Nonlocal Optical Real Image Formation Theory

Transcription

1 See discussions, stats, and author profiles for this publication at: Nonlocal Optical Real Image Formation Theor Article December 010 Source: arxiv READS 59 1 author: Greson Gilson Mulith Inc. 5 PUBLICATIONS 5 CITATIONS SEE PROFILE Available from: Greson Gilson Retrieved on: 10 Ma 016

2 Nonlocal Optical Real Image Formation Theor Greson Gilson Mulith Inc. 30 Chestnut Street # 3 Nashua, New Hampshire , USA greson.gilson@mulithinc.com ABSTRACT A nonlocal theor of optical real image formation is developed from the basic phsics associated with an optical real image formation apparatus. The theor shows how two separated illuminated object points are required for real image formation. When the distance between the points eceeds twice the wavelength of the light used two equi-amplitude plane waves propagate independentl awa from the object plane toward the imaging sstem. Quantum amplitude components with spatial frequencies that are within the passband of the imaging sstem pass through the imaging sstem. Each plane wave pair that propagates through the imaging sstem and ultimatel arrives at the image plane contributes to real image formation. Object and image field depths are treated and found to be ver different than the are ordinaril thought to be. Resolution limits related to real image formation are investigated. Reference distribution real image formation (RIF) is introduced. Real image formation with no fundamental resolution limit is predicted to be achievable b means of RIF.

3 PART I: FUNDAMENTALS INTRODUCTION A new theor of optical real image formation is developed from fundamental phsics in this paper. The theor shows how a previousl unknown cooperative phenomenon arises and how the phenomenon underlies the formation of optical real images. In accord with the new theor, optical real images with a ver large field depth (object and image) and no fundamental resolution limit can be formed. During the process of real image formation, light wave components propagate awa from an object, through an imaging sstem and ultimatel form an image. Image forming light necessaril begins its journe b propagating awa from the object plane. A generalied optical real image formation apparatus is illustrated in Figure 1. The image formation apparatus is an ordinar diffraction-limited linear space-invariant imaging sstem. Initiall, light that is incident upon the object plane is transmitted through or reflected from phsical features in the object plane. Such light is distributed in a definite configuration on the side of the object plane nearest to the imaging sstem. A portion of this light propagates from the object plane to the imaging sstem. Some of the light that arrives at OBJECT PLANE OPTICAL AXIS IMAGING SYSTEM Figure 1. Generalied optical real image formation apparatus. IMAGE PLANE the imaging sstem propagates through it. Subsequentl, the light that propagates through the imaging sstem propagates from the imaging sstem to the image plane. The resulting configuration of light on the image plane closel approimates a magnified (enlarged, unchanged or reduced and perhaps inverted) version of the configuration of light on the object plane. As shown later, two separated illuminated object points are required to form each light wave component that propagates awa from the object plane. This finding is in distinct contrast to the ordinaril accepted assumption that onl one object point is involved.

4 3 LIGHT No one knows what light is. Nevertheless, light has certain properties that can be treated mathematicall. In particular, light associated with optical image formation displas wave properties and particle properties. The particles associated with light are known as photons. Images are formed b the apparentl random arrival of one photon at a time 1,. An image that is formed with sufficientl feeble light shows no evidence of having wave properties. Rather, images are built up b individual photons that act independentl In addition to its particle properties, light has wave properties. An image that is formed with sufficientl strong light shows no evidence of being built up b individual photons. Rather, the entire image appears to be formed as a single event. The wave and particle properties of light coeist. The wave properties define the probabilit densit that describes the probabilit of arrival of a photon at a particular location. Each photon is associated with the entire probabilit distribution. This coeistence is known as wave-particle dualit; quantum phsics is needed to cope with it. The wave properties of light are common to phsical waves in a lossless medium. The defining epression for phsical waves in a lossless medium is the differential wave equation 3 1 v t (1) In this equation is the wave function, v is the speed of propagation of the wave, t is time, and is the Laplacian operator. The wave function for optics is known as quantum amplitude. Optics is often understood as a branch of electromagnetism and its fundamental laws are derived from Mawell's equations 4. An electromagnetic field is characteried b an electric field E and a magnetic induction B. These fields are linked together b means of Mawell's equations. In free space and in the absence of sources, each Cartesian component of E and B satisfies equation (1) independentl 5. The waves involved are known as electromagnetic waves. Mawell used his theor to deduce the speed of propagation of electromagnetic waves 6. This speed was found to coincide with the eperimentall determined speed of light. The etremel 1 Eugene Hecht, Optics (4 th Ed.) p. 8, pp , and pp (Addison Wesle, San Francisco, 00). A. P. French and Edwin F. Talor, An Introduction to Quantum Phsics, pp (W. W. Norton & Compan Inc., New York 1978). 3 Eugene Hecht, Optics (4 th Ed.) pp and pp. 7-8 (Addison Wesle, San Francisco, 00). 4 David Hallida and Robert Resnick, Phsics, p. 986 (John Wile & Sons, Inc., New York, 1960). 5 John David Jackson, Classical Electrodnamics, pp (John Wile & Sons, Inc., New York, 1967). 6 David Hallida and Robert Resnick, Phsics, p. 986 (John Wile & Sons, Inc., New York, 1960).

5 4 successful electromagnetic theor of light was born. The electromagnetic wave function is known as comple amplitude. Wave-particle dualit cannot be understood within the framework of the electromagnetic theor of light. The electromagnetic theor is unable to treat photons and the use of electromagnetic concepts can be inappropriate and misleading. Quantum phsics is needed. Quantum amplitude (or probabilit amplitude) is the term used to denote the wave function in quantum phsics. A quantum amplitude, when multiplied b its comple conjugate, defines a relative probabilit densit. The relative probabilit densit describes the probabilit of arrival of a photon at a particular location. The concepts of quantum amplitude and probabilit densit are used in quantum phsics to accommodate the eistence of wave-particle dualit. Equation (1), with the wave function identified as quantum amplitude, is known as the optical wave equation. Optical real image formation can be understood in terms of the optical wave equation and its solutions. OVERVIEW OF OPTICAL REAL IMAGES As a result of interacting with phsical features in the object plane, a definite configuration of light eists on the side of the object plane nearest to the imaging sstem. This configuration of light consists of photons that are transmitted through or reflected from the object plane at the instant the leave the object plane. Photons cannot be observed directl; what is known of them comes from observing the results of their being either created or destroed 7. A configuration of light can eist whether or not it is observed. A real image ma or ma not be observed. A right-handed rectangular coordinate sstem can be convenientl introduced to facilitate treating real image formation mathematicall. In this coordinate sstem, let the object plane be the o o -plane and let the image plane be the i i -plane. The subscripts o and i designate the object side and the image side, respectivel, of the imaging sstem. Let the -ais coincide with the optical ais defined b the imaging sstem. Let the positive - direction be the direction from the object plane toward the imaging sstem. Finall, let the -ais intersect the o o -plane at the origin of coordinates. Let u, ; o be the quantum amplitude of the photons linked with light of wavelength on the object plane. The relative probabilit densit, given b o, ; *, ;, ; p u u () o 0 7 Eugene Hecht, Optics, (4 th Ed.) p. 5 (Addison Wesle, San Francisco, 00).

