Abstract

Silica glass exhibits a permanent anisotropic response, polarization-induced birefringence (PIB), when exposed to short-wavelength polarized light. This behavior has been correlated with the OH content of the glass. In this paper we describe polarized infrared studies of silica glasses of different OH content exposed with polarized 157nm laser light. Changes in the fundamental OH band as a consequence of exposure are shown. We find differential bleaching of a particular OH band where OH species that are oriented parallel to the incident exposing polarization undergo greater bleaching than those oriented perpendicular. The preferential bleaching as a function of exposure time correlates strongly with the evolution of PIB, leading to a bleaching model of OH that is causally linked to PIB.

© 2006 Optical Society of America

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References

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  1. C. M. Smith and N. F. Borrelli, 'Behavior of 157 nm excimer laser-induced refractive changes in silica,' J. Opt. Soc. Am. B 26, 1815-1821 (2006).
    [CrossRef]
  2. U. Neukirch, D. C. Allan, N. F. Borrelli, C. E. Heckle, M. Mlejnek, J. Moll, and C. M. Smith, 'Laser induced birefringence in fused silica from polarized lasers,' in Optical Microlithography XVII, B.Smith, ed., Proc. SPIE 5754, 638-645 (2005).
  3. C. K. Van Peski, R. Morton, and Z. Bor, 'Behavior of fused silica irradiated by low level 193-nm excimer laser for tens of billions of pulses,' J. Non-Cryst. Solids 265, 285-289 (2000).
    [CrossRef]
  4. N. F. Borrelli, C. M. Smith, J. J. Price, and D. C. Allan, 'Polarized excimer laser-induced birefringence in silica,' Appl. Phys. Lett. 80, 219-221 (2002).
    [CrossRef]
  5. B. Kuhn, S. Kaiser, I. Radosevic, B. Uebbing, and S. Thomas, 'Synthetic fused silica tailored for 193 nm immersion lithography,' presented at the Sematech 2nd International Symposium on Immersion Lithography, September 12-15, 2005, Bruges, Belgium.
  6. V. G. Plotnichenko, V. O. Sokolov, and E. M. Dianov, 'Hydroxyl groups in high-purity silica glass,' J. Non-Cryst. Solids 261, 186-194 (2000).
    [CrossRef]
  7. K. M. Davis and M. Tomozawa, 'An infrared spectroscopic study of water-related species in silica glasses,' J. Non-Cryst. Solids 201, 177-198 (1996).
    [CrossRef]
  8. Y. Morimoto and S. Nozawa, 'Effect of Xe2* light (7.2 eV) on the infrared and vacuum ultraviolet absorption properties of hydroxyl groups in silica glass,' Phys. Rev. B 59, 4066-4073 (1999).
    [CrossRef]
  9. K. Kajihara, Y. Ikuta, M. Hirano, T. Ichimura, and H. Hosono, 'Interaction of F2 excimer laser pulses with hydroxy groups in SiO2 glass: hydrogen bond formation and bleaching of vacuum ultraviolet absorption edge,' J. Chem. Phys. 115, 9473-9476 (2001).
    [CrossRef]
  10. U. Natura, O. Sohr, R. Martin, M. Kahlke, and G. Fasold, 'Mechanism of radiation-induced defect generation in fused silica,' in Laser-Induced Damage in Optical Materials, G.J.Exharos, A.H.Guentheer, N.Kaiser, K.L.Lewis, M.J.Soileau, and C.J.Stolz, eds., Proc. SPIE 5273, 155-164 (2003).
  11. J. E. Shelby, 'Protonic species in vitreous silica,' J. Non-Cryst. Solids 179, 138-147 (1994).
    [CrossRef]
  12. T. Erdogan and V. Mizrahi, 'Characterization of UV-induced birefringence in photosensitive Ge-doped optical fibers,' J. Opt. Soc. Am. B 11, 2100-2105 (1994).
    [CrossRef]
  13. J. Canning, H. J. Deyerl, H. R. Sorensen, and M. Kristensen, 'Ultraviolet-induced birefringence in hydrogen-loaded optical fiber,' J. Appl. Phys. 97, 053104 (2005).
    [CrossRef]

2006 (1)

C. M. Smith and N. F. Borrelli, 'Behavior of 157 nm excimer laser-induced refractive changes in silica,' J. Opt. Soc. Am. B 26, 1815-1821 (2006).
[CrossRef]

