Abstract

In the early stages of optical design, the consideration of low-order imaging properties significantly simplifies the identification of promising configurations. This process is investigated here for plane-symmetric systems composed of two off-axis sections of conic mirrors. In particular, constraints are derived that ensure sharp imagery through second order for such systems. For completeness, a number of special cases must be treated. In the cases that admit of realizable solutions, the requirement of sharp imagery through second order reduces the dimensionality of the resulting configuration spaces from thirteen to between four and six.

© 1994 Optical Society of America

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References

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  1. B. D. Stone, G. W. Forbes, “Foundations of first-order layout for asymmetric systems: an application of Hamilton’s methods,” J. Opt. Soc. Am. A 9, 96–110 (1992).
    [CrossRef]
  2. B. D. Stone, G. W. Forbes, “Foundations of second-order layout for asymmetric systems,” J. Opt. Soc. Am. A 9, 2067–2082 (1992).
    [CrossRef]
  3. See Sec. 3 of B. D. Stone, G. W. Forbes, “Illustration of second-order design methods: global merit function plots for a class of projection systems,” J. Opt. Soc. Am. A 11, 3308–3321 (1994).
    [CrossRef]
  4. For a description of Hamilton’s characteristic functions see, for example, H. A. Buchdahl, Introduction to Hamiltonian Optics (Dover, New York, 1993), Chap. 2.
  5. For an axially symmetric system, the base ray typically coincides with the axis of symmetry. In such a case, the terms of degree three in the Taylor expansion of any of Hamilton’s characteristic functions vanish on account of the symmetry. For an asymmetric system, the base ray is chosen to be near the center of the bundle of rays passed by the system, and, in general, the terms of degree three in the Taylor expansion of any of Hamilton’s characteristic functions are nonzero.
  6. B. D. Stone, G. W. Forbes, “First-order layout of asymmetric systems: sharp imagery of a single plane object,” J. Opt. Soc. Am. A 9, 832–843 (1992).
    [CrossRef]
  7. This convention is more appropriate for this analysis than the more general parameterization that is used in Refs. 1, 2, and 6. In those works, the systems of interest were not necessarily plane symmetric, and only angles of incidence between 0 and π/2 were considered. However, the situation corresponding to a negative angle of incidence here corresponds to a change in the value of a separate variable that specifies the angle between the planes of incidence associated with the base ray at two consecutive surfaces. Since all these planes of incidence are coincident in a plane-symmetric system, these variables are abandoned here, and the information they contained is now carried by the signs of the angles of incidence.
  8. An expression for s(y) appears in Eq. (6.1) of G. W. Forbes, B. D. Stone, “Hamilton’s angle characteristic in closed form for generally configured conic and toric interfaces,” J. Opt. Soc. Am. A 10, 1270–1278 (1993). In that work, the daggered coordinate system corresponds to the coordinate system located at the basal point of the mirror, and ∊has been replaced here by (κ+ 1). Also note that in that work, his taken to be positive, but, to keep all the Zaxes parallel, his a signed entity here. Note that Eq. (6.1) of this reference is also valid for negative h.
    [CrossRef]
  9. The refractive indices in object and image space are also generally required for determination of a system’s first-order imaging properties, but these indices have been taken to be unity here.
  10. For example, in Eq. (2.10), if F is not symmetric, it can be replaced by a symmetric tensor with components ⅙(Fijk+Fikj+Fjik+Fjki+Fkij+Fkji) without altering the value of the characteristic function.
  11. See, for example, B. D. Stone, G. W. Forbes, “Characterization of first-order optical properties for asymmetric systems,” J. Opt. Soc. Am. A 9, 478–489 (1992), and the references cited therein.
    [CrossRef]
  12. For a description of the Scheimpflug condition see, for example, R. Kingslake, Optical System Design (Academic, Orlando, Fla., 1983), Chap. 5, Sec. VI.
  13. The expression for M˜11 follows from Eq. (3.4b) of Ref. 2, while the expressions for L˜111,L˜221, and L˜122, follow from Eqs. (3.4e) of Ref. 2.
  14. To apply the expressions that appear in Ref. 2, the following correspondence must be made here: In Eqs. (A1), θscorresponds to θ1, whereas θs′ corresponds to (π− θ1), tcorresponds to d0, the matrix S is diagonal, and the diagonal elements correspond to S11and S22associated with mirror 1 (which are denoted in what follows by S11I and S22I). The sum of t′ that appears in Eqs. (A1) and tthat appears in Eqs. (A5) corresponds to d1. In Eqs. (A5), θscorresponds to θ2, θs′ corresponds to (π− θ2), t′ corresponds to d2, the matrix S is diagonal, and the elements are denoted in what follows by S11II and S22II. Finally, all refractive indices that appear in Eqs. (A5) and (A1) are taken to be unity here.

1994 (1)

1993 (1)

1992 (4)

Buchdahl, H. A.

For a description of Hamilton’s characteristic functions see, for example, H. A. Buchdahl, Introduction to Hamiltonian Optics (Dover, New York, 1993), Chap. 2.

Forbes, G. W.

