Abstract

We generalize the transfer matrix ABCD theorem for paraxial rays of the optical system to skew rays propagated off axis, whether or not the system possesses rotational symmetry. Furthermore, we apply the generalized ABCD theorem to evaluate the diffraction integral matrix elements AD expressed in terms of the angle eikonal T, with the primary aberrations included. Finally, analysis and numerical calculation are given for propagation of a light beam through the optical system in the case in which spherical aberration and coma are present.

© 2004 Optical Society of America

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References

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  1. W. Brouwer, E. L. O’Neill, A. Walther, “The role of eikonal and matrix methods in contrast transfer calculations,” Appl. Opt. 2, 1239–1246 (1963).
    [CrossRef]
  2. W. Brouwer, Matrix Methods in Optical Instrument Design (Benjamin, New York, 1964).
  3. H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 154, 1312–1329 (1966).
    [CrossRef]
  4. A. Maitland, M. H. Dann, Laser Physics (North-Holland, 1969), p. 161.
  5. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).
    [CrossRef]
  6. W. Shaomin, L. Ronchi, Progress in Optics, Vol. XXV, E. Wolf, ed. (Elsevier, Amsterdam, 1988), p. 281.
  7. G. Piquero, P. M. Mejias, R. Martinez-Herrero, “Generalized ABCD matrix of thin pure phase transmittances: applications to thick spherically aberrated lenses,” Optik 105, 20–23 (1997).
  8. M. Born, E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999), p. 143 and p. 517.
  9. H. T. Yura, S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987).
    [CrossRef]

1997 (1)

G. Piquero, P. M. Mejias, R. Martinez-Herrero, “Generalized ABCD matrix of thin pure phase transmittances: applications to thick spherically aberrated lenses,” Optik 105, 20–23 (1997).

1987 (1)

1970 (1)

1966 (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 154, 1312–1329 (1966).
[CrossRef]

1963 (1)

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999), p. 143 and p. 517.

Brouwer, W.

Collins, S. A.

Dann, M. H.

A. Maitland, M. H. Dann, Laser Physics (North-Holland, 1969), p. 161.

Hanson, S. G.

Kogelnik, H.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 154, 1312–1329 (1966).
[CrossRef]

Li, T.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 154, 1312–1329 (1966).
[CrossRef]

Maitland, A.

A. Maitland, M. H. Dann, Laser Physics (North-Holland, 1969), p. 161.

Martinez-Herrero, R.

G. Piquero, P. M. Mejias, R. Martinez-Herrero, “Generalized ABCD matrix of thin pure phase transmittances: applications to thick spherically aberrated lenses,” Optik 105, 20–23 (1997).

Mejias, P. M.

G. Piquero, P. M. Mejias, R. Martinez-Herrero, “Generalized ABCD matrix of thin pure phase transmittances: applications to thick spherically aberrated lenses,” Optik 105, 20–23 (1997).

O’Neill, E. L.

Piquero, G.

G. Piquero, P. M. Mejias, R. Martinez-Herrero, “Generalized ABCD matrix of thin pure phase transmittances: applications to thick spherically aberrated lenses,” Optik 105, 20–23 (1997).

Ronchi, L.

W. Shaomin, L. Ronchi, Progress in Optics, Vol. XXV, E. Wolf, ed. (Elsevier, Amsterdam, 1988), p. 281.

Shaomin, W.

W. Shaomin, L. Ronchi, Progress in Optics, Vol. XXV, E. Wolf, ed. (Elsevier, Amsterdam, 1988), p. 281.

Walther, A.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999), p. 143 and p. 517.

Yura, H. T.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

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

Optik (1)

G. Piquero, P. M. Mejias, R. Martinez-Herrero, “Generalized ABCD matrix of thin pure phase transmittances: applications to thick spherically aberrated lenses,” Optik 105, 20–23 (1997).

Proc. IEEE (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 154, 1312–1329 (1966).
[CrossRef]

Other (4)

A. Maitland, M. H. Dann, Laser Physics (North-Holland, 1969), p. 161.

W. Brouwer, Matrix Methods in Optical Instrument Design (Benjamin, New York, 1964).

M. Born, E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999), p. 143 and p. 517.

W. Shaomin, L. Ronchi, Progress in Optics, Vol. XXV, E. Wolf, ed. (Elsevier, Amsterdam, 1988), p. 281.

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of the optical system.

Fig. 2
Fig. 2

(a) Amplitude distributions of the ideal case (solid curve), primary spherical aberration (dashed curve), and coma (dashed–dotted curve). (b) Phase distributions of the ideal case (solid curve), primary spherical aberration (dashed curve), and coma (dashed–dotted curve).

Fig. 3
Fig. 3

Amplitude distribution (a) of the ideal case, (b) in the presence of primary spherical aberration, (c) in the presence of primary coma, and (d) in the presence of simultaneous spherical aberration and coma.

Fig. 4
Fig. 4

Phase distribution (a) of the ideal case, (b) in the presence of primary spherical aberration, (c) in the presence of primary coma, and (d) in the presence of simultaneous spherical aberration and coma.

