Abstract

We describe the focusing region associated with transmittances, analyzing its associated phase function. We show that generic features can be studied from the differential equation for focusing geometry, which is obtained through angular representation for diffraction fields. With the treatment, we recover the results for circular zone plates, and by introducing a linear transformation into the transmittance function we generate structures that keep the ability to generate focusing. According to the choice of the parameters involved, the diffraction field presents new focusing regions, whose three-dimensional geometry and spatial evolution can be described in a selective fashion with analysis of only the phase singularities associated with the diffraction field and avoidance of the integral representation. The treatment is also applied to a simple lens. We recover the theoretical predictions obtained by Berry and Upstill [M. V. Berry and C. Upstill, in Progress in Optics, E. Wolf, ed. (North–Holland, Amsterdam, 1980), Vol. XVIII, p. 259], and these predictions are corroborated experimentally. The results obtained are shown.

© 2001 Optical Society of America

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References

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  1. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), p. 109.
  2. R. Courant, D. Hillbert, Methods of Mathematical Physics (Interscience, Berlin, 1989), Vol. II, p. 10.
  3. M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” in Progress in Optics, E. Wolf, ed. (North–Holland, Amsterdam, 1980), Vol. XVIII, p. 259.
  4. I. Freund, A. Belenkiy, “Higher-order extrema in two-dimensional wave fields,” J. Opt. Soc. Am. A 17, 434–445 (2000).
    [CrossRef]
  5. N. M. Ceglio, D. W. Sweeney, “Zone plate-coded imaging: theory and applications,” in Progress in Optics XXII, E. Wolf, ed. (North–Holland, Amsterdam, 1982), Chap. IV, pp. 289–353.
  6. A. W. Lohman, Optical Information Processing (University of Erlangen–Nurnberg, Erlangen, Germany, 1978), p. 113.
  7. M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), p. 237.

2000

Belenkiy, A.

Berry, M. V.

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” in Progress in Optics, E. Wolf, ed. (North–Holland, Amsterdam, 1980), Vol. XVIII, p. 259.

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), p. 237.

Ceglio, N. M.

N. M. Ceglio, D. W. Sweeney, “Zone plate-coded imaging: theory and applications,” in Progress in Optics XXII, E. Wolf, ed. (North–Holland, Amsterdam, 1982), Chap. IV, pp. 289–353.

Courant, R.

R. Courant, D. Hillbert, Methods of Mathematical Physics (Interscience, Berlin, 1989), Vol. II, p. 10.

Freund, I.

Hillbert, D.

R. Courant, D. Hillbert, Methods of Mathematical Physics (Interscience, Berlin, 1989), Vol. II, p. 10.

Lohman, A. W.

A. W. Lohman, Optical Information Processing (University of Erlangen–Nurnberg, Erlangen, Germany, 1978), p. 113.

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), p. 109.

Sweeney, D. W.

N. M. Ceglio, D. W. Sweeney, “Zone plate-coded imaging: theory and applications,” in Progress in Optics XXII, E. Wolf, ed. (North–Holland, Amsterdam, 1982), Chap. IV, pp. 289–353.

Upstill, C.

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” in Progress in Optics, E. Wolf, ed. (North–Holland, Amsterdam, 1980), Vol. XVIII, p. 259.

Wolf, E.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), p. 109.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), p. 237.

J. Opt. Soc. Am. A

Other

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), p. 109.

R. Courant, D. Hillbert, Methods of Mathematical Physics (Interscience, Berlin, 1989), Vol. II, p. 10.

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” in Progress in Optics, E. Wolf, ed. (North–Holland, Amsterdam, 1980), Vol. XVIII, p. 259.

N. M. Ceglio, D. W. Sweeney, “Zone plate-coded imaging: theory and applications,” in Progress in Optics XXII, E. Wolf, ed. (North–Holland, Amsterdam, 1982), Chap. IV, pp. 289–353.

A. W. Lohman, Optical Information Processing (University of Erlangen–Nurnberg, Erlangen, Germany, 1978), p. 113.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), p. 237.

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Figures (4)

Fig. 1
Fig. 1

Scaled zone plate for α=1 and β=0.8.

Fig. 2
Fig. 2

Experimental results for the spatial evolution of the focusing region. The transmittance was illuminated with a coherent plane wave.

Fig. 3
Fig. 3

Experimental results for the focusing of a simple lens illuminated with a spherical wave.

Fig. 4
Fig. 4

Theoretical prediction for the focusing of a simple lens. Mapping of the curves on the exit pupil, generate the focusing region.

Equations (39)

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ϕ(x, y, z)=-|A(u, v)|×exp{i2π[xu+yv+zp+φ(u, v)]}dudv,
u2+v2+p2=1/λ2,
u=cos αλ,
v=cos βλ,
p=cos γλ,
|A(u, v)|exp[i2piφ(u, v)]
=-ϕ(x, y, z=0) exp[-i2π(xu+yv)]dxdy.
L(x, y, u, v)=xu+yv+pf(x, y).
f(x, y)=ax+B(a)y+C(a),
fx(x, y)=a,fy(x, y)=β(a)=β(fx),
fxy(x, y)=fxfxx β(fx)=fxxfx β(fx),
fyy(x, y)=fxfxy β(fx)=fxyfx β(fx).
2fx22fy-2fxy2=0,
2Lx22Ly2-2Lxy2=0.
Lx=0,Ly=0.
t(x, y)=- anexpi2π(x2+y2)nd2,
L(x, y, z)
=2π(x2+y2)nd2+k x2+y22z-k xx0z-k yy0z.
xa1x+β1,yα2y+β2,
t(x, y)=- anexpi2π[(α1x+β1)2+(α2y+β2)2]nd2,
L(x, y, z)=2πn[(α1x+β1)2+(α2y+β2)2]d2+π x2+y2λz-2π xx0λz-2π yy0λz.
Lx=4πn(α1x+β)α1d2+2πxλz-2πx0λz=0,
Ly=4πn(α2y+β2)α2d2+2πyλz-2πy0λz=0,
2Lx22Ly2-2Lxy2
=4πnα12d2+2πλz4πnα22d2+2πλz=0,
z1=-d22nα12λ,z2=-d22nα22λ.
|z2-z1|=Δz=-d22nα22λ+d22nα12λ=d22nλ1α22-1α12.
x0=x2nα12λzd2+1,y0=y2nα22λzd2+1,
x2+y2=sd2,s=1, 2,,
(α1x)2+(α2y)2=sd2,s=1, 2,,
x=sd/α1cos t,y=sd/α2sin t,
x0=sdα12nα12λzd2+1cos t;
y0=sdα22nα22λzd2+1sin t,
f(x, y, x0, y0)=-14 B(x2+y2)2-C(x0x+yy0)2-D(x02+y02)(x2+y2)+E(x02+y02)(x0x+yy0)+F(x2+y2)(x0x+yy0),
L(x0, y0, x, y, X, Y)=z-f(x0, y0, x, y)+12z [(X-x)2+(Y-y)2],
X=x+zfx,
Y=y+zfy,
(fxx-1/z)(fyy-1/z)-fxy2=0.
1z=fx+fy2±[(fxx-fyy)-fxy2]1/2.

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