See Sec. 3 of B. D. Stone, G. W. Forbes, “Illustration of second-order design methods: global merit function plots for a class of projection systems,” J. Opt. Soc. Am. A 11, 3308–3321 (1994).
[CrossRef]

An expression for s(y) appears in Eq. (6.1) of G. W. Forbes, B. D. Stone, “Hamilton’s angle characteristic in closed form for generally configured conic and toric interfaces,” J. Opt. Soc. Am. A 10, 1270–1278 (1993). In that work, the daggered coordinate system corresponds to the coordinate system located at the basal point of the mirror, and ∊has been replaced here by (κ+ 1). Also note that in that work, his taken to be positive, but, to keep all the Zaxes parallel, his a signed entity here. Note that Eq. (6.1) of this reference is also valid for negative h.
[CrossRef]

See, for example, B. D. Stone, G. W. Forbes, “Characterization of first-order optical properties for asymmetric systems,” J. Opt. Soc. Am. A 9, 478–489 (1992), and the references cited therein.
[CrossRef]

B. D. Stone, G. W. Forbes, “First-order layout of asymmetric systems: sharp imagery of a single plane object,” J. Opt. Soc. Am. A 9, 832–843 (1992).
[CrossRef]

B. D. Stone, G. W. Forbes, “Foundations of first-order layout for asymmetric systems: an application of Hamilton’s methods,” J. Opt. Soc. Am. A 9, 96–110 (1992).
[CrossRef]

B. D. Stone, G. W. Forbes, “Foundations of second-order layout for asymmetric systems,” J. Opt. Soc. Am. A 9, 2067–2082 (1992).
[CrossRef]

Kingslake, R.

For a description of the Scheimpflug condition see, for example, R. Kingslake, Optical System Design (Academic, Orlando, Fla., 1983), Chap. 5, Sec. VI.

Stone, B. D.

See Sec. 3 of B. D. Stone, G. W. Forbes, “Illustration of second-order design methods: global merit function plots for a class of projection systems,” J. Opt. Soc. Am. A 11, 3308–3321 (1994).
[CrossRef]

An expression for s(y) appears in Eq. (6.1) of G. W. Forbes, B. D. Stone, “Hamilton’s angle characteristic in closed form for generally configured conic and toric interfaces,” J. Opt. Soc. Am. A 10, 1270–1278 (1993). In that work, the daggered coordinate system corresponds to the coordinate system located at the basal point of the mirror, and ∊has been replaced here by (κ+ 1). Also note that in that work, his taken to be positive, but, to keep all the Zaxes parallel, his a signed entity here. Note that Eq. (6.1) of this reference is also valid for negative h.
[CrossRef]

See, for example, B. D. Stone, G. W. Forbes, “Characterization of first-order optical properties for asymmetric systems,” J. Opt. Soc. Am. A 9, 478–489 (1992), and the references cited therein.
[CrossRef]

B. D. Stone, G. W. Forbes, “Foundations of second-order layout for asymmetric systems,” J. Opt. Soc. Am. A 9, 2067–2082 (1992).
[CrossRef]

B. D. Stone, G. W. Forbes, “Foundations of first-order layout for asymmetric systems: an application of Hamilton’s methods,” J. Opt. Soc. Am. A 9, 96–110 (1992).
[CrossRef]

B. D. Stone, G. W. Forbes, “First-order layout of asymmetric systems: sharp imagery of a single plane object,” J. Opt. Soc. Am. A 9, 832–843 (1992).
[CrossRef]

J. Opt. Soc. Am. A (6)

B. D. Stone, G. W. Forbes, “Foundations of first-order layout for asymmetric systems: an application of Hamilton’s methods,” J. Opt. Soc. Am. A 9, 96–110 (1992).
[CrossRef]

B. D. Stone, G. W. Forbes, “Foundations of second-order layout for asymmetric systems,” J. Opt. Soc. Am. A 9, 2067–2082 (1992).
[CrossRef]

See Sec. 3 of B. D. Stone, G. W. Forbes, “Illustration of second-order design methods: global merit function plots for a class of projection systems,” J. Opt. Soc. Am. A 11, 3308–3321 (1994).
[CrossRef]

B. D. Stone, G. W. Forbes, “First-order layout of asymmetric systems: sharp imagery of a single plane object,” J. Opt. Soc. Am. A 9, 832–843 (1992).
[CrossRef]

An expression for s(y) appears in Eq. (6.1) of G. W. Forbes, B. D. Stone, “Hamilton’s angle characteristic in closed form for generally configured conic and toric interfaces,” J. Opt. Soc. Am. A 10, 1270–1278 (1993). In that work, the daggered coordinate system corresponds to the coordinate system located at the basal point of the mirror, and ∊has been replaced here by (κ+ 1). Also note that in that work, his taken to be positive, but, to keep all the Zaxes parallel, his a signed entity here. Note that Eq. (6.1) of this reference is also valid for negative h.
[CrossRef]

See, for example, B. D. Stone, G. W. Forbes, “Characterization of first-order optical properties for asymmetric systems,” J. Opt. Soc. Am. A 9, 478–489 (1992), and the references cited therein.
[CrossRef]

Other (8)

For a description of the Scheimpflug condition see, for example, R. Kingslake, Optical System Design (Academic, Orlando, Fla., 1983), Chap. 5, Sec. VI.

The expression for M˜11 follows from Eq. (3.4b) of Ref. 2, while the expressions for L˜111,L˜221, and L˜122, follow from Eqs. (3.4e) of Ref. 2.

To apply the expressions that appear in Ref. 2, the following correspondence must be made here: In Eqs. (A1), θscorresponds to θ1, whereas θs′ corresponds to (π− θ1), tcorresponds to d0, the matrix S is diagonal, and the diagonal elements correspond to S11and S22associated with mirror 1 (which are denoted in what follows by S11I and S22I). The sum of t′ that appears in Eqs. (A1) and tthat appears in Eqs. (A5) corresponds to d1. In Eqs. (A5), θscorresponds to θ2, θs′ corresponds to (π− θ2), t′ corresponds to d2, the matrix S is diagonal, and the elements are denoted in what follows by S11II and S22II. Finally, all refractive indices that appear in Eqs. (A5) and (A1) are taken to be unity here.

