Abstract

We calculated the intensity distribution behind a thermal lens by using a numerical quadrature of the Fresnel diffraction integral and compared it to several given approximate models for laser light detection in the center behind a thermal lens, which includes a new approximate solution of the diffraction integral with applicability to strong thermal lenses. Consideration of the aberrant nature of the thermal lens is crucial even if the thermal lens is weak. A simple approximate formula for the position of the most intense interference ring stating a linear dependence of the thermal lens strength is given. The transverse profile of a weak thermal lens is discussed. It is shown that spherical aberration modifies the central intensity even if a Gaussian profile is observed.

© 1995 Optical Society of America

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  1. J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
    [CrossRef]
  2. C. Hu, J. R. Whinnery, “New thermooptical measurement method and a comparison with other methods,” Appl. Opt. 12, 72–79 (1973).
    [CrossRef] [PubMed]
  3. H. L. Fang, R. L. Swofford, “The thermal lens in absorption spectroscopy,” in Ultrasensitive Laser Spectroscopy, D. S. Kliger, ed. (Academic, New York, 1983), pp. 175–232.
  4. S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, R. V. Khokhlov, “Thermal self-actions of laser beams,” IEEE. J. Quantum Electron QE-8, 568–575 (1968).
    [CrossRef]
  5. S. J. Sheldon, L. V. Knight, J. M. Thorne, “Laser-induced thermal lens effect: a new theoretical model,” Appl. Opt. 21, 1663–1669 (1982).
    [CrossRef] [PubMed]
  6. J. Shen, R. D. Lowe, R. D. Snook, “A model for cw laser induced mode-mismatched dual-beam thermal lens spectrometry,” Chem. Phys. 165, 385–396 (1992).
    [CrossRef]
  7. J. F. Power, “Pulsed mode thermal lens effect detection in the near field via thermally induced probe beam spatial phase modulation: a theory,” Appl. Opt. 29, 52–63 (1990).
    [CrossRef] [PubMed]
  8. S. Wu, N. J. Dovichi, “Fresnel diffraction theory for steady-state thermal lens measurements in thin films,” J. Appl. Phys. 67, 1170–1182 (1990).
    [CrossRef]
  9. C. A. Carter, J. M. Harris, “Comparison of models describing the thermal lens effect,” Appl. Opt. 23, 476–481 (1984).
    [CrossRef] [PubMed]
  10. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).
  11. H. Kogelnik, “Propagation of laser beams,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1979), Vol. 7, pp. 155–190.
  12. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), Chap. 5.
  13. Wolfram Research, Inc., Mathematica, Ver. 2.2 (Wolfram Research, Inc., Champaign, Ill., 1992).
  14. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972).
  15. A. Erdélyi, ed., Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 1.

1992

J. Shen, R. D. Lowe, R. D. Snook, “A model for cw laser induced mode-mismatched dual-beam thermal lens spectrometry,” Chem. Phys. 165, 385–396 (1992).
[CrossRef]

1990

S. Wu, N. J. Dovichi, “Fresnel diffraction theory for steady-state thermal lens measurements in thin films,” J. Appl. Phys. 67, 1170–1182 (1990).
[CrossRef]

J. F. Power, “Pulsed mode thermal lens effect detection in the near field via thermally induced probe beam spatial phase modulation: a theory,” Appl. Opt. 29, 52–63 (1990).
[CrossRef] [PubMed]

1984

1982

1973

1968

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, R. V. Khokhlov, “Thermal self-actions of laser beams,” IEEE. J. Quantum Electron QE-8, 568–575 (1968).
[CrossRef]

1965

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Akhmanov, S. A.

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, R. V. Khokhlov, “Thermal self-actions of laser beams,” IEEE. J. Quantum Electron QE-8, 568–575 (1968).
[CrossRef]

Carter, C. A.

Dovichi, N. J.

S. Wu, N. J. Dovichi, “Fresnel diffraction theory for steady-state thermal lens measurements in thin films,” J. Appl. Phys. 67, 1170–1182 (1990).
[CrossRef]

Fang, H. L.