6 5 where the superscript asterisk denotes comple conjugation, is linked to u, ; o. The relative probabilit densit specifies the instantaneous presence of a photon of wavelength within an infinitesimal area surrounding the point, on the object plane. Each quantum amplitude component can be characteried as being monochromatic and coherent. Some of the photons in the configuration of light on the object plane propagate awa from the object plane and pass through the imaging sstem illustrated in Figure 1. In turn, some of these photons ultimatel form a real image on the image plane. Let, ; of wavelength. Furthermore, let u, ; be the coherent impulse response of the real image formation apparatus for light i be the quantum amplitude of the photons linked with light of wavelength on the image plane. This quantum amplitude is given b 1 ui, ; u o, ;, ; m m m m m (3) where denotes the convolution operator and m denotes the lateral magnification of the image relative to the object; m ma be either negative or positive, depending upon whether the real image is inverted or not inverted relative to the object. The relative probabilit densit is linked to u, ;, ; *, ;, ; p u u (4) i i i i. This relative probabilit densit specifies the instantaneous presence of a photon of wavelength within an infinitesimal area surrounding the point, on the image plane. A ver large number of photons contribute when acceptable real image formation occurs. The inherent granularit associated with the object and the image configurations of light vanishes. Consequentl the densit of photons that contribute to real image formation is proportional to the probabilit densit. Man authors provide treatments concerning the derivation of equation (3); see, for eample, Collier, Burkhardt and Lin 8, Gaskill 9, Cathe 10 or Goodman 11. In addition, equation (3) is derived in this paper where it is presented as equation (13). Equation (3) is the coherent real image equation for light of wavelength. 8 Robert J. Collier, Christoph B. Burkhardt and Lawrence H. Lin, Optical Holograph, p (Academic Press, Inc., Orlando, 1971). 9 Jack D. Gaskill, Linear Sstems, Fourier Transforms, and Optics, pp (John Wile & Sons, Inc., New York, 1978). 10 W. Thomas Cathe, Optical Information Processing and Holograph, pp (John Wile & Sons, Inc., New York, 1974). 11 Joseph W. Goodman, Introduction to Fourier Optics, pp (McGraw-Hill, Inc., New York, 1968).

7 6 Derivations of equation (3) are ordinaril based upon the Hugens-Fresnel principle. Following the Hugens-Fresnel principle, it has been widel assumed that onl one illuminated point in an object is needed to support image-forming light. However, as shown in this paper, real image formation is dependent upon a previousl unknown cooperative phenomenon. This cooperative phenomenon is such that two separate illuminated object points are required to support imageforming light. The Hugens-Fresnel principle is not a law of phsics and is furthermore known to be problematic 1. In addition, Ad hoc approimations (such as initial approimations, the Fresnel approimations, the Fraunhofer approimation, etc.) are invoked in derivations of equation (3) that are based upon the Hugens-Fresnel principle. None of these derivations are based on fundamental phsics. A new theor of optical real image formation is developed in this paper. The Hugens-Fresnel principle and its attendant approimations are avoided in this development. Rather, the optical wave equation is solved subject to boundar conditions. Salient boundar conditions are imposed b the object and b the imaging sstem. OPTICAL WAVE EQUATION Light that propagates awa from the object plane and subsequentl forms a real image is associated with a linear superposition of quantum amplitude components. Each quantum amplitude component propagates in a direction such that its -component is non-negative. Let the quantum amplitude component associated with the wavelength be represented b the comple valued scalar function,,, t, where,, and are the coordinates of a point in,,, t satisfies the optical wave equation space and t represents time. The function 1 v t,,, t,,, t (5) in the region between the object plane and the image plane and outside the imaging sstem. In accord with the method of separation of variables, solutions of equation (5) in the form,,, t,, t (6) are sought. After substituting equation (6) into equation (5) and then dividing both sides of the,, t result b 1 Eugene Hecht, Optics, (4 th Ed.) p. 445, p. 456, and p. 489 (Addison Wesle, San Francisco, 00).

8 7 1 1,,,, v t t t (7) follows. The three variables on the left hand side of equation (7) are all independent of the single variable on the right hand side of equation (7). This can be true onl when each side of the equation is equal to the same constant. Consequentl, 1,, k,, (8) and 1 v t t t k (9) where the separation constant has been designated, with modest foresight, as and (9) can be written as k. Equations (8) k,, 0 (10) which is known as the Helmholt equation, and t t t k v 0 (11) respectivel. The spatial dependence of,,, t dependence of,,, t satisfies equation (11). TIME DEPENDENCE satisfies equation (10) while the time Equation (11) is a partial differential equation that involves onl one variable and can consequentl be written as the ordinar differential equation d dt t Equation (1) can be solved to ield the solution t k v 0 (1)

9 8 t C ep ikvt C ep ikvt (13) 1 where C 1 and C are arbitrar constants. Equation (13) can be verified b direct substitution into equation (1). Oscillations of temporal frequenc are described b equation (13). kv (14) As observed at a particular point in space each quantum amplitude component oscillates at a frequenc that is independent of the medium involved. In addition, the quantum amplitude component is associated with a wavelength (introduced previousl) that is dependent upon the medium involved. The wavelength and frequenc are related b the equation v (15) where v is the speed of light in the medium involved. Equation (15) is a fundamental equation of wave motion. The wave number k (16) can be obtained after equation (15) has been substituted into equation (14). Furthermore, t C ep i t C ep i t (17) 1 results when equation (14) is substituted into equation (13). The constants C 1 and C will be treated later. PROPAGATION DIRECTION The three-dimensional Fourier transform of,,,,,, ep i d d d (18) can now be convenientl introduced. Here, the reciprocal variables, and, commonl known as spatial frequencies, are needed to define the three-dimensional Fourier transform of,,. The corresponding three-dimensional inverse Fourier transform

10 9,,,, ep i d d d (19) and,, as linear combinations of three-dimensional comple eponential functions. can also be convenientl introduced. The Fourier transform pair epresses,, Equation (19) describes the spatial dependence of a linear superposition of plane waves that propagate in the direction associated with direction cosines cos cos cos (0) where, and are the angles between the directions of wave propagation and the -, - and -aes, respectivel. The equation cos cos cos 1 (1) constitutes a well-known fundamental propert of direction cosines. Upon substitution of equation (0) into equation (1), 1 () follows readil. As indicated earlier, the -component of the direction of wave propagation is non-negative. Consequentl, and, since 0 (the wavelength of light is a positive entit), cos 0 (3) 0 (4) follows when inequalit (3) is substituted into the third row of equation (0). Substitution of equations (17) and (19) into equation (6) ields,,, t C ep i t C ep i t 1,, ep X i d d d (5)

11 10 The first term in equation (5) describes waves that propagate in the negative -direction. This possibilit is precluded because the -component of the direction of wave propagation is necessaril non-negative. Consequentl is required. As a result equations (17) and (5) reduce to and C1 0 (6) t C i t ep (7),,, t,, ep C i t d d d (8) respectivel. When wave propagation occurs in a direction such that its -component is non-ero is required. SPATIAL DEPENDENCE C 0 (9) Two-dimensional Fourier transform pairs defined on arbitrar planes parallel to the object plane can be used to begin treating the spatial dependence of,,, t. The method used here closel parallels a method Goodman 13 has used to treat the angular spectrum of plane waves. Accordingl, let, ;,, ep i d d (30) be the two-dimensional Fourier transform of,, value of ; on the plane defined b an arbitrar,,, ; ep i d d (31) is the corresponding two-dimensional inverse Fourier transform. Substitution of equation (31) into equation (10) ields 13 Joseph W. Goodman, Introduction to Fourier Optics, pp (McGraw-Hill, Inc., New York, 1968).

12 11 k, ; ep i d d 0 (3) which can be written as k, ; ep i d d 0 (33) because the integration variables differ from the differentiation variables. Equation (33) reduces to, ; 4 4, ; k X ep i d d 0 (34) or, equivalentl,, ; 4, ; ep i d d 0 (35) after equations (16) and () are invoked. Each component function in the integrand of equation (35) satisfies the equation independentl and the partial differential equation, ; 4, ; 0 (36) necessaril follows. Equation (36) involves onl one independent variable and can therefore be written as d, ; 4, ; 0 d (37) which is an ordinar differential equation. Equation (37) can be solved to ield the solution, ; C3ep i C4ep i (38) where C 3 and C 4 are arbitrar constants. Equation (38) can be verified b direct substitution into equation (37).