The refractive indices in object and image space are also generally required for determination of a system’s first-order imaging properties, but these indices have been taken to be unity here.

For example, in Eq. (2.10), if F is not symmetric, it can be replaced by a symmetric tensor with components ⅙(Fijk+Fikj+Fjik+Fjki+Fkij+Fkji) without altering the value of the characteristic function.

This convention is more appropriate for this analysis than the more general parameterization that is used in Refs. 1, 2, and 6. In those works, the systems of interest were not necessarily plane symmetric, and only angles of incidence between 0 and π/2 were considered. However, the situation corresponding to a negative angle of incidence here corresponds to a change in the value of a separate variable that specifies the angle between the planes of incidence associated with the base ray at two consecutive surfaces. Since all these planes of incidence are coincident in a plane-symmetric system, these variables are abandoned here, and the information they contained is now carried by the signs of the angles of incidence.

For a description of Hamilton’s characteristic functions see, for example, H. A. Buchdahl, Introduction to Hamiltonian Optics (Dover, New York, 1993), Chap. 2.

For an axially symmetric system, the base ray typically coincides with the axis of symmetry. In such a case, the terms of degree three in the Taylor expansion of any of Hamilton’s characteristic functions vanish on account of the symmetry. For an asymmetric system, the base ray is chosen to be near the center of the bundle of rays passed by the system, and, in general, the terms of degree three in the Taylor expansion of any of Hamilton’s characteristic functions are nonzero.

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Figures (4)

Fig. 1
Fig. 1

Schematic representation of a plane-symmetric system of two conic mirrors. The figure illustrates a cross section in the plane of symmetry. The base ray through the system is shown, along with object and image planes. Each mirror’s axis of revolution is shown as a thick, dashed line.

Fig. 2
Fig. 2

Representation of a single conic mirror.

Fig. 3
Fig. 3

Schematic representation of a two-mirror, plane-symmetric system for which M11 is positive. The base ray is represented by the thick line segments. A second ray is shown as thin line segments. If M22 is also positive, then this system does not form an internal image.

Fig. 4
Fig. 4

For completeness, we treated a number of special cases in order to investigate all systems of two conic mirrors that possesses sharp imagery to second order. This chart summarizes the special cases and shows the section in which each case is discussed. The number of degrees of freedom (DOF) that remain for each case is also shown. The boxes that are outlined by thick black lines represent cases for which no realizable systems exist.

Equations (126)

