Exam Preparation Guide Geometrical optics (TN3313)

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1 Exam Preparation Guide Geometrical optics (TN3313) Lectures: September - December 2001 Version of When preparing for the exam, check on Blackboard for a possible newer version of this guide. For an oral exam appointment, contact the examiner Dr. F. Bociort via bociort@optica.tn.tudelft.nl or phone: (015) or come along at room E020 (physics building). Notes: 1.Most questions during the oral exam are related to the issues below. Those unrelated are "bonus questions". You don't lose anything if you don't answer them well, but you gain if you do! 2.Only short mathematical derivations are requested, there where they enlighten the physical content. The starting point is always given by the examiner. You may ask the examiner - no penalty - for other earlier formulas that might be necessary. 3. The "intellectual independence" of the student will be respected. You don't have to answer the questions in the way presented during the course. If you have read somewhere else a more appealing way to explain the subject, or if you want to present the ideas in your own way, feel free to do it. Chapter 1. Fundamentals of geometrical optics 1.1. a) The student is able to explain qualitatively how geometrical optics can be considered as a limiting case of wave optics. - What is the essence of the approximation of geometrical optics? - When is it valid? - Give examples of situations where this approximation is not valid b) The student is able to explain the notion of the "eikonal function". - How can the equation for this quantity be derived? (just the general ideas, derivation details are not requested.) c) The student is able to explain the relationship between rays and wavefronts. - Define the notion of a wavefront 1.2. a) Derive and briefly discuss the differential ray equation in inhomogeneous media b) Define the optical path length 1

2 c) Define the optical direction cosines -How do the transverse coordinates x and y change during propagation in homogeneous media? d) How is the ray path curved in inhomogeneous media? Chapter 2. Ray tracing - Describe the purpose of ray tracing and the general procedure. 2.1 Indicate how Snell's law can be derived in the general vector form. Be able to write it in the simple scalar form. 2.2 Compute the ray direction after refraction. (Some necessary formulas are given by the examiner.) 2.3 Briefly discuss total internal reflection - When does it occur? - What happens then? 2.4 Indicate two ways in which the reflection law can be written. 2.5 a) Be able to reproduce the general formula (using the gradient) for the normal to a general surface. b) Derive the components of the normal vector to a spherical surface. c) What is the "generalized power"? 2.6 Qualitatively discuss how we can find the ray transfer equations between two surfaces. (The second one may be curved!) - What can go wrong? (Make a drawing of a possible ray failure situation) 2.7. Briefly discuss aspherical surfaces - How do you compute the normal? - How can you write the surface equations? - Why are they useful? 2

3 Chapter 3. The paraxial approximation - Discuss the basic difference between paraxial and finite rays - How many ray parameters do you need to characterize a finite ray? - Indicate some possible choices of those parameters - Name some advantages and disadvantages of using paraxial rays, as compared to real rays 3.1 a)derive the paraxial ray tracing equations from the real ray tracing ones. (Some necessary formulas are given by the examiner.) - Refraction - Transfer - Comment on mirrors: How do you obtain reflection formulas from refraction formulas? b)compare spherical and aspherical surfaces in the paraxial approximation. 3.2 a) The student is able to describe refraction and transfer in terms of matrix formalism. - What are the special properties of the determinants of the matrices involved? b) Write the system matrix in the general case. - Describe the transformation performed by this matrix. 3.3 a) Describe the ideal imaging in terms of matrix properties. - What are the characteristics of ideal imaging? b) Derive the transverse magnification from matrix properties. c) Define the notion of conjugate planes and discuss their relationship with the value of the transverse magnification (general lines only). - When is the above discussion not applicable? - Give the physical interpretation of the situations when the object or image position change signs. e) Define and compute the axial magnification. - When is the axial magnification zero? f) Define and compute the front focal length and the back focal length. 3.4 a) Define the principal planes. b) Write the imaging equation referred to principal planes. c) Compare the imaging referred to principal planes with imaging through a single surface. d) Define the effective focal length. e) For objects at infinity, show the relationship between the effective focal length and image height. 3.5 a) Define a telescopic system. 3

