Abstract

A measurement system for quantitative determination of both surface and bulk contributions to the photo-thermal absorption in DUV optics was developed. It is based upon a Hartmann-Shack wavefront sensor with a sensitivity of ~λ/10000 rms, accomplishing precise on-line monitoring of wavefront deformations of a collimated test laser beam transmitted perpendicular through the excimer laser-irradiated site of a cuboid sample. Caused by the temperature dependence of the refractive index as well as thermal expansion, the initially plane wavefront of the test laser is distorted into a cylindrical shape, with bending ends towards the surface. Sign and magnitude depend on index change and expansion. By comparison with thermal theory, this transient wavefront distortion yields a quantitative absolute measure of bulk and surface absorption losses in the sample. First rresults for fused silica are presented.

©2010 Optical Society of America

1. Introduction

Although considerable improvements of the optical properties of fused silica at 193nm have been achieved in recent years, the combination of high photon energies, increasing fluencies and high radiation doses represents still a limiting factor for the performance of modern DUV wafer steppers. Apart from degradation mechanisms such as color center formation [1] or compaction and rarefaction [2,3], the absorption within the bulk, at the surfaces and in dielectric coatings raises the temperature of an optical element, which leads to local changes of the refractive index as well as thermal expansion, resulting in the development of a thermal lens. Unfortunately, the most commonly used techniques in this field, i.e. laser calorimetry [4,6], photothermal deflection [5] and ratiometric transmission measurements, suffer from several problems, as e.g. long measurement times, missing spatial resolution, or limited accuracy at low absorption levels. This can be avoided by direct inspection of the laser-induced wavefront deformation a test beam picks up when passing the sample almost parallel to the heating beam, as described in [7]. The advantages of this photo-thermal approach are the fast measurement, flexible sample dimensions, good spatial resolution and the direct measurement of an optically relevant quantity. Moreover, the method facilitates calibration, the temporal dynamics of the process can be evaluated and, in contrast to a transmission measurement, only the absorptance and not the scatter losses contribute to the signal. However, due to the accumulation in test beam direction, bulk and surface contributions are mixed up in the signal, and, unless either of them is known a priori, a thickness series of samples is required to obtain quantitative values of bulk and surface absorption.

In this paper we present a modified setup for quantitative determination of laser-induced wavefront deformations, which enables the separation of bulk and surface contributions to absorption in DUV optics. Following a brief section on fundamentals of thermal lensing, the experimental setup is presented, and results for DUV grade fused silica are discussed.

2. Theory

2.1 Thermal lensing

The transient temperature change δT(r,t) of an cuboid sample G (Fig. 1 ) irradiated by a circular laser beam traveling in z-direction is obtained by the solution of the heat equation with appropriate boundary conditions on ∂G:

 

Fig. 1 Geometry and notations characterizing an orthorhombic optical element during laser irradiation.

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cpρtδT(r,t)+(λδT(r,t))+μIP(r,t)=0           rGn(λδT(r,t)+κδT(r,t)nβIp(r,t)ez)=0             rG.

In Eq. (1) μ and β denote the bulk and surface absorption coefficients, respectively, Ip(r,t) denotes the laser power density distribution, κ the coefficient of heat transfer, λ the thermal conductivity, cp the specific heat, ρ the density, and n the outward directed surface normal.

Usually μ⋅ls << 1 and β << 1 holds, and from a linear superposition the solution δT(x,y,z,t) can be expressed as:

δT(x,y,z,t)=μls,β<<1P[μV(x,y,z,t)+βS(x,y,z,t)],
where P is the average laser power. In (2) the bulk and surface terms V(x,y,z,t) and S(x,y,z,t) depend on sample dimension and laser beam profile, as well as on λ/cpρ and κ, respectively. However, they are independent of P, μ and β.

