Section 10 Vignetting Vignetting the Stop Determines Determines the Stop the Size of the Bundle of Rays That Propagates On-Axis an the System for Through Object

Home , Entrance pupil, Vignetting

OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp z z 10-2 10-1 Rays Vignetted Aperture Aperture Ray Ray Bundle Bundle Stop Stop Off-Axis On-Axis Section 10 Vignetting Vignetting The stop determines determines The stop the size of the bundle of rays that propagates on-axis an the system for through object. As the object height increases, apertures the other in the of one system (such as a lens clear of all or part limit aperture) may the This bundle of rays. is known as vignetting. OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2018 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 10-4 10-3 z z Plane Image y =0 Chief Ray is z y y y s, and is comprised s, and is comprised sections of of skew e circular the cross section the ray pupil, of tationally-symmetric spindle made up of up made spindle tationally-symmetric section is defined by the pupil and the object e individual cone sections match sections at the e individual cone match up y Stop fined by fined by the object pupil and or image point in that optical y Stop y =0  MAX yyy y -y y Pupil Pupil y remains circular with a radius equal to the radius of the axial bundle. The The the axial of bundle. remains circular to the radius a radius equal with z , the cross section of the bundle is circular, and the radius of the bundle is bundle the the radius of and is circular, the bundle the cross section of , z Since the base of the cone is defined by th by is defined the cone Since the base of bundle at any off-axis bundle is centered about the chief ray height. is centered the chief ray about bundle off-axis at any bundle ray the extent of radial maximum The the marginal ray value. The ray bundle is centered the optical is centered on axis. bundle ray The value. ray the marginal At any For an off-axis object point, the ray skew bundle the ray object off-axis point, an For circular which are still cones de space. The ray bundle for an on-axis object is a ro on-axis an for bundle ray The elements. surfaces and sections of right circular cones. Each cone Each circular cones. right sections of that optical or image point in space. Th Ray Bundle – Bundle Ray Axis Off Ray Bundle – Bundle Ray On-Axis OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 10-6 10-5    ay ay y  and ay y Fully Vignetted: nge if the system the system (chief nge if FOV Unvignetted: rs when an aperture completely rs when an aperture completely blocks res that the aperture passes the marginal res that the aperture passes the marginal e, vignetting occurs when other apertures occurs when vignetting e, x x block all or part of an off-axis ray bundle. ray off-axis an part of block all or res pass the entire ray bundle from res the object from the entire pass ray bundle on, vignetting cannot occur at the aperture at the aperture stop vignetting cannot occur on, Aperture Aperture a a must equal or exceed the maximum the ray bundle must equal or exceed the maximum height of y y y a e required apertures size will cha y y y Ray Bundle Ray Bundle ray and is not the system definiti By the system stop. is not and ray at a pupil. or The second part of this vignetting condition ensu this vignetting condition part of The second The maximum FOV supported by the system the system occu by supported FOV maximum The the ray bundle from the object point. from bundle the ray While defines the axial While the stop alone ray bundl the system, clear as a lens aperture, in such No vignetting occurs when all of the apertu aperture Each radius point. at the aperture. that Note th changed. is ray) Fully Vignetted Unvignetted OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 10-8 10-7 z Plane Image ), the     ay ay and Half Vignetted: Chief Ray is z y y -y -y aperture passes about half of the ray the ray half of aperture passes about x tector, different amounts of vignetting can occur can vignetting of amounts different tector, ll to the and lower upper ray bundles.  Aperture  Stop a y y MAX yyy y Pupil Ray Bundle If a system rectangular with a If a system is used de in the horizontal, vertical and diagonal directions. diagonal and vertical horizontal, the in The maximum radial extent of both ray bundles at any at any bundles ray both extent of radial maximum The Because the Field of View of an optical system optical an system View ofBecause the Field of is symmetric (i.e. or ± ±h A third vignetting condition is defined when an defined when is third condition vignetting A bundle from an object point. vignetting conditions apply equally equally we vignetting conditions apply Ray Bundles and Aperture Ray Bundles and Aperture Diameter Half Vignetted OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp z 10-9 10-10    y ay y   yy   y  y  2  L  ay ay EP The irradiance will then begin to fall off, at has full irradiance irradiance at has full to a or brightness out maximum description ignores This possible. maximum Stop the system the system will with a prescribed support amount XP 1 L sfer, law. as the cosine fourth such sfer, y ng condition, the required aperture diameters aperture the required diameters ng condition, be can -vignetted FOV, and decreasing and decreasing to zero at the fully-vignetted -vignetted FOV,    ay ay y ay y Marginal Ray  yy y The required clear aperture radius at a given z: given a required clear at aperture radius The Unvignetted: Fully Vignetted:Half Vignetted: and and Vignetting Summary Example System – System Example Extent Bundle Ray going to about half at the half to about going This is the fully-vignetted absolute FOV FOV. the obliquity radiative factors of tran The vignetting conditions are used in two different manners: different in two are used conditions vignetting The - For a given set of apertures, that the FOV different chief ray defines each FOV. A determined. vignetting can be of - and vignetti For a given FOV determined. system with vignetting will have an image A th limit. FOV radius corresponding to the unvignetted 10-11 OPTI-502Optical Designand Instrumentation I

An object is located 100 mm to the left of a 50 mm focal length thin lens. The object has a height of 10 mm above the optical axis of the lens. The lens diameter is 20 mm, and the lens serves as the system stop. An aperture is placed 50 mm to the right of the lens. What is the required diameter of the aperture so that the system operates without vignetting?