13 1 Substitution of equation (38) into equation (31) ields,, 3 ep 4 ep C i d d C i d d (39) Furthermore,,, 3 ep 4 ep t C C i t d d C C i t d d (40) follows after equations (7) and (39) are substituted into equation (6). The second term in equation (40) describes waves that propagate in the negative -direction. This possibilit is precluded because the -component of the direction of wave propagation is necessaril non-negative. Consequentl is required. This reduces to CC 4 0 (41) C4 0 (4) after substituting equation (9) into equation (41). As a result equations (38), (39) and (40) reduce to and respectivel. The value of C 3, given b, ; C3 ep i (43),, C3 ep i d d (44),,, t CC3 ep i td d (45) p p C3, ; ep i (46)

14 13 can be obtained b evaluating equation (43) on an arbitraril chosen reference plane, designated as the p -plane. Furthermore,, ;, ; ep i p p (47) results when equation (46) is substituted into equation (43). Equation (47) provides the value of, ; on an arbitrar plane in terms of its value on a parallel reference plane p. Substitution of equation (47) into equation (31) leads to,,, ; ep p i d d X ep i p quite simpl. Subsequent substitution of equations (7) and (48) into equation (6) leads to,,,, ; ep ep t C p i d d X i p t (48) (49) easil. The value of C, given b ep C t i t (50) 0 0 can be obtained b evaluating equation (7) at the arbitraril chosen time t 0. Furthermore, t t ep i t t 0 0 (51) results when equation (50) is substituted into equation (7). Equation (51) provides the value of t at an arbitrar time t in terms of its value at another time t 0. Substitution of equation (50) into equation (49) leads to,,, t t0, ; p ep i d d X ep i p t t0 (5)

15 14 quite easil. Equation (5) describes the quantum amplitude,,, t as a linear combination of plane wave components that propagate from the object plane toward the imaging sstem. Z o PLANE Z l- PLANE IMAGING SYSTEM Z l+ PLANE Z i WAVE COMPONENT PROPAGATION Refer to the optical real image formation apparatus illustrated in Figure 1. Propagation of wave components in the region between the object plane and the image plane and outside the imaging sstem can be treated in terms of the planes, linked to real image formation, illustrated in Figure. In Figure, the object and image planes are designated as the o - and i sstem are designated as the l - and l - planes, respectivel. Furthermore, the entrance and eit planes of the imaging - planes, respectivel. Spatial frequencies associated with the wave components when the enter the imaging sstem ma differ from those associated with the same wave components when the eit the imaging sstem. Accordingl, let unprimed spatial frequencies be used in the region where o l and let primed spatial frequencies be used in the region where l i. Treatment of the region where o l can be separated from treatment of the region where l. i On the object side of the imaging sstem (in the region where o l ), the wavelength of light used for image formation has been designated as. The possibilit eists that this wavelength will be different on the image side of the imaging sstem due to a possible change in refractive inde. To accommodate this possibilit, the wavelength of light used will be designated as ' on the image side of the imaging sstem (in the region where l ). i PROPAGATION THROUGH THE IMAGE FORMATION APPARATUS The value of, ; OPTICAL AXIS Figure. Image formation apparatus planes. on an arbitrar plane in terms of its value on a parallel reference plane p is given b equation (47), i.e.,, ;, ; ep i p p (53) in the open interval o l (inside the region where o l ). Similarl, the value of ' ', ' ; on an arbitrar plane in terms of its value on a parallel plane p' is given b i ' ', ' ; ' ', ' ; p' ep p' ' (54)

16 15 in the open interval l (inside the region where i l ). The reference planes i p and are determined independentl of each other. p' The open intervals o l and l i include the limiting values that occur on the various planes identified in Figure. These limiting values are given b lim, ; o, ; o lim, ;, ; l l lim ' ', ' ; ' ', ' ; l l lim ' ', ' ; ' ', ' ; i i (55) In all cases, the limiting process occurs inside the region between the object plane and the image plane and outside the imaging sstem. The relationship follows when, ;, ; ep i l o l o (56) l p o (57) is substituted into equation (53). Similarl, the relationship follows when ' i ' ', ' ; i ', ' ; l ep i l ' (58) i p' l (59) is substituted into equation (54). After introducing do l o d i i l (60) equations (56) and (58) can be written as

17 16, ; l, ; o ep i do (61) and ' ', ' ; i ' ', ' ; l ep i di ' (6) respectivel. NEW NOTATION New notation such that ' ' uo, ;, ; o ul, ;, ; l ul ', ' ; ' ', ' ; l ui ', ' ; ' ', ' ; i (63) can now be introduced to accommodate treating the quantum amplitudes on each of the planes identified in Figure. Equations (61) and (6) can be written as and, ;, ; ep ul uo i do (64) ', ' ; ' ', ' ; ' ep ' u u i d (65) i l i respectivel, b using the new notation. Two-dimensional Fourier transform relationships, treated below, can be used to obtain corresponding relationships in ordinar space. FOURIER TRANSFORM PAIRS Fourier transform pairs are associated with the quantum amplitudes on each of the planes, linked to real image formation, illustrated in Figure. Thus, the two-dimensional Fourier transform of u, ; is given b o, ;, ; ep (66) uo uo i d d while, ;, ; ep (67) u u i d d o o

18 17 u. The twodimensional Fourier transform pair given b equations (66) and (67) is applicable on the object plane. is the corresponding two-dimensional inverse Fourier transform of o, ; The two-dimensional Fourier transform of u, ; while l is, ;, ; ep (68) ul ul i d d, ;, ; ep (69) u u i d d l l is the corresponding two-dimensional inverse Fourier transform of ul, ;. The twodimensional Fourier transform pair given b equations (68) and (69) is applicable on the imaging sstem's entrance plane. The two-dimensional Fourier transform of u, ; ' while l is ', ' ; ', ; ' ep ' ' (70) ul ul i d d, ; ' ', ' ; ' ep ' ' ' ' (71) u u i d d l l is the corresponding two-dimensional inverse Fourier transform of ul ', ' ; '. The twodimensional Fourier transform pair given b equations (70) and (71) is applicable on the imaging sstem's eit plane. The Fourier transform of, ; ' while u is i ', ' ; ', ; ' ep ' ' (7) ui ui i d d, ; ' ', ' ; ' ep ' ' ' ' (73) u u i d d i i u. The twodimensional Fourier transform pair given b equations (7) and (73) is applicable on the image plane. is the corresponding two-dimensional inverse Fourier transform of i ', ' ; '

19 18 OBJECT PLANE QUANTUM AMPLITUDE COMPONENTS A definite configuration of light eists on the side of the object plane nearest to the imaging sstem. The quantum amplitude associated with light of wavelength on the object plane has been designated as uo, ;. Ever point in uo, ; is specified relative to the origin of coordinates. The origin of coordinates is determined b the intersection of the -ais with the o o -plane. The location of the -ais coincides with the optical ais. This location is arbitrar relative to the location of uo, ; and is unrelated to the object being imaged. The two-dimensional Fourier transform of uo, ; is defined in terms of the arbitrar location of the origin of coordinates. Undesirable consequences of such arbitrariness can be dealt with b introducing the double delta u, ;. function decomposition of o Equivalent two-dimensional Dirac delta function decompositions of u, ; and o, ;, ;, o are given b o (74) u u d d o, ;, ;, o (75) u u d d where, and, Addition of equations (74) and (75) leads to are arbitrar points within the quantum amplitude, ; 1 uo uo, ;, ;,,, ;,, o u d d d d u. (76) readil. This result epresses the quantum amplitude u, ; as a linear combination of twodimensional Dirac delta function pairs. o After minor manipulation that includes changing the order of integration and recalling equation (66), the two-dimensional Fourier transform of equation (76), i.e., 1 uo, ; o, ;,, u (77) uo, ;,, ep i d d d d d d can be found. Equation (77) reduces to o

20 19 1 uo uo i, ;, ;, ep o, ;, ep u i d d d d (78) following integration over and. In turn, equation (78) can be written as u, ; o 1 uo, ;, ep i uo, ;, ep i X ep i d d d d (79) after minor manipulation. Each component in the integrand of equation (79) describes a twodimensional comple eponential periodic function multiplied b a two-dimensional comple eponential phase factor. INTERPRETATION The two-dimensional comple eponential phase factor in equation (79) eists because the intersection of the -ais (the optical ais) and the o o -plane is arbitrar. The distance between the -ais and the midpoint between the points, and, is given b M (80) where (81) and (8) are the - and - coordinates, respectivel, of the midpoint. The distance M is not a phsical propert of the distribution of light.