H. L. Fang, R. L. Swofford, “The thermal lens in absorption spectroscopy,” in Ultrasensitive Laser Spectroscopy, D. S. Kliger, ed. (Academic, New York, 1983), pp. 175–232.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), Chap. 5.

Gordon, J. P.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).

Harris, J. M.

Hu, C.

Khokhlov, R. V.

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, R. V. Khokhlov, “Thermal self-actions of laser beams,” IEEE. J. Quantum Electron QE-8, 568–575 (1968).
[CrossRef]

Knight, L. V.

Kogelnik, H.

H. Kogelnik, “Propagation of laser beams,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1979), Vol. 7, pp. 155–190.

Krindach, D. P.

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, R. V. Khokhlov, “Thermal self-actions of laser beams,” IEEE. J. Quantum Electron QE-8, 568–575 (1968).
[CrossRef]

Leite, R. C. C.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Lowe, R. D.

J. Shen, R. D. Lowe, R. D. Snook, “A model for cw laser induced mode-mismatched dual-beam thermal lens spectrometry,” Chem. Phys. 165, 385–396 (1992).
[CrossRef]

Migulin, A. V.

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, R. V. Khokhlov, “Thermal self-actions of laser beams,” IEEE. J. Quantum Electron QE-8, 568–575 (1968).
[CrossRef]

Moore, R. S.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Porto, S. P. S.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Power, J. F.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).

Sheldon, S. J.

Shen, J.

J. Shen, R. D. Lowe, R. D. Snook, “A model for cw laser induced mode-mismatched dual-beam thermal lens spectrometry,” Chem. Phys. 165, 385–396 (1992).
[CrossRef]

Snook, R. D.

J. Shen, R. D. Lowe, R. D. Snook, “A model for cw laser induced mode-mismatched dual-beam thermal lens spectrometry,” Chem. Phys. 165, 385–396 (1992).
[CrossRef]

Sukhorukov, A. P.

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, R. V. Khokhlov, “Thermal self-actions of laser beams,” IEEE. J. Quantum Electron QE-8, 568–575 (1968).
[CrossRef]

Swofford, R. L.

H. L. Fang, R. L. Swofford, “The thermal lens in absorption spectroscopy,” in Ultrasensitive Laser Spectroscopy, D. S. Kliger, ed. (Academic, New York, 1983), pp. 175–232.

Thorne, J. M.

Whinnery, J. R.

C. Hu, J. R. Whinnery, “New thermooptical measurement method and a comparison with other methods,” Appl. Opt. 12, 72–79 (1973).
[CrossRef] [PubMed]

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Wu, S.

S. Wu, N. J. Dovichi, “Fresnel diffraction theory for steady-state thermal lens measurements in thin films,” J. Appl. Phys. 67, 1170–1182 (1990).
[CrossRef]

Appl. Opt.

Chem. Phys.

J. Shen, R. D. Lowe, R. D. Snook, “A model for cw laser induced mode-mismatched dual-beam thermal lens spectrometry,” Chem. Phys. 165, 385–396 (1992).
[CrossRef]

IEEE. J. Quantum Electron

S. A. Akhmanov, D. P. Krindach, A. V. Migulin, A. P. Sukhorukov, R. V. Khokhlov, “Thermal self-actions of laser beams,” IEEE. J. Quantum Electron QE-8, 568–575 (1968).
[CrossRef]

J. Appl. Phys.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

S. Wu, N. J. Dovichi, “Fresnel diffraction theory for steady-state thermal lens measurements in thin films,” J. Appl. Phys. 67, 1170–1182 (1990).
[CrossRef]

Other

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).

H. Kogelnik, “Propagation of laser beams,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1979), Vol. 7, pp. 155–190.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), Chap. 5.

Wolfram Research, Inc., Mathematica, Ver. 2.2 (Wolfram Research, Inc., Champaign, Ill., 1992).

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972).

A. Erdélyi, ed., Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 1.

H. L. Fang, R. L. Swofford, “The thermal lens in absorption spectroscopy,” in Ultrasensitive Laser Spectroscopy, D. S. Kliger, ed. (Academic, New York, 1983), pp. 175–232.