First consider the imaging: f = 50 mm h = 10 mm Image z Object h' z = -100 mm z'

Stop at Lens

DSTOP = 20 mm

zmm100 111   zmm 100 f  50mm zzf'

Lens is operating at 1:1 conjugates hmm  10

10-12 OPTI-502Optical Designand Instrumentation I

Draw the Marginal and Chief Rays and evaluate ray heights at the aperture:

h = 10 mm 50 mm Aperture Image yA y z Object A h′ = -10 mm Stop z = -100 mm z′ = 100 mm

aSTOP = 10 mm

At the aperture:

ya/2 5 mm ASTOP For no vignetting:

ayymm10 yhA  /2 5 mm APERTURE A A

DmmAPERTURE  20 10-13 OPTI-502Optical Designand Instrumentation I

The ray bundle with no vignetting:

D = 20 mm h = 10 mm 50 mm APERTURE Image z Object Stop h = -10mm

z = -100 mm z′ = 100 mm

The full ray bundle is passed by the aperture.

10-14 OPTI-502Optical Designand Instrumentation I

Given this aperture diameter of 20 mm, what is the object height that will be imaged with half vignetting? The marginal ray does not change, but the chief ray changes with object height. Start with an arbitrary object height. The chief ray goes through the center of the stop. h = ? D = 20 mm 50 mm APERTURE Image u u z Object yA Stop h = -h z = -100 mm z′ = 100 mm

h hh At the aperture: ymmumm50 50 u  A  zz z

For half vignetting: aymmAPERTURE A 10

hh Equating: ymmmmmm50 50  10 A zmm100

hmm 20 Half Vignetted Object Height 10-15 OPTI-502Optical Designand Instrumentation I

The ray bundle at half vignetting

h = 20 mm D = 20 mm 50 mm APERTURE Image u z Object yA Stop h = -20mm z = -100 mm z′ = 100 mm

aymmAPERTURE A 10

Half of the ray bundle is blocked by the aperture. The chief ray goes through the edge of the aperture and the center of the stop.

10-16 OPTI-502Optical Designand Instrumentation I

The following reverse telephoto objective is comprised of two thin lenses in air. The system stop is located between the two lenses. The system operates at f/4. The object is at infinity. The maximum image size is +/- 30 mm.

Stop Image f = -200 mm f = 100 mm 1 2 Plane (+/- 30 mm)

40 mm 40 mm

Determine the folowing: - Entrance pupil and exit pupil locations and sizes. - System focal length and back focal distance. - Stop diameter. - Angular field of view (in object space). - Required diameters for the two lenses for the system to be unvignetted over the specified maximum image size. 10-17 OPTI-502Optical Designand Instrumentation I

First set up the raytrace sheet.

Trace a potential Chief Ray starting at the center of the Stop. The Pupils are located where this ray crosses the axis in object space and image space.

Object EP L1 Stop L2 XP Image Surface 0123456 f -200 - 100 - 0.005 - -0.01 t -33.33 40 40 -66.67 Potential Chief Ray: y 0 -4.00 0 4.00 0 u .12 0.1* 0.1* 0.06

Entrance Pupil: Located 33.33 mm to the Right of L1 Exit Pupil: Located 66.67 mm to the Left of L2

Both Pupils are virtual.

* Arbitrary

10-18 OPTI-502Optical Designand Instrumentation I

In order to determine the Focal Length and the Back Focal Distance, trace a potential Marginal Ray. Since the object is at infinity, this ray is parallel to the axis in object space. The Rear Focal Point is located where this ray crosses the axis in image space. F'

Object EP L1 Stop L2 XP Image Surface 0123456 f -200 - 100 - 0.005 - -0.01 t ∞ -33.33 40 40 -66.67 222.22 Potential Marginal Ray: y 1* 1 1 1.2 1.4 2.0 0 u 0 0 0.005 0.005 -0.009 -0.009

The image or the rear focal point is located 222.22 mm to the Right of the XP.