According to the temperature variation ∂n/∂T of the refractive index and the sample elongation in direction of the test laser beam, the wavefront of a well collimated probe beam, traveling perpendicular to the heating beam (parallel to the specimen y-axis), picks up a wavefront deformation δw(x,z,t):

δw(x,z,t)=0ds(n01)uyy(x,y,z,t)+nTδT(x,y,z,t)dy~linearityP[μV'(x,z,t)+βS'(x,z,t)],
where n0 denotes the refractive index at ambient temperature and ds the specimen width in direction of the test laser beam. The displacements u = (ux, uy, uz)T are determined by the condition of elastic equilibrium [7]. As in Eq. (2), the form factors V’ and S’ are independent of P, μ and β.

3. Experimental

3.1 Setup

The setup used for photo-thermal measurements on ArF laser optics @ 193nm is shown in Fig. 2 . It consists of a nitrogen purged Al chamber which contains all optically relevant components. Cuboids of tetragonal symmetry with both square and two opposite rectangular surfaces polished to high optical quality are used as specimens. The latter were placed on four small polyamide posts (λ = 0.23 W/mK) for good thermal isolation. A collimated ArF excimer laser (Novaline A2010, Coherent/LambdaPhysik) irradiates the sample collinearly with respect to the fourfold axis; for defined conditions the excimer laser beam is expanded and confined by an aperture to a circular flat-top profile of Ø 3mm. The probe beam, a 639 nm fiber coupled diode laser, intersects the specimen perpendicular to the excimer laser (cf. Figure 1). For homogeneous illumination the probe laser is expanded 10 times, and, after passing the sample, demagnified again to fit the detector area of the Hartmann-Shack sensor [8]. This consists of a 12bit digital CCD camera with 1280 x 1024 pixels behind an orthogonal quartz microlens array (plano-convex lenslets, pitch 0.3mm, focal length f = 40mm). Both camera data and laser power readings obtained from two power monitors (Ophir) are fed into a PC for online data processing. Furthermore, the beam line is equipped with a PC controlled shutter in order to accomplish variable heating and cooling intervals. For noise reduction up to 32 frames per record are sampled. The oxygen pressure is monitored with a respective sensor (Zirox) and kept below 100ppm during the measurements.

 

Fig. 2 Setup for measurement of the laser-induced photo-thermal wavefront deformation.

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The amount of the photo-thermal effect is quantified from the reconstructed wavefront w(x, z) evaluated at the Hartmann-Shack grid positions. Reconstruction is performed via a modal expansion (degree M) in Legendre polynomials Pl with expansion coefficients cl, i.e.:

w(x,z)=l=0MclPl(x,z),
where the coefficients are obtained from the displacement of the Hartmann-Shack spots with respect to reference positions by a least square approach and standard techniques of linear theory [7,9].

The measurement protocol starts with an equilibration period of 54s; during the next 6 seconds a reference is taken by sampling and averaging 32 camera frames for noise reduction. Thereafter, the ArF laser is switched on and the photo-thermal signal is recorded during the time interval from 15sec to 21sec after the beginning of irradiation. All in all, a complete cycle lasts about 75s. This short measurement time compared to calorimetry stems from the fact that only wave-front gradients are detected, and thus no equilibrium conditions are required as has been shown previously [7].

3.2 Calibration

In order to gain absolute values of absorption coefficients the setup has to be calibrated. This can be achieved from the results of a numerical simulation, since the form factor S’ (cf. Equation (3) is mainly concentrated at the surfaces, while V’ spreads over the complete (x, z)-plane, as shown in Fig. 3 .

 

Fig. 3 Normalized stationary form functions V’ (left) and S’ (right), corresponding to pure bulk and pure surface absorption of fused silica (sample dimension 25x25x45mm3, top hat beam of 3mm diameter, heat transfer coefficient κ = 10). The isolines represent equidistant values between [0, 1].

