Nonlocal Optical Real Image Formation Theory

Transcription

1 Nnlcal Optical Real Image Frmatin Ther Gresn Gilsn Mulith Inc. 30 Chestnut Street # 3 Nashua, New Hampshire , USA gresn.gilsn@mulithinc.cm ABSTRACT A nnlcal ther f ptical real image frmatin is develped frm the basic phsics assciated with an ptical real image frmatin apparatus. The ther shws hw tw separated illuminated bject pints are required fr real image frmatin. When the distance between the pints eceeds twice the wavelength f the light used tw equi-amplitude plane waves prpagate independentl awa frm the bject plane tward the imaging sstem. Quantum amplitude cmpnents with spatial frequencies that are within the passband f the imaging sstem pass thrugh the imaging sstem. Each plane wave pair that prpagates thrugh the imaging sstem and ultimatel arrives at the image plane cntributes t real image frmatin. Object and image field depths are treated and fund t be ver different than the are rdinaril thught t be. Reslutin limits related t real image frmatin are investigated. Reference distributin real image frmatin (RIF) is intrduced. Real image frmatin with n fundamental reslutin limit is predicted t be achievable b means f RIF.

2 PART I: FUNDAMENTALS INTRODUCTION A new ther f ptical real image frmatin is develped frm fundamental phsics in this paper. The ther shws hw a previusl unknwn cperative phenmenn arises and hw the phenmenn underlies the frmatin f ptical real images. In accrd with the new ther, ptical real images with a ver large field depth (bject and image) and n fundamental reslutin limit can be frmed. During the prcess f real image frmatin, light wave cmpnents prpagate awa frm an bject, thrugh an imaging sstem and ultimatel frm an image. Image frming light necessaril begins its jurne b prpagating awa frm the bject plane. A generalied ptical real image frmatin apparatus is illustrated in Figure 1. The image frmatin apparatus is an rdinar diffractin-limited linear space-invariant imaging sstem. Initiall, light that is incident upn the bject plane is transmitted thrugh r reflected frm phsical features in the bject plane. Such light is distributed in a definite cnfiguratin n the side f the bject plane nearest t the imaging sstem. A prtin f this light prpagates frm the bject plane t the imaging sstem. Sme f the light that arrives at OBJECT PLANE OPTICAL AXIS IMAGING SYSTEM Figure 1. Generalied ptical real image frmatin apparatus. IMAGE PLANE the imaging sstem prpagates thrugh it. Subsequentl, the light that prpagates thrugh the imaging sstem prpagates frm the imaging sstem t the image plane. The resulting cnfiguratin f light n the image plane clsel apprimates a magnified (enlarged, unchanged r reduced and perhaps inverted) versin f the cnfiguratin f light n the bject plane. As shwn later, tw separated illuminated bject pints are required t frm each light wave cmpnent that prpagates awa frm the bject plane. This finding is in distinct cntrast t the rdinaril accepted assumptin that nl ne bject pint is invlved.

3 3 LIGHT N ne knws what light is. Nevertheless, light has certain prperties that can be treated mathematicall. In particular, light assciated with ptical image frmatin displas wave prperties and particle prperties. The particles assciated with light are knwn as phtns. Images are frmed b the apparentl randm arrival f ne phtn at a time 1,. An image that is frmed with sufficientl feeble light shws n evidence f having wave prperties. Rather, images are built up b individual phtns that act independentl In additin t its particle prperties, light has wave prperties. An image that is frmed with sufficientl strng light shws n evidence f being built up b individual phtns. Rather, the entire image appears t be frmed as a single event. The wave and particle prperties f light ceist. The wave prperties define the prbabilit densit that describes the prbabilit f arrival f a phtn at a particular lcatin. Each phtn is assciated with the entire prbabilit distributin. This ceistence is knwn as wave-particle dualit; quantum phsics is needed t cpe with it. The wave prperties f light are cmmn t phsical waves in a lssless medium. The defining epressin fr phsical waves in a lssless medium is the differential wave equatin 3 1 v t (1) In this equatin is the wave functin, v is the speed f prpagatin f the wave, t is time, and is the Laplacian peratr. The wave functin fr ptics is knwn as quantum amplitude. Optics is ften understd as a branch f electrmagnetism and its fundamental laws are derived frm Mawell's equatins 4. An electrmagnetic field is characteried b an electric field E and a magnetic inductin B. These fields are linked tgether b means f Mawell's equatins. In free space and in the absence f surces, each Cartesian cmpnent f E and B satisfies equatin (1) independentl 5. The waves invlved are knwn as electrmagnetic waves. Mawell used his ther t deduce the speed f prpagatin f electrmagnetic waves 6. This speed was fund t cincide with the eperimentall determined speed f light. The etremel 1 Eugene Hecht, Optics (4 th Ed.) p. 8, pp , and pp (Addisn Wesle, San Francisc, 00). A. P. French and Edwin F. Talr, An Intrductin t Quantum Phsics, pp (W. W. Nrtn & Cmpan Inc., New Yrk 1978). 3 Eugene Hecht, Optics (4 th Ed.) pp and pp. 7-8 (Addisn Wesle, San Francisc, 00). 4 David Hallida and Rbert Resnick, Phsics, p. 986 (Jhn Wile & Sns, Inc., New Yrk, 1960). 5 Jhn David Jacksn, Classical Electrdnamics, pp (Jhn Wile & Sns, Inc., New Yrk, 1967). 6 David Hallida and Rbert Resnick, Phsics, p. 986 (Jhn Wile & Sns, Inc., New Yrk, 1960).