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x ν = c y ν 2 1 + [ 1 - ( κ + 1 ) c 2 y ν 2 ] 1 / 2 ,
s ( y ) = ½ ( S 11 y 2 + S 22 z 2 ) + y ( S 111 y 2 + 3 S 122 z 2 ) + O ( 4 ) ,
S 11 = c ( 1 - c 2 κ h 2 ) - 3 / 2 ,
S 22 = c ( 1 - c 2 κ h 2 ) - 1 / 2 ,
S 111 = 3 c 3 κ h [ 1 - c 2 ( κ + 1 ) h 2 ] 1 / 2 ( 1 - c 2 κ h 2 ) - 3 ,
S 122 = c 3 κ h [ 1 - c 2 ( κ + 1 ) h 2 ] 1 / 2 ( 1 - c 2 κ h 2 ) - 2 .
c = S 22 S 22 S 11 ,
κ = S 122 2 + S 22 2 ( S 11 - S 22 ) 2 S 22 3 ( S 11 - S 22 ) ,
h = sgn ( S 122 ) ( S 11 - S 22 ) [ S 122 2 + S 22 2 ( S 11 - S 22 ) 2 ] 1 / 2 .
S 111 = 3 S 122 S 11 S 22 .
x ν ± = 1 c ( 1 ± - κ ) .
tan ( θ ) = ± c h - κ = ± ( S 22 / S 11 ) - 1 ,
t = - sgn ( S 11 ) [ S 22 ( S 22 - S 11 ) ] 1 / 2 [ S 22 2 S 11 ( S 22 - S 11 ) ] 1 / 2 - sgn ( θ ) S 122 .
t = - sgn ( S 11 ) [ S 22 ( S 22 - S 11 ) ] 1 / 2 [ S 22 2 S 11 ( S 22 - S 11 ) ] 1 / 2 + sgn ( θ ) S 122 .
C 01 ( y , p ) = c 0 + ½ y T F y + y T M p + ½ p T B p + ( y i y j y k F i j k + ½ y i y j p k L i j k + ½ y i p j p k N i j k + p i p j p k B i j k ) + O ( 4 ) .
F = [ F 11 0 0 F 22 ] ,
M = [ M 11 0 0 M 22 ] ,
B = [ B 11 0 0 B 22 ] .
p = - M - 1 p .
y = - C 01 p ,
p = - C 01 y ,
y k = - ( y i M i k + p i B i k + ½ y i y j L i j k + y i p j N i j k + ½ p i p j B i j k ) + O ( 3 ) .
tan ( θ obj ) = N 111 - N 122 M 11 2 - M 22 2 ,
tan ( θ im ) = M 11 2 N 122 - M 22 2 N 111 M 11 ( M 11 2 - M 22 2 ) .
M 11 2 tan ( θ obj ) + M 11 tan ( θ im ) = N 111 .
y im = - ( M ˜ 11 y obj + ½ L ˜ 11 y obj 2 + ½ L ˜ 221 z obj 2 ) + O ( 3 ) ,
z im = - ( M ˜ 22 z obj + L ˜ 122 y obj z obj ) + O ( 3 ) ,
M ˜ 11 = cos ( θ obj ) cos ( θ im ) M 11 ,
M ˜ 22 = M 22 ,
L ˜ 111 = cos 2 ( θ obj ) cos ( θ im ) [ L 111 - 2 F 11 M 11 tan ( θ obj ) ] ,
L ˜ 122 = cos ( θ obj ) [ L 122 - F 22 M 22 tan ( θ obj ) ] ,
L ˜ 221 = L 221 cos ( θ im ) .
B = - d 2 - ( { d 0 [ + 2 d 0 cos ( θ 1 ) A 1 - 1 S 1 A 1 - 1 ] - 1 + d 1 } - 1 + 2 cos ( θ 2 ) A 2 - 1 S 2 A 2 - 1 ) - 1 ,
M = - { 2 d 0 cos ( θ 2 ) A 2 - 1 S 2 A 2 - 1 + [ 2 d 1 cos ( θ 2 ) A 2 - 1 S 2 A 2 - 1 + ] × [ 2 d 0 cos ( θ 1 ) A 1 - 1 S 1 A 1 - 1 + ] } - 1 ,
A i = [ cos ( θ i ) 0 0 1 ] ,
S 1 = [ S 11 I 0 0 S 22 I ] ,             S 2 = [ S 11 II 0 0 S 22 II ] .
S 1 = 1 2 d 0 d 1 cos ( θ 1 ) A 1 2 [ d 2 M - 1 - ( d 0 + d 1 ) ] ,
S 2 = 1 2 d 1 d 2 cos ( θ 0 ) A 2 2 [ d 0 M - ( d 1 + d 2 ) ] .
F = ( d 2 - d 1 M + d 0 M 2 ) / ( d 0 d 2 ) .
[ M 11 d 0 - ( d 1 + d 2 ) ] [ M 22 d 0 - ( d 1 + d 2 ) ] > 0 ,
[ d 2 M 11 - ( d 0 + d 1 ) ] [ d M 22 - ( d 0 + d 1 ) ] > 0.
( S 22 I / S 11 I ) - 1 = d 2 [ M 22 cos 2 ( θ 1 ) - M 11 ] + M 11 M 22 sin 2 ( θ 1 ) ( d 0 + d 1 ) M 22 [ M 11 ( d 0 + d 1 ) - d 2 ] cos 2 ( θ 1 ) ,
( S 22 II / S 11 II ) - 1 = d 0 [ M 11 cos 2 ( θ 2 ) - M 22 ] + sin 2 ( θ 2 ) ( d 1 + d 2 ) [ ( d 1 + d 2 ) - M 11 d 0 ] cos 2 ( θ 2 ) .
B 111 = 3 B 122 ,
( N 111 - N 122 ) = 2 N 221 .