21 0 Each two-dimensional comple eponential periodic function in the integrand of equation (79) is periodic in two dimensions. The - and - components of this quantum amplitude spatial period are given b T (83) and T (84) respectivel. The two-dimensional comple eponential periodic functions each represent a phsical propert of the distribution of light. The quantum amplitude spatial period is given b T T T (85) which can be written as T (86) b invoking equations (83) and (84). The distance that separates, and, is given b (87) where and (88) (89) are the associated distance components in the - and - directions, respectivel. After substituting equations (88) and (89) into equation (87) (90)

22 1 follows readil. In turn, T (91) can be obtained b substituting equation (90) into equation (86). Thus, the quantum amplitude spatial period associated with two points in a distribution of light is equivalent to one-half the distance between the two points. The quantum amplitude spatial frequenc F 1 (9) T can now be defined. After substituting equation (91) into equation (9) F (93) can be obtained. Thus, the quantum amplitude spatial frequenc associated with two points in a distribution of light is equivalent to twice the reciprocal of the distance between them. ILLUMINATION Consider the quantum amplitude associated with the two separated points, and, in a configuration of light on the side of the object plane nearest to the imaging sstem. This quantum amplitude is given b o, ;,, u A B (94) where A and B are constants, for both points taken together. The individual quantum amplitudes are given b and for the two separated points. o, ;, u A (95) o, ;, u B (96) The two-dimensional Fourier transform of equation (94) is given b

23 , ; ep uo i d d,, ep A B i d d (97) which ields, ; ep ep uo A i B i (98) upon integration. Equation (98) can be written as u, ; ep i o X A ep i B ep i (99) for eas comparison with equation (79). Equation (99) reduces to, ; ep X Aep i T T Bep i T T u i o (100) after,, T and T, given b equations (81), (8), (83) and (84), respectivel, are recalled. Equation (100) can be written as, ; ep u i o cos sin X A B T T i A B T T (101) b invoking Euler's formula and using trigonometric notation. The periodicit of o, ; nonero. Consequentl u is dependent upon u, ; and, ; o u both being o A 0 B 0 (10) Is required when uo, ; is periodic.

24 3 The special case where is of modest interest. Equation (101) reduces to A B (103), ; ep cos uo A i T T (104) for this special case. Thoroughgoing investigation of light that is suitable for real image formation illumination is beond the scope of this paper. Nevertheless, light that is either coherent or partiall coherent where both illuminated points eist in the same area of coherence can be used satisfactoril. INDIVIDUAL PHOTONS Two illuminated object plane points are required, as a minimum, to form an optical real image. This is true either when the image is formed b a large number of photons in a single event or when the image is built up b individual photons. Evidentl each individual photon is associated with illumination of both of the requisite object plane points. PLANE WAVE PROPAGATION Equation (5) reduces to,,, 0, ; ep ep o t t o i d d X i t t0 (105) in the region where o l. Equation (105) describes a linear combination of plane waves that propagate awa from the object plane toward the imaging sstem. The spatial frequencies associated with each individual plane wave component in the integrand of equation (105) are given b cos cos cos (106)

25 4 a result that follows from equation (0) triviall. Each spatial frequenc is associated with a spatial period given b 1 T 1 T T 1 (107) b definition. Combination of equations (106) and (107) leads to T cos cos T cos T (108) The propagation direction and a wavefront that is perpendicular to the propagation direction are illustrated in Figure 3 for an individual plane wave component. The wavefront is an infinite plane that is perpendicular to the plane of the figure and that etends out of the plane of the figure; onl the trace of the wavefront on the plane of the figure is shown in the figure. OBJECT PLANE (XY-PLANE) WAVEFRONT T / PROPAGATION DIRECTION OPTICAL AXIS (Z-AXIS) Figure 3. Plane wave propagation.

26 Propagation Angle (Degrees) 5 In Figure 3, as elsewhere in this paper, the -ais coincides with the optical ais defined b the imaging sstem. The object plane is rotated about the -ais so that the line defined b the points, and, is parallel to the plane of the figure. This line is associated with the spatial period T. The spatial period T is shown as the vertical ais in the figure and is also shown as the hpotenuse of a right triangle. The wavelength of light used,, is depicted in the figure as the distance between the origin and the wavefront introduced previousl. In addition, is shown as the angle between the direction of wave propagation and the optical ais;, T and have been introduced previousl. MINIMUM SEPARATION REQUIREMENT The relationship sin (109) T can be established b inspecting Figure 3 and appling the definition of an angle s sine. Consequentl, 1 sin T (110) is the propagation angle associated with the spatial period T. Equation (110) can be written as Separation Distance (Light Wavelengths) Figure 4. Propagation angle as a function of the separation distance between two points in the object.

27 6 1 sin (111) where equation (91) has been invoked. The propagation angle (in degrees) is plotted in Figure 4 as a function of the separation distance (in wavelengths of the light used) between the points, and,. Upon considering equation (111) or eamining the curve in Figure 4, it can be determined readil that the minimum separation distance required for light to propagate awa from the object plane eceeds two wavelengths of the light used. Thus, satisfaction of the inequalit (11) is required for light to propagate awa from the object plane. As a consequence, propagation of light awa from a single point (as hpothesied b the Hugens-Fresnel principle) does not occur. Two illuminated points in the object are required to form light waves that contribute to real image formation. PLANE WAVE PAIRS The equation sin cos cos (113) which is equivalent to sin cos cos (114) follows from equation (1) readil. Thus, as illustrated in Figure 5, two equi-amplitude plane wave components are associated with each pair of illuminated points in the object distribution. For each plane wave component that propagates at an angle + relative to the optical ais, an equi-amplitude plane wave component propagates at an angle - relative to the optical ais. PARAMETER CHANGES Some properties of the light that emerges from an imaging sstem differ from those that enter the imaging sstem. As introduced previousl, let unprimed parameters be used in the region between the object plane and the imaging sstem. Also, let primed parameters be used in the region between the imaging sstem and the image plane. Let n be the inde of refraction in the unprimed region and let n ' be the inde of refraction in the primed region. Furthermore, let S be the wavelength of the light used as observed in vacuum. Then

28 7 OBJECT PLANE PROPAGATION DIRECTION T / WAVE FRONT OPTICAL AXIS / T WAVE FRONT PROPAGATION DIRECTION Figure 5. Plane wave pair. S n ' S n ' (115) where is the wavelength of the light used in the unprimed region and wavelength of light in the primed region. Substitution of equation (115) into equation (109) leads to ' is the corresponding

29 8 S sin nt sin ' S nt ' ' (116) readil. In turn 1 S sin nt ' 1 S sin nt ' ' (117) can be obtained from equation (116) quite easil. The angle between the direction of wave propagation and the -ais is changed from to ' as a result of the light passing through the imaging sstem. The spatial period at the image plane of a plane wave that contributes to real image formation is given b T ' mt (118) where the lateral magnification m has been recalled. Substitution of equation (118) into the bottom row of equation (116) ields which can be combined with the top row of equation (116) to obtain a result known as the optical invariant. The lateral magnification can be obtained from equation (10) b simpl rearranging terms. IMAGE FORMATION Let S sin ' (119) n' mt mn'sin ' nsin (10) n sin m (11) n ' sin '

30 9, ;, ; ep i d d (1) be the coherent transfer function of the imaging sstem;, ; coherent impulse response. The equation ', ' ; l, ;, ; l is the corresponding (13) relates the two-dimensional Fourier transform of the quantum amplitude on the eit plane of the imaging sstem to the two-dimensional Fourier transform of the quantum amplitude on the entrance plane of the imaging sstem. Simultaneous satisfaction of equations (61), (6) and (13) ields readil. ', ' ; i, ;, ; o ep i do di ' (14) New notation, introduced in equation (63), such that, ; o uo, ; ', ' ; i ui ', ' ; (15) can now be used. Using this notation, equation (14) can be written as ', ' ;, ;, ; ep ' ui uo i do di (16) convenientl. Advanced imaging sstems are designed such that (ideall) occurs. Thus, the change of variables d d' 0 (17) o i m ' ' m ' do d i (18)