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As S’ and V’ can be calculated from Eq. (1) for any given set of parameters (for values used in numerical calculations see [7]), μ and β can be obtained from δw[τ]/P evaluated for a number N of z-positions (zi) via a least square approach, yielding the solution:

(βμa)T=A+1P(δw[τ](z1)...δw[τ](zN))T
with the N-data vector δw[τ]/P and A+ the pseudo inverse of the 3xN form factor matrix A:

A=(S'[τ](z1)V'[τ](z1)T(z1):::S'[τ](zN)V'[τ](zN)T(zN)).

In Eq. (5), [τ] signifies an average over the recording time interval Δt with respect to the heating sequence, whereas the coefficient a, respectively the linear form function T(z), account for any tilt difference between simulation and experiment. The heat transfer coefficient, needed for determination of the form functions S’[τ] and V’[τ], can be estimated from conductive and convective heat flow. Using the properties of nitrogen, the specimen dimension and mass ms, its average distance from the chamber walls as well as its mean temperature δT = P·t/(cp·ms) after an irradiation time t, this results in 1<κ<5. Within these bounds, however, both form functions change by no more than 1%. Thus, by setting κ = 2.5, the related uncertainty in surface and bulk absorption stays well below 2%, as verified in the next subsection.

4. Results

Figure 4 shows signals from two 25x25x45mm3 quartz samples irradiated with 193nm laser pulses of 20ns at 800 Hz and 0.59W (8.3 W/cm2), corresponding to a fluence of 10.4 mJ/cm2. Both wavefronts show the expected shape of combined surface and bulk contributions, yielding a distinct valley with bending end towards the specimen front surface. Reconstruction was performed by a Legendre expansion of 8th degree from 19x19 = 361 Hartmann-Shack spots in the (x, z)-plane, covering a specimen area of [-8.55, 8.55] x [0, 17.1] mm2 with respect to the center of the front surface. The central section along the bottom of the valley (the top of the ridge in terms of optical path difference) as shown in Fig. 5 , is then taken as input data for determination of bulk and surface absorption coefficients according to Eq. (5) with a linear form function T(z)=a(z8.55).

 

Fig. 4 Laser induced photo-thermal wavefront deformation for two 25x25x45 mm3 quartz samples S1 (left) and S2 (right) with different absorption (cf text).

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Fig. 5 Time averaged normalized optical path length δw/P (x = 0, z) for two different fused silica samples irradiated with 8.3 W/cm2. The solid lines represent the least square approximation for κ = 2.5 W/m2K with bulk, surface and tilt form functions (parameters according to Table 1).

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The approximation fits the experimental data extraordinary well, leading to values of R2 > 99%. As expected, the bulk and surface absorption coefficients k and β, respectively, shown in Table1 show virtually no dependence of the heat transfer coefficient used in the linear model. The tilt contribution stays well below 10% for both specimens, indicating sufficiently stable measurement conditions during the signal recording time of 6sec. The sensitivity of the model was checked by calculating the variances of the parameter estimates from a singular value decomposition of A and assuming 1nm/W standard deviation of the experimental data, yielding (σ2(k), σ2(β)) = (2.2·10−6, 1.3·10−7). Correspondingly, R2 responds more sensitive to variation of k; values calculated for a 10% deviation of the parameters k and β from the best fit results (cf. Table 1) gave R2(k ± 0.1k, β, a) = 0.78, R2(k, β ± 0.1β, a) = 0.98 for sample S1 and R2(k ± 0.1k, β, a) = −0.77, R2(k, β ± 0.1β, a) = 0.99 for sample S2, respectively.

Tables Icon

Table 1. Bulk absorption k factors (k = μ·lg e), surface absorption β, and tilt a as well as coefficient of determination R2 according to Eq. (5) for two samples of fused silica and two different heat transfer coefficients κ used in the numerical calibration. The two lower rows show the result for fixed tilt parameter a = 0

5. Conclusion

The photo-thermal measurement technique introduced in [7] was improved in order to separate bulk and surface absorption in monitoring laser-induced thermal lens effects. Besides a highly sensitive Hartmann-Shack sensor the technique uses a crossed arrangement of test and heating beam. The distinguished shape of bulk and surface contributions facilitates reliable fitting for the corresponding absorption parameters. An absolute calibration of the measured data is possible by comparison with a numerical simulation. The advantages of the modified setup in comparison to calorimetry are the much faster measurement (no equilibrium, only wavefront gradients), flexible sample dimensions and the possibility to investigate the dynamics in imaging quality. In comparison to transmission measurements the setup yields more stable values, and there is no bias from scattering. Thus, the modified method may extend the application field of the photo-thermal approach as an alternative to both techniques.

