Abstract

Hybrid elements, containing optical power with both diffractive (holographic) and refractive components, are shown to be able to eliminate the effect of propagation time difference. The consideration is provided through a paraxial approximation of diffraction theory.

© 1995 Optical Society of America

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References

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  1. Z. Bor, “Distortion of femtosecond laser pulses in lenses and lens systems,” J. Mod. Opt. 35, 1907–1918 (1988).
    [CrossRef]
  2. M. Kempe, U. Stamm, B. Wilhelmi, W. Rudolph, “Spatial and temporal transformation of femtosecond laser pulses by lenses and lens systems,” J. Opt. Soc. Am. B 9, 1158–1165 (1992).
    [CrossRef]
  3. M. Kempe, U. Stamm, B. Wilhelmi, “Spatial and temporal transformation of femtosecond laser pulses by lenses with annular aperture,” Opt. Commun. 89, 119–125 (1992).
    [CrossRef]
  4. T. E. Sharp, P. J. Wilsoff, “Analysis of lens and zone plate focusing of ultrashort laser pulses,” Appl. Opt. 31, 2765–2769 (1992).
    [CrossRef] [PubMed]
  5. A. J. Taylor, R. G. Gibson, J. P. Roberts, C. R. Tallman, “Nonlinear absorption in ultraviolet window materials,” in Conference on Lasers and Electro-Optics, Vol. 7 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), paper WD1.
  6. S. Szamari, F. P. Shafer, “Simplied laser system for the generation of 60 fs pulses at 248 nm,” Opt. Commun. 68, 196–202 (1988).
    [CrossRef]
  7. T. Stone, N. George, “Hybrid diffractive–refractive lenses and achromats,” Appl. Opt. 27, 2960–2971 (1988).
    [CrossRef] [PubMed]
  8. E. Ibragimov, E. A. Volynkina, “Impact of initial space distribution on the temporal form of ultrashort light pulses in the focal plane of the lens,” Opt. Lett. 19, 2140–2142 (1994).
    [CrossRef] [PubMed]
  9. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 2, p. 27.
  10. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), Chap. 6, p. 717.

1994 (1)

1992 (3)

1988 (3)

Z. Bor, “Distortion of femtosecond laser pulses in lenses and lens systems,” J. Mod. Opt. 35, 1907–1918 (1988).
[CrossRef]

S. Szamari, F. P. Shafer, “Simplied laser system for the generation of 60 fs pulses at 248 nm,” Opt. Commun. 68, 196–202 (1988).
[CrossRef]

T. Stone, N. George, “Hybrid diffractive–refractive lenses and achromats,” Appl. Opt. 27, 2960–2971 (1988).
[CrossRef] [PubMed]

Bor, Z.

Z. Bor, “Distortion of femtosecond laser pulses in lenses and lens systems,” J. Mod. Opt. 35, 1907–1918 (1988).
[CrossRef]

George, N.

Gibson, R. G.

A. J. Taylor, R. G. Gibson, J. P. Roberts, C. R. Tallman, “Nonlinear absorption in ultraviolet window materials,” in Conference on Lasers and Electro-Optics, Vol. 7 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), paper WD1.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 2, p. 27.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), Chap. 6, p. 717.

Ibragimov, E.

Kempe, M.

M. Kempe, U. Stamm, B. Wilhelmi, “Spatial and temporal transformation of femtosecond laser pulses by lenses with annular aperture,” Opt. Commun. 89, 119–125 (1992).
[CrossRef]

M. Kempe, U. Stamm, B. Wilhelmi, W. Rudolph, “Spatial and temporal transformation of femtosecond laser pulses by lenses and lens systems,” J. Opt. Soc. Am. B 9, 1158–1165 (1992).
[CrossRef]

Roberts, J. P.

A. J. Taylor, R. G. Gibson, J. P. Roberts, C. R. Tallman, “Nonlinear absorption in ultraviolet window materials,” in Conference on Lasers and Electro-Optics, Vol. 7 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), paper WD1.

Rudolph, W.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), Chap. 6, p. 717.

Shafer, F. P.

S. Szamari, F. P. Shafer, “Simplied laser system for the generation of 60 fs pulses at 248 nm,” Opt. Commun. 68, 196–202 (1988).
[CrossRef]

Sharp, T. E.

Stamm, U.