4 4 successful electrmagnetic ther f light was brn. The electrmagnetic wave functin is knwn as cmple amplitude. Wave-particle dualit cannt be understd within the framewrk f the electrmagnetic ther f light. The electrmagnetic ther is unable t treat phtns and the use f electrmagnetic cncepts can be inapprpriate and misleading. Quantum phsics is needed. Quantum amplitude (r prbabilit amplitude) is the term used t dente the wave functin in quantum phsics. A quantum amplitude, when multiplied b its cmple cnjugate, defines a relative prbabilit densit. The relative prbabilit densit describes the prbabilit f arrival f a phtn at a particular lcatin. The cncepts f quantum amplitude and prbabilit densit are used in quantum phsics t accmmdate the eistence f wave-particle dualit. Equatin (1), with the wave functin identified as quantum amplitude, is knwn as the ptical wave equatin. Optical real image frmatin can be understd in terms f the ptical wave equatin and its slutins. OVERVIEW OF OPTICAL REAL IMAGES As a result f interacting with phsical features in the bject plane, a definite cnfiguratin f light eists n the side f the bject plane nearest t the imaging sstem. This cnfiguratin f light cnsists f phtns that are transmitted thrugh r reflected frm the bject plane at the instant the leave the bject plane. Phtns cannt be bserved directl; what is knwn f them cmes frm bserving the results f their being either created r destred 7. A cnfiguratin f light can eist whether r nt it is bserved. A real image ma r ma nt be bserved. A right-handed rectangular crdinate sstem can be cnvenientl intrduced t facilitate treating real image frmatin mathematicall. In this crdinate sstem, let the bject plane be the -plane and let the image plane be the i i -plane. The subscripts and i designate the bject side and the image side, respectivel, f the imaging sstem. Let the -ais cincide with the ptical ais defined b the imaging sstem. Let the psitive - directin be the directin frm the bject plane tward the imaging sstem. Finall, let the -ais intersect the -plane at the rigin f crdinates. Let u, ; be the quantum amplitude f the phtns linked with light f wavelength n the bject plane. The relative prbabilit densit, given b, ; *, ;, ; p u u () 0 7 Eugene Hecht, Optics, (4 th Ed.) p. 5 (Addisn Wesle, San Francisc, 00).