References and links

1. R. E. Schenker, L. Eichner, H. Vaidya, S. Vaidya, P. M. Schermerhorn, D. R. Fladd, and W. G. Oldham, “Ultraviolet damage properties of various fused silica materials,” Proc. SPIE 2428, 458–468 (1994).

2. W. Primak and R. Kampwirth, “The Radiation Compaction of Vitreous Silica,” J. Appl. Phys. 39(12), 5651–5658 (1968).

3. C. Van Peski, “Behavior of Fused Silica under 193nm Irradiation,” Technology Transfer # 00073974A-TR, International SEMATECH (2000)

4. E. Eva and K. Mann, “Calorimetric measurement of two-photon absorption and color-center formation in ultraviolet-window materials,” Appl. Phys., A Mater. Sci. Process. 62(2), 143–149 (1996).

5. M. Guntau and W. Triebel, “Novel method to measure bulk absorption in optically transparent materials,” Rev. Sci. Instrum. 71(6), 2279–2282 (2000).

6. C. Görling, U. Leinhos, and K. Mann, “Comparative studies of absorptance behaviour of alkaline-earth fluorides at 193 nm and 157nm,” Appl. Phys. B 74(3), 259–265 (2002).

7. B. Schäfer, J. Gloger, U. Leinhos, and K. Mann, “Photo-thermal measurement of absorptance losses, temperature induced wavefront deformation and compaction in DUV-optics,” Opt. Express 17(25), 23025–23036 (2009).

8. D. R. Neal, W. J. Alford, J. K. Gruetzner, and M. E. Warren, “Amplitude and phase beam characterization using a two-dimensional wavefront sensor,” Proc. SPIE 2870, 72 (1996).

9. B. Schäfer and K. Mann, “Determination of beam parameters and coherence properties of laser radiation by use of an extended Hartmann-Shack wave-front sensor,” Appl. Opt. 41(15), 2809–2817 (2002). [PubMed]  

References

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  1. R. E. Schenker, L. Eichner, H. Vaidya, S. Vaidya, P. M. Schermerhorn, D. R. Fladd, and W. G. Oldham, “Ultraviolet damage properties of various fused silica materials,” Proc. SPIE 2428, 458–468 (1994).
  2. W. Primak and R. Kampwirth, “The Radiation Compaction of Vitreous Silica,” J. Appl. Phys. 39(12), 5651–5658 (1968).
  3. C. Van Peski, “Behavior of Fused Silica under 193nm Irradiation,” Technology Transfer # 00073974A-TR, International SEMATECH (2000)
  4. E. Eva and K. Mann, “Calorimetric measurement of two-photon absorption and color-center formation in ultraviolet-window materials,” Appl. Phys., A Mater. Sci. Process. 62(2), 143–149 (1996).
  5. M. Guntau and W. Triebel, “Novel method to measure bulk absorption in optically transparent materials,” Rev. Sci. Instrum. 71(6), 2279–2282 (2000).
  6. C. Görling, U. Leinhos, and K. Mann, “Comparative studies of absorptance behaviour of alkaline-earth fluorides at 193 nm and 157nm,” Appl. Phys. B 74(3), 259–265 (2002).
  7. B. Schäfer, J. Gloger, U. Leinhos, and K. Mann, “Photo-thermal measurement of absorptance losses, temperature induced wavefront deformation and compaction in DUV-optics,” Opt. Express 17(25), 23025–23036 (2009).
  8. D. R. Neal, W. J. Alford, J. K. Gruetzner, and M. E. Warren, “Amplitude and phase beam characterization using a two-dimensional wavefront sensor,” Proc. SPIE 2870, 72 (1996).
  9. B. Schäfer and K. Mann, “Determination of beam parameters and coherence properties of laser radiation by use of an extended Hartmann-Shack wave-front sensor,” Appl. Opt. 41(15), 2809–2817 (2002).
    [PubMed]

