Abstract

The contribution made to the propagation of a Gaussian beam by the spherical aberrations introduced by the lens used to focus the beam into the test sample is considered as a possible explanation of the multivestige structure observed in laser-induced damage of transparent materials. The paraxial approximation for the wave equation is used to determine the intensity of the beam on the beam axis when the aberrations are present. The inclusion of spherical aberrations modifies the form expected for a diffraction-limited Gaussian beam, shifting the main peak in intensity away from the focus and suppressing it. More importantly, oscillations are introduced in the intensity prior to (past) the focus when the spherical aberrations focus off-axis rays prior to (past) the geometrical focus. Although the spatial arrangement of the peaks in the intensity appears consistent with some of the experimental results for the vestige structure, the spacing between peaks does not correspond to the observed spacing between damage sites. These findings and other possible explanations of the vestige structure are considered critically.

© 1981 Optical Society of America

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References

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  1. Yu. K. Danileiko and et al., “Investigation of mechanisms of damage to semiconductors by high-power infrared laser radiation,” Zh. Eksp. Teor. Fiz. 74, 765–771 (1978) [Sov. Phys. JETP 47, 401–404 (1978)].
  2. W. L. Smith, J. H. Bechtel, and N. Bloembergen, “Picosecond laser-induced damaged morphology: spatially resolved microscopic plasma sites,” Opt. Commun. 18, 592–596 (1976).
    [Crossref]
  3. W. L. Smith, J. H. Bechtel, and N. Bloembergen, “Picosecond laser-induced breakdown at 5321 and 3547 Å: observation of frequency dependent behavior,” Phys. Rev. B 15, 4039–4055 (1977).
    [Crossref]
  4. J. P. Anthes and M. Bass, “Direct observation of the dynamics of picosecond-pulse optical breakdown,” Appl. Phys. Lett. 31, 412–414 (1977).
    [Crossref]
  5. L. R. Evans and C. G. Morgan, “Lens aberration effects in optical-frequency breakdown of gases,” Phys. Rev. Lett. 22, 1099–1102 (1969).
    [Crossref]
  6. M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [Crossref]
  7. F. Tappert, “Diffractive ray tracing of laser beams,” J. Opt. Soc. Am. 66, 1368–1373 (1976).
    [Crossref]
  8. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, England, 1975).
  9. C. L. M. Ireland and et al., “Focal-length dependence of air breakdown by a 20-psec laser pulse,” Appl. Phys. Lett. 24, 175–177 (1974).
    [Crossref]
  10. A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), Chap. 6.
  11. W. L. Smith, J. H. Bechtel, and N. Bloembergen, “Delectric-breakdown threshold and nonlinear-refractive-index measurement with picosecond laser pulses,” Phys. Rev. B 12, 706–714 (1975).
    [Crossref]

1978 (1)

Yu. K. Danileiko and et al., “Investigation of mechanisms of damage to semiconductors by high-power infrared laser radiation,” Zh. Eksp. Teor. Fiz. 74, 765–771 (1978) [Sov. Phys. JETP 47, 401–404 (1978)].

1977 (2)

W. L. Smith, J. H. Bechtel, and N. Bloembergen, “Picosecond laser-induced breakdown at 5321 and 3547 Å: observation of frequency dependent behavior,” Phys. Rev. B 15, 4039–4055 (1977).
[Crossref]

J. P. Anthes and M. Bass, “Direct observation of the dynamics of picosecond-pulse optical breakdown,” Appl. Phys. Lett. 31, 412–414 (1977).
[Crossref]

1976 (2)

W. L. Smith, J. H. Bechtel, and N. Bloembergen, “Picosecond laser-induced damaged morphology: spatially resolved microscopic plasma sites,” Opt. Commun. 18, 592–596 (1976).
[Crossref]

F. Tappert, “Diffractive ray tracing of laser beams,” J. Opt. Soc. Am. 66, 1368–1373 (1976).
[Crossref]

1975 (2)

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[Crossref]

W. L. Smith, J. H. Bechtel, and N. Bloembergen, “Delectric-breakdown threshold and nonlinear-refractive-index measurement with picosecond laser pulses,” Phys. Rev. B 12, 706–714 (1975).
[Crossref]

1974 (1)

C. L. M. Ireland and et al., “Focal-length dependence of air breakdown by a 20-psec laser pulse,” Appl. Phys. Lett. 24, 175–177 (1974).
[Crossref]

1969 (1)

L. R. Evans and C. G. Morgan, “Lens aberration effects in optical-frequency breakdown of gases,” Phys. Rev. Lett. 22, 1099–1102 (1969).
[Crossref]

Anthes, J. P.