4 b) Why do we have to treat paraxial imaging in this case separately? c) Define the angular magnification. 3.6 a) Give (at least) two reasons why in a real optical system the aperture and the field cannot be arbitrarily large. b) Why should aperture and field not be too small? c) What is vignetting? - In the image plane, how is the light intensity off-axis, compared to the on-axis situation? Why? d) What are the aperture and field stops? (stop=diaphragm) e) What are the entrance and exit pupils? 3.7 a) Define the marginal and chief rays. -Specify for each of them the points in the aperture and field through which they pass. b) On a drawing showing the paths of these two rays, - How can you determine the positions of the entrance and exit pupil? - How can you see whether we have an intermediate real image or not? 3.8 a) Show that a certain paraxial quantity remains unchanged throughout the system (the Lagrange invariant). b) Write the Lagrange invariant in special cases: - in the object and image plane - in the entrance and exit pupil - at the stop position c) Derive the relation between the marginal ray angles in the object and image space. d) Define the F number and derive its relationship with the marginal ray angle in the image space (for objects at infinity). e) Explain the physical significance (in terms of energy) of the Lagrange invariant. 3.9 a) Compute the total power of a system with object at infinity in terms of surface powers and marginal ray data - Indicate situations when the power of a given surface does not influence the total power. b) Express the power of a thin lens in terms of surface powers Define a telecentric system (image side, object side, both sides). - Explain its practical utility in the case of defocussing. - What are the special positions of the entrance and exit pupils? - What special property has the chief ray path in this case? - What is the power of a double telecentric system? Why? 4

5 Chapter 4. Aberrations a) Define monochromatic and chromatic aberrations b) Give an example of a nonlinear law that causes aberrations and compare it with its paraxial equivalent. 4.1 a) Define the vector of the transverse ray aberration. b) Define the wavefront aberration (make a drawing therefore). - Within the frame of wave optics, discuss briefly and qualitatively the influence of the wavefront aberration on the image formation. c) What is a diffraction limited system? 4.2 a) What is the relationship between transverse and wave aberration? b) How can the wave aberration be computed from ray tracing? (Mention only general lines.) 4.3 a) Write the power series expansion for the wave aberration for arbitrary optical systems (without symmetry) b) For rotationally symmetric systems: - Which combinations of ray coordinates are left unchanged by a rotation around the optical axis? Why? - Write the power series expansion for the wave aberration c) Why is rotational symmetry useful for practical purposes? 4.4 a) Write the lowest order nonvanishing terms in the wavefront aberration - when the image plane is placed at its paraxial position - when the image plane is placed somewhere else b) Derive the power series expansion for the transverse aberration (first and third order terms) c) How many first-order terms do we have? - When do they vanish? - What is their influence on the image quality? d) Name a special type of optical system when in the case of defocussing we have only one nonzero linear term. e) Comment on the wavefront in the case of defocussing. 4.5 a) How many Seidel aberrations do we have? Why? b) What is the order of the Seidel terms in the series expansion for the - wave aberration - transverse aberration 5

6 c) When do the Seidel terms give a quantitatively accurate description of the image quality? - Why are the Seidel aberrations useful also for more general types of systems? d) Give an example of a mathematical formula used for finite ray tracing, write a power series expansion for it and show which terms lead to Seidel aberrations. 4.6 a) Give the dependence of spherical aberration on aperture and field coordinates b) Name a situation when in an optical system all aberrations vanish, except this one. c) Describe the effect of spherical aberration on the image quality. d) Describe the longitudinal spherical aberration and define the marginal focus. e) Be prepared to explain what you have seen experimentally! f) Where is the light energy concentrated in the case of spherical aberration? 4.7 Discuss the balancing of spherical aberration through defocussing. - Plot the spherical aberration as a function of aperture when the image plane is at its paraxial position. - Plot the transverse aberration (spherical + defocus) at optimal defocus (i.e. when the spot radius is minimal) -Referred to the paraxial and marginal focus, where is the optimal image plane position situated? 4.8 a) Give the dependence of coma on aperture and field coordinates b) Describe the effect of coma on the image quality. - Describe the shape of the coma image. - Why is the coma image asymmetric? c) Be prepared to explain what you have seen experimentally! 4.9 a) Give the dependence of astigmatism on aperture and field coordinates b) Describe the effect of astigmatism on the image quality. - Describe the astigmatic focal lines? c) Be prepared to explain what you have seen experimentally! d) Describe the astigmatic wavefront a) Give the dependence of field curvature on aperture and field coordinates 6