S 122 I = [ M ( d 0 + d 1 ) - d 2 ] [ M ( d 0 - d 1 ) + d 2 ] tan ( θ 1 ) + 2 d 2 ( M d 0 - d 1 - d 2 ) tan ( θ 2 ) 4 M 2 d 0 2 d 1 2 ,
S 122 II = 2 M d 0 [ M ( d 0 + d 1 ) - d 2 ] tan ( θ 1 ) + ( M d 0 - d 1 - d 2 ) ( M d 0 - d 1 + d 2 ) tan ( θ 2 ) 4 d 1 2 d 2 2 ,
S 1 = [ d 2 - M ( d 0 + d 1 ) ] 2 M d 0 d 1 cos ( θ 1 ) A 1 2 ,
S 2 = ( M d 0 - d 1 - d 2 ) 2 d 1 d 2 cos ( θ 2 ) A 2 2 .
M tan ( θ obj ) + tan ( θ im ) = 2 d 1 { [ M ( d 0 + d 1 ) - d 2 ] tan ( θ 1 ) + ( M d 0 - d 1 - d 2 ) tan ( θ 2 ) } .
L ˜ 111 = M cos 2 ( θ obj ) 2 d 0 d 2 cos ( θ im ) [ 4 ( d 1 + d 2 - M d 0 ) tan ( θ 2 ) + ( 3 M d 1 - 4 d 2 ) tan ( θ obj ) + ( 4 M d 0 - d 1 ) tan ( θ im ) ] ,
L ˜ 221 = M [ 4 ( d 1 + d 2 - M d 0 ) tan ( θ 2 ) + M d 1 tan ( θ obj ) + d 1 tan ( θ im ) ] 2 d 0 d 2 cos ( θ im ) ,
L ˜ 122 = M cos ( θ obj ) [ ( M d 1 - 2 d 2 ) tan ( θ obj ) + ( 2 M d 0 - d 1 ) tan ( θ im ) ] 2 d 0 d 2 .
S 122 I = [ M ( d 0 + d 1 ) + d 2 ] [ M ( d 0 - d 1 ) + d 2 ] tan ( θ 1 ) + 2 d 2 ( M d 0 + d 1 + d 2 ) tan ( θ 2 ) 4 M 2 d 0 2 d 1 2 ,
S 122 II = ( M d 0 + d 1 + d 2 ) ( M d 0 - d 1 - d 2 ) tan ( θ 2 ) - 2 M d 0 [ M ( d 0 + d 1 ) + d 2 ] tan ( θ 1 ) 4 d 1 2 d 2 2 ,
tan ( θ 1 ) = - ( M d 0 + d 1 + d 2 ) [ M ( d 0 + d 1 ) + d 2 ] tan ( θ 2 ) .
S 1 = - d 2 2 M d 0 ( d 0 + d 2 ) cos ( θ 1 ) × A 1 [ ( M + 1 ) 0 0 ( M - 1 ) ] A 1 ,
S 2 = - d 0 2 d 2 ( d 0 + d 2 ) cos ( θ 2 ) A 2 [ ( M + 1 ) 0 0 - ( M - 1 ) ] A 2 .
S 122 I = [ M 11 ( d 0 - d 1 ) + d 2 ] [ M 22 ( d 0 + d 1 ) - d 2 ] 4 M 11 M 22 d 0 2 d 1 2 tan ( θ 1 ) + d 2 ( M 22 d 0 - d 1 - d 2 ) 2 M 11 M 22 d 0 2 d 1 2 tan ( θ 2 ) ,
S 122 II = M 11 d 0 [ M 22 ( d 0 + d 1 ) - d 2 ] 2 d 1 2 d 2 2 tan ( θ 1 ) + ( M 11 d 0 - d 1 + d 2 ) ( M 22 d 0 - d 1 + d 2 ) 4 d 1 2 d 2 2 tan ( θ 2 ) .
3 d 0 d 2 ( M 11 - M 22 ) ( d 0 + d 1 + d 2 ) d 1 × { tan ( θ 1 ) ( M 22 d 0 - d 1 - d 2 ) + tan ( θ 2 ) [ M 22 ( d 0 + d 1 ) - d 2 ] } = 0.
tan ( θ 1 ) = - ( M 22 d 0 - d 1 - d 2 ) [ M 22 ( d 0 + d 1 ) - d 2 ] tan ( θ 2 ) .
S 122 I = [ M 11 ( d 1 - d 0 ) + d 2 ] ( M 22 d 0 - d 1 - d 2 ) 4 M 11 M 22 d 0 2 d 1 2 tan ( θ 2 ) ,
S 122 II = ( - M 11 d 0 - d 1 + d 2 ) ( M 22 d 0 - d 1 - d 2 ) 4 d 1 2 d 2 2 tan ( θ 2 ) .
tan ( θ obj ) = - M 11 ( d 0 + d 1 + d 2 ) ( M 11 + M 22 ) [ M 22 ( d 0 + d 1 ) - d 2 ] tan ( θ 2 ) ,
tan ( θ im ) = - M 22 2 ( d 0 + d 1 + d 2 ) ( M 11 + M 22 ) [ M 22 ( d 0 + d 1 ) - d 2 ] tan ( θ 2 ) .
L ˜ 111 = 2 M 11 cos 2 ( θ obj ) tan ( θ 2 ) d 0 d 2 ( M 11 + M 22 ) [ M 22 ( d 0 + d 1 ) - d 2 ] cos ( θ im ) × { M 11 d 0 d 2 ( 1 + M 22 ) 2 + M 11 M 22 d 1 ( d 0 + d 1 + d 2 ) + M 22 d 2 ( M 22 d 0 - d 1 - d 2 ) - M 11 2 d 0 [ M 22 ( d 0 + d 1 ) - d 2 ] } ,
L ˜ 221 = - ( M 11 + M 22 ) ( M 22 d 0 - d 1 - d 2 ) tan ( θ 2 ) d 0 d 2 cos ( θ im ) ,
L ˜ 122 = M 22 ( M 22 d 1 - d 2 - M 22 2 d 0 ) sin ( θ obj ) d 0 d 2 .
M ˜ 11 = M 11 { ( M 11 + M 22 ) 2 [ M 22 ( d 0 + d 1 ) - d 2 ] 2 + M 22 4 ( d 0 + d 1 + d 2 ) 2 tan 2 ( θ 2 ) ( M 11 + M 22 ) 2 [ M 22 ( d 0 + d 1 ) - d 2 ] 2 + M 11 2 ( d 0 + d 1 + d 2 ) 2 tan 2 ( θ 2 ) } 1 / 2 .