5 5 where the superscript asterisk dentes cmple cnjugatin, is linked t u, ;. The relative prbabilit densit specifies the instantaneus presence f a phtn f wavelength within an infinitesimal area surrunding the pint, n the bject plane. Each quantum amplitude cmpnent can be characteried as being mnchrmatic and cherent. Sme f the phtns in the cnfiguratin f light n the bject plane prpagate awa frm the bject plane and pass thrugh the imaging sstem illustrated in Figure 1. In turn, sme f these phtns ultimatel frm a real image n the image plane. Let, ; f wavelength. Furthermre, let u, ; be the cherent impulse respnse f the real image frmatin apparatus fr light i be the quantum amplitude f the phtns linked with light f wavelength n the image plane. This quantum amplitude is given b 1 ui, ; u, ;, ; m m m m m (3) where dentes the cnvlutin peratr and m dentes the lateral magnificatin f the image relative t the bject; m ma be either negative r psitive, depending upn whether the real image is inverted r nt inverted relative t the bject. The relative prbabilit densit is linked t u, ;, ; *, ;, ; p u u (4) i i i i. This relative prbabilit densit specifies the instantaneus presence f a phtn f wavelength within an infinitesimal area surrunding the pint, n the image plane. A ver large number f phtns cntribute when acceptable real image frmatin ccurs. The inherent granularit assciated with the bject and the image cnfiguratins f light vanishes. Cnsequentl the densit f phtns that cntribute t real image frmatin is prprtinal t the prbabilit densit. Man authrs prvide treatments cncerning the derivatin f equatin (3); see, fr eample, Cllier, Burkhardt and Lin 8, Gaskill 9, Cathe 10 r Gdman 11. In additin, equatin (3) is derived in this paper where it is presented as equatin (13). Equatin (3) is the cherent real image equatin fr light f wavelength. 8 Rbert J. Cllier, Christph B. Burkhardt and Lawrence H. Lin, Optical Hlgraph, p (Academic Press, Inc., Orland, 1971). 9 Jack D. Gaskill, Linear Sstems, Furier Transfrms, and Optics, pp (Jhn Wile & Sns, Inc., New Yrk, 1978). 10 W. Thmas Cathe, Optical Infrmatin Prcessing and Hlgraph, pp (Jhn Wile & Sns, Inc., New Yrk, 1974). 11 Jseph W. Gdman, Intrductin t Furier Optics, pp (McGraw-Hill, Inc., New Yrk, 1968).

6 6 Derivatins f equatin (3) are rdinaril based upn the Hugens-Fresnel principle. Fllwing the Hugens-Fresnel principle, it has been widel assumed that nl ne illuminated pint in an bject is needed t supprt image-frming light. Hwever, as shwn in this paper, real image frmatin is dependent upn a previusl unknwn cperative phenmenn. This cperative phenmenn is such that tw separate illuminated bject pints are required t supprt imagefrming light. The Hugens-Fresnel principle is nt a law f phsics and is furthermre knwn t be prblematic 1. In additin, Ad hc apprimatins (such as initial apprimatins, the Fresnel apprimatins, the Fraunhfer apprimatin, etc.) are invked in derivatins f equatin (3) that are based upn the Hugens-Fresnel principle. Nne f these derivatins are based n fundamental phsics. A new ther f ptical real image frmatin is develped in this paper. The Hugens-Fresnel principle and its attendant apprimatins are avided in this develpment. Rather, the ptical wave equatin is slved subject t bundar cnditins. Salient bundar cnditins are impsed b the bject and b the imaging sstem. OPTICAL WAVE EQUATION Light that prpagates awa frm the bject plane and subsequentl frms a real image is assciated with a linear superpsitin f quantum amplitude cmpnents. Each quantum amplitude cmpnent prpagates in a directin such that its -cmpnent is nn-negative. Let the quantum amplitude cmpnent assciated with the wavelength be represented b the cmple valued scalar functin,,, t, where,, and are the crdinates f a pint in,,, t satisfies the ptical wave equatin space and t represents time. The functin 1 v t,,, t,,, t (5) in the regin between the bject plane and the image plane and utside the imaging sstem. In accrd with the methd f separatin f variables, slutins f equatin (5) in the frm,,, t,, t (6) are sught. After substituting equatin (6) int equatin (5) and then dividing bth sides f the,, t result b 1 Eugene Hecht, Optics, (4 th Ed.) p. 445, p. 456, and p. 489 (Addisn Wesle, San Francisc, 00).

7 7 1 1,,,, v t t t (7) fllws. The three variables n the left hand side f equatin (7) are all independent f the single variable n the right hand side f equatin (7). This can be true nl when each side f the equatin is equal t the same cnstant. Cnsequentl, 1,, k,, (8) and 1 v t t t k (9) where the separatin cnstant has been designated, with mdest fresight, as and (9) can be written as k. Equatins (8) k,, 0 (10) which is knwn as the Helmhlt equatin, and t t t k v 0 (11) respectivel. The spatial dependence f,,, t dependence f,,, t satisfies equatin (11). TIME DEPENDENCE satisfies equatin (10) while the time Equatin (11) is a partial differential equatin that invlves nl ne variable and can cnsequentl be written as the rdinar differential equatin d dt t Equatin (1) can be slved t ield the slutin t k v 0 (1)