M. Kempe, U. Stamm, B. Wilhelmi, W. Rudolph, “Spatial and temporal transformation of femtosecond laser pulses by lenses and lens systems,” J. Opt. Soc. Am. B 9, 1158–1165 (1992).
[CrossRef]

M. Kempe, U. Stamm, B. Wilhelmi, “Spatial and temporal transformation of femtosecond laser pulses by lenses with annular aperture,” Opt. Commun. 89, 119–125 (1992).
[CrossRef]

Stone, T.

Szamari, S.

S. Szamari, F. P. Shafer, “Simplied laser system for the generation of 60 fs pulses at 248 nm,” Opt. Commun. 68, 196–202 (1988).
[CrossRef]

Tallman, C. R.

A. J. Taylor, R. G. Gibson, J. P. Roberts, C. R. Tallman, “Nonlinear absorption in ultraviolet window materials,” in Conference on Lasers and Electro-Optics, Vol. 7 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), paper WD1.

Taylor, A. J.

A. J. Taylor, R. G. Gibson, J. P. Roberts, C. R. Tallman, “Nonlinear absorption in ultraviolet window materials,” in Conference on Lasers and Electro-Optics, Vol. 7 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), paper WD1.

Volynkina, E. A.

Wilhelmi, B.

M. Kempe, U. Stamm, B. Wilhelmi, W. Rudolph, “Spatial and temporal transformation of femtosecond laser pulses by lenses and lens systems,” J. Opt. Soc. Am. B 9, 1158–1165 (1992).
[CrossRef]

M. Kempe, U. Stamm, B. Wilhelmi, “Spatial and temporal transformation of femtosecond laser pulses by lenses with annular aperture,” Opt. Commun. 89, 119–125 (1992).
[CrossRef]

Wilsoff, P. J.

Appl. Opt. (2)

J. Mod. Opt. (1)

Z. Bor, “Distortion of femtosecond laser pulses in lenses and lens systems,” J. Mod. Opt. 35, 1907–1918 (1988).
[CrossRef]

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

Opt. Commun. (2)

M. Kempe, U. Stamm, B. Wilhelmi, “Spatial and temporal transformation of femtosecond laser pulses by lenses with annular aperture,” Opt. Commun. 89, 119–125 (1992).
[CrossRef]

S. Szamari, F. P. Shafer, “Simplied laser system for the generation of 60 fs pulses at 248 nm,” Opt. Commun. 68, 196–202 (1988).
[CrossRef]

Opt. Lett. (1)

Other (3)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 2, p. 27.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), Chap. 6, p. 717.

A. J. Taylor, R. G. Gibson, J. P. Roberts, C. R. Tallman, “Nonlinear absorption in ultraviolet window materials,” in Conference on Lasers and Electro-Optics, Vol. 7 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), paper WD1.

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Figures (4)

Fig. 1
Fig. 1

Position of the pulse fronts for the focusing process: (a) the refractive lens, (b) the diffractive lens. The solid curves represent an ideal lens (the lens without chromatic aberration), and the dashed curves represent a real lens.

Fig. 2
Fig. 2

Amplitude at the center (r = 0) of an ultrashort pulse in a focal plane, a, without dispersion and, b with dispersion. A single lens is denoted by a dotted curve, and a hybrid achromat is denoted by a solid curve; a = 0.8 cm, f r = 2 cm, τ0 = 55 fs, λ = 589 nm.

Fig. 3
Fig. 3

Values of parameter q that are needed to compensate for the PTD effect, plotted as functions of the radius of the beam (λ = 589 nm). Curves 1–6 correspond to the following values of the focal length of the refractive element: 1, 1.25, 2, 3, 5, and 15 cm, respectively.

Fig. 4
Fig. 4

Normalized intensity distribution U n (r 1, f 1, t) = 2f 1 U(r 1, f 1, t)/k 0 a 2 in the focal plane of (a) an ideal lens, (b) a combination of refractive and diffractive elements (hybrid achromat), and (c) a single lens. Initial duration of the pulse is τ0 = 25 fs at the 1/e level.