J. P. Anthes and M. Bass, “Direct observation of the dynamics of picosecond-pulse optical breakdown,” Appl. Phys. Lett. 31, 412–414 (1977).
[Crossref]

Bass, M.

J. P. Anthes and M. Bass, “Direct observation of the dynamics of picosecond-pulse optical breakdown,” Appl. Phys. Lett. 31, 412–414 (1977).
[Crossref]

Bechtel, J. H.

W. L. Smith, J. H. Bechtel, and N. Bloembergen, “Picosecond laser-induced breakdown at 5321 and 3547 Å: observation of frequency dependent behavior,” Phys. Rev. B 15, 4039–4055 (1977).
[Crossref]

W. L. Smith, J. H. Bechtel, and N. Bloembergen, “Picosecond laser-induced damaged morphology: spatially resolved microscopic plasma sites,” Opt. Commun. 18, 592–596 (1976).
[Crossref]

W. L. Smith, J. H. Bechtel, and N. Bloembergen, “Delectric-breakdown threshold and nonlinear-refractive-index measurement with picosecond laser pulses,” Phys. Rev. B 12, 706–714 (1975).
[Crossref]

Bloembergen, N.

W. L. Smith, J. H. Bechtel, and N. Bloembergen, “Picosecond laser-induced breakdown at 5321 and 3547 Å: observation of frequency dependent behavior,” Phys. Rev. B 15, 4039–4055 (1977).
[Crossref]

W. L. Smith, J. H. Bechtel, and N. Bloembergen, “Picosecond laser-induced damaged morphology: spatially resolved microscopic plasma sites,” Opt. Commun. 18, 592–596 (1976).
[Crossref]

W. L. Smith, J. H. Bechtel, and N. Bloembergen, “Delectric-breakdown threshold and nonlinear-refractive-index measurement with picosecond laser pulses,” Phys. Rev. B 12, 706–714 (1975).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, England, 1975).

Danileiko, Yu. K.

Yu. K. Danileiko and et al., “Investigation of mechanisms of damage to semiconductors by high-power infrared laser radiation,” Zh. Eksp. Teor. Fiz. 74, 765–771 (1978) [Sov. Phys. JETP 47, 401–404 (1978)].

Evans, L. R.

L. R. Evans and C. G. Morgan, “Lens aberration effects in optical-frequency breakdown of gases,” Phys. Rev. Lett. 22, 1099–1102 (1969).
[Crossref]

Ireland, C. L. M.

C. L. M. Ireland and et al., “Focal-length dependence of air breakdown by a 20-psec laser pulse,” Appl. Phys. Lett. 24, 175–177 (1974).
[Crossref]

Lax, M.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[Crossref]

Louisell, W. H.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[Crossref]

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[Crossref]

Morgan, C. G.

L. R. Evans and C. G. Morgan, “Lens aberration effects in optical-frequency breakdown of gases,” Phys. Rev. Lett. 22, 1099–1102 (1969).
[Crossref]

Smith, W. L.

W. L. Smith, J. H. Bechtel, and N. Bloembergen, “Picosecond laser-induced breakdown at 5321 and 3547 Å: observation of frequency dependent behavior,” Phys. Rev. B 15, 4039–4055 (1977).
[Crossref]

W. L. Smith, J. H. Bechtel, and N. Bloembergen, “Picosecond laser-induced damaged morphology: spatially resolved microscopic plasma sites,” Opt. Commun. 18, 592–596 (1976).
[Crossref]

W. L. Smith, J. H. Bechtel, and N. Bloembergen, “Delectric-breakdown threshold and nonlinear-refractive-index measurement with picosecond laser pulses,” Phys. Rev. B 12, 706–714 (1975).
[Crossref]

Tappert, F.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, England, 1975).