2005 (1)

J. Canning, H. J. Deyerl, H. R. Sorensen, and M. Kristensen, 'Ultraviolet-induced birefringence in hydrogen-loaded optical fiber,' J. Appl. Phys. 97, 053104 (2005).
[CrossRef]

2002 (1)

N. F. Borrelli, C. M. Smith, J. J. Price, and D. C. Allan, 'Polarized excimer laser-induced birefringence in silica,' Appl. Phys. Lett. 80, 219-221 (2002).
[CrossRef]

2001 (1)

K. Kajihara, Y. Ikuta, M. Hirano, T. Ichimura, and H. Hosono, 'Interaction of F2 excimer laser pulses with hydroxy groups in SiO2 glass: hydrogen bond formation and bleaching of vacuum ultraviolet absorption edge,' J. Chem. Phys. 115, 9473-9476 (2001).
[CrossRef]

2000 (2)

V. G. Plotnichenko, V. O. Sokolov, and E. M. Dianov, 'Hydroxyl groups in high-purity silica glass,' J. Non-Cryst. Solids 261, 186-194 (2000).
[CrossRef]

C. K. Van Peski, R. Morton, and Z. Bor, 'Behavior of fused silica irradiated by low level 193-nm excimer laser for tens of billions of pulses,' J. Non-Cryst. Solids 265, 285-289 (2000).
[CrossRef]

1999 (1)

Y. Morimoto and S. Nozawa, 'Effect of Xe2* light (7.2 eV) on the infrared and vacuum ultraviolet absorption properties of hydroxyl groups in silica glass,' Phys. Rev. B 59, 4066-4073 (1999).
[CrossRef]

1996 (1)

K. M. Davis and M. Tomozawa, 'An infrared spectroscopic study of water-related species in silica glasses,' J. Non-Cryst. Solids 201, 177-198 (1996).
[CrossRef]

1994 (2)

Allan, D. C.

N. F. Borrelli, C. M. Smith, J. J. Price, and D. C. Allan, 'Polarized excimer laser-induced birefringence in silica,' Appl. Phys. Lett. 80, 219-221 (2002).
[CrossRef]

U. Neukirch, D. C. Allan, N. F. Borrelli, C. E. Heckle, M. Mlejnek, J. Moll, and C. M. Smith, 'Laser induced birefringence in fused silica from polarized lasers,' in Optical Microlithography XVII, B.Smith, ed., Proc. SPIE 5754, 638-645 (2005).

Bor, Z.

C. K. Van Peski, R. Morton, and Z. Bor, 'Behavior of fused silica irradiated by low level 193-nm excimer laser for tens of billions of pulses,' J. Non-Cryst. Solids 265, 285-289 (2000).
[CrossRef]

Borrelli, N. F.

C. M. Smith and N. F. Borrelli, 'Behavior of 157 nm excimer laser-induced refractive changes in silica,' J. Opt. Soc. Am. B 26, 1815-1821 (2006).
[CrossRef]

N. F. Borrelli, C. M. Smith, J. J. Price, and D. C. Allan, 'Polarized excimer laser-induced birefringence in silica,' Appl. Phys. Lett. 80, 219-221 (2002).
[CrossRef]

U. Neukirch, D. C. Allan, N. F. Borrelli, C. E. Heckle, M. Mlejnek, J. Moll, and C. M. Smith, 'Laser induced birefringence in fused silica from polarized lasers,' in Optical Microlithography XVII, B.Smith, ed., Proc. SPIE 5754, 638-645 (2005).

Canning, J.

J. Canning, H. J. Deyerl, H. R. Sorensen, and M. Kristensen, 'Ultraviolet-induced birefringence in hydrogen-loaded optical fiber,' J. Appl. Phys. 97, 053104 (2005).
[CrossRef]

Davis, K. M.

K. M. Davis and M. Tomozawa, 'An infrared spectroscopic study of water-related species in silica glasses,' J. Non-Cryst. Solids 201, 177-198 (1996).
[CrossRef]

Deyerl, H. J.

J. Canning, H. J. Deyerl, H. R. Sorensen, and M. Kristensen, 'Ultraviolet-induced birefringence in hydrogen-loaded optical fiber,' J. Appl. Phys. 97, 053104 (2005).
[CrossRef]

Dianov, E. M.