8 8 t C ep ikvt C ep ikvt (13) 1 where C 1 and C are arbitrar cnstants. Equatin (13) can be verified b direct substitutin int equatin (1). Oscillatins f tempral frequenc are described b equatin (13). kv (14) As bserved at a particular pint in space each quantum amplitude cmpnent scillates at a frequenc that is independent f the medium invlved. In additin, the quantum amplitude cmpnent is assciated with a wavelength (intrduced previusl) that is dependent upn the medium invlved. The wavelength and frequenc are related b the equatin v (15) where v is the speed f light in the medium invlved. Equatin (15) is a fundamental equatin f wave mtin. The wave number k (16) can be btained after equatin (15) has been substituted int equatin (14). Furthermre, t C ep i t C ep i t (17) 1 results when equatin (14) is substituted int equatin (13). The cnstants C 1 and C will be treated later. PROPAGATION DIRECTION The three-dimensinal Furier transfrm f,,,,,, ep i d d d (18) can nw be cnvenientl intrduced. Here, the reciprcal variables, and, cmmnl knwn as spatial frequencies, are needed t define the three-dimensinal Furier transfrm f,,. The crrespnding three-dimensinal inverse Furier transfrm

9 9,,,, ep i d d d (19) and,, as linear cmbinatins f three-dimensinal cmple epnential functins. can als be cnvenientl intrduced. The Furier transfrm pair epresses,, Equatin (19) describes the spatial dependence f a linear superpsitin f plane waves that prpagate in the directin assciated with directin csines cs cs cs (0) where, and are the angles between the directins f wave prpagatin and the -, - and -aes, respectivel. The equatin cs cs cs 1 (1) cnstitutes a well-knwn fundamental prpert f directin csines. Upn substitutin f equatin (0) int equatin (1), 1 () fllws readil. As indicated earlier, the -cmpnent f the directin f wave prpagatin is nn-negative. Cnsequentl, and, since 0 (the wavelength f light is a psitive entit), cs 0 (3) 0 (4) fllws when inequalit (3) is substituted int the third rw f equatin (0). Substitutin f equatins (17) and (19) int equatin (6) ields,,, t C ep i t C ep i t 1,, ep X i d d d (5)

10 10 The first term in equatin (5) describes waves that prpagate in the negative -directin. This pssibilit is precluded because the -cmpnent f the directin f wave prpagatin is necessaril nn-negative. Cnsequentl is required. As a result equatins (17) and (5) reduce t and C1 0 (6) t C i t ep (7),,, t,, ep C i t d d d (8) respectivel. When wave prpagatin ccurs in a directin such that its -cmpnent is nn-er is required. SPATIAL DEPENDENCE C 0 (9) Tw-dimensinal Furier transfrm pairs defined n arbitrar planes parallel t the bject plane can be used t begin treating the spatial dependence f,,, t. The methd used here clsel parallels a methd Gdman 13 has used t treat the angular spectrum f plane waves. Accrdingl, let, ;,, ep i d d (30) be the tw-dimensinal Furier transfrm f,, value f ; n the plane defined b an arbitrar,,, ; ep i d d (31) is the crrespnding tw-dimensinal inverse Furier transfrm. Substitutin f equatin (31) int equatin (10) ields 13 Jseph W. Gdman, Intrductin t Furier Optics, pp (McGraw-Hill, Inc., New Yrk, 1968).

11 11 k, ; ep i d d 0 (3) which can be written as k, ; ep i d d 0 (33) because the integratin variables differ frm the differentiatin variables. Equatin (33) reduces t, ; 4 4, ; k X ep i d d 0 (34) r, equivalentl,, ; 4, ; ep i d d 0 (35) after equatins (16) and () are invked. Each cmpnent functin in the integrand f equatin (35) satisfies the equatin independentl and the partial differential equatin, ; 4, ; 0 (36) necessaril fllws. Equatin (36) invlves nl ne independent variable and can therefre be written as d, ; 4, ; 0 d (37) which is an rdinar differential equatin. Equatin (37) can be slved t ield the slutin, ; C3ep i C4ep i (38) where C 3 and C 4 are arbitrar cnstants. Equatin (38) can be verified b direct substitutin int equatin (37).