31 30 results as a consequence of an optical disturbance s interaction with an ideal imaging sstem. Equation (16) reduces to ui, ; uo, ;, ; m m (19) after the change of variables has been invoked. Man practical imaging sstems have been designed to achieve ver close approimations to the change of variables given b equation (18). Treatment of these imaging sstems is beond the scope of this paper. The two-dimensional inverse Fourier transform of equation (19) is given b ui, ; ep i d d m m, ;, ; ep uo i d d (130) which ields, ;, ;, ; m ui m m uo upon evaluation. Equation (131) can be written as (131) 1 ui, ; u o, ;, ; m m m m m (13) which relates the quantum amplitude of the optical disturbance on the image plane to the quantum amplitude of the optical disturbance on the object plane. This result, the coherent real image equation, is identical with equation (3). Equation (13) describes an optical real image as an ideal (undistorted) optical real image convolved with the impulse response linked to an imaging sstem. Ecept for special cases, the convolution operation leads to image distortion 14. Thus, features in the image are (ordinaril) approimatel as wide as the sum of the widths of the two functions being convolved; fine features become indistinguishable (washed out). The distortion occurs as a consequence of the imaging sstem's impulse response and related transfer function. 14 Jack D. Gaskill, Linear Sstems, Fourier Transforms, and Optics, pp (John Wile & Sons, Inc., New York, 1978).

32 31 PART II. OPTICAL IMAGE RESOLUTION INTRODUCTION The coherent impulse response associated with an arbitrar imaging sstem is given b, ;, ; ep i d d (133) for light of wavelength. Equation (133) is the two-dimensional inverse Fourier transform of the imaging sstem s coherent transfer function. The reciprocal variables and, commonl known as spatial frequencies, are needed to define the two-dimensional inverse Fourier transform. The imaging sstem s coherent transfer function is given b, ;, ; ep i d d (134) for light of wavelength.. Equation (134) is the two-dimensional Fourier transform of the imaging sstem s coherent impulse response. Equations (133) and (134) constitute a twodimensional Fourier transform pair. Consider a diffraction-limited imaging sstem that has a clear aperture of arbitrar shape and sie. For light of wavelength this imaging sstem is associated with a passband and a coherent transfer function. The coherent transfer function is given b 1 inside the passband, ; (135) 0 outside the passband Quantum amplitude components with spatial frequencies that are inside the passband of the imaging sstem pass through the imaging sstem and contribute to image formation. Quantum amplitude components with spatial frequencies that are outside the passband of the imaging sstem do not pass through the imaging sstem and do not contribute to image formation. Image distortion due to missing quantum amplitude components occurs. Most imaging sstems are circular. Although non-circular imaging sstems eist, little would be gained b considering them in the present contet. Henceforth, unless otherwise indicated, attention will be restricted to circular imaging sstems. Man authors provide treatments concerning coherent impulse responses and coherent transfer functions; see, for eample, Gaskill 15 or Goodman 16. A general treatment of coherent impulse responses and coherent transfer functions is beond the scope of this paper. 15 Jack D. Gaskill, Linear Sstems, Fourier Transforms, and Optics, pp (John Wile & Sons, Inc., New York, 1978). 16 Joseph W. Goodman, Introduction to Fourier Optics, pp (McGraw-Hill, Inc., New York, 1968).

33 3 PASSBAND Each plane wave component that arrives at the imaging sstem is associated with the spatial frequencies, and. These spatial frequencies are not independent. Thus 1 (136) follows readil b combining equations () and (4). A maimum spatial frequenc c such that (137) c is included in the passband that is linked to a circular imaging sstem. This spatial frequenc is known as the spatial frequenc cutoff of the imaging sstem. The inequalit 1 c (138) is easil obtained b substituting equation (136) into inequalit (137). The spatial frequenc cutoff of an imaging sstem is the value of the spatial frequenc beond which the associated transfer function is ero. Inside the imaging sstem's passband 1 (139) 0 c while 1 c (140) outside the imaging sstem's passband. Accordingl c, ; 1 0 c (141)

34 33 can be obtained b substituting equations (139) and (140) into equation (135). Imaging sstems with non-circular apertures are, in general, associated with multiple spatial frequenc cutoffs. General treatment of such imaging sstems is beond the scope of this paper. The transfer functions associated with the imaging sstems considered in this paper are ver closel approimated b equation (141). An imaging sstem that is associated with the spatial frequenc cutoff c is also associated with a spatial period cutoff c. The imaging sstem's spatial period cutoff, given b 1 c (14) is equal to the minimum spatial period of an quantum amplitude that can be transferred through the imaging sstem. This minimum spatial period is given b c T c (143) c Consequentl, images of periodic quantum amplitude components with spatial periods T such that T c (144) can be formed while images of quantum amplitude components with smaller spatial periods cannot be formed. All periodic quantum amplitude components with spatial periods that eceed pass through the imaging sstem and consequentl contribute to real image formation. c Substitution of equation (91) into inequalit (144) leads to c (145) readil. Accordingl, an image of two points in the object plane can be formed (the two points can be resolved) provided the distance between them equals or eceeds the minimum distance (146) c Two points in the object plane are, b definition, well-separated when inequalit (145) is satisfied. Inequalit (145) constitutes a fundamental two-point optical resolution criterion. Circular imaging sstems are endowed with a definite diameter. Accordingl, a circular imaging sstem restricts the propagation angle of the light that can enter it to a maimum allowed value c. Similarl, a circular imaging sstem restricts the propagation angle of the light that can leave it to a maimum allowed value ' c. The cutoff propagation angles cand ' c are the c

35 34 entrance angle (also known as the acceptance angle) and the eit angle, respectivel, of the imaging sstem. Equation (116) becomes sin sin ' c c S nt c S nt ' ' c (147) when evaluated at cutoff. After rearranging equation (147) and substituting equation (143) into the result S nsin c c ' c S n 'sin ' c (148) can be obtained. The imaging sstem's object side numerical aperture NA o and image side numerical aperture NA i, given b NA sin o n c NA n'sin ' i c (149) can be convenientl introduced. Equation (11) becomes n sinc m (150) n ' sin ' when evaluated at cutoff. Substitution of equation (149) into equation (150) ields easil. Substitution of equation (149) into equation (148) leads to c NA o m (151) NA i

36 35 S NA c o ' c S NAi (15) readil. In addition NA o c S ' c NA i S (153) follows after substitution of equation (14) into equation (15). As shown b equations (15) and (153), knowledge of a circularl smmetric imaging sstem's object side and image side numerical apertures is sufficient to determine the imaging sstem's spatial period and spatial frequenc cutoffs, respectivel, for an particular wavelength of light. An imaging sstem is often characteried in terms of its numerical aperture without distinguishing the object side numerical aperture from the image side numerical aperture. Rather, the term numerical aperture is used genericall to treat either numerical aperture. The fundamental limits of performance for a circular imaging sstem are often epressed in terms of its numerical aperture without specifing the intended numerical aperture. Confusion can result. RESOLUTION CRITERIA A minimum separation distance between two distinguishable image features eists. Two image features that are sufficientl near each other merge together to form a single image feature. Two image features that can be distinguished as two image features are said to be resolved. Comparison of the fundamental resolution criterion given b inequalit (145) with two wellknown classical resolution criteria can be made. These resolution criteria are the Abbė resolution criterion (ordinaril used for microscopes) and the Raleigh resolution criterion (ordinaril used for telescopes and photolithograph). The Raleigh resolution criterion is often used to identif what is known as the classical diffraction limit of optical sstem performance. Neither of these criteria is based on fundamental phsics; nevertheless, the are both roughl consistent with the fundamental resolution criterion introduced in this paper as inequalit (145). An overview of optical resolution has been published b A.J. den Decker and A. van den Bos 17. These authors point out that optical resolution is not unambiguousl defined and is interpreted in 17 A. J. den Decker and A. van den Bos, Resolution: a Surve. J. Opt. Soc. Am. A, Vol 14, No. 3, pp (March 1997).