V. G. Plotnichenko, V. O. Sokolov, and E. M. Dianov, 'Hydroxyl groups in high-purity silica glass,' J. Non-Cryst. Solids 261, 186-194 (2000).
[CrossRef]

Erdogan, T.

Fasold, G.

U. Natura, O. Sohr, R. Martin, M. Kahlke, and G. Fasold, 'Mechanism of radiation-induced defect generation in fused silica,' in Laser-Induced Damage in Optical Materials, G.J.Exharos, A.H.Guentheer, N.Kaiser, K.L.Lewis, M.J.Soileau, and C.J.Stolz, eds., Proc. SPIE 5273, 155-164 (2003).

Heckle, C. E.

U. Neukirch, D. C. Allan, N. F. Borrelli, C. E. Heckle, M. Mlejnek, J. Moll, and C. M. Smith, 'Laser induced birefringence in fused silica from polarized lasers,' in Optical Microlithography XVII, B.Smith, ed., Proc. SPIE 5754, 638-645 (2005).

Hirano, M.

K. Kajihara, Y. Ikuta, M. Hirano, T. Ichimura, and H. Hosono, 'Interaction of F2 excimer laser pulses with hydroxy groups in SiO2 glass: hydrogen bond formation and bleaching of vacuum ultraviolet absorption edge,' J. Chem. Phys. 115, 9473-9476 (2001).
[CrossRef]

Hosono, H.

K. Kajihara, Y. Ikuta, M. Hirano, T. Ichimura, and H. Hosono, 'Interaction of F2 excimer laser pulses with hydroxy groups in SiO2 glass: hydrogen bond formation and bleaching of vacuum ultraviolet absorption edge,' J. Chem. Phys. 115, 9473-9476 (2001).
[CrossRef]

Ichimura, T.

K. Kajihara, Y. Ikuta, M. Hirano, T. Ichimura, and H. Hosono, 'Interaction of F2 excimer laser pulses with hydroxy groups in SiO2 glass: hydrogen bond formation and bleaching of vacuum ultraviolet absorption edge,' J. Chem. Phys. 115, 9473-9476 (2001).
[CrossRef]

Ikuta, Y.

K. Kajihara, Y. Ikuta, M. Hirano, T. Ichimura, and H. Hosono, 'Interaction of F2 excimer laser pulses with hydroxy groups in SiO2 glass: hydrogen bond formation and bleaching of vacuum ultraviolet absorption edge,' J. Chem. Phys. 115, 9473-9476 (2001).
[CrossRef]

Kahlke, M.

U. Natura, O. Sohr, R. Martin, M. Kahlke, and G. Fasold, 'Mechanism of radiation-induced defect generation in fused silica,' in Laser-Induced Damage in Optical Materials, G.J.Exharos, A.H.Guentheer, N.Kaiser, K.L.Lewis, M.J.Soileau, and C.J.Stolz, eds., Proc. SPIE 5273, 155-164 (2003).

Kaiser, S.

B. Kuhn, S. Kaiser, I. Radosevic, B. Uebbing, and S. Thomas, 'Synthetic fused silica tailored for 193 nm immersion lithography,' presented at the Sematech 2nd International Symposium on Immersion Lithography, September 12-15, 2005, Bruges, Belgium.

Kajihara, K.

K. Kajihara, Y. Ikuta, M. Hirano, T. Ichimura, and H. Hosono, 'Interaction of F2 excimer laser pulses with hydroxy groups in SiO2 glass: hydrogen bond formation and bleaching of vacuum ultraviolet absorption edge,' J. Chem. Phys. 115, 9473-9476 (2001).
[CrossRef]

Kristensen, M.

J. Canning, H. J. Deyerl, H. R. Sorensen, and M. Kristensen, 'Ultraviolet-induced birefringence in hydrogen-loaded optical fiber,' J. Appl. Phys. 97, 053104 (2005).
[CrossRef]

Kuhn, B.

B. Kuhn, S. Kaiser, I. Radosevic, B. Uebbing, and S. Thomas, 'Synthetic fused silica tailored for 193 nm immersion lithography,' presented at the Sematech 2nd International Symposium on Immersion Lithography, September 12-15, 2005, Bruges, Belgium.

Martin, R.

