Abstract

The eye can rotate to accommodate the angular position of an object and the distance of the object from it. The rotation of the eye inside its socket to align its visual axis in the direction of an off-axis image may introduce full or partial vignetting when one is looking through a visual instrument with a real exit pupil. We analyze the effects of vignetting owing to rotation of the eye in visual instruments with real exit pupils.

© 2002 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 6, p. 239.
  2. W. T. Welford, Geometrical Optics (North-Holland, Amsterdam, 1962), Chap. 8, p. 136.
  3. D. H. Jacobs, Fundamentals of Optical Engineering (McGraw-Hill, New York, 1943), Chap. XII, p. 173.
  4. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Chap. 4.

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Chap. 4.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 6, p. 239.

Jacobs, D. H.

D. H. Jacobs, Fundamentals of Optical Engineering (McGraw-Hill, New York, 1943), Chap. XII, p. 173.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Chap. 4.

Welford, W. T.

W. T. Welford, Geometrical Optics (North-Holland, Amsterdam, 1962), Chap. 8, p. 136.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 6, p. 239.

Other

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 6, p. 239.

W. T. Welford, Geometrical Optics (North-Holland, Amsterdam, 1962), Chap. 8, p. 136.

D. H. Jacobs, Fundamentals of Optical Engineering (McGraw-Hill, New York, 1943), Chap. XII, p. 173.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Chap. 4.

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Figures (10)

Fig. 1
Fig. 1

Interpretation of specifications when (a) there is no vignetting for any object within the field of view and the eye remains static, (b) there is 50% vignetting at the extreme field of view and the eye remains static, and (c) there is 50% vignetting at the extreme field of view and the eye rotates to align its visual axis with the object.

Fig. 2
Fig. 2

The eye rotates to observe off-axis objects. There is vignetting over the whole field of view if the telescope pupil is not sufficiently large and is not properly located relative to the eye pupil.

Fig. 3
Fig. 3

Region of interference of the eye’s entrance pupil and the telescope’s exit pupil.

Fig. 4
Fig. 4

Geometric construction for deriving an expression for vignetting.

Fig. 5
Fig. 5

(a) Radius of the exit pupil of the instrument R TP (in millimeters) versus vignetting V α,ϕ T given by Eq. (22). (b) Separation between the pupils X (in millimeters) versus vignetting V α,ϕ T given by Eq. (23). R IP is the radius of the eye’s entrance pupil.

Fig. 6
Fig. 6

Relative position and orientation between the telescope’s pupil and the eye’s pupil for three cases: bottom, when the visual axis is in the direction of the telescope axis; center, when the visual axis makes an angle ϕ = 0.7ϕ T with the telescope axis; and top, when the visual axis makes an angle ϕ = ϕ T with the telescope axis. (a) Dashed lines, the rays that have to be traced through the telescope for three object positions: at -ϕ T , on axis, and at ϕ T . (b) Solid lines, the rays that pass through the eye’s entrance pupil for three object positions: at -ϕ T (0.5 vignetting), on axis (no vignetting), and at ϕ T (0.5 vignetting).

Fig. 7
Fig. 7

Bottom, relative position between the telescope’s and the eye’s pupils. Top, orientation of the eye when it is looking at an object at the extreme field of view. These telescopes were designed to have (a) 0.5 vignetting at ϕ T (number 3 in Table 1) and (b) 0 vignetting at ϕ T (number 4 in Table 1).

Fig. 8
Fig. 8

(a) Semifield of view ϕ T (in degrees) versus the radius of the exit pupil of the instrument R TP (in millimeters) given by Eq. (36). (b) Separation between the pupils X (in millimeters) versus R TP. R IP is the radius of the eye’s entrance pupil.

Fig. 9
Fig. 9

Positions of the exit pupil of the visual instrument and the entrance pupil of the static eye. The rotation point of the eye, R, is chosen as the center of the coordinate system.

Fig. 10
Fig. 10

The eye is rotated by an angle ϕ measured from the optical axis of the visual instrument.