37 36 man was. The go on to review the concept of optical resolution and man of its interpretations. FUNDAMENTAL RESOLUTION CRITERION An image of two points in the object plane can be formed (the two points can be resolved) provided the distance between them equals or eceeds the cutoff distance. This fundamental twopoint optical resolution criterion has been introduced previousl b means of inequalit (145). Equation (147) can be rearranged so that S T nsin c c T ' c S n 'sin ' c (154) results. Substitution of equation (149) into equation (154) ields S NA Tc o T ' c S NAi (155) readil. When epressed in terms of both unprimed and primed parameters, inequalities (144) and (145) lead to S NA T o T ' S NAi (156) and S NA o ' S NAi (157) respectivel. Inequalit (157) constitutes object side and image side fundamental two-point optical resolution criteria.

38 37 ABBÉ RESOLUTION CRITERION The Abbė resolution criterion can be written as S NA p o p ' S NAi (158) and is applicable to a coherentl illuminated periodic amplitude object of period p as measured on the object plane and period p ' as measured on the image plane. Inequalit (158) is mathematicall equivalent to inequalit (156) but it is fundamentall linked to spatial periods rather than two-point separations. An amplitude object where the spatial periods are phsicall identifiable is needed to appl the Abbė resolution criterion. B wa of contrast, the fundamental resolution criterion is applicable to general objects. RAYLEIGH RESOLUTION CRITERION The Raleigh resolution criterion, which can be written as 1. S NA R o R ' 1. S NAi (159) is a measure of the optical imaging sstem s performance. According to this criterion, an image of two points in the object plane can be formed (the two points can be resolved) provided the distance between them satisfies the Raleigh criterion. CONVENTIONAL OPTICAL RESOLUTION The conventional rationale that underlies the Raleigh resolution criterion is presented in Appendi A. According to this rationale a tpical image consists of man overlapping distributions of light. Each distribution of light is formed b the diffraction of coherent light from the imaging sstem s aperture. The image consists of the incoherent superposition of the coherentl formed individual distributions of light. In accord with the Raleigh criterion, two identical component distributions of light are barel resolved when the are separated b the distance R (object side) or b the distance R ' (image side). The distance R ' is the laterall magnified version of the distance R where the magnification is produced b the imaging sstem. In accord with conventional real image formation theor, resolution and field depth can be understood (in disagreement with the nonlocal optical real image formation theor) in connection

39 with an isolated three-dimensional distribution of light. Resolution is related to the width of the distribution of light in the focal plane while image depth is related to the length of the distribution of light perpendicular to the focal plane. As a consequence of this model, conventional resolution and field depth are closel linked to one another. According to the nonlocal optical real image formation theor, no such close linkage occurs. 38

40 39 PART III: FIELD DEPTH INTRODUCTION A configuration of light that closel approimates the configuration of light on the object (image) plane eists on various auiliar planes that are parallel to and include the object (image) plane. The etreme range of distances over which these planes can be identified is the unobstructed object (image) field depth. Unobstructed field depth occurs when the object (image) plane is transparent. Obstructed field depth occurs when the object (image) plane is opaque, and is ordinaril one-half the unobstructed field depth. Separate treatment of obstructed field depth is not included in this paper. In the following analsis the object plane and the image plane are both treated as being transparent. The associated unobstructed field depth etends on both sides of the object (image) plane. AUXILIARY PLANE T' ' s IMAGE PLANE Figure 7. Plane wave propagation through the image plane and an auiliar plane.

41 40 Table I. Parameters identified in Figure 7. ' Wavelength of light used (image side) Angle between the direction of propagation and a perpendicular to the image and auiliar planes Quantum amplitude spatial period of the sinusoidal component associated with the light wave (image side) Distance between the auiliar plane and the image plane s Longitudinal displacement of a spatial period on the auiliar plane Distance in the direction defined b IMAGE FIELD DEPTH Consider optical real image formation where the image plane is transparent. Each plane wave pair that contributes to real image formation propagates through the image plane. These plane wave pairs also propagate through various planes that are parallel to the image plane. The plane wave pairs overlap on each of these planes. As a result, sinusoidal components that closel approimate corresponding sinusoidal components on the image plane are formed on auiliar planes that are parallel to and sufficientl near the image plane. These sinusoidal components eist on both sides of the image plane and are considered to be part of the real image. Superposition of the two members of a plane wave pair that overlap on a plane parallel to the image plane (an auiliar plane) and a distance from it can now be considered. Geometric relationships defined b one member of the plane wave pair that contributes to real image formation are illustrated in Figure 7. Parameters identified in the figure are described in table I. Let A/ be the wave amplitude of each member of the plane wave pair and let amplitude spatial period of each member of the plane wave pair. Then T ' be the quantum A 1 cos ( s) T ' (160) represents one member of the plane wave pair while A cos ( s) T ' (161) represents the other member of the plane wave pair. The superposition of both members of the plane wave pair can be described b or 1 (16)

42 41 A A cos ( s) cos ( s) T' T' (163) which reduces to A cos cos T' s T' (164) with little effort. The relationships s tan (165) sin tan (166) cos ' sin (167) T ' and T ' ' cos (168) T ' can be obtained b inspecting Figure 7. Hence, tan ' T ' ' (169) and s T ' ' ' (170) follow readil. Substitution of equation (170) into equation (164) ields

43 4 A ' cos cos ' ' T' T' T (171) easil. The first cosine function on the right hand side of equation (171) is spatiall periodic where T' T' ' (17) ' is the function s spatial period. The range of values of for which changes in the value of are acceptable can be epressed as T' T' ' (173) K ' where K is a subjectivel determined constant. The depth of image field can now be defined as where D' (174) T' T' ' (175) K ' and T' T' ' (176) K ' have been introduced. Substitution of equations (175) and (176) into equation (174) ields T' T' ' D' (177) K ' easil. The depth of image field is given b equation (177).

44 43 OBJECT FIELD DEPTH The analsis used to derive equation (177) for the depth of image field is also applicable to deriving analogous epressions for the depth of object field. Primed parameters are used for the image side analsis; unprimed counterparts are used for the object side analsis. Thus, the object side wavelength of light used is designated as (unprimed) and the object side quantum amplitude spatial period is designated as T (unprimed) in the treatment rather than their image side (primed) counterparts. The object side counterpart of equation (177) is T T D (178) K The depth of object field is given b equation (178). VALUE OF K As introduced previousl, K is a subjectivel determined constant. Selecting a suitable value for K is perhaps best done on the basis of eperience. Possibl the value K 16 ma be suitable as a tentative useful value for K. TWO POINT SEPARATION The quantum amplitude spatial period associated with two points in a configuration of light is equivalent to one-half the distance between the two points. This distance is given b equation (91) for points on the object plane. Consequentl, T T ' ' (179) when the object side and its image side (primed) counterpart are both considered. Substitution of equation (179) into equations (178) and (177) leads to 4 D K D ' ' ' 4 ' K ' (180) The depth of object (image) field is independent of the imaging sstem.

45 44 CONVENTIONAL IMAGE FIELD DEPTH Equations that are commonl used to estimate the field depth (object and image) associated with real image formation are dependent upon the numerical aperture designated as NA (without regard for the side of the imaging sstem involved) of the imaging sstem that is used. Specificall D C (181) NA is the equation conventionall used to estimate the depth of object field associated with real image formation 18. Similarl, D' C ' (18) NA is the equation that is conventionall used to estimate the depth of image field associated with real image formation These equations are substantiall different than the counterpart equations derived in this paper. 18 Chris Mack, Fundamental Principles of Optical Lithograph: The Science of Microfabrication, p. 105 (John Wile & Sons, Ltd., Chichester, England, 007).