U. Natura, O. Sohr, R. Martin, M. Kahlke, and G. Fasold, 'Mechanism of radiation-induced defect generation in fused silica,' in Laser-Induced Damage in Optical Materials, G.J.Exharos, A.H.Guentheer, N.Kaiser, K.L.Lewis, M.J.Soileau, and C.J.Stolz, eds., Proc. SPIE 5273, 155-164 (2003).

Mizrahi, V.

Mlejnek, M.

U. Neukirch, D. C. Allan, N. F. Borrelli, C. E. Heckle, M. Mlejnek, J. Moll, and C. M. Smith, 'Laser induced birefringence in fused silica from polarized lasers,' in Optical Microlithography XVII, B.Smith, ed., Proc. SPIE 5754, 638-645 (2005).

Moll, J.

U. Neukirch, D. C. Allan, N. F. Borrelli, C. E. Heckle, M. Mlejnek, J. Moll, and C. M. Smith, 'Laser induced birefringence in fused silica from polarized lasers,' in Optical Microlithography XVII, B.Smith, ed., Proc. SPIE 5754, 638-645 (2005).

Morimoto, Y.

Y. Morimoto and S. Nozawa, 'Effect of Xe2* light (7.2 eV) on the infrared and vacuum ultraviolet absorption properties of hydroxyl groups in silica glass,' Phys. Rev. B 59, 4066-4073 (1999).
[CrossRef]

Morton, R.

C. K. Van Peski, R. Morton, and Z. Bor, 'Behavior of fused silica irradiated by low level 193-nm excimer laser for tens of billions of pulses,' J. Non-Cryst. Solids 265, 285-289 (2000).
[CrossRef]

Natura, U.

U. Natura, O. Sohr, R. Martin, M. Kahlke, and G. Fasold, 'Mechanism of radiation-induced defect generation in fused silica,' in Laser-Induced Damage in Optical Materials, G.J.Exharos, A.H.Guentheer, N.Kaiser, K.L.Lewis, M.J.Soileau, and C.J.Stolz, eds., Proc. SPIE 5273, 155-164 (2003).

Neukirch, U.

U. Neukirch, D. C. Allan, N. F. Borrelli, C. E. Heckle, M. Mlejnek, J. Moll, and C. M. Smith, 'Laser induced birefringence in fused silica from polarized lasers,' in Optical Microlithography XVII, B.Smith, ed., Proc. SPIE 5754, 638-645 (2005).

Nozawa, S.

Y. Morimoto and S. Nozawa, 'Effect of Xe2* light (7.2 eV) on the infrared and vacuum ultraviolet absorption properties of hydroxyl groups in silica glass,' Phys. Rev. B 59, 4066-4073 (1999).
[CrossRef]

Plotnichenko, V. G.

V. G. Plotnichenko, V. O. Sokolov, and E. M. Dianov, 'Hydroxyl groups in high-purity silica glass,' J. Non-Cryst. Solids 261, 186-194 (2000).
[CrossRef]

Price, J. J.

N. F. Borrelli, C. M. Smith, J. J. Price, and D. C. Allan, 'Polarized excimer laser-induced birefringence in silica,' Appl. Phys. Lett. 80, 219-221 (2002).
[CrossRef]

Radosevic, I.

B. Kuhn, S. Kaiser, I. Radosevic, B. Uebbing, and S. Thomas, 'Synthetic fused silica tailored for 193 nm immersion lithography,' presented at the Sematech 2nd International Symposium on Immersion Lithography, September 12-15, 2005, Bruges, Belgium.

Shelby, J. E.

J. E. Shelby, 'Protonic species in vitreous silica,' J. Non-Cryst. Solids 179, 138-147 (1994).
[CrossRef]

Smith, C. M.

C. M. Smith and N. F. Borrelli, 'Behavior of 157 nm excimer laser-induced refractive changes in silica,' J. Opt. Soc. Am. B 26, 1815-1821 (2006).
[CrossRef]

N. F. Borrelli, C. M. Smith, J. J. Price, and D. C. Allan, 'Polarized excimer laser-induced birefringence in silica,' Appl. Phys. Lett. 80, 219-221 (2002).
[CrossRef]

U. Neukirch, D. C. Allan, N. F. Borrelli, C. E. Heckle, M. Mlejnek, J. Moll, and C. M. Smith, 'Laser induced birefringence in fused silica from polarized lasers,' in Optical Microlithography XVII, B.Smith, ed., Proc. SPIE 5754, 638-645 (2005).