12 1 Substitutin f equatin (38) int equatin (31) ields,, 3 ep 4 ep C i d d C i d d (39) Furthermre,,, 3 ep 4 ep t C C i t d d C C i t d d (40) fllws after equatins (7) and (39) are substituted int equatin (6). The secnd term in equatin (40) describes waves that prpagate in the negative -directin. This pssibilit is precluded because the -cmpnent f the directin f wave prpagatin is necessaril nn-negative. Cnsequentl is required. This reduces t CC 4 0 (41) C4 0 (4) after substituting equatin (9) int equatin (41). As a result equatins (38), (39) and (40) reduce t and respectivel. The value f 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)

13 13 can be btained b evaluating equatin (43) n an arbitraril chsen reference plane, designated as the p -plane. Furthermre,, ;, ; ep i p p (47) results when equatin (46) is substituted int equatin (43). Equatin (47) prvides the value f, ; n an arbitrar plane in terms f its value n a parallel reference plane p. Substitutin f equatin (47) int equatin (31) leads t,,, ; ep p i d d X ep i p quite simpl. Subsequent substitutin f equatins (7) and (48) int equatin (6) leads t,,,, ; ep ep t C p i d d X i p t (48) (49) easil. The value f C, given b ep C t i t (50) 0 0 can be btained b evaluating equatin (7) at the arbitraril chsen time t 0. Furthermre, t t ep i t t 0 0 (51) results when equatin (50) is substituted int equatin (7). Equatin (51) prvides the value f t at an arbitrar time t in terms f its value at anther time t 0. Substitutin f equatin (50) int equatin (49) leads t,,, t t0, ; p ep i d d X ep i p t t0 (5)

14 14 quite easil. Equatin (5) describes the quantum amplitude,,, t as a linear cmbinatin f plane wave cmpnents that prpagate frm the bject plane tward the imaging sstem. Z PLANE Z l- PLANE IMAGING SYSTEM Z l+ PLANE Z i WAVE COMPONENT PROPAGATION Refer t the ptical real image frmatin apparatus illustrated in Figure 1. Prpagatin f wave cmpnents in the regin between the bject plane and the image plane and utside the imaging sstem can be treated in terms f the planes, linked t real image frmatin, illustrated in Figure. In Figure, the bject and image planes are designated as the - and i sstem are designated as the l - and l - planes, respectivel. Furthermre, the entrance and eit planes f the imaging - planes, respectivel. Spatial frequencies assciated with the wave cmpnents when the enter the imaging sstem ma differ frm thse assciated with the same wave cmpnents when the eit the imaging sstem. Accrdingl, let unprimed spatial frequencies be used in the regin where l and let primed spatial frequencies be used in the regin where l i. Treatment f the regin where l can be separated frm treatment f the regin where l. i On the bject side f the imaging sstem (in the regin where l ), the wavelength f light used fr image frmatin has been designated as. The pssibilit eists that this wavelength will be different n the image side f the imaging sstem due t a pssible change in refractive inde. T accmmdate this pssibilit, the wavelength f light used will be designated as ' n the image side f the imaging sstem (in the regin where l ). i PROPAGATION THROUGH THE IMAGE FORMATION APPARATUS The value f, ; OPTICAL AXIS Figure. Image frmatin apparatus planes. n an arbitrar plane in terms f its value n a parallel reference plane p is given b equatin (47), i.e.,, ;, ; ep i p p (53) in the pen interval l (inside the regin where l ). Similarl, the value f ' ', ' ; n an arbitrar plane in terms f its value n a parallel plane p' is given b i ' ', ' ; ' ', ' ; p' ep p' ' (54)

15 15 in the pen interval l (inside the regin where i l ). The reference planes i p and are determined independentl f each ther. p' The pen intervals l and l i include the limiting values that ccur n the varius planes identified in Figure. These limiting values are given b lim, ;, ; lim, ;, ; l l lim ' ', ' ; ' ', ' ; l l lim ' ', ' ; ' ', ' ; i i (55) In all cases, the limiting prcess ccurs inside the regin between the bject plane and the image plane and utside the imaging sstem. The relatinship fllws when, ;, ; ep i l l (56) l p (57) is substituted int equatin (53). Similarl, the relatinship fllws when ' i ' ', ' ; i ', ' ; l ep i l ' (58) i p' l (59) is substituted int equatin (54). After intrducing d l d i i l (60) equatins (56) and (58) can be written as