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Figures (8)

Fig. 1
Fig. 1

Stationary phase shift per unit strength θ given by Eq. (10) [solid curve (4)] versus u = r2H2 compared to the approximations: (3) 0.817 u - 0.164 [approximation (32)], (2) Eq. (10) using the first two terms of the series expansion in Eq. (2), (1) traditional parabolic model [Eq. (10)] using the first term of the series expansion in Eq. (2).

Fig. 2
Fig. 2

Geometry of a TL experiment: (a) Definition of the geometrical parameters of a focused TEM00 beam. ωG(z) is the e−2 diameter of the problem beam behind the sample. R(z) is the radius of wave-front curvature of the undisturbed beam behind the sample. R(z) is related to zc, which is the confocal parameter [Eqs. (22)]. A similar figure could be used for the heating beam, which can be but need not be identical to the probing beam. (b) Illustration of the diffraction geometry between the TL and the detector. The field at the detector (x, y) is the sum of the spherical wavelets coming from all points (x′, y′) of the input plane, which is situated behind the sample. D is the distance between the detector position and the point in the input plane. (c) The phase shift that is relative to the beam center just behind the TL, which consists of the part ΔΦG that is due to the Gaussian beam phase curvature and the part ΔΦTL that is due to the index-of-refraction gradient induced by the heating beam.

Fig. 3
Fig. 3

Intensity half-profiles that were calculated by using Eq. (28). x ^ is defined in Eq. (27) and is a measure of the distance from the optical axis. (Note that, as the beam diverges, the beam power is spread over a larger area, which causes the flattening.) (a) Steady-state (τ = 0, t = ∞) profiles for strong TL (θ ≫ 1). The outer maximum is always the strongest. For h = 0, h is a position parameter from Eq. (23) and the position of the ring is given by approximation (41). (b) The intensity at different times t after the heating has started; τ = 1/(1 + 2t/tc).

Fig. 4
Fig. 4

(a) Stationary central intensities that depend on the TL strength. The dots are exact values calculated using Eq. (28). The parabolic model (1) is incorrect even for the smallest θ. The Sheldon model (2) loses track at values of θ larger than 0.2, whereas the modified version (3) diverges nonphysically at θ > 1.5 (a relative error of 0.5% is already reached at θ = 0.5). Equation (35) (4) shows the best overall behavior (a relative error of 0.5% is reached at θ = 1.6). (b) Central intensities at an early stage of TL buildup (t = tc/2, a typical order of magnitude for tc is 50 ms).

Fig. 5
Fig. 5

(a) Time dependence of the central intensities with small θ: (1a) parabolic model, Eq. (37); (1b) modified parabolic model [tc is replaced by 2tc in Eq. (37)]; (2) aberrant model by Sheldon et al.,5 Eq. (38); (3) modified aberrant model by Shen et al.,6 Eq. (38) without the ln term; (4) new model, Eq. (35). The dots are exact values calculated using Eq. (28). The modification does not eliminate the lack of accuracy of the parabolic model. The model by Shen et al.6 (3) is the most accurate. (b) Time dependence of the central intensities with large θ. The modified parabolic model is (1b), both (2) and (3) show nonphysical divergence, (4) shows the best overall behavior.

Fig. 6
Fig. 6

(a) Position dependence of the relative change of central intensities between t = 0 (τ = 1) and t = ∞ for a weak TL; θ = 0.2 and h corresponds to the sample position according to Eq. (23). (1) Parabolic model, Eq. (37); (2) aberrant model by Sheldon et al.5 Eq. (38); (3) modified aberrant model by Shen et al.,6 Eq. (38) without the ln term; (4) new model, Eq. (35). The dots are exact values calculated using Eq. (28). The parabolic model gives a wrong optimum signal position. (b) Position dependence of the relative change of central intensities between t = 0 (τ = 1) and t = ∞ for strong TL; θ = 2. Only the new model (4) gives a reasonable prediction.

Fig. 7
Fig. 7

Position of the main interference ring in the intensity profile that depends on TL strength. The dots indicate the position from numerical calculation of the diffraction integral [Eq. 28)]. The solid curve gives the position that is linearly proportional to θ according to Eq. (42). The coefficients were obtained using a least-squares fit to the exact values.

