Copright 2018 John E. Greivenkamp 11-1 Section 11 Vignetting
Vignetting The stop determines the sie of the bundle of ras that propagates through the sstem for an on-axis object. As the object height increases, one of the other apertures in the sstem (such as a lens clear aperture) ma limit part or all of the bundle of ras. This is known as vignetting. On-Axis Off-Axis Stop Ra Bundle Ra Bundle Aperture 11-2 Vignetted Ras Copright 2018 John E. Greivenkamp Stop Aperture
Ra Bundle On-Axis The ra bundle for an on-axis object is a rotationall-smmetric spindle made up of sections of right circular cones. Each cone section is defined b the pupil and the object or image point in that optical space. The individual cone sections match up at the surfaces and elements. Pupil Stop = 0 11-3 Copright 2018 John E. Greivenkamp At an, the cross section of the bundle is circular, and the radius of the bundle is the marginal ra value. The ra bundle is centered on the optical axis.
Ra Bundle Off Axis For an off-axis object point, the ra bundle skews, and is comprised of sections of skew circular cones which are still defined b the pupil and object or image point in that optical space. Pupil - Stop = 0 Chief Ra Image Plane 11-4 Copright 2018 John E. Greivenkamp Since the base of the cone is defined b the circular pupil, the cross section of the ra bundle at an remains circular with a radius equal to the radius of the axial bundle. The off-axis bundle is centered about the chief ra height. The maximum radial extent of the ra bundle at an is MAX
Unvignetted While the stop alone defines the axial ra bundle, vignetting occurs when other apertures in the sstem, such as a lens clear aperture, block all or part of an off-axis ra bundle. No vignetting occurs when all of the apertures pass the entire ra bundle from the object point. Each aperture radius a must equal or exceed the maximum height of the ra bundle at the aperture. Note that the required apertures sie will change if the sstem FOV (chief ra) is changed. Ra Bundle Unvignetted: 11-5 Copright 2018 John E. Greivenkamp a x Aperture a
Full Vignetted The maximum FOV supported b the sstem occurs when an aperture completel blocks the ra bundle from the object point. Ra Bundle a x Aperture Full Vignetted: and a a 11-6 Copright 2018 John E. Greivenkamp The second part of this vignetting condition ensures that the aperture passes the marginal ra and is not the sstem stop. B definition, vignetting cannot occur at the aperture stop or at a pupil.
Half Vignetted A third vignetting condition is defined when an aperture passes about half of the ra bundle from an object point. Ra Bundle a x Aperture Half Vignetted: and a a 11-7 Copright 2018 John E. Greivenkamp
Ra Bundles and Aperture Diameter Because the Field of View of an optical sstem is smmetric (i.e. ±h or ± ), the vignetting conditions appl equall well to the upper and lower ra bundles. Pupil Stop - - Chief Ra Image Plane 11-8 Copright 2018 John E. Greivenkamp The maximum radial extent of both ra bundles at an is MAX If a sstem is used with a rectangular detector, different amounts of vignetting can occur in the horiontal, vertical and diagonal directions.
Vignetting Summar The required clear aperture radius at a given : Unvignetted: Full Vignetted: Half Vignetted: a a and a a and a The vignetting conditions are used in two different manners: - For a given set of apertures, the FOV that the sstem will support with a prescribed amount of vignetting can be determined. A different chief ra defines each FOV. - For a given FOV and vignetting condition, the required aperture diameters can be determined. 11-9 Copright 2018 John E. Greivenkamp A sstem with vignetting will have an image that has full irradiance or brightness out to a radius corresponding to the unvignetted FOV limit. The irradiance will then begin to fall off, going to about half at the half-vignetted FOV, and decreasing to ero at the full-vignetted FOV. This full-vignetted FOV is the absolute maximum possible. This description ignores the obliquit factors of radiative transfer, such as the cosine fourth law.