16 16, ; l, ; ep i d (61) and ' ', ' ; i ' ', ' ; l ep i di ' (6) respectivel. NEW NOTATION New ntatin such that ' ' u, ;, ; ul, ;, ; l ul ', ' ; ' ', ' ; l ui ', ' ; ' ', ' ; i (63) can nw be intrduced t accmmdate treating the quantum amplitudes n each f the planes identified in Figure. Equatins (61) and (6) can be written as and, ;, ; ep ul u i d (64) ', ' ; ' ', ' ; ' ep ' u u i d (65) i l i respectivel, b using the new ntatin. Tw-dimensinal Furier transfrm relatinships, treated belw, can be used t btain crrespnding relatinships in rdinar space. FOURIER TRANSFORM PAIRS Furier transfrm pairs are assciated with the quantum amplitudes n each f the planes, linked t real image frmatin, illustrated in Figure. Thus, the tw-dimensinal Furier transfrm f u, ; is given b, ;, ; ep (66) u u i d d while, ;, ; ep (67) u u i d d

17 17 u. The twdimensinal Furier transfrm pair given b equatins (66) and (67) is applicable n the bject plane. is the crrespnding tw-dimensinal inverse Furier transfrm f, ; The tw-dimensinal Furier transfrm f u, ; while l is, ;, ; ep (68) ul ul i d d, ;, ; ep (69) u u i d d l l is the crrespnding tw-dimensinal inverse Furier transfrm f ul, ;. The twdimensinal Furier transfrm pair given b equatins (68) and (69) is applicable n the imaging sstem's entrance plane. The tw-dimensinal Furier transfrm f u, ; ' while l is ', ' ; ', ; ' ep ' ' (70) ul ul i d d, ; ' ', ' ; ' ep ' ' ' ' (71) u u i d d l l is the crrespnding tw-dimensinal inverse Furier transfrm f ul ', ' ; '. The twdimensinal Furier transfrm pair given b equatins (70) and (71) is applicable n the imaging sstem's eit plane. The Furier transfrm f, ; ' while u is i ', ' ; ', ; ' ep ' ' (7) ui ui i d d, ; ' ', ' ; ' ep ' ' ' ' (73) u u i d d i i u. The twdimensinal Furier transfrm pair given b equatins (7) and (73) is applicable n the image plane. is the crrespnding tw-dimensinal inverse Furier transfrm f i ', ' ; '

18 18 OBJECT PLANE QUANTUM AMPLITUDE COMPONENTS A definite cnfiguratin f light eists n the side f the bject plane nearest t the imaging sstem. The quantum amplitude assciated with light f wavelength n the bject plane has been designated as u, ;. Ever pint in u, ; is specified relative t the rigin f crdinates. The rigin f crdinates is determined b the intersectin f the -ais with the -plane. The lcatin f the -ais cincides with the ptical ais. This lcatin is arbitrar relative t the lcatin f u, ; and is unrelated t the bject being imaged. The tw-dimensinal Furier transfrm f u, ; is defined in terms f the arbitrar lcatin f the rigin f crdinates. Undesirable cnsequences f such arbitrariness can be dealt with b intrducing the duble delta u, ;. functin decmpsitin f Equivalent tw-dimensinal Dirac delta functin decmpsitins f u, ; and, ;, ;, are given b (74) u u d d, ;, ;, (75) u u d d where, and, Additin f equatins (74) and (75) leads t are arbitrar pints within the quantum amplitude, ; 1 u u, ;, ;,,, ;,, u d d d d u. (76) readil. This result epresses the quantum amplitude u, ; as a linear cmbinatin f twdimensinal Dirac delta functin pairs. After minr manipulatin that includes changing the rder f integratin and recalling equatin (66), the tw-dimensinal Furier transfrm f equatin (76), i.e., 1 u, ;, ;,, u (77) u, ;,, ep i d d d d d d can be fund. Equatin (77) reduces t

19 19 1 u u i, ;, ;, ep, ;, ep u i d d d d (78) fllwing integratin ver and. In turn, equatin (78) can be written as u, ; 1 u, ;, ep i u, ;, ep i X ep i d d d d (79) after minr manipulatin. Each cmpnent in the integrand f equatin (79) describes a twdimensinal cmple epnential peridic functin multiplied b a tw-dimensinal cmple epnential phase factr. INTERPRETATION The tw-dimensinal cmple epnential phase factr in equatin (79) eists because the intersectin f the -ais (the ptical ais) and the -plane is arbitrar. The distance between the -ais and the midpint between the pints, and, is given b M (80) where (81) and (8) are the - and - crdinates, respectivel, f the midpint. The distance M is nt a phsical prpert f the distributin f light.