Fig. 5
Fig. 5

(a) Intensity distribution in the presence of primary coma. (b) Spot diagram in the presence of primary coma. (c) Ray distribution in the presence of primary coma.

Equations (103)

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Lx=ABCDLx
V(x0, y0, z0; x1, y1, z1)=P0P1nds,
T=V+p0x0-p1x1,
x0-p0m0 z0=Tp0,x1-p1m1 z1=-Tp1,
y0-q0m0 z0=Tq0,y1-q1m1 z1=-Tq1,
m0=n0-1n0 u-12n03 u2+=n0-1n0 u+O0(u2),
m1=n1-1n1 v-12n13 v2+=n1-1n1 v+O1(v2),
V=T-p0x0-q0y0-m0z0+p1x1+q1y1+m1z1=-n0z0+n1z1+T+z0n0p02+q022-z1n1p12+q122-z0O0(u2)+z1O1(v2)-p0x0-q0y0+p1x1+q1y1.
V=V0+T+z0n0p02+q022-z1n1p12+q122-p0x0-q0y0+p1x1+q1y1.
x0-p0n0 z0=Tp0,x1-p1n1 z1=-Tp1,
y0-q0n0 z0=Tq0,y1-q1n1 z1=-Tq1,
T=T(p0, q0; p1, q1)=T(0)+T(2)+T(4)+,
T(0)=n1a1-n0a0,
T(2)=au+bv+cw,
T(4)=du2+ev2+fw2+guv+huw+jvw,
x0-p0n0 z0=ap0+cp1,
x1-p1n1 z1=-bp1-cp0;
p1x1=-z0n0+ac1c-c-1cz0n0+az1n1-bz1n1-bcp0x0=ABCDp0x0,
Tp0=Tuup0+Twwp0=Tup0+Twp1,
Tp1=Tvvp1+Twwp1=Tvp1+Twp0.
Tq0=Tuq0+Twq1,Tq1=Tvq1+Twq0.
x0-p0n0 z0=Tup0+Twp1,
x1-p1n1 z1=-Tvp1-Twp0,p1
=Tw-1x0-z0n0+TuTw-1p0,
x1=-Twp0+z1n1-Tvp1=-Twp0+z1n1-TvTw-1x0-z0n0+TuTw-1p0;
  p1x1=-z0n0+TuTw-1Tw-1-Tw-z0n0+Tuz1n1-TvTw-1z1n1-TvTw-1p0x0,ABCD=-z0n0+Tuz1n1-TvTw-2+1+z0n0+Tuz1n1-TvTw-2=1.
u=p02+q022,v=p12+q122,w=p0p1+q0q1,
χ=p0p1-q0q1,
T=T(0)+T(2)+T(4)+,
T(0)=n1a1-n0a0,
T(2)=au+bv+cw+dχ,
T(4)=T(4)(u, v, w, χ).
p1x1=-z0n0+ac+d1c+d-(c+d)-1c+dz0n0+az1n1-bz1n1-bc+dp0x0,
q1y1=-z0n0+ac-d1c-d-(c-d)-1c-dz0n0+az1n1-bz1n1-bc-dq0y0.
p1x1=-z0n0+Tu(Tw+Tχ)-1(Tw+Tχ)-1-(Tw+Tχ)-z0n0+Tuz1n1-Tv(Tw+Tχ)-1z1n1-Tv(Tw+Tχ)-1p0x0,
q1y1=-z0n0+Tu(Tw-Tχ)-1(Tw-Tχ)-1-(Tw-Tχ)-z0n0+Tuz1n1-Tv(Tw-Tχ)-1z1n1-Tv(Tw-Tχ)-1q0y0,
p1x1=ABCDp0x0,
E(x1, y1)=-ik2πCexp(ikL0)E(x0, y0)×expik2C [D(x02+y02)-2(x0x1+y0y1)+A(x12+y12)]dx0dy0.
 