2009 (1)

2002 (2)

C. Görling, U. Leinhos, and K. Mann, “Comparative studies of absorptance behaviour of alkaline-earth fluorides at 193 nm and 157nm,” Appl. Phys. B 74(3), 259–265 (2002).

B. Schäfer and K. Mann, “Determination of beam parameters and coherence properties of laser radiation by use of an extended Hartmann-Shack wave-front sensor,” Appl. Opt. 41(15), 2809–2817 (2002).
[PubMed]

2000 (1)

M. Guntau and W. Triebel, “Novel method to measure bulk absorption in optically transparent materials,” Rev. Sci. Instrum. 71(6), 2279–2282 (2000).

1996 (2)

E. Eva and K. Mann, “Calorimetric measurement of two-photon absorption and color-center formation in ultraviolet-window materials,” Appl. Phys., A Mater. Sci. Process. 62(2), 143–149 (1996).

D. R. Neal, W. J. Alford, J. K. Gruetzner, and M. E. Warren, “Amplitude and phase beam characterization using a two-dimensional wavefront sensor,” Proc. SPIE 2870, 72 (1996).

1994 (1)

R. E. Schenker, L. Eichner, H. Vaidya, S. Vaidya, P. M. Schermerhorn, D. R. Fladd, and W. G. Oldham, “Ultraviolet damage properties of various fused silica materials,” Proc. SPIE 2428, 458–468 (1994).

1968 (1)

W. Primak and R. Kampwirth, “The Radiation Compaction of Vitreous Silica,” J. Appl. Phys. 39(12), 5651–5658 (1968).

Alford, W. J.

D. R. Neal, W. J. Alford, J. K. Gruetzner, and M. E. Warren, “Amplitude and phase beam characterization using a two-dimensional wavefront sensor,” Proc. SPIE 2870, 72 (1996).

Eichner, L.

R. E. Schenker, L. Eichner, H. Vaidya, S. Vaidya, P. M. Schermerhorn, D. R. Fladd, and W. G. Oldham, “Ultraviolet damage properties of various fused silica materials,” Proc. SPIE 2428, 458–468 (1994).

Eva, E.

E. Eva and K. Mann, “Calorimetric measurement of two-photon absorption and color-center formation in ultraviolet-window materials,” Appl. Phys., A Mater. Sci. Process. 62(2), 143–149 (1996).

Fladd, D. R.

R. E. Schenker, L. Eichner, H. Vaidya, S. Vaidya, P. M. Schermerhorn, D. R. Fladd, and W. G. Oldham, “Ultraviolet damage properties of various fused silica materials,” Proc. SPIE 2428, 458–468 (1994).

Gloger, J.

Görling, C.

C. Görling, U. Leinhos, and K. Mann, “Comparative studies of absorptance behaviour of alkaline-earth fluorides at 193 nm and 157nm,” Appl. Phys. B 74(3), 259–265 (2002).

Gruetzner, J. K.

D. R. Neal, W. J. Alford, J. K. Gruetzner, and M. E. Warren, “Amplitude and phase beam characterization using a two-dimensional wavefront sensor,” Proc. SPIE 2870, 72 (1996).

Guntau, M.

M. Guntau and W. Triebel, “Novel method to measure bulk absorption in optically transparent materials,” Rev. Sci. Instrum. 71(6), 2279–2282 (2000).

Kampwirth, R.

W. Primak and R. Kampwirth, “The Radiation Compaction of Vitreous Silica,” J. Appl. Phys. 39(12), 5651–5658 (1968).

Leinhos, U.