tan 2 ( θ 2 ) = ( M 11 + M 22 ) 2 [ M 22 ( d 0 + d 1 ) - d 2 ] 2 ( M 11 2 - M ˜ 11 2 ) M 11 2 ( d 0 + d 1 + d 2 ) 2 ( M ˜ 11 2 - M 22 4 ) .
M 11 < M ˜ 11 when ( M ˜ 11 2 < M 22 4 ) , M 11 > M ˜ 11 when ( M ˜ 11 2 > M 22 4 ) .
cos 2 ( θ 1 ) = M 11 [ M 22 ( d 0 + d 1 ) - d 2 ] M 22 [ M 11 ( d 0 + d 1 ) - d 2 ] .
tan ( θ 2 ) = - [ M 22 ( d 0 + d 1 ) - d 2 ] [ M 11 ( d 0 - d 1 ) + d 2 ] 2 d 3 ( M 22 d 0 - d 1 - d 2 ) tan ( θ 1 ) ,
3 M 11 d 0 ( M 11 - M 22 ) ( d 0 + d 1 + d 2 ) [ M 11 ( d 1 - d 0 ) + d 2 ] 2 d 2 ( M 22 d 0 - d 1 - d 2 ) × tan ( θ 1 ) = 0.
d 1 = d 0 - d 2 M 11 ,
tan ( θ 1 ) = ± [ d 2 ( M 22 - M 11 ) d 2 ( M 11 + M 22 ) - 2 d 0 M 11 M 22 ] 1 / 2 ,
tan ( θ 2 ) = sgn [ d 2 θ 1 ( M 22 - M 11 ) ] × { d 2 ( M 22 - M 11 ) [ d 2 ( M 11 + M 22 ) - 2 d 0 M 11 M 22 ] } 1 / 2 d 2 ( 1 - M 11 ) + M 11 d 0 ( M 22 - 1 ) ,
S 11 I = S 22 I = - 1 d 0 [ 2 M 11 M 22 d 0 - d 2 ( M 11 + M 22 ) 2 M 22 ( M 11 d 0 - d 2 ) ] 1 / 2 ,
S 11 II = ( M 11 - 1 ) 2 d 2 cos ( θ 2 ) ,
S 22 II = [ M 11 d 0 ( M 22 - 1 ) - d 2 ( M 11 - 1 ) ] 2 d 2 ( M 11 d 0 - d 2 ) cos ( θ 2 ) .
d 0 > max [ d 2 M 11 , d 2 ( M 11 - 1 ) M 11 ( M 22 - 1 ) ]             when ( M 11 - 1 ) ( M 22 - 1 ) > 0 , d 2 M 11 < d 0 < d 2 ( M 11 - 1 ) M 11 ( M 22 - 1 )             when ( M 11 - 1 ) ( M 22 - 1 ) < 0.
S 122 II = - sgn [ ( M 22 - M 11 ) θ 1 ] ( M 11 + 1 ) 4 d 2 ( M 11 d 0 - d 2 ) × { ( M 22 - M 11 ) [ d 2 ( M 11 + M 22 ) - 2 d 0 M 11 M 22 ] d 2 } 1 / 2 .
tan ( θ obj ) = M 11 [ 2 M 11 d 0 + d 2 ( M 11 - 1 ) ] tan ( θ 1 ) ( M 11 + M 22 ) [ M 11 d 0 ( M 22 - 1 ) - d 2 ( M 11 - 1 ) ] ,
tan ( θ im ) = - M 22 2 [ 2 M 11 d 0 + d 2 ( M 11 - 1 ) ] tan ( θ 1 ) ( M 11 + M 22 ) [ M 11 d 0 ( M 22 - 1 ) - d 2 ( M 11 - 1 ) ] .
L ˜ 111 = - 2 M 11 cos 2 ( θ obj ) tan ( θ 1 ) d 0 d 2 ( M 11 + M 22 ) [ M 11 d 0 ( M 22 - 1 ) - d 2 ( M 11 - 1 ) ] cos ( θ im ) [ M 11 d 0 d 2 ( M 11 + 1 ) ( M 11 + M 22 2 ) - 2 M 22 ( M 11 - 1 ) ( M 11 d 0 - d 2 ) 2 ] ,
L ˜ 221 = sgn [ ( M 11 - M 22 ) θ 1 ] ( M 11 + M 22 ) M 11 d 0 cos ( θ im ) { ( M 22 - M 11 ) [ d 2 ( M 11 + M 22 ) - 2 d 0 M 11 M 22 ] d 2 } 1 / 2 ,
L ˜ 122 = M 22 [ M 11 M 22 d 0 ( 1 - M 22 ) - d 2 ( M 11 + M 22 ) ] sin ( θ obj ) M 11 d 0 d 2 ,
M ˜ 11 = M 11 ( a 0 M 22 4 + a 1 a 0 M 11 2 + a 1 ) 1 / 2 ,
a 0 = ( M 22 - M 11 ) [ 2 M 11 ( d 0 / d 2 ) - ( M 11 - 1 ) ] 2 ,
a 1 = ( M 11 + M 22 ) 2 [ M 11 + M 22 - 2 M 11 M 22 ( d 0 / d 2 ) ] × [ M 11 - 1 - ( d 0 / d 2 ) M 11 ( M 22 - 1 ) ] 2 .
y obj = - [ 1 M ˜ 11 y im + 1 2 ( L ˜ 111 M ˜ 11 3 ) y im 2 + 1 2 ( L ˜ 221 M ˜ 11 M 22 2 ) z im 2 ] + O ( 3 ) ,
z obj = - [ 1 M 22 z im + ( L ˜ 122 M ˜ 11 M 22 2 ) y im z im ] + O ( 3 ) ,
B 111 = - 6 d 0 3 M 11 2 M 22 [ M 11 ( d 0 + d 1 ) - d 2 ] [ M 22 ( d 0 + d 1 ) - d 2 ] S 122 I + 6 d 2 3 ( M 11 d 0 - d 1 - d 2 ) ( M 22 d 0 - d 1 - d 2 ) S 122 II + 3 M 11 d 0 [ M 11 ( d 0 + d 1 ) - d 2 ] [ M 11 ( d 0 - d 1 ) - d 2 ] 2 d 1 2 tan ( θ 1 ) + 3 d 2 ( M 11 d 0 - d 1 - d 2 ) ( M 11 d 0 + d 1 - d 2 ) 2 d 1 2 tan ( θ 2 ) ,