46 45 PART IV: REFERENCE DISTRIBUTION REAL IMAGE FORMATION INTRODUCTION Reference distribution real image formation (RIF) is achieved b using a reference distribution of light in addition to the usual subject distribution. This assures that two well-separated illuminated points are used to form each light wave component that propagates awa from the object plane and ultimatel contributes to real image formation. One point lies within the subject distribution and the other point lies within the reference distribution. The distance between these points is designed to assure that it eceeds the minimum needed for real image formation. Thus, inequalit (145) is satisfied intentionall for ever pair of points such that one point lies within the subject distribution and the other point lies within the reference distribution. An image of both distributions of light is formed. The spatial frequencies of the component light waves that contribute to RIF lie within a welldefined finite bandwidth. The imaging sstem is designed such that it transfers the wave components with spatial frequencies that lie within this bandwidth to the image plane. These wave components are superposed on the image plane to form a real image. RIF REAL IMAGE EQUATION All spatial frequencies linked to the combined reference and subject distributions are included in uo, ; when RIF is used. The imaging sstem is chosen to assure that these spatial frequencies lie within its passband. Thus u, ;, ; u, ; o o (183) for reference distribution image formation. Consequentl, equation (19) reduces to ui, ; uo, ; m m (184) and equation (13) reduces to 1 ui, ; uo, ; m m m (185) for RIF. Equation (185) is the RIF real image equation for light of wavelength. No convolution operation is associated with the RIF real image equation. The RIF real image equation defines a magnified (enlarged, unchanged or reduced and perhaps inverted) real image of both the subject distribution and the reference distribution. Magnified versions of all spatiall periodic components linked to one point in the object distribution and

47 46 another point in the reference distribution are included in the real image. No real image distortion due to missing spatiall periodic components occurs. Resolution is complete. OBJECT PLANE CONFIGURATION A configuration of light that is suitable for achieving RIF is illustrated in Figure 8. The configuration of light eists on the side of the object plane nearest to the imaging sstem and is confined to the interiors of the circles shown in the figure. Light throughout the configuration is coherent or partiall coherent. All points in both the subject distribution and the reference distribution are in the same area of coherence. As indicated in the figure, the subject distribution eists inside a circular area of diameter while the reference distribution eists inside a circular area of diameter D R. As introduced previousl, let s be wavelength of light used, as measured in vacuum, and let NA o be the object side numerical aperture of the imaging sstem. Ideal RIF occurs when the criteria D S S S (186) NA o AREA OF ILLUMINATION PERIMETER SUBJECT DISTRIBUTION REFERENCE DISTRIBUTION C/L D S S D R Figure 8. RIF object plane configuration: plan and front elevation views.

48 47 D S s (187) NA o and D R s (188) NA o are realied phsicall. Nois RIF occurs when either satisf the foregoing inequalities. D S or D R or both D S and D R do not The fundamental resolution criterion, given b equation (157), is satisfied when inequalit (186) is satisfied; an image of ever object point and ever image point is formed. The fundamental resolution criterion is not satisfied when inequalities (187) and (188) are satisfied. The RIF configuration that has been presented is b no means the onl possible RIF configuration that can be devised. Treatment of other possible RIF configurations is beond the scope of this paper. NOISY RIF CONCEPT Consider the conceptual illustration of transmission nois RIF provided in Figure 9. Initiall, light is incident (from the left) upon an opaque screen with two apertures in it. The side of the screen nearest to the imaging sstem (shown as a lens) serves as an object plane. Coherent or partiall coherent light passes through the apertures to form two distributions of light on the object plane. These distributions of light are, ideall, the same sie and shape as the apertures. The distributions of light are designated as the subject distribution and the reference distribution. Taken separatel, the propagation angles of the light that travels awa from either of the two distributions of light are larger than the acceptance angle of the imaging sstem. Such light does not pass through the imaging sstem and consequentl does not contribute to image formation. No two points in either the subject distribution or the reference distribution are sufficientl separated to contribute to image formation. Ever pair of points such that one point is in the subject distribution and one point is in the reference distribution is sufficientl separated to contribute to image formation. Taken together, the propagation angles of the light that travels awa from the combined distributions of light are smaller than the acceptance angle of the imaging sstem. This light defines a finite bandwidth and is transferred, without amplitude or phase distortion, through the

49 48 LIGHT THAT PASSES THROUGH ONE APERTURE SUBJECT IMAGE FORMING LIGHT PASSES THROUGH BOTH APERTURES REFERENCE IMAGE SUBJECT IMAGE REFERENCE LIGHT THAT PASSES THROUGH ONE APERTURE Figure 9. Conceptual nois RIF process. imaging sstem. Consequentl, light that propagates awa from the combined distributions of light contributes to undistorted real image formation.

50 49 PART V: CONCLUSION A nonlocal theor of optical real image formation has been developed from the fundamental phsics associated with an optical real image formation apparatus. The theor shows how two separated illuminated object points are fundamentall required for real image formation. This finding is in distinct contrast to the ordinaril accepted assumption that onl one object point is involved. In accord with the new theor, optical real images with a ver large field depth (object and image) and no fundamental resolution limit can be formed. Each quantum amplitude component that contributes to real image formation is associated with a pair of illuminated points in the object. Both illuminated points are inside the area of coherence associated with the quantum amplitude component. Image-forming light is either coherent or partiall coherent. Single photons are capable of providing image-forming light. A fundamental two-point resolution criterion was derived as part of the new theor and compared with the Abbė and Raleigh resolution criteria. Although neither of these criteria is based on fundamental phsics, the are both roughl consistent with the fundamental resolution criterion introduced in this paper. Field depths for both the object and image were shown to be independent of the imaging sstem used and to be substantiall different than the are traditionall thought to be. Reference distribution real image formation (RIF) has been introduced. RIF is achieved b using a reference distribution of light in addition to the usual subject distribution. This assures that two well-separated illuminated points are used to form each light wave component that propagates awa from the object plane and ultimatel contributes to real image formation. One point lies within the subject distribution and the other point lies within the reference distribution. The spatial frequencies of the component light waves that contribute to RIF lie within a welldefined finite bandwidth. The imaging sstem transfers the wave components with spatial frequencies that lie within this bandwidth to the image plane. These wave components are superposed on the image plane to form a real image. Real images formed b means of RIF are described b the RIF real image equation. The RIF real image equation defines a magnified (enlarged, unchanged or reduced and perhaps inverted) real image of both the subject distribution and the reference distribution. No convolution operation is involved. Magnified versions of all spatiall periodic components included in the object are included in the real image. No real image distortion due to missing spatiall periodic components occurs. Resolution is complete when real images are formed b means of RIF.

51 50 APPENDIX A: CONVENTIONAL OPTICAL REAL IMAGE RESOLUTION INTRODUCTION APERTURE PLANE FOCAL PLANE A brief review of the generall accepted treatment of resolution associated with FOCUS optical real image formation (in disagreement with the nonlocal optical IMAGING SYSTEM real image formation theor) is provided in this appendi. Thus, resolution associated with a spherical wave that converges to a geometric focus after passing through a circular aperture, as Figure 10. Apertured converging spherical wave. conventionall understood, is reviewed. Such a wave is often created b a focusing element, where the edge of the focusing element constitutes the edge of the aperture. For this conventional treatment, the angle at the geometric focus subtended b the circular aperture is restricted to being ver small. Referring to Figure 10, a uniform spherical monochromatic light wave of wavelength ' converges toward a geometric focus after passing through a circular aperture. The phase of the wave is the same at all points in the aperture; i.e., the light is coherent. As shown in Figure 11, the -ais is chosen to pass through the aperture s center and the geometric focus. The aperture radius a, the distance from the aperture s center to the geometric focus f, and the angle ' c that is subtended b the aperture s radius at the FOCAL PLANE geometric focus are shown in the figure. The angle is the focusing element s eit angle. ' c RAYLEIGH RESOLUTION CRITERION OPTICAL AXIS APERTURE A three-dimensional distribution of light that is dominated b a bright central region surrounded b a set of alternating dark and bright threedimensional ovals envelops the focal point 19. A tpical image consists of man of these distributions of light, some of which ma (and probabl do) overlap. The center-to-center separation (in the focal plane) of two resolved oval-shaped distributions of light defines the Raleigh resolution criterion. In accord with the OPTICAL AXIS Figure 11. Geometric parameters. FOCUS 19 Ma Born and Emil Wolf, Principles of Optics (Sith Corrected Edition) pp (Pergamon Press, Oford, 1980).