Yariv, A.

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), Chap. 6.

Appl. Phys. Lett. (2)

J. P. Anthes and M. Bass, “Direct observation of the dynamics of picosecond-pulse optical breakdown,” Appl. Phys. Lett. 31, 412–414 (1977).
[Crossref]

C. L. M. Ireland and et al., “Focal-length dependence of air breakdown by a 20-psec laser pulse,” Appl. Phys. Lett. 24, 175–177 (1974).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

W. L. Smith, J. H. Bechtel, and N. Bloembergen, “Picosecond laser-induced damaged morphology: spatially resolved microscopic plasma sites,” Opt. Commun. 18, 592–596 (1976).
[Crossref]

Phys. Rev. A (1)

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[Crossref]

Phys. Rev. B (2)

W. L. Smith, J. H. Bechtel, and N. Bloembergen, “Picosecond laser-induced breakdown at 5321 and 3547 Å: observation of frequency dependent behavior,” Phys. Rev. B 15, 4039–4055 (1977).
[Crossref]

W. L. Smith, J. H. Bechtel, and N. Bloembergen, “Delectric-breakdown threshold and nonlinear-refractive-index measurement with picosecond laser pulses,” Phys. Rev. B 12, 706–714 (1975).
[Crossref]

Phys. Rev. Lett. (1)

L. R. Evans and C. G. Morgan, “Lens aberration effects in optical-frequency breakdown of gases,” Phys. Rev. Lett. 22, 1099–1102 (1969).
[Crossref]

Zh. Eksp. Teor. Fiz. (1)

Yu. K. Danileiko and et al., “Investigation of mechanisms of damage to semiconductors by high-power infrared laser radiation,” Zh. Eksp. Teor. Fiz. 74, 765–771 (1978) [Sov. Phys. JETP 47, 401–404 (1978)].

Other (2)

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, England, 1975).

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), Chap. 6.

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Figures (5)

Fig. 1
Fig. 1

Location of the lens at zL and the sample at zS. The coordinate directions x ˆ and z ˆ are indicated. Three light rays are shown: (a) a near-axial ray for which spherical aberrations are unimportant and off-axis rays (b) without and (c) with spherical aberration. zL + Rg is the focus inside the sample, and Δ is the spread introduced by spherical aberrations.

Fig. 2
Fig. 2

On-axis intensity for propagation in free space when λ = 1.06 μm and B = 0.01 cm−3. The region shown is prior to the focus zf. The solid lines are for Gaussian beams without aberration (a0 = 0.1 and 0.3 cm for the lower and upper curves, respectively, the maximum intensity when a0 = 0.3 cm is 1.7 × 104). The dashed curves include the aberrations (a0 = 0.1, 0.2, and 0.3 cm, respectively, for the curves in order of increasing magnitude, the maximum intensity when a0 = 0.2 cm is 1.3 × 103 and when a0 = 0.3 cm is 2.3 × 103 and 1.1 × 103 for the first two peaks).

Fig. 3
Fig. 3

On-axis intensity for propagation in free space when λ = 1.06 μm and B = 0.05 cm−3. The solid lines are for beams without aberration (a0 = 0.1 and 0.3 cm for the lower and upper curves, respectively). The dashed curves include the aberrations (a0 = 0.1, 0.2, and 0.3 cm, respectively, for the curves in order of increasing magnitude).

Fig. 4
Fig. 4

On-axis intensity for propagation in free space when λ = 1.06 μm and B = 0.1 cm−3. The solid lines are for beams without aberration (a0 = 0.1 and 0.3 cm for the lower and upper curves, respectively). The dashed curves include the aberrations (a0 = 0.1, 0.12, 0.2, and 0.3 cm, respectively, for the curves in order of increasing magnitude).