Copright 2018 John E. Greivenkamp Example Sstem Ra Bundle Extent L 1 11-10 L 2 Stop Marginal Ra XP EP a
Vignetting Example Page 1 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 sstem stop. An aperture is placed 50 mm to the right of the lens. What is the required diameter of the aperture so that the sstem operates without vignetting? First consider the imaging: h = 10 mm Object = -100 mm f = 50 mm ' Image h' 11-11 Copright 2018 John E. Greivenkamp Stop at Lens D STOP = 20 mm 100mm f 50mm 1 1 1 ' f 100mm Lens is operating at 1:1 conjugates h 10mm
Vignetting Example Page 2 Draw the Marginal and Chief Ras and evaluate ra heights at the aperture: h = 10 mm Object At the aperture: = -100 mm Stop 50 mm A a STOP = 10 mm Aperture A = 100 mm Image h = -10 mm 11-12 Copright 2018 John E. Greivenkamp a /2 5mm A STOP h /2 5mm A For no vignetting: a 10mm APERTURE A A D APERTURE 20mm
Vignetting Example Page 3 The ra bundle with no vignetting: h = 10 mm Object = -100 mm Stop 50 mm D APERTURE = 20 mm = 100 mm Image h = -10mm 11-13 Copright 2018 John E. Greivenkamp The full ra bundle is passed b the aperture.
Vignetting Example Page 4 Given this aperture diameter of 20 mm, what is the object height that will be imaged with half vignetting? The marginal ra does not change, but the chief ra changes with object height. Start with an arbitrar object height. The chief ra goes through the center of the stop. h =? Object u = -100 mm Stop 50 mm u D APERTURE = 20 mm A = 100 mm Image h = -h 11-14 Copright 2018 John E. Greivenkamp u h h At the aperture: For half vignetting: h A 50 mm u 50 mm a 10mm APERTURE A h h A 50 mm 50 mm 10mm 100mm Equating: h 20mm Half Vignetted Object Height
Vignetting Example Page 5 The ra bundle at half vignetting h = 20 mm Object = -100 mm u Stop 50 mm D APERTURE = 20 mm A = 100 mm Image h = -20mm 11-15 Copright 2018 John E. Greivenkamp a 10mm APERTURE A Half of the ra bundle is blocked b the aperture. The chief ra goes through the edge of the aperture and the center of the stop.
Example Sstem Pupils and Vignetting Page 1 The following reverse telephoto objective is comprised of two thin lenses in air. The sstem stop is located between the two lenses. The sstem operates at f/4. The object is at infinit. The maximum image sie is +/- 30 mm. f 1 = -200 mm Stop f 2 = 100 mm Image Plane (+/- 30 mm) 11-16 Copright 2018 John E. Greivenkamp 40 mm 40 mm Determine the folowing: - Entrance pupil and exit pupil locations and sies. - Sstem focal length and back focal distance. - Stop diameter. - Angular field of view (in object space). - Required diameters for the two lenses for the sstem to be unvignetted over the specified maximum image sie.
Example Sstem Pupils and Vignetting Page 2 First set up the ratrace sheet. Trace a potential Chief Ra starting at the center of the Stop. The Pupils are located where this ra crosses the axis in object space and image space. Object EP L 1 Stop L 2 XP Image Surface 0 1 2 3 4 5 6 f -200-100 - 0.005 - -0.01 t -33.33 40 40-66.67 Potential Chief Ra: 0-4.00 0 4.00 0 u.12 0.1* 0.1* 0.06 11-17 Copright 2018 John E. Greivenkamp Entrance Pupil: Exit Pupil: Located 33.33 mm to the Right of L1 Located 66.67 mm to the Left of L2 Both Pupils are virtual. * Arbitrar
Example Sstem Pupils and Vignetting Page 3 In order to determine the Focal Length and the Back Focal Distance, trace a potential Marginal Ra. Since the object is at infinit, this ra is parallel to the axis in object space. The Rear Focal Point is located where this ra crosses the axis in image space. F' Object EP L 1 Stop L 2 XP Image Surface 0 1 2 3 4 5 6 f -200-100 - 0.005 - -0.01 t -33.33 40 40-66.67 222.22 Potential Marginal Ra: 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 Sstem Power and Focal Length: u u 0.009 1 1.0 mm 1 1 * Arbitrar 0.009 / mm f 111.11 mm 11-18 Copright 2018 John E. Greivenkamp