BFD L2 XP  XP  F 66.67 mm  222.22 mm BFD155.56 mm  System Power and Focal Length: u  uymm 0.0091  1.0 y1 1 * Arbitrary  0.009 /mm f 111.11 mm  10-19 OPTI-502Optical Designand Instrumentation I

To determine the Stop and Pupil sizes, make use of the fact that the system operates at f/4. f f / # 4fmmDmmrmm  111.11EP  27.78 EP  13.89 DEP Scale the Potential Marginal Ray to obtain this ray height at the EP: rmm13.89 Scale Factor EP 13.89 ymm 1 EP F'

Object EP L1 Stop L2 XP Image Surface 0123456 f -200 - 100 - 0.005 - -0.01 t ∞ -33.33 40 40 -66.67 222.22 Potential Marginal Ray: y 1 1 1 1.2 1.4 2.0 0 u 0 0 0.005 0.005 -0.009 -0.009 Marginal Ray: y 13.89 13.89 13.89 16.67 19.45 27.78 0 u 0 0 0.0695 0.0695 -0.125 -0.125

rySTOP STOP16.67 mmD STOP  33.33 mm

ryXP XP27.78 mmD XP  55.56 mm

10-20 OPTI-502Optical Designand Instrumentation I

For the Field of View calculation, the Potential Chief Ray is extended to the image plane.

The Potential Chief Ray is scaled to the required image height of 30 mm: y 30 mm Scale Factor IMAGE 2.25  ymmIMAGE 13.33 F'

Object EP L1 Stop L2 XP Image Surface 0123456 f -200 - 100 - 0.005 - -0.01 t ∞ -33.33 40 40 -66.67 222.22 Potential Chief Ray: y 0 -4.00 0 4.00 0 13.33 u .12 .12 0.1 0.1 0.06 0.06 Chief Ray: y 0 -9.00 0 9.00 0 30.0 u 0.2700.270 0.225 0.225 0.135 0.135

1 Object Space Chief Ray: u0  0.270 HFOV tan u0   15.1 FOV 30.2 or 15.1 10-21 OPTI-502Optical Designand Instrumentation I

Summarizing the completed Marginal and Chief Rays: F'

Object EP L1 Stop L2 XP Image Surface 0123456 f -200 - 100 - 0.005 - -0.01 t ∞ -33.33 40 40 -66.67 222.22 Marginal Ray: y 13.89 13.89 13.89 16.67 19.45 27.78 0 u 0 0 0.0695 0.0695 -0.125 -0.125 Chief Ray: y 0 -9.00 0 9.00 0 30.0 u 0.2700.270 0.225 0.225 0.135 0.135

For No Vignetting: ayy L1:ymmamm11 13.89 22.89

ymmDmm119.0  45.78

L2 :ymmamm22 19.45 28.45

ymmD229.0 56.9 mm

10-22 OPTI-502Optical Designand Instrumentation I

Marginal and Chief Rays: STOP XP

L1 EP L2

Ray Bundles:

STOP

L1 L2 10-23 OPTI-502Optical Designand Instrumentation I

In a raytrace, zero-power surfaces can be inserted at any location (any z) in order to examine the ray properties (or image quality in the case of real rays). These are called “Dummy Surfaces.”

An example of their use would be determining the required size of the hole in the primary mirror of a Cassegrain objective. It is nothing more than a vignetting problem with a dummy surface placed at the location of the hole at the primary mirror vertex. R  200mm R 1 1 tmm80 R2 R2 50mm V z V F nn1 1 n2 1 nn 1 t WD 3 The hole is located to the right of Surface 0 Primary Secondary Hole F the secondary mirror. R -200 -50 Marginal and chief rays are traced, t ∞ -80 80 WD and the vignetting condition is applied to determine the hole size. n 1.0 -1.0 1.0 1.0 By using the dummy surface, the - -.01 .04 0 working distance (WD) is also directly computed on the raytrace t/n ∞ 80 80 WD sheet.

10-24 OPTI-502Optical Designand Instrumentation I

In addition to using actual angles instead of paraxial approximations, a real raytrace must use the actual surfaces instead of just the vertex planes. (It is a 3D problem).

1.) Start at a point on one surface and transfer to a target (vertex) plane for the next surface. The direction cosines of the initial ray are known.

(xs, ys, zs) (xt, yt) (k, l, m) z OPTI-502 Optical Design and Instrumentation I OPTI-502 Optical Design and Instrumentation I © Copyright 2019 John E. Greivenkamp © Copyright 2019 John E. Greivenkamp 10-26 10-25   U z   R  LR  ()       I IUIU  U  U I sin  sin R sin sin 222 2 180  zxyR   n'   R  ()  LR UIIU U' z (4) (3) L A  e new direction e new cosines in the optical space C allow L'U' and This calculated. to be to the actual surface. the Find intersection of to determine to determine the intersection point. pheric surfaces. The expressions may get may expressions The pheric surfaces. L' R R CA  ) I' s I R , z U s I sin , y s (x s R  z LR () y R  n  CA  L ) t I () sin sin  , y t sin sin nInI  (x U sin defines the refracted ray. Given R, L and U, these equations four Given R, L (1) (2) 2.) Transfer from the target (vertex) plane the Transfer from 2.) the ray with the surface. the surface 3.) Find normal at the intersection point. 4.) Refract Snell’s using law to determine th surface.after the 5.) Transfer the next surface. Repeat. to A real raytrace can also be done with as complicated, and iteration may be needed Used for manual forUsed manual raytracing. Rays restricted the meridional to plane. Trigonometric Raytrace Trigonometric Real Raytrace Real Raytrace - Continued