B. Schäfer, J. Gloger, U. Leinhos, and K. Mann, “Photo-thermal measurement of absorptance losses, temperature induced wavefront deformation and compaction in DUV-optics,” Opt. Express 17(25), 23025–23036 (2009).

C. Görling, U. Leinhos, and K. Mann, “Comparative studies of absorptance behaviour of alkaline-earth fluorides at 193 nm and 157nm,” Appl. Phys. B 74(3), 259–265 (2002).

Mann, K.

B. Schäfer, J. Gloger, U. Leinhos, and K. Mann, “Photo-thermal measurement of absorptance losses, temperature induced wavefront deformation and compaction in DUV-optics,” Opt. Express 17(25), 23025–23036 (2009).

C. Görling, U. Leinhos, and K. Mann, “Comparative studies of absorptance behaviour of alkaline-earth fluorides at 193 nm and 157nm,” Appl. Phys. B 74(3), 259–265 (2002).

B. Schäfer and K. Mann, “Determination of beam parameters and coherence properties of laser radiation by use of an extended Hartmann-Shack wave-front sensor,” Appl. Opt. 41(15), 2809–2817 (2002).
[PubMed]

E. Eva and K. Mann, “Calorimetric measurement of two-photon absorption and color-center formation in ultraviolet-window materials,” Appl. Phys., A Mater. Sci. Process. 62(2), 143–149 (1996).

Neal, D. R.

D. R. Neal, W. J. Alford, J. K. Gruetzner, and M. E. Warren, “Amplitude and phase beam characterization using a two-dimensional wavefront sensor,” Proc. SPIE 2870, 72 (1996).

Oldham, W. G.

R. E. Schenker, L. Eichner, H. Vaidya, S. Vaidya, P. M. Schermerhorn, D. R. Fladd, and W. G. Oldham, “Ultraviolet damage properties of various fused silica materials,” Proc. SPIE 2428, 458–468 (1994).

Primak, W.

W. Primak and R. Kampwirth, “The Radiation Compaction of Vitreous Silica,” J. Appl. Phys. 39(12), 5651–5658 (1968).

Schäfer, B.

Schenker, R. E.

R. E. Schenker, L. Eichner, H. Vaidya, S. Vaidya, P. M. Schermerhorn, D. R. Fladd, and W. G. Oldham, “Ultraviolet damage properties of various fused silica materials,” Proc. SPIE 2428, 458–468 (1994).

Schermerhorn, P. M.

R. E. Schenker, L. Eichner, H. Vaidya, S. Vaidya, P. M. Schermerhorn, D. R. Fladd, and W. G. Oldham, “Ultraviolet damage properties of various fused silica materials,” Proc. SPIE 2428, 458–468 (1994).

Triebel, W.

M. Guntau and W. Triebel, “Novel method to measure bulk absorption in optically transparent materials,” Rev. Sci. Instrum. 71(6), 2279–2282 (2000).

Vaidya, H.

R. E. Schenker, L. Eichner, H. Vaidya, S. Vaidya, P. M. Schermerhorn, D. R. Fladd, and W. G. Oldham, “Ultraviolet damage properties of various fused silica materials,” Proc. SPIE 2428, 458–468 (1994).

Vaidya, S.

R. E. Schenker, L. Eichner, H. Vaidya, S. Vaidya, P. M. Schermerhorn, D. R. Fladd, and W. G. Oldham, “Ultraviolet damage properties of various fused silica materials,” Proc. SPIE 2428, 458–468 (1994).

Warren, M. E.

D. R. Neal, W. J. Alford, J. K. Gruetzner, and M. E. Warren, “Amplitude and phase beam characterization using a two-dimensional wavefront sensor,” Proc. SPIE 2870, 72 (1996).

Appl. Opt. (1)

Appl. Phys. B (1)

C. Görling, U. Leinhos, and K. Mann, “Comparative studies of absorptance behaviour of alkaline-earth fluorides at 193 nm and 157nm,” Appl. Phys. B 74(3), 259–265 (2002).