Sohr, O.

U. Natura, O. Sohr, R. Martin, M. Kahlke, and G. Fasold, 'Mechanism of radiation-induced defect generation in fused silica,' in Laser-Induced Damage in Optical Materials, G.J.Exharos, A.H.Guentheer, N.Kaiser, K.L.Lewis, M.J.Soileau, and C.J.Stolz, eds., Proc. SPIE 5273, 155-164 (2003).

Sokolov, V. O.

V. G. Plotnichenko, V. O. Sokolov, and E. M. Dianov, 'Hydroxyl groups in high-purity silica glass,' J. Non-Cryst. Solids 261, 186-194 (2000).
[CrossRef]

Sorensen, H. R.

J. Canning, H. J. Deyerl, H. R. Sorensen, and M. Kristensen, 'Ultraviolet-induced birefringence in hydrogen-loaded optical fiber,' J. Appl. Phys. 97, 053104 (2005).
[CrossRef]

Thomas, S.

B. Kuhn, S. Kaiser, I. Radosevic, B. Uebbing, and S. Thomas, 'Synthetic fused silica tailored for 193 nm immersion lithography,' presented at the Sematech 2nd International Symposium on Immersion Lithography, September 12-15, 2005, Bruges, Belgium.

Tomozawa, M.

K. M. Davis and M. Tomozawa, 'An infrared spectroscopic study of water-related species in silica glasses,' J. Non-Cryst. Solids 201, 177-198 (1996).
[CrossRef]

Uebbing, B.

B. Kuhn, S. Kaiser, I. Radosevic, B. Uebbing, and S. Thomas, 'Synthetic fused silica tailored for 193 nm immersion lithography,' presented at the Sematech 2nd International Symposium on Immersion Lithography, September 12-15, 2005, Bruges, Belgium.

Van Peski, C. K.

C. K. Van Peski, R. Morton, and Z. Bor, 'Behavior of fused silica irradiated by low level 193-nm excimer laser for tens of billions of pulses,' J. Non-Cryst. Solids 265, 285-289 (2000).
[CrossRef]

Appl. Phys. Lett. (1)

N. F. Borrelli, C. M. Smith, J. J. Price, and D. C. Allan, 'Polarized excimer laser-induced birefringence in silica,' Appl. Phys. Lett. 80, 219-221 (2002).
[CrossRef]

J. Appl. Phys. (1)

J. Canning, H. J. Deyerl, H. R. Sorensen, and M. Kristensen, 'Ultraviolet-induced birefringence in hydrogen-loaded optical fiber,' J. Appl. Phys. 97, 053104 (2005).
[CrossRef]

J. Chem. Phys. (1)

K. Kajihara, Y. Ikuta, M. Hirano, T. Ichimura, and H. Hosono, 'Interaction of F2 excimer laser pulses with hydroxy groups in SiO2 glass: hydrogen bond formation and bleaching of vacuum ultraviolet absorption edge,' J. Chem. Phys. 115, 9473-9476 (2001).
[CrossRef]

J. Non-Cryst. Solids (4)

V. G. Plotnichenko, V. O. Sokolov, and E. M. Dianov, 'Hydroxyl groups in high-purity silica glass,' J. Non-Cryst. Solids 261, 186-194 (2000).
[CrossRef]

K. M. Davis and M. Tomozawa, 'An infrared spectroscopic study of water-related species in silica glasses,' J. Non-Cryst. Solids 201, 177-198 (1996).
[CrossRef]

C. K. Van Peski, R. Morton, and Z. Bor, 'Behavior of fused silica irradiated by low level 193-nm excimer laser for tens of billions of pulses,' J. Non-Cryst. Solids 265, 285-289 (2000).
[CrossRef]

J. E. Shelby, 'Protonic species in vitreous silica,' J. Non-Cryst. Solids 179, 138-147 (1994).
[CrossRef]

J. Opt. Soc. Am. B (2)

T. Erdogan and V. Mizrahi, 'Characterization of UV-induced birefringence in photosensitive Ge-doped optical fibers,' J. Opt. Soc. Am. B 11, 2100-2105 (1994).
[CrossRef]

C. M. Smith and N. F. Borrelli, 'Behavior of 157 nm excimer laser-induced refractive changes in silica,' J. Opt. Soc. Am. B 26, 1815-1821 (2006).
[CrossRef]