20 0 Each tw-dimensinal cmple epnential peridic functin in the integrand f equatin (79) is peridic in tw dimensins. The - and - cmpnents f this quantum amplitude spatial perid are given b T (83) and T (84) respectivel. The tw-dimensinal cmple epnential peridic functins each represent a phsical prpert f the distributin f light. The quantum amplitude spatial perid is given b T T T (85) which can be written as T (86) b invking equatins (83) and (84). The distance that separates, and, is given b (87) where and (88) (89) are the assciated distance cmpnents in the - and - directins, respectivel. After substituting equatins (88) and (89) int equatin (87) (90)

21 1 fllws readil. In turn, T (91) can be btained b substituting equatin (90) int equatin (86). Thus, the quantum amplitude spatial perid assciated with tw pints in a distributin f light is equivalent t ne-half the distance between the tw pints. The quantum amplitude spatial frequenc F 1 (9) T can nw be defined. After substituting equatin (91) int equatin (9) F (93) can be btained. Thus, the quantum amplitude spatial frequenc assciated with tw pints in a distributin f light is equivalent t twice the reciprcal f the distance between them. ILLUMINATION Cnsider the quantum amplitude assciated with the tw separated pints, and, in a cnfiguratin f light n the side f the bject plane nearest t the imaging sstem. This quantum amplitude is given b, ;,, u A B (94) where A and B are cnstants, fr bth pints taken tgether. The individual quantum amplitudes are given b and fr the tw separated pints., ;, u A (95), ;, u B (96) The tw-dimensinal Furier transfrm f equatin (94) is given b

22 , ; ep u i d d,, ep A B i d d (97) which ields, ; ep ep u A i B i (98) upn integratin. Equatin (98) can be written as u, ; ep i X A ep i B ep i (99) fr eas cmparisn with equatin (79). Equatin (99) reduces t, ; ep X Aep i T T Bep i T T u i (100) after,, T and T, given b equatins (81), (8), (83) and (84), respectivel, are recalled. Equatin (100) can be written as, ; ep u i cs sin X A B T T i A B T T (101) b invking Euler's frmula and using trignmetric ntatin. The peridicit f, ; nner. Cnsequentl u is dependent upn u, ; and, ; u bth being A 0 B 0 (10) Is required when u, ; is peridic.

23 3 The special case where is f mdest interest. Equatin (101) reduces t A B (103), ; ep cs u A i T T (104) fr this special case. Thrughging investigatin f light that is suitable fr real image frmatin illuminatin is bend the scpe f this paper. Nevertheless, light that is either cherent r partiall cherent where bth illuminated pints eist in the same area f cherence can be used satisfactril. INDIVIDUAL PHOTONS Tw illuminated bject plane pints are required, as a minimum, t frm an ptical real image. This is true either when the image is frmed b a large number f phtns in a single event r when the image is built up b individual phtns. Evidentl each individual phtn is assciated with illuminatin f bth f the requisite bject plane pints. PLANE WAVE PROPAGATION Equatin (5) reduces t,,, 0, ; ep ep t t i d d X i t t0 (105) in the regin where l. Equatin (105) describes a linear cmbinatin f plane waves that prpagate awa frm the bject plane tward the imaging sstem. The spatial frequencies assciated with each individual plane wave cmpnent in the integrand f equatin (105) are given b cs cs cs (106)

24 4 a result that fllws frm equatin (0) triviall. Each spatial frequenc is assciated with a spatial perid given b 1 T 1 T T 1 (107) b definitin. Cmbinatin f equatins (106) and (107) leads t T cs cs T cs T (108) The prpagatin directin and a wavefrnt that is perpendicular t the prpagatin directin are illustrated in Figure 3 fr an individual plane wave cmpnent. The wavefrnt is an infinite plane that is perpendicular t the plane f the figure and that etends ut f the plane f the figure; nl the trace f the wavefrnt n the plane f the figure is shwn in the figure. OBJECT PLANE (XY-PLANE) WAVEFRONT T / PROPAGATION DIRECTION OPTICAL AXIS (Z-AXIS) Figure 3. Plane wave prpagatin.

25 