11-19 Example Sstem Pupils and Vignetting Page 4 To determine the Stop and Pupil sies, make use of the fact that the sstem operates at f/4. f f / # 4 f 111.11 mm DEP 27.78 mm rep 13.89 mm D EP Scale the Potential Marginal Ra to obtain this ra height at the EP: rep 13.89 mm Scale Factor 13.89 1 mm Object EP L 1 Stop L 2 XP Image Surface 0 1 2 3 4 5 6 f -200-100 - 0.005 - -0.01 t -33.33 40 40-66.67 222.22 Potential Marginal Ra: 1 1 1 1.2 1.4 2.0 0 u 0 0 0.005 0.005-0.009-0.009 Marginal Ra: 13.89 13.89 13.89 16.67 19.45 27.78 0 u 0 0 0.0695 0.0695-0.125-0.125 EP F' Copright 2018 John E. Greivenkamp r 16.67 mm D 33.33 mm STOP STOP STOP r 27.78 mm D 55.56 mm XP XP XP
Example Sstem Pupils and Vignetting Page 5 For the Field of View calculation, the Potential Chief Ra is extended to the image plane. The Potential Chief Ra is scaled to the required image height of 30 mm: 30 mm Scale Factor 2.25 IMAGE IMAGE 13.33 mm F' Object EP L 1 Stop L 2 XP Image Surface 0 1 2 3 4 5 6 f -200-100 - 0.005 - -0.01 t -33.33 40 40-66.67 222.22 Potential Chief Ra: 0-4.00 0 4.00 0 13.33 u.12.12 0.1 0.1 0.06 0.06 Chief Ra: 0-9.00 0 9.00 0 30.0 u 0.270 0.270 0.225 0.225 0.135 0.135 11-20 Copright 2018 John E. Greivenkamp Object Space Chief Ra: u0 0.270 HFOV tan 15.1 1 u FOV 30.2 or 15.1 0
Example Sstem Pupils and Vignetting Page 6 Summariing the completed Marginal and Chief Ras: F' Object EP L 1 Stop L 2 XP Image Surface 0 1 2 3 4 5 6 f -200-100 - 0.005 - -0.01 t -33.33 40 40-66.67 222.22 Marginal Ra: 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 Ra: 0-9.00 0 9.00 0 30.0 u 0.270 0.270 0.225 0.225 0.135 0.135 11-21 Copright 2018 John E. Greivenkamp For No Vignetting: a L1: 13.89 mm a 22.89 mm 1 1 9.0 mm D 45.78 mm 1 1 L2 : 19.45 mm a 28.45 mm 2 2 9.0 mm D 56.9 mm 2 2
Copright 2018 John E. Greivenkamp Example Sstem Pupils and Vignetting Page 7 Marginal and Chief Ras: XP STOP 11-22 L1 EP L2 Ra Bundles: STOP L1 L2
Dumm Surfaces In a ratrace, ero-power surfaces can be inserted at an location (an ) in order to examine the ra properties (or image qualit in the case of real ras). These are called Dumm Surfaces. An example of their use would be determining the required sie of the hole in the primar mirror of a Cassegrain objective. It is nothing more than a vignetting problem with a dumm surface placed at the location of the hole at the primar mirror vertex. R 2 V t R 1 V WD Surface 0 Primar Secondar Hole R -200-50 t n 1.0-80 -1.0 80 1.0 WD 1.0 - t/n -.01 80.04 80 0 F WD F R1 200mm t 80mm R2 50mm n1 n 1 n2 1 n n 1 3 11-23 The hole is located to the right of the secondar mirror. Marginal and chief ras are traced, and the vignetting condition is applied to determine the hole sie. B using the dumm surface, the working distance (WD) is also directl computed on the ratrace sheet. Copright 2018 John E. Greivenkamp
Real Ratrace In addition to using actual angles instead of paraxial approximations, a real ratrace 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 ra are known. (x s, s, s ) (k, l, m) (x t, t ) 11-24 Copright 2018 John E. Greivenkamp
Real Ratrace - Continued (x s, s, s ) (x t, t ) s R ( R ) x R 2 2 2 2 11-25 Copright 2018 John E. Greivenkamp 2.) Transfer from the target (vertex) plane to the actual surface. Find the intersection of the ra with the surface. 3.) Find the surface normal at the intersection point. 4.) Refract using Snell s law to determine the new direction cosines in the optical space after the surface. 5.) Transfer to the next surface. Repeat. A real ratrace can also be done with aspheric surfaces. The expressions ma get complicated, and iteration ma be needed to determine the intersection point.
11-26 Trigonometric Ratrace Used for manual ratracing. Ras restricted to the meridional plane. n I A n' I' U' U R C L' L CA CA sinu sin I 180 I U I U ( L R) R (3) U I I U ( L R) (1) sin I sinu ( L R) R sin I sinu R Rsin I (2) n sin I nsin I (4) L R sinu Copright 2018 John E. Greivenkamp Given R, L and U, these four equations allow L' and U' to be calculated. This defines the refracted ra.