Equations (21)

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F ( r , q ) = cos ( q r 2 ) ,
U ( r , 0 , ω ) = A ¯ ( ω ) exp ( i k l d ) exp [ - ( r 2 / a 2 ) ( 1 + i p a 2 ) ] × cos ( q r 2 ) ,
U ( r , z , ω ) = 0 U ¯ ( α , ω ) J 0 ( α r ) exp [ i ( k z - α 2 2 k z ) ] α d α ,
U ¯ ( α , ω ) = 0 U ( r , 0 , ω ) J 0 ( α r ) r d r .
U ¯ ( α , ω ) = U ¯ 1 ( α , ω ) exp [ i Φ 1 ( α , ω ) ] + U ¯ 2 ( α , ω ) exp [ i Φ 2 ( α , ω ) ] ,
U ¯ 1 , 2 ( α , ω ) = a 2 4 [ 1 + i ( p ± q ) a 2 ] × exp { - α 2 a 2 4 [ 1 + ( p ± q ) 2 a 4 ] } , Φ 1 , 2 ( α , ω ) = α 2 a 4 ( p ± q ) 4 [ 1 + ( p ± q ) 2 α 4 ] + i k l d .
U ( r , z , t ) = 1 2 π - + A ¯ ( ω ) exp [ - i ( ω t - k z ) ] d ω × m = 1 2 0 U ¯ m ( α , ω ) × exp [ i Φ m ( α , ω ) + i k l d - i z 2 k α 2 ] × J 0 ( α r m ) α d α .
Φ 1 ( α , Δ ω ) - z 2 k α 2 = α 2 4 [ p 0 ( 1 + b 1 Δ ω ) + q ] - z α 2 2 k 0 ( 1 + Δ ω / ω 0 ) .
q = p 0 ω 0 n - 1 d n d ω | ω = ω 0 ,             f 1 = k 0 2 ( p 0 + q ) ,
1 f d = 1 f r ω 0 n - 1 d n d ω | ω = ω 0 = 1 f r k ,             f 1 = f r f d f r + f d .
U ( r m , z , t ) - i k 0 a 2 m = 1 2 0 1 2 f m exp ( - η 2 ) × exp [ i η 2 β m ( z ) ] A [ t + η 2 τ m ( z ) ] J 0 ( η r ) η d η ,
β m ( z ) = a 2 k 0 2 f m ( 1 - z f m ) ,             η = α f 1 , 2 k 0 a ,             r m = k a f m r , τ m ( z ) = a 2 k 0 2 f m ( b 1 - z f m ω 0 ) .
U ( r , z , t ) = A 0 exp ( - t 2 τ 0 2 ) m = 1 2 π τ 0 k 0 a 2 2 τ m ( z ) f m × exp ( { t τ 0 + [ 1 - i β m ( z ) ] τ 0 2 τ m ( z ) } 2 ) × ( 1 - Φ { t τ 0 + [ 1 - i β m ( z ) ] τ 0 2 τ m ( z ) } ) ,
Φ 1 ( α , Δ ω ) - z 2 k α 2 = α 2 a 4 ( p + q ) 4 { 1 + [ p 0 ( 1 + b 1 Δ ω ) + q ] 2 a 4 } - z α 2 2 k 0 ( 1 + Δ ω / ω 0 ) .
k 0 ω 0 ( p 0 + q ) + k 0 p 0 b 1 = 4 z ( p 0 + q ) p 0 b 1 ,
z = a 4 k 0 ( p 0 + q ) 2 [ 1 + a 4 ( p 0 + q ) 2 ] .
Δ Φ = Φ 1 ( α , ω ) + i k l d - z 2 k α 2 = α 2 4 [ p 0 ( 1 + b 1 Δ ω + b 2 Δ ω 2 ) + q ] - z α 2 2 k 0 ( 1 + Δ ω / ω 0 ) + i k l d ,
b 2 = 1 ω 0 ( n 0 - 1 ) d n d ω 0 | ω = ω 0 + 1 2 ( n 0 - 1 ) d 2 n d ω 2 | ω = ω 0 .
Δ Φ = - α 2 p 0 b 2 16 ( p 0 + q ) 2 Δ ω 2 + k 0 b 2 d ( n 0 - 1 ) Δ ω 2 .
U ( r 1 , z , t ) k 0 a 2 i 2 f 1 2 π - + A ¯ ( Δ ω ) d ( Δ ω ) 0 exp ( - η 2 ) × exp ( i { [ k 0 d ( n 0 - 1 ) - η n 2 p 0 ] b 2 Δ ω 2 - t Δ ω } ) × J 0 ( η r 1 ) η d η ,
U ( r 1 , f 1 , t ) k 0 a 2 i 2 f 1 0 1 [ 1 - i 4 ( 1 - η 2 ) δ τ 0 2 ] - 1 / 2 × exp { - ( t / τ 0 ) 2 [ 1 - i 4 ( 1 - η 2 ) ( δ / τ 0 2 ) ] } × exp ( - η 2 ) J 0 ( η r 1 ) η d η ,

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