B 122 = - 2 d 0 3 M 11 M 22 2 S 122 I + 2 d 2 3 S 122 II + d 0 [ M 22 ( d 0 + d 1 ) - d 2 ] [ M 11 M 22 ( d 0 - d 1 ) - d 2 ( M 22 - 2 M 11 ) ] 2 d 1 2 tan ( θ 1 ) + d 2 ( M 22 d 0 - d 1 - d 2 ) [ d 0 ( 2 M 22 - M 11 ) + d 1 - d 2 ] 2 d 1 2 tan ( θ 2 ) ,
N 221 = - 2 d 0 2 M 11 M 22 2 ( M 22 d 0 - d 1 ) d 2 S 122 I + 2 d 2 2 S 122 II + [ M 22 ( d 0 + d 1 ) - d 2 ] { M 22 [ M 11 ( d 0 - d 1 ) + d 2 ] ( M 22 d 0 - d 1 ) - 2 M 11 M 22 d 0 d 2 } 2 d 1 2 d 2 tan ( θ 1 ) + M 22 ( M 22 d 0 - d 1 - d 2 ) [ d 0 ( 2 M 22 - M 11 ) - d 1 - d 2 ] 2 d 1 2 tan ( θ 2 ) ,
N 111 = 6 M 11 2 M 22 d 0 2 ( d 1 - M 11 d 0 ) [ M 11 ( d 0 + d 1 ) - d 2 ] d 2 [ M 22 ( d 0 + d 1 ) - d 2 ] S 122 I + 6 M 11 d 2 2 ( M 11 d 0 - d 1 - d 2 ) ( M 22 d 0 - d 1 - d 2 ) S 122 II + M 11 [ M 11 ( d 0 + d 1 ) - d 2 ] { 3 ( M 11 d 0 - d 1 ) [ M 11 ( d 0 - d 1 ) - d 2 ] - 2 d 1 d 2 } 2 d 1 2 d 2 tan ( θ 1 ) + M 11 ( M 11 d 0 - d 1 - d 2 ) ( 3 M 11 d 0 + d 1 - 3 d 2 ) 2 d 1 2 tan ( θ 2 ) ,
N 122 = 2 M 11 M 122 2 d 0 2 ( d 1 - M 11 d 0 ) d 2 S 122 I + 2 M 11 d 2 2 S 122 II + [ M 22 ( d 0 + d 1 ) - d 2 ] { M 11 ( M 11 d 0 - d 1 ) [ M 22 ( d 0 - d 1 ) - 2 d 2 ] + M 22 d 2 ( M 11 d 0 + d 1 ) } 2 d 1 2 d 2 tan ( θ 1 ) + M 11 ( M 22 d 0 - d 1 - d 2 ) [ d 0 ( 2 M 22 - M 11 ) + 3 d 1 - d 2 ] 2 d 1 2 tan ( θ 2 ) ,
L 111 = 6 M 11 2 { M 22 d 0 ( M 11 d 0 - d 1 ) 2 [ d 2 - M 11 ( d 0 + d 1 ) ] d 2 2 [ M 22 ( d 0 + d 1 ) - d 2 ] S 122 I + d 2 ( M 11 d 0 - d 1 - d 2 ) ( M 22 d 0 - d 1 - d 2 ) S 122 II } + [ 3 M 11 ( d 1 - d 0 ) ( M 11 d 0 - d 1 ) + d 2 ( 3 M 11 d 0 + d 1 ) ] { M 11 ( M 11 d 0 - d 1 ) [ d 2 - M 11 ( d 0 + d 1 ) ] 2 d 0 d 1 2 d 2 2 } tan ( θ 1 ) + [ M 11 2 ( M 11 d 0 - d 1 - d 2 ) ( 3 M 11 d 0 - d 1 - d 2 ) 2 d 1 2 d 2 ] tan ( θ 2 ) ,
L 221 = 2 M 22 2 [ - M 11 d 0 ( M 22 d 0 - d 1 ) 2 d 2 2 S 122 I + d 2 S 122 II ] + { M 22 [ M 11 ( d 0 - d 1 ) + d 2 ] ( M 22 d 0 - d 1 ) - 2 M 11 d 2 ( M 22 d 0 + d 1 ) } × { ( M 22 d 0 - d 1 ) [ M 22 ( d 0 + d 1 ) - d 2 ] 2 d 0 d 1 2 d 2 2 } tan ( θ 1 ) + { M 22 2 ( M 22 d 0 - d 1 - d 2 ) [ d 0 ( 2 M 22 - M 11 ) - 3 d 1 - d 2 ] 2 d 1 2 d 2 } tan ( θ 2 ) ,
L 122 = 2 M 11 M 22 [ - M 22 d 0 ( M 11 d 0 - d 1 ) ( M 22 d 0 - d 1 ) d 2 2 S 122 I + d 2 S 122 II ] + { ( M 11 d 0 - d 1 ) ( M 22 d 0 - d 1 ) [ d 2 - M 11 ( d 0 - d 1 ) ] + 2 M 11 ( M 11 - M 22 ) d 0 2 d 1 } × { ( M 22 [ d 2 - M 22 ( d 0 + d 1 ) ] 2 d 0 d 1 2 d 2 2 } tan ( θ 1 ) + { M 11 M 22 ( M 22 d 0 - d 1 - d 2 ) [ d 0 ( 2 M 22 - M 11 ) + d 1 - d 2 ] 2 d 1 2 d 2 } tan ( θ 2 ) .
d 1 = - ( d 0 + d 2 ) .
S 1 = - d 2 2 d 0 ( d 0 + d 2 ) cos ( θ 1 ) A 1 ( M - 1 + ) A 1 ,
S 2 = - d 0 2 d 2 ( d 0 + d 2 ) cos ( θ 2 ) A 2 ( M + ) A 2 .
S 122 I = d 2 ( M 22 + 1 ) 4 M 11 M 22 d 0 2 ( d 0 + d 2 ) 2 { 2 d 0 tan ( θ 2 ) - [ 2 M 11 d 0 + d 2 ( M 11 + 1 ) ] tan ( θ 1 ) } ,