Fig. 8
Fig. 8

Deviation of the transverse intensity profile from a Gaussian-type profile (a) as defined in Eq. (54) for various values of θ; for θ < 0.1 the deviation is of the order of O (0.01); (b) for various values of h and small θ; (c) for larger values of θ; an interference ring pattern emerges.

Equations (56)

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Δ T ( r , t ) = P H 4 π κ [ E i ( 2 r 2 ω H 2 ) - E i ( - 2 r 2 ω H 2 1 1 + 2 t / t c ) ] ,
E i ( x ) = - - x exp ( - t ) t d t = γ + ln ( - x ) + k = 1 x k k k ! ,             x < 0
f ( t ) = π κ ω 2 P H l n / T ( 1 + t c / 2 t ) ,
Δ opt . path ( r ) = l [ ( n ( r , t ) - n ( 0 , t ) ] = l ( n / T ) [ Δ T ( r , t ) - Δ T ( 0 , t ) ] .
lim x 0 E i ( x ) = γ + ln ( - x ) ,             x < 0 ,
lim r 0 E i ( - 2 r 2 ω H 2 ) - E i ( - 2 r 2 ω H 2 τ ) = - ln τ = ln 1 / τ ,
Δ T ( 0 , t ) = ( P H / 4 π κ ) ln 1 / τ ,
Δ T ( r , t ) - Δ T ( 0 , t ) = ( P H / 4 π κ ) [ E i ( - 2 r 2 ω H 2 ) - E i ( - 2 r 2 ω H 2 τ ) - ln ( 1 / τ ) ] .
Δ T ( r , ) - Δ T ( 0 , ) = ( P H / 4 π κ ) [ E i ( - 2 r 2 ω H 2 ) - ln ( 2 r 2 ω H 2 ) - γ ] ,
Δ Φ TL = k × opt . path = θ 2 [ E i ( - 2 r 2 ω H 2 τ ) - E i ( 2 r 2 ω H 2 ) - ln τ ] ,
θ = - ( n / T ) l P H λ κ
U ^ ( x , y , 0 ) = U ^ 0 exp ( - x 2 + y 2 ω G 2 ) exp [ i ( Δ Φ G + Δ Φ TL ) ] ,
U ( x , y , z D ) = C U ^ 0 input plane exp ( - x 2 + y 2 ω G 2 ) × exp i ( k D + Δ Φ G + Δ Φ TL ) D d x d y .
D = [ Z D 2 + ( x - x ) 2 + ( y - y ) 2 ] 1 / 2 z D + ( x - x ) + ( y - y ) 2 2 z D - O ( x 4 8 z D 3 ) .
D D 0 - x x + y y z D + x 2 + y 2 2 z D ,
U ( x , z D ) = C U ^ 0 exp ( i k D 0 ) D 0 × 0 2 π 0 exp ( - r 2 / ω G 2 ) × exp [ i ( k r 2 / 2 z D + Δ Φ TL + Δ Φ G ) ] × exp ( - i k x r cos φ / z D ) r d r d φ .
J 0 ( z ) = 1 2 π 0 2 π exp ( ± i z cos φ ) d φ ,
U ( x , z D ) = C 0 exp ( - r 2 / ω G 2 ) exp [ i ( k r 2 / 2 z D + Δ Φ G + Δ Φ TL ) ] J 0 ( r k x z D ) r d r ,
C = C U ^ 0 exp ( i k D 0 ) D 0 2 π .
R + Δ Φ G / k = R 2 + r 2 R ( 1 + r 2 / 2 R 2 ) ,
Δ Φ G = k r 2 / 2 R .
h = k ω G 2 2 ( 1 / R + 1 / z D ) ,
R = 1 / z ( z 2 + z c 2 ) ,             z c = ω 0 2 π / λ , ω G 2 = ω 0 2 ( 1 + z 2 / z c 2 ) ,
h = z c ( 1 + z 2 z c 2 ) ( z ( z 2 + z c 2 ) + 1 z D ) = z z c + z z D ( z c z + z z c ) .