Tables (3)

Tables Icon

Table 1 Radius of the Exit Pupil of the Instrument RTP and Its Position Relative to the Eye’s Pupil X for Several Amounts of Vignetting Vα,ϕT and Semifields of View ϕT a

Tables Icon

Table 2 Semifield of View ϕT Covered by the Instrument and Separation X between Pupils for Five Telescopesa

Tables Icon

Table 3 Vignetting Vα,ϕ T for Five Telescopes That Cover the Same Field of Viewa

Equations (59)

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V=AD/AB.
V=ADAB=AB-DBAB=1-DBAB.
AB=2RIP cosα-ϕ,
DB=DO cos α,
DO=DJ+JO,
DJ=RTP-L,
JO=QO-QJ.
QO=CS=BCcos α=RIP cosα-ϕcos α,
QJ=JC tanα.
x2+L2=K2,
x=K2-L21/2.
JC=K--X-x2+L-L21/2.
JC=K+X-K2-L21/2.
L=K sin ϕ,
JC=K+X-K2-K2 sin2 ϕ1/2=K+X-K cos ϕ.
QJ=K+X-K cos ϕtan α.
JO=RIP cosα-ϕcos α+K cos ϕ-K-Xtan α.
DO=RTP-K sin ϕ+RIP cosα-ϕcos α+K cos ϕ-K-Xtan α,
DB=RTP cos α+K sinα-ϕ+RIP cosα-ϕ-K+Xsin α.
V=1-DBAB=1-DB2RIP cosα-ϕ,
Vϕ,α=1-RTP cos α+K sinα-ϕ+RIP cosα-ϕ-K+Xsin α2RIP cosα-ϕ.
VϕT,α=0=1-RTP-K sin ϕT+RIP cos ϕT2RIP cos ϕT.
RTP=K sin ϕT+RIP cos ϕT1-2VϕT,α=0.
X=-K+K cos ϕT-1-2VϕT,αRIP sin ϕT.
RTP=K sin ϕT,
X=Kcos ϕT-1,
VϕT,α=VϕT,α=0, α.
1-2VϕT,α=0RIP cos ϕT+K sin ϕT-RIP=0
A cos ϕT+B sin ϕT+C=0,
A=1-2VϕT,α=0RIP, B=K, C=-RIP.
tanu/2=z,
sin u=2z1+z2, cos u=1-z21+z2.
A1-z21+z2+B2z1+z2+C=0
z2C-A+2Bz+C+A=0.
z=-2B±4B2-4C2-A21/22C-A=-B±B2-C2+A21/2C-A.
z=-K+K2-RTP2+RIP21-2VϕT,α=021/2-RTP-1-2VϕT,α=0RIP,
z=tanϕT/2.
ϕT=2 tan-1-K+[K2-RTP2+RIP21-2VϕT,α=02-RTP-1-2VϕT,α=0RIP.
X=-K+K cos ϕT-1-2VϕT,αRIP sin ϕT.
ŝ=sin-αey-cos-αex,
rA=yAêy+xAêx,
yA=HE cos ϕ+xE sin ϕ,
xA=-HE sin ϕ+xE cos ϕ.
rB=rA+λs,
λ=xA-xTcos-α.
yBu=yAu+xAu-xTcos-αsin-α.
yBl=yAl+xAl-xTcos-αsin-α,
rAl=yAêy+xAêx,
yAl=-HE cos ϕ+xE sin ϕ,
xAl=HE sin ϕ+xE cos ϕ.
Δyu=yBu-HT,
Δyl=-yBl-HT.
Vα, ϕ=Vu+Vl =Δyu cos α2HE cosϕ-α ΘΔyu+Δyl cos α2HE cosϕ-α ΘΔyl,
Θx=1x>00x0.
0V1,
Vu=12+xE sinϕ-α-HT cos α+xT sin α2HE cosϕ-α×ΘΔyu,
Vl=12+-xE sinϕ-α-HT cos α+xT sin α2HE cosϕ-α×ΘΔyl.
Vα, ϕ=12+K sinϕ-α-RTP cos α+K+Xsin α2RIP cosϕ-α
Vα, ϕ=RIP cosα-ϕ+K sinϕ-α-RTP cos α+K+Xsin α2RIP cosα-ϕ,

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