S 122 II = d 0 ( M 22 + 1 ) 4 d 2 2 ( d 0 + d 2 ) 2 { [ 2 d 2 + d 0 ( M 11 + 1 ) ] tan ( θ 2 ) - 2 d 2 tan ( θ 1 ) } .
tan ( θ obj ) = ( 1 + 2 M 11 + M 22 ) ( M 11 + M 22 ) ( d 0 + d 2 ) × [ d 2 tan ( θ 1 ) - d 0 tan ( θ 2 ) ] ,
tan ( θ im ) = ( M 11 + 2 M 22 + M 11 M 22 ) ( M 11 + M 22 ) ( d 0 + d 2 ) × [ d 2 tan ( θ 1 ) - d 0 tan ( θ 2 ) ] .
L ˜ 111 = M 11 ( M 11 + 1 ) cos 2 ( θ obj ) d 0 d 2 ( d 0 + d 2 ) cos ( θ im ) { d 2 [ d 0 ( 1 + 3 M 11 ) - ( d 2 + M 11 d 0 ) ( 2 + 3 M 11 + M 22 ) ( M 11 + M 22 ) ] tan ( θ 1 ) + d 0 [ ( d 2 + M 11 d 0 ) ( 2 + M 11 - M 22 ) ( M 11 + M 22 ) - d 0 ( 3 + M 11 ) ] tan ( θ 2 ) } ,
L ˜ 221 = - ( M 22 + 1 ) d 0 d 2 cos ( θ im ) [ M 11 d 2 tan ( θ 1 ) + M 22 d 0 tan ( θ 2 ) ] ,
L ˜ 122 = M 22 ( 1 + M 11 ) ( 1 + M 22 ) ( M 11 d 0 - d 2 ) [ d 2 tan ( θ 1 ) - d 0 tan ( θ 2 ) ] d 0 d 2 ( d 0 + d 2 ) ( M 11 + M 22 ) cos ( θ obj ) .
tan ( θ 2 ) = d 2 d 0 tan ( θ 1 ) ± 1 d 0 [ ( M 11 + M 22 ) 2 ( d 0 + d 2 ) 2 ( M ˜ 11 2 - M 11 2 ) M 11 2 ( M 11 + 2 M 22 + M 11 M 22 ) 2 - M ˜ 11 2 ( 1 + 2 M 11 + M 22 ) 2 ] 1 / 2 .
{ ( M ˜ 11 - M 22 ) + sgn ( M 22 ) [ ( M ˜ 11 + M 22 ) ( M ˜ 11 + 1 ) ] 1 / 2 }
tan ( θ 1 ) = ± [ M 22 - M 11 M 11 ( M 22 + 1 ) ] 1 / 2 ,
tan ( θ 2 ) = sgn ( θ 1 ) [ 2 M 11 d 0 + d 2 ( M 11 + 1 ) ] 2 d 0 × [ M 22 - M 11 M 11 ( M 22 + 1 ) ] 1 / 2 .
S 1 = - sgn [ M 11 ( M 11 + 1 ) ] d 2 2 d 0 ( d 0 + d 2 ) [ ( M 11 + 1 ) ( M 22 + 1 ) M 11 M 22 ] 1 / 2 ,
S 2 = - d 0 2 d 2 ( d 0 + d 2 ) cos ( θ 2 ) A 2 ( M + ) A 2 ,
S 122 II = sgn ( θ 1 ) × ( M 11 + 1 ) ( M 22 + 1 ) [ d 2 ( d 0 + 2 d 2 ) + M 11 d 0 ( d 2 + 2 d 0 ) ] 8 d 2 2 ( d 0 + d 2 ) 2 × [ M 22 - M 11 M 11 ( M 22 + 1 ) ] 1 / 2 .
tan ( θ obj ) = sgn ( θ 1 ) × [ d 2 ( 1 - M 11 ) - 2 M 11 d 0 ] ( 1 + 2 M 11 + M 22 ) 2 ( d 0 + d 2 ) ( M 11 + M 22 ) × [ M 22 - M 11 M 11 ( M 22 + 1 ) ] 1 / 2 ,
tan ( θ im ) = sgn ( θ 1 ) × [ d 2 ( 1 - M 11 ) - 2 M 11 d 0 ] ( M 11 + 2 M 22 + M 11 M 22 ) 2 ( d 0 + d 2 ) ( M 11 + M 22 ) × [ M 22 - M 11 M 11 ( M 22 + 1 ) ] 1 / 2 .
L ˜ 111 = M 11 ( M 11 + 1 ) cos 2 ( θ obj ) tan ( θ 1 ) 2 d 0 d 2 ( M 11 + M 22 ) ( d 0 + d 2 ) cos ( θ im ) { d 0 d 2 ( M 11 + 1 ) ( M 11 - M 22 - 2 M 11 M 22 ) - d 2 2 [ 2 + 3 ( M 11 + M 22 ) - M 11 ( M 11 - M 22 ) ] - 2 M 11 d 0 2 ( M 11 + 3 M 22 + 2 M 11 M 22 ) } ,
L ˜ 221 = - ( M 22 + 1 ) [ 2 M 11 M 22 d 0 + d 2 ( 2 M 11 + M 22 + M 11 M 22 ) ] tan ( θ 1 ) 2 d 0 d 2 cos ( θ im ) ,
L ˜ 122 = - M 22 ( M 11 + 1 ) ( M 22 + 1 ) ( d 2 - M 11 d 0 ) [ 2 M 11 d 0 + d 2 ( M 11 - 1 ) ] cos ( θ obj ) tan ( θ 1 ) 2 d 0 d 2 ( M 11 + M 22 ) ( d 0 + d 2 ) ,
M ˜ 11 = M 11 [ b 0 ( M 11 + 2 M 22 + M 11 M 22 ) 2 + b 1 b 0 ( 1 + 2 M 11 + M 22 ) 2 + b 1 ] 1 / 2 ,
b 0 = ( M 22 - M 11 ) [ 1 - M 11 - 2 M 11 ( d 0 / d 2 ) ] 2 ,
b 1 = 2 M 11 ( M 22 + 1 ) ( M 11 + M 22 ) 2 [ 1 + ( d 0 / d 2 ) ] 2 .

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