexible Infrared Waveguides for High-Power Transmission,” IEEE J. Quantum Electron. QE-16, 23–32 (1980).
[CrossRef]

A. Hongo, K. Morosawa, T. Shiota, Y. Matsuura, M. Miyagi, “Transmission Characteristics of Germanium Thin Film Coated Metallic Hollow Waveguides for High-Power CO2 Laser Light,” IEEE J. Quantum Electron., 26, 1510–1515 (1990).
[CrossRef]

IEEE/OSA J. Lightwave Technol. (1)

M. Miyagi, S. Kawakami, “Design Theory of Dielectric-Coated Circular Metallic Waveguides for Infrared Transmission,” IEEE/OSA J. Lightwave Technol. LT-2, 116–126 (1984).
[CrossRef]

J. Am. Ceram. Soc. (1)

R. C. Turnbull, W. G. Lawrence, “The Role of Titania in Silica Glass,” J. Am. Ceram. Soc. 35, 48–53 (1952).
[CrossRef]

J. Appl. Phya. (1)

T. Hidaka, T. Morikawa, J. Shimada, “Hollow-Core Oxide-Glass Cladding Optical Fibers for Middle-Infrared Region,” J. Appl. Phya. 32, 44G7–4471 (1981).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Phys. D (1)

C. A. Worrell, V. Skarda, “CO2 Laser Waveguides from Germania-Based Glasses,” J. Phys. D 22, 535–541 (1989).
[CrossRef]

Opt. Laser Technol. (1)

Y. Matsuura, M. Miyagi, A. Hongo, “Loss Reduction of Dielectric-Coated Metallic Hollow Waveguides for CO2 Laser Light Transmission,” Opt. Laser Technol. 22, 141–145 (1990).
[CrossRef]

Phys. Chem. Glasses (1)

B. V. Janakirama Rao, “The Dual Role of Titanium in the System K2O.SiO2.TiO2,” Phys. Chem. Glasses 4, 22–34 (1963).

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

J. A. Harrington, C. C. Gregory, R. Nubling, “Hollow Waveguides for CO2 Laser Delivery System,” Proc. Soc. Photo-Opt. Instrum. Eng. 1048, 117–121 (1989).

Other (3)

T. Kano, Y. Saito, A. Nomura, “A Long-Path Differential Monitoring Using a Hollow-Core Fiber in the Infrared Region,” in Proceedings, Symposium on Guided Wave Technology and Its Application for Mid-Infrared, Sendai, Japan (13–14 Mar. 1986) pp. 165–172, in Japanese.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, Orlando, FL, 1985), Part 2, pp. 749–768.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1985), pp. 36–66.

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Figures (9)

Fig. 1
Fig. 1

Compositions in wt% of melted (a) TiO2–SiO2–Na2O and (b) TiO2–SiO2–K2O mixtures. Open circles and closed circles correspond to materials that were vitrified and crystallized, respectively.

Fig. 2
Fig. 2

Optical system for reflection measurement.

Fig. 3
Fig. 3

Real and imaginary parts n and κ of the refractive index of SiO2 glass. Circles denote the data that were measured in this work and lines denote data taken from Ref. 15.

Fig. 4
Fig. 4

Refractive indices of TiO2–SiO2 based glasses with various compositions: (a), (b) glasses including 30-wt% Na2O and 30-wt% K2O, respectively; a, b, c, and d, TiO2 content of 10, 20, 30, and 40 wt%, respectively.

Fig. 5
Fig. 5

Spectral regions where n is less than unity in TiO2–SiO2 based glasses. Mixed glasses include Na2O (dot–dash lines) or K2O (solid lines) of 30 mol% and a pure SiO2 glass (dashed line) includes no alkaline oxide component.

Fig. 6
Fig. 6

Loss spectra of TiO2–SiO2 hollow waveguides of 1 mm diameter calculated theoretically by using the experimental data. Dashed and solid lines correspond to the glasses containing Na2O and K2O, respectively.

Fig. 7
Fig. 7

Minimum waveguide loss and wavelength λmin where the minimum loss is attained.

Fig. 8
Fig. 8

Dependence of a waveguide loss at 10.6 μm on TiO2 content.

Fig. 9
Fig. 9

Optical loss of a CO2 laser in a hollow waveguide calculated as a function of refractive indices n and κ. The inner diameter of the waveguide is assumed to be 1 mm.

Equations (14)

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

R s = ( cos θ X ) 2 + Y 2 ( cos θ + X ) 2 + Y 2 ,
R p = [ ( X 2 Y 2 + sin 2 θ ) cos θ X ] 2 + ( 1 2 X cos θ ) 2 Y 2 [ ( X 2 Y 2 + sin 2 θ ) cos θ + X ] 2 + ( 1 + 2 X cos θ ) 2 Y 2 ,
2 X 2 = [ ( n 2 κ 2 sin 2 θ ) 2 + 4 n 2 κ 2 ] 1 / 2 + ( n 2 κ 2 sin 2 θ ) ,
2 Y 2 = [ ( n 2 κ 2 sin 2 θ ) 2 + 4 n 2 κ 2 ] 1 / 2 ( n 2 κ 2 sin 2 θ ) ,
X = ( 1 R p / R s ) cos 2 θ / cos θ 2 ( 1 + R p / R s ) + 4 cos 2 θ ( 1 R p ) / ( 1 R s ) ,
Y = { [ R s ( cos θ + X ) 2 ( cos θ X ) 2 ] / ( 1 R s ) } 1 / 2 .
n = { ( X 2 Y 2 + sin 2 θ ) + [ ( X 2 Y 2 + sin 2 θ ) 2 + ( 2 X Y ) 2 ] 1 / 2 2 } 1 / 2 ,
κ = X Y n .
I = I o T o R .
I 1 = I o T o R o ,
I 2 = I o T o R o 3
R o = ( I 2 / I 1 ) 1 / 2 .
R = ( I / I 1 ) R o = ( I / I 1 ) ( I 2 / I 1 ) 1 / 2 .
α = ( 2 . 4048 2 π ) 2 λ 2 T 3 Re { 1 / 2 ( ν 2 + 1 ) ( ν 2 1 ) 1 / 2 } ,

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