Appl. Phys., A Mater. Sci. Process. (1)

E. Eva and K. Mann, “Calorimetric measurement of two-photon absorption and color-center formation in ultraviolet-window materials,” Appl. Phys., A Mater. Sci. Process. 62(2), 143–149 (1996).

J. Appl. Phys. (1)

W. Primak and R. Kampwirth, “The Radiation Compaction of Vitreous Silica,” J. Appl. Phys. 39(12), 5651–5658 (1968).

Opt. Express (1)

Proc. SPIE (2)

D. R. Neal, W. J. Alford, J. K. Gruetzner, and M. E. Warren, “Amplitude and phase beam characterization using a two-dimensional wavefront sensor,” Proc. SPIE 2870, 72 (1996).

R. E. Schenker, L. Eichner, H. Vaidya, S. Vaidya, P. M. Schermerhorn, D. R. Fladd, and W. G. Oldham, “Ultraviolet damage properties of various fused silica materials,” Proc. SPIE 2428, 458–468 (1994).

Rev. Sci. Instrum. (1)

M. Guntau and W. Triebel, “Novel method to measure bulk absorption in optically transparent materials,” Rev. Sci. Instrum. 71(6), 2279–2282 (2000).

Other (1)

C. Van Peski, “Behavior of Fused Silica under 193nm Irradiation,” Technology Transfer # 00073974A-TR, International SEMATECH (2000)

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Figures (5)

Fig. 1
Fig. 1 Geometry and notations characterizing an orthorhombic optical element during laser irradiation.
Fig. 2
Fig. 2 Setup for measurement of the laser-induced photo-thermal wavefront deformation.
Fig. 3
Fig. 3 Normalized stationary form functions V’ (left) and S’ (right), corresponding to pure bulk and pure surface absorption of fused silica (sample dimension 25x25x45mm3, top hat beam of 3mm diameter, heat transfer coefficient κ = 10). The isolines represent equidistant values between [0, 1].
Fig. 4
Fig. 4 Laser induced photo-thermal wavefront deformation for two 25x25x45 mm3 quartz samples S1 (left) and S2 (right) with different absorption (cf text).
Fig. 5
Fig. 5 Time averaged normalized optical path length δw/P (x = 0, z) for two different fused silica samples irradiated with 8.3 W/cm2. The solid lines represent the least square approximation for κ = 2.5 W/m2K with bulk, surface and tilt form functions (parameters according to Table 1).

Tables (1)

Tables Icon

Table 1 Bulk absorption k factors (k = μ·lg e), surface absorption β, and tilt a as well as coefficient of determination R2 according to Eq. (5) for two samples of fused silica and two different heat transfer coefficients κ used in the numerical calibration. The two lower rows show the result for fixed tilt parameter a = 0

Equations (6)

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c p ρ t δ T ( r , t ) + ( λ δ T ( r , t ) ) + μ I P ( r , t ) = 0             r G n ( λ δ T ( r , t ) + κ δ T ( r , t ) n β I p ( r , t ) e z ) = 0               r G .
δ T ( x , y , z , t ) = μ l s , β < < 1 P [ μ V ( x , y , z , t ) + β S ( x , y , z , t ) ] ,
δ w ( x , z , t ) = 0 d s ( n 0 1 ) u y y ( x , y , z , t ) + n T δ T ( x , y , z , t ) d y ~ l i n e a r i t y P [ μ V ' ( x , z , t ) + β S ' ( x , z , t ) ] ,
w ( x , z ) = l = 0 M c l P l ( x , z ) ,
( β μ a ) T = A + 1 P ( δ w [ τ ] ( z 1 ) ... δ w [ τ ] ( z N ) ) T
A = ( S ' [ τ ] ( z 1 ) V ' [ τ ] ( z 1 ) T ( z 1 ) : : : S ' [ τ ] ( z N ) V ' [ τ ] ( z N ) T ( z N ) ) .

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