52 51 Raleigh resolution criterion, two identical oval-shaped distributions of light are barel resolved (distinguishable) when the center of one of them coincides (on the focal plane) with the first dark oval of the other one. Let s R ' be the radius (on the focal plane) of the first dark oval associated with a single oval shaped distribution of light. As illustrated in Appendi B, Figure 1 ' s ' 1.0 c R (189) so that s R 0.61 ' ' c (190) when the Raleigh resolution criterion is satisfied. In accord with equation (153) ' c (NA) s i (191) where (NA) i is the image side numerical aperture of the focusing element and s is the wavelength (in vacuum) of the light used. Substitution of equation (191) into equation (190) leads to s R 0.61s ' (NA) i (19) readil. The center-to-center separation (on the focal plane) of the two identical oval-shaped distributions of light when the are barel resolved is given b ' s ' R R (193) Consequentl, two points are resolved (barel or otherwise) when the image plane Raleigh resolution criterion 1. R ' NA S i (194) is satisfied. The projection of the image side Raleigh resolution criterion through the imaging sstem onto the object plane is the object side Raleigh resolution criterion. Consequentl, the Raleigh resolution criteria are given b

53 5 for both the object plane (unprimed) and the image plane (primed). 1. S NA R o R ' 1. S NAi (195) Raleigh resolution criteria are commonl 0 applied far outside their ver restricted range 1 of validit. Such application has led to heroic attempts to use ever decreasing wavelengths of light and increasing numerical apertures to improve resolution. This approach has met with a measure of success without a true understanding of the phsics involved. The approach constitutes a fight against nature. 0 Chris Mack, Fundamental Principles of Optical Lithograph: The Science of Microfabrication (John Wile & Sons, Ltd., Chichester, England, 007). 1 Colin J.R. Sheppard, The Optics of Microscop, J. Opt. A: Pure Appl. Opt. 9, pp. S1 S6 (007).

54 53 APPENDIX B. CIRCULARLY SYMMETRIC IMAGING SYSTEMS INTRODUCTION Most imaging sstems are circularl smmetric. A change of variables from rectangular coordinates to plane polar coordinates in both the -plane and the -plane is advantageous for treating circularl smmetric imaging sstems. Thus rcos rsin (196) and cos sin (197) can be appropriatel introduced. In addition and, ; g r, ; (198), ; g, ; (199) can be introduced to epress the imaging sstem's coherent impulse response and coherent transfer function, respectivel, in polar coordinates. The Jacobians linked to the transformation from rectangular to polar coordinates are and which reduce to cos r sin Jr, (00) sin r cos cos sin J, (01) sin cos J r, r (0)

55 54 and J, (03) respectivel. Consequentl and d d r dr d (04) d d d d (05) follow. Substitution of the foregoing equations into equations (133) and (134), as appropriate, leads to and, ;, ; ep cos cos sin sin g r g i r d d (06) 0 0, ;, ; ep cos cos sin sin g g r i r r dr d (07) 0 0 respectivel. After introducing the trigonometric identit cos cos sin sin cos (08) equations (06) and (07) can be written as and respectivel., ;, ; ep cos g r g i r d d (09) 0 0, ;, ; ep cos g g r i r r dr d (10) 0 0 Integral representations of the Bessel function of the first kind of order ero, i.e., Frank Bowman, Introduction to Bessel Functions, p. 57 (Dover Publications Inc., New York, 1958).

56 55 1 J 0 r ep i r cos d 0 (11) and 1 J0 r ep i r cos d 0 (1) can now be advantageousl introduced. Equations (11) and (1) can be rearranged to ield and respectivel. ep i r cos d J 0 0 r (13) ep i r cos d J 0 0 r (14) Substitution of equation (13) into equation (09) leads to while, ;, ; 0 0 (15) g r g J r d, ;, ; g g r J r r dr (16) 0 can be obtained after substituting equation (14) into equation (10). The Bessel function of the first kind of order ero is an even function. Consequentl 0 0 can be substituted into equation (15) to obtain triviall. 0 J r J r (17), ;, ; g r g J r d (18) CIRCULARLY SYMMETRIC COHERENT TRANSFER FUNCTION 0 0

57 56 Coherent transfer functions which are circularl smmetric are necessaril independent of ; consequentl g, ; h ; (19) can be appropriatel introduced. Substitution of equation (19) into equation (18) ields, ; ; g r h J r d (0) The right hand side of equation (0) is independent of. Accordingl 0, ; hr; g r 0 (1) can be introduced to reflect the fact that the left hand side of equation (0) is necessaril also independent of. Substitution of equation (1) into equation (0) ields ; ; h r h J r d () Substitution of equations (19) and (1) into equation (16) ields 0 ; ; 0 0 h h r J r d (3) The circularl smmetric coherent transfer function ; coherent impulse response ; (3). 0 h and the circularl smmetric hr are related to one another b means of equations () and Substitution of equation (19) into equation (199) and then substituting the result into equation (135) leads to 1 inside the passband h; (4) 0 outside the passband which is the circularl smmetric coherent transfer function. Equation (4) can be written as 1 h ; 0 c c (5) where c is the spatial frequenc cutoff of the imaging sstem.

58 57 CIRCULARLY SYMMETRIC COHERENT IMPULSE RESPONSE Equation () relates the circularl smmetric coherent impulse response ; circularl smmetric coherent transfer function ; hr to the h. Substitution of the circularl smmetric transfer function given b equation (5) into equation () Leads to readil. Equation (6) can be rewritten as ; c ; h r J r d (6) c hr; r J 0 r ; r d r (7) 0 advantageousl. Man authors, together with Hecht 3 and Born and Wolf 4, treat circularl smmetric coherent impulse response functions and related topics. Let J r ; 1 be the Bessel function of the first kind of order one. The recursion relation 1 d r J1 r ; r Jo r ; r d (8) is introduced and discussed b man authors, including Boas 5, Hecht 6, and Born and Wolf 7. Equation (8) can be rearranged and written as d r J r ; r J r ; r d (9) 1 o readil. Substitution of equation (9) into equation (7) ields 1 c hr; d r J r ; r immediatel. Upon evaluation, equation (30) ields 0 1 (30) 1 hr; r J r; r c 1 c (31) 3 Eugene Hecht, Optics, (4 th Ed.) pp (Addison Wesle, San Francisco, 00). 4 Ma Born and Emil Wolf, Principles of Optics (Sith Corrected Edition) pp (Pergamon Press, Oford, 1980). 5 Mar L. Boas, Mathematical Methods in the Phsical Sciences, p (John Wile & Sons, Inc., New York, 1966). 6 Eugene Hecht, Optics, (4 th Ed) p. 468 (Addison Wesle, San Francisco, 00). 7 Ma Born and Emil Wolf, Principles of Optics (Sith Corrected Edition), p. 396 (Pergamon Press, Oford, 1980).

59 Figure 1. Standardied Air formula. in a straightforward manner. Equation (31) can be simplified to obtain J1 c r; hr ; c c r (3) readil. Equation (3) describes the coherent impulse response for a circularl smmetric imaging sstem. The relative probabilit densit associated with the impulse response provided b equation (3) is given b ; * ; ; Substitution of equation (3) into equation (33) ields I r h r h r (33) ; c I r r J1 c ; c r (34) triviall. This result is known as the Air formula. AIRY DISK A graph of the standardied Air formula, i.e.

60 59 Figure 13. Air pattern. Source: Wikipedia, Air Disk. c I r; J1 c r; c r (35) is provided in Figure 1. As illustrated in Figure 1, the optical real image linked to the standardied Air formula is an Air pattern. As shown in Figures 1 and 13, an Air pattern consists of a central bright region that is surrounded b a number of much fainter rings. The central bright region is known as the Air disk associated with the imaging sstem. An Air disk is bounded b a dark ring that eists at a location that corresponds to the first ero of the Bessel function J1 r ; c 8. As indicated in Figure 1, this occurs when the value of the independent variable in equation (35) has the value 1.0, i.e. when 1.0 (36) r c 0 where the radius of the Air disk r0 (37) c 8 Ma Born and Emil Wolf, Principles of Optics (Sith Corrected Edition), p. 397 (Pergamon Press, Oford, 1980).