Abstract

A general method is presented of optimizing one-dimensional holograms to realize an arbitrary diffracted field distribution over the propagation volume. A computationally simpler approach is developed for the case in which only the transverse distribution in one plane and the longitudinal distribution on one axis are of interest. Scaling effects in the optimized beams are studied and compared with those of canonical beams.

© 1995 Optical Society of America

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References

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  1. W. H. Lee, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), Vol. XVI, pp. 119–173.
    [CrossRef]
  2. J. Rosen, A. Yariv, “Synthesis of an arbitrary axial field profile by computer-generated holograms,” Opt. Lett. 19, 843–845 (1994).
    [CrossRef] [PubMed]
  3. M. D. Levenson, “Phase-shifting mask strategies: isolated bright contacts,” Microlithography World (September–October 1992).
  4. E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5, 454–458 (1969).
    [CrossRef]
  5. J. Rosen, B. Salik, A. Yariv, H. K. Liu, “Pseudo non-diffracting slitlike beam and its analogy to the pseudo nondispersing pulse,” Opt. Lett. 20, 423–425 (1995).
    [CrossRef] [PubMed]
  6. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 59–61.
  7. R. Piestun, B. Spector, J. Shamir, “Control of wavefront propagation with diffractive elements,” Opt. Lett. 19, 771–773 (1994).
    [CrossRef] [PubMed]
  8. C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. 54, 240–244 (1964).
    [CrossRef]
  9. L. R. Foulds, Optimization Techniques (Springer-Verlag, New York, 1981), pp. 329–335.
  10. A. Yariv, Optical Electronics (Saunders, Philadelphia, 1991), pp. 48–49.

1995

1994

1992

M. D. Levenson, “Phase-shifting mask strategies: isolated bright contacts,” Microlithography World (September–October 1992).

1969

E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5, 454–458 (1969).
[CrossRef]

1964

Foulds, L. R.

L. R. Foulds, Optimization Techniques (Springer-Verlag, New York, 1981), pp. 329–335.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 59–61.

Lee, W. H.

W. H. Lee, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), Vol. XVI, pp. 119–173.
[CrossRef]

Levenson, M. D.

M. D. Levenson, “Phase-shifting mask strategies: isolated bright contacts,” Microlithography World (September–October 1992).

Liu, H. K.

McCutchen, C. W.

Piestun, R.

Rosen, J.

Salik, B.

Shamir, J.

Spector, B.

Treacy, E. B.

E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5, 454–458 (1969).
[CrossRef]

Yariv, A.

IEEE J. Quantum Electron.

E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5, 454–458 (1969).
[CrossRef]

J. Opt. Soc. Am.

Microlithography World

M. D. Levenson, “Phase-shifting mask strategies: isolated bright contacts,” Microlithography World (September–October 1992).

Opt. Lett.

Other

W. H. Lee, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), Vol. XVI, pp. 119–173.
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 59–61.

L. R. Foulds, Optimization Techniques (Springer-Verlag, New York, 1981), pp. 329–335.

A. Yariv, Optical Electronics (Saunders, Philadelphia, 1991), pp. 48–49.

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Figures (5)

Fig. 1
Fig. 1

System for realizing one transverse and one axial constraint.

Fig. 2
Fig. 2

Mask that generated the beam in Fig. 3(d) below. The mask was fabricated on tone-developed film for a cylindrical lens with focal length f = 30 cm, a HeNe laser at λ = 633 nm, and spatial frequency 206 cm−1 (θ = 13 mrad, mask width of 1.24 cm).

Fig. 3
Fig. 3

(a) Transverse constraint for PND beam, (b) axial constraint, (c) simulated PND beam axial intensity and three transverse profiles, and (d) experimentally obtained PND beam (top) compared with an ordinary focused beam over Δz = 15 cm. The mask used, along with relevant parameters, is shown in Fig. 2.

Fig. 4
Fig. 4

(a) Two-dimensional constraint for the second example discussed in Section 4; white denotes the field maximum, black denotes the field minimum, and gray denotes unconstrained areas. (b) Simulated intensity distribution. (c) Observed intensity distribution over Δz = 8 cm. The lens has f = 30 cm, λ = 633 nm, and mask spatial frequency of 206 cm−1 (mask width of 1.24 cm).

Fig. 5
Fig. 5

Log(error) versus focal depth and beam width of optimized PND beam.

Equations (13)

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u ( x , y ) = exp ( i k z ) i λ z - g ( x 1 ) exp [ i k 2 z ( x - x 1 ) 2 ] d x 1 ,
E [ g ( x 1 ) ] = ( x , z ) R 0 e [ u ( x , z ) ] d x d z ,
E [ g ( x 1 ) ] = ( x , z ) R 0 [ I 0 ( x , y ) - u ( x , z ) 2 ] 2 d x d z = ( x , z ) R 0 { I 0 ( x , z ) - 1 λ z | - g ( x 1 ) × exp [ i k 2 z ( x - x 1 ) 2 ] d x 1 | 2 } 2 d x d z .
d E = A E · d A + ϕ E · d ϕ ,
u ( x , z ) = exp [ i k ( z + 2 f ) ] i λ f - g ( x ) × exp [ - i k ( z x 2 + 2 f x x ) 2 f 2 ] d x ,
u 1 ( z ) = u ( x = 0 , z ) = exp [ i k ( z + 2 f ) ] i λ f - g ( x ) exp ( - i k z x 2 2 f 2 ) d x .
g ( x ) = i λ f exp ( - 2 i k f ) - u t ( x ) exp ( i k x x f ) d x .
T [ u t ( x ) ] = u 1 ( z ) = exp ( i k z ) λ f - - u t ( x ) × exp [ i k ( 2 f x x - z x 2 ) 2 f 2 ] d x d x .
u 1 ( z ) = u ( 0 , z ) = exp [ i k ( z + 2 f ) ] i λ f - g ( x ˜ ) × exp { - i k [ x ˜ 2 2 f - x ˜ 2 2 ( z + f ) + x ˜ 4 8 ( z + f ) 3 ] } d x ˜ ,
g ( x ) = 1 + Re [ g ( x ) exp ( i 2 π sin θ λ x ) ] ,
u ( x , z ) = exp [ i k ( z + 2 f ) ] i λ f - g ( x ) exp [ i k sin ( θ ) x ] × exp [ - i k ( z x 2 + 2 f x x ) 2 f 2 ] d x = exp [ i k ( z + 2 f ) ] i λ f - g ( x ) × exp [ - i k ( z x 2 + 2 f x ˜ x ) 2 f 2 ] d x ,
E [ u t ( x ) ] = z Δ z [ u 1 ( z ) - I 1 ( z ) ] 2 d z ,
E [ g ( x 1 ) ] = ( x , z ) R 0 [ I 0 ( x , z ) - u ( x , z ) ] 2 d x d z ,

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