r 2 / ω G 2 = u .
g 2 = ω G 2 / ω H 2
Δ Φ TL = ( θ / 2 ) [ E i ( - 2 u g 2 τ ) - E i ( - 2 u g 2 ) - ln τ ] .
x ^ = k ω G x / z D
U ( x ^ ) = C 0 exp ( - u ) exp [ i ( u h + Δ Φ TL ) ] J 0 ( x ^ u ) d u ,
I ( x ^ , τ , θ , h ) = U 2 .
s = 1 - i h ,
U ( x ^ ) = C L [ exp ( i Δ Φ TL ) J 0 ( x ^ u ) ] .
Δ Φ TL ( u ) / θ g a 0 u + c ,
L [ exp ( i α u ) ] = 1 s + α π 2 s 3 / 2 exp ( - α 2 / 4 s ) erfc ( - i α 2 s ) .
α = θ g a ( h , τ ) = θ g ( 0.40 exp { - [ ( h + 0.5 ) / 7 ] 2 } + 0.45 ) × ( 1 - 1.69 τ + 1.02 τ 2 - 0.34 τ 3 ) .
I ( 0 ) = U ( 0 ) 2 = I 00 1 + z w ( z ) 2 ,
I 00 = | C s | 2 = | C 1 - i h | 2 = C 2 1 + h 2
I ( t ) / I ( 0 ) = [ 1 + θ 1 + t c / 2 t ( 2 γ 1 + γ 2 ) + ( θ 1 + t c / 2 t ) 2 1 1 + γ 2 ] - 1 ,
I ( t ) = C | 1 1 + i h | 2 × { ( 1 - θ 2 arctan { 2 m h [ ( 1 + 2 m ) 2 + h 2 ] ( t c / 2 t ) + 1 + 2 m + h 2 } ) 2 + ( θ 4 ln { [ 1 + 2 m / ( 1 + 2 t / t c ) ] 2 + h 2 ( 1 + 2 m ) 2 + h 2 } ) 2 } ,
J 0 ( x ) 2 π x cos ( x - π 4 ) ,
x ^ max ( Δ Φ TL + π / 4 ) / u Δ Φ TL u ,             ( Δ Φ TL π / 4 ) .
x ^ max a ( 0 , τ ) θ g + c ,
x ^ max = 0.847 θ g - 1.46.
θ g = x max z D 7.418 ω G λ + 1.72 ± 0.03 ,             ( h = 0 ) ,
z h 0 = z D 2 [ 1 - 2 ( z c z D ) 2 - 1 ] = - z c 2 z D .
L [ J 0 ( x ^ u ) ] = 1 s exp ( - x ^ 2 4 s ) .
I ( x ^ ) = I 00 exp [ - x ^ 2 2 ( 1 + h 2 ) ] ,             ( θ = 0 t = 0 ) ,
L [ u n f ( u ) ] = ( - 1 ) n d n d s n L [ ( f ( u ) ]
L [ cos ( Δ Φ TL ) J 0 ( x ^ u ) ] + i L [ sin ( Δ Φ TL ) J 0 ( x ^ u ) ] ,
I ( x ^ ) = C 2 | n = 0 n max ( a n + i b n ) ( - 1 ) n d n d s n ( 1 s exp - x ^ 2 4 s ) | 2 .
L n ( w ) = m = 0 n ( n m ) ( - w ) m m ! ,
d n d y n 1 y exp - a y = ( - 1 ) n exp ( - a y ) L n ( a y ) n ! y n + 1 ,
I ( x ^ ) = | C s | 2 exp [ - x ^ 2 2 ( 1 + h 2 ) ] × | n = 0 n max [ a n ( θ , τ ) + i b n ( θ , τ ) ] L n ( x ^ 2 4 s ) n ! s n | 2 ,
I ( x ^ ) = I 00 exp [ - x ^ 2 2 ( 1 + h 2 ) ] × v ( x ^ , ) .
I ( x ^ ) = I * ( x ^ ) + Δ I ( x ^ ) = I 00 exp [ - x ^ 2 2 ( 1 + h 2 ) ] { v ( 0 ) + [ v ( x ) - v ( 0 ) ] } = exp [ - x ^ 2 2 ( 1 + h 2 ) ] I ( 0 ) I * ( x ^ ) + Δ I ( x ^ ) .
I ( x ^ ) I * ( x ^ ) ,             ( θ 1 ) .

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