Phys. Rev. B (1)

Y. Morimoto and S. Nozawa, 'Effect of Xe2* light (7.2 eV) on the infrared and vacuum ultraviolet absorption properties of hydroxyl groups in silica glass,' Phys. Rev. B 59, 4066-4073 (1999).
[CrossRef]

Other (3)

U. Natura, O. Sohr, R. Martin, M. Kahlke, and G. Fasold, 'Mechanism of radiation-induced defect generation in fused silica,' in Laser-Induced Damage in Optical Materials, G.J.Exharos, A.H.Guentheer, N.Kaiser, K.L.Lewis, M.J.Soileau, and C.J.Stolz, eds., Proc. SPIE 5273, 155-164 (2003).

U. Neukirch, D. C. Allan, N. F. Borrelli, C. E. Heckle, M. Mlejnek, J. Moll, and C. M. Smith, 'Laser induced birefringence in fused silica from polarized lasers,' in Optical Microlithography XVII, B.Smith, ed., Proc. SPIE 5754, 638-645 (2005).

B. Kuhn, S. Kaiser, I. Radosevic, B. Uebbing, and S. Thomas, 'Synthetic fused silica tailored for 193 nm immersion lithography,' presented at the Sematech 2nd International Symposium on Immersion Lithography, September 12-15, 2005, Bruges, Belgium.

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Figures (11)

Fig. 1
Fig. 1

Polarization-induced birefringence (PIB) as a function of pulses for silica glasses of different OH contents. Exposure fluence was 0.24 mJ cm 2 . Figure taken from Ref. [1]; used with permission.

Fig. 2
Fig. 2

Fundamental OH band of silica (thick solid curve) with its components shown as dotted curves. The band ν 1 is assigned to the “free” (non-H-bonded) OH species (schematic structure shown at right). Bands labeled ν 2 4 are due to OH species in various H-bonded environments. Figure taken from Ref. [6]; used with permission.

Fig. 3
Fig. 3

Unexposed, exposed and difference IR spectrum (exposed minus unexposed) of 1200 ppm OH glass exposed with polarized 157 - nm light. IR measurements were made (a) using polarized light, with the polarization parallel to the incident exposing polarization and (b) with light polarized perpendicular to the incident exposing polarization. (c) Comparison of IR difference spectra from (a) and (b) for measurements made with light polarized parallel to incident exposing polarization (dotted curve) and perpendicular (solid curve).

Fig. 4
Fig. 4

(Parallel OH minus perpendicular OH) as a function of pulses for 1200 ppm OH glass exposed at 0.24 mJ cm 2 .

Fig. 5
Fig. 5

(a) Δ O H F and (b) δ O H as a function of number of exposure pulses for glasses with different OH contents.

Fig. 6
Fig. 6

Comparison of Δ O H F (left axis) and PIB (right axis) as a function of number of exposure pulses for glass with 500 ppm OH.

Fig. 7
Fig. 7

PIB at saturation as a function of Δ O H F at steady state for glasses with different OH content. The line is a guide for the eye.

Fig. 8
Fig. 8

Δ O H F normalized to total OH content as a function of number of exposure pulses for glasses with different OH content.

Fig. 9
Fig. 9

Fits (curves) to the experimental IR data (points) using bleaching model. Δ O H F and Δ O H F are amounts of free OH bleached, obtained by taking the difference spectra between exposed and unexposed glass, for each measurement polarization. All quantities are normalized to total initial OH content.

Fig. 10
Fig. 10

Difference spectra (exposed minus unexposed) for low OH glass exposed with polarized 193 nm light. Measurement made with polarized light parallel to incident exposure polarization is shown with dotted curve; perpendicular orientation is solid curve.

Fig. 11
Fig. 11

PIB as a function of Δ O H F for low OH glass exposed with polarized 157 nm light. Linear fit to the data is shown as dotted line; equation of the line is given in the figure.

Equations (10)

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

W = k 1 λ NA ,
O H k I O H B ,
O H k I O H B ,
O H B k r I O H F .
N 0 = N + N x + N y + N z = N + N + N .
d N d I t = k N + k r N 3 ,
d N d I t = k N + 2 k r N 3 ,
d N d I t = + k N + k N k r N .
at t = 0 { N = N 2 N = N i } .
PIB = n x n y = N x α N y α ,

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