7 b) Describe the effect of field curvature on the image quality. - Discuss the difference between sharpness aberrations and position aberrations. c) Be prepared to explain what you have seen experimentally! 4.11 a) Give the dependence of distorsion on aperture and field coordinates b) Describe the effect of distorsion on the image quality. - Comment the relationship between distortion and transverse magnification. - Describe the possible forms of the distorted image of a rectangular grid and relate them with the sign of the distortion coefficient. c) Be prepared to explain what you have seen experimentally! 4.12 a) Why do chromatic aberrations occur? b) What is their order in aperture and field coordinates? c) How many primary chromatic aberrations do we have? d) Describe their effect on the image quality a) Name a situation when the most important aberrations are spherical aberration and coma. b) Name a situation when the most important aberration are distortion, astigmatism and field curvature When are the spherical aberration and coma contributions of a given surface zero? (Necessary formulas are given by the examiner.) 4.15 Where should we place an asphere so that we influence only spherical aberration? - How can we maximize its effect? (Necessary formulas are given by the examiner.) 4.16 a) Describe the axial color. - In the thin-lens approximation, the change of which paraxial lens parameter is associated with chromatic aberrations? b) What is the general behavior of the refractive index when the wavelength increases? c) How can we correct chromatic aberrations? d) Define the Abbe number e) What is an achromatic doublet? f) Compute the individual powers of the two lenses that form an achromatic doublet. - How must glasses be chosen in order to avoid excessively large surface curvatures? 7

8 Chapter 5. Optimization of optical systems a) What does optimization mean? - How does the computer software achieve this goal? b) What must the optical designer specify for a successful optimization? (4 elements!) - Give some examples of optimization variables c) What is the error function? - If the image quality must be improved, why is an error function needed (instead of, for instance, solving a system of nonlinear equations)? - Write the formula for the error function in terms of operands - Give some examples of possible operands d) Write the mathematical condition for an (unconstrained) minimum in the N- dimensional parameter space e) Describe a simple local optimization algorithm (steepest descent method) f) What does constrained optimization mean? - State the problem mathematically (equality and inequality constraints) - Give some examples of constraints in optical design - In a one-dimensional case, how many types of minima can occur? - What is the difference between active and inactive inequality constraints? g) How can constraints be integrated in the optimization algorithm? h) Write the mathematical condition for a constrained minimum (using the gradient of the constraint) - in a 2-dimensional parameter space, when one constraint is present - in the N-dimensional parameter space, when several constraints are present Recommended literature 1. J. Braat, Diktaat Geometrische Optica, TU Delft 1991 (in Dutch; English- speaking students should use Born and Wolf (Ref 5) instead) 2. J. Braat, Paraxial Optics Handout (obtainable from the secretary of Group Optica, a version scanned with fax quality can be found on Blackboard) 3. W.T. Welford, Aberrations of Optical Systems, Adam Hilger, 1986 (or the earlier version Aberrations of the Symmetrical Optical System,1974) 4. F. Bociort, Optimization of optical systems (can be found on Blackboard) Supplementary reading (not required for the exam, just if you want extra depth on some subject) 5. M. Born and E. Wolf, Principles of Optics 6. R.R. Shannon, The Art and Science of Optical Design, Cambridge University Press, D. Sinclair, Optical Design Software, Handbook of Optics, Chapter 34 8

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