L0+12C [D(x02+y02)-2(x0x1+y0y1)+A(x12+y12)]
VV0+a+z0n0p022+a+z0n0q022+b-z1n1p122+b-z1n1q122+c(p0p1+q0q1)-p0x0-q0y0+p1x1+q1y1=V0+V1+V2,
V1=a+z0n0p022+b-z1n1p122+cp0p1-p0x0+p1x1.
p1=1C (Ax1-x0),p0=1C (x1-Dx0),
V1=12BC2 [-A(x1-Dx0)2-D(Ax1-x0)2+2(x1-Dx0)(Ax1-x0)-2BC(x1-Dx0)x0+2BC(Ax1-x0)x1]=12C (Ax12+Dx02-2x0x1).
V2=a+z0n0q022+b-z1n1q122+cq0q1-q0y0+q1y1=12C (Ay12+Dy02-2y0y1).
V=V0+12C [A(x12+y12)+D(x02+y02)-2(x0x1+y0y1)],
E(x, y)=ANE(x0, y0)exp(ikV)dx0dy0=ANexp(ikV0)E(x0, y0)×expik2C [D(x02+y02)-2(xx0+yy0)+A(x2+y2)]dx0dy0.
V1=12C (Ax12+Dx02-2x0x1)=12 Twz0n0+TuTw-1p02-z1n1-TvTw-1p12+2p0p1-p0x0+p1x1,
V1+V2=z0n0 u-z1n1 v+Tuu+Tvv+Tww-p0x0+p1x1-q0y0+q1y1.
V1+V2=z0n0 u-z1n1 v-p0x0+p1x1-q0y0+q1y1+au+bv+cw+2(du2+ev2+fw2+guv+huw+jvw)+ .
V-V0=z0n0 u-z1n1 v-p0x0+p1x1-q0y0+q1y1+au+bv+cw+du2+ev2+fw2+guv+huw+jvw+ .
T(2n)=1n (Tu(2n)u+Tv(2n)v+Tw(2n)w).
V-V0=z0n0 u-z1n1 v+T-p0x0+p1x1-q0y0+q1y1=z0n0 u-z1n1 v+1n (Tu(2n)u+Tv(2n)v+Tw(2n)w)-p0x0+p1x1-q0y0+q1y1.
A˜=-z0n0+T˜uT˜w-1,D˜=z1n1-T˜vT˜w-1,
B˜=T˜w-1,
T˜=T˜(2n),T˜(2n)=1n T(2n),
(T˜u(2n)u+T˜v(2n)v+T˜w(2n)w)=T.
V=V0+T+z0n0 u-z1n1 v-p0x0+p1x1-q0y0+q1y1.
T=au+bv+cw+du2+ev2+jvw,
T˜au+bv+cw+12(du2+ev2+jvw).
x0-z0n0 p0=T˜p0=(a+du)p0+c+12 jvp1,
x1-z1n1 p1=-T˜p1=-c+12 jvp0
-b+ev+12 jwp1;
p1=-(z0/n0+a+du)p0c+12jv+x0c+12jv,
x1=-(c+12jv)-(z1/n1-b-ev-jw/2)z0n0+a+duc+12jvp0+(z1/n1-b-ev-jw/2)c+12jv x0,
A=-(z0/n0+a+du)c+12jv,
D=(z1/n1-b-ev-jw/2)c+12jv,B=1c+12jv,
C=-(c+12jv)-(z0/n0+a+du)z1n1-b-ev-jw/2c+12jv.
p=Ap0+Bx0=AC (x-Dx0)+Bx0=AC x-1C x0,
p0=xC-DC x0.
E(x, y)= ik2πC E(x0, y0)expikB2D (x2+y2)expik2CxD-Dx02×expik2CyD-Dy02dx0dy0.
x0=x/D,y0=y/D.
E(x, y)E(x/D, y/D)D expik2BD (x2+y2).
u=0,
v=12C2 A-1D2(x2+y2)=12 B2D2 (x2+y2).
A=A0,B=B0,D=D0-eB0v,
C=AD-1B=C0-A0ev.
x0=xD=xD0-eB0v,y0=yD=yD0-eB0v.
Δx=x0-xD0=xD0-eB0v-xD0eB0vD02 x=e2 B03D04 (x2+y2)x.
Δy=e2 B03D04 (x2+y2)y.
B0D0 x,B0D0 yρ cos θ, ρ sin θ,
A=A01+j B02 v,B=B01+j B02 v,D=D0-j B02 w1+j B02 v.
x01+j B02 v=xD0-j B02 w,
x01-j B02 vxD0 1+B0D0 j w2,
Δx=x0-xD0=j B02 vx0+w xD02.
vx0=12 B02D02 (x2+y2)x0=12 x0ρ2.
pp0=d0d11D0,
w xD02=pp0 xD02p2 xD0=B0D02x2x0ρ2 cos2 θx0,
w yD02=pp0 yD02=B0D02xyx0x0ρ2 sin θ cos θ.
Δx=jB04 xbρ2(1+2 cos2 θ),
Δy=j2 B0vyb+w yD02=j B02 xbρ2 sin θ cos θ.
A0=1-d0+Δd0f0,B0=1f1=-1f0,
D0=1-d1f1,
C0=A0D0-1B0=1-d0+Δd0f01-d1f1/B0
-1/B0=Δd0D0.
Δd0 p02+q022=C02D0 -D0C0 (x0-xb)2+-D0C0 y02=D02C0 [(x0-xb)2+y02]
E(x0, y0)=exp-ikD02C0 [(x0-xb)2+y02],
E(x, y)=-ik2π 1C expik2C [A(x2+y2)-2(xx0+yy0)+D(x02+y02)]-ikD02C0 [(x0-xb)2+y02]dx0dy0.
f1=-f0=20mmd0=-39.8mm,
d1=40mm,
Δd0=200μ,xb=40μ,x0=-2020μ,
y0=-2020μ,λ=1μ,
(x02+y02)1/220μ.

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