Fig. 5
Fig. 5

On-axis intensity for propagation in free space when λ = 0.53 μm and B = 0.1 cm−3. The solid curve is for a beam without aberration (a0 = 0.1 cm). The dashed curves include the aberrations (a0 = 0.1 and 0.2 cm, respectively, for the lower and upper curves, the maximum intensity when a0 = 0.2 cm is 600). The dotted curve includes aberrations when a0 = 0.3 cm and is scaled by 10−1.

Equations (23)

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E ( r , t ) = Re { ( z , x ) exp [ i ( k z - ω t ) ] } .
i z + 1 2 k 2 = 0
( z , x ) = k 2 π i ( z - z p ) d 2 x exp [ i k x - x 2 / 2 ( z - z p ) ] ( z p , x ) ,
( z S , x ) = k 0 2 π i ( z S - z l ) d 2 x exp [ i k 0 x - x 2 / 2 ( z S - z L ) ] ( z L , x ) ,
( z , x ) = k 2 π i ( z - z S ) d 2 x exp [ i k x - x 2 / 2 ( z - z S ) ] ( z S , x ) ,
( z S + 0 + , x ) = exp [ i ( k 0 - k ) z S ] ( z S - 0 + , x ) ,
( z , x ) = k k 0 ( 2 π i ) 2 ( z - z S ) ( z S - z L ) d 2 x d 2 x             exp [ i k x - x 2 / 2 ( z - z S ) ] × exp [ i k 0 x - x 2 / 2 ( z S - z L ) ] ( z L , x ) .
( z , x ) = k e 2 π i ( z - z L ) d 2 x exp [ i k e x - x 2 / 2 ( z - z L ) ] ( z L , x ) ,
k e = n k 0 ( z - z L ) n ( z S - z L ) + z - z S .
k e z - z L = k 0 z ˜ - z L ,
z ˜ = z / n - ( 1 - n ) z S / n .
( z L , x ) = E 0 exp ( - x 2 2 a 0 2 ) exp [ - i k 0 ( x 2 2 R 0 + B x 4 ) ] .
I = n c 8 π { z ˆ ( z , x ) 2 + 1 2 i k [ * ( z , x ) ( z , x ) - ( z , x ) * ( z , x ) ] } .
I ( z ) = c s U ,
I ( z ) = c n k e 2 E 0 2 32 π 3 ( z - z L ) 2 d 2 x d 2 x × exp { i k e 2 ( z - z L ) ( x 2 - x 2 ) - ( x 2 + x 2 ) 2 a 0 2 - i k 0 [ x 2 - x 2 2 R 0 + B ( x 4 - x 4 ) ] } ,
I ( z ) = c n k e 2 E 0 2 32 π ( z - z L ) 2 0 0 u v d u d v exp [ - ( u 2 + v 2 ) 4 a 0 2 ] × J 0 { u v k e 2 ( z - z L ) [ 1 - z - z L R e - B e ( z - z L ) ( u 2 + v 2 ) ] } ,
I ( z ) = c n k e E 0 2 16 π ( z - z L ) 0 d ρ × ρ exp ( - ρ 2 / 4 a 0 2 ) sin { k e ρ 2 4 ( z - z L ) [ 1 - ( z - z L ) R e - ( z - z L ) B e ρ 2 ] } [ 1 - ( z - z L ) R e - ( z - z L ) B e ρ 2 ] .
Δ = 4 B a 0 2 R 0 2 1 + 4 B a 0 2 R 0 .
Δ z ˜ = Δ z / n .
I ( z ) = c k 0 E 0 2 16 π ( z - z L ) 2 × 0 d ρ ρ exp ( - ρ 2 / 4 a 0 2 ) sin [ k 0 ρ 2 4 ( 1 z - z L - 1 R 0 - B ρ 2 ) ] ( 1 z - z L - 1 R 0 - B ρ 2 ) .
v - c v 2 = m π ,
v = 1 ± ( 1 - 4 π m c ) 1 / 2 2 c .
z m = z L + ( k 0 16 π B m ) 1 / 2 / [ 1 + ( k 0 16 π B R 0 2 m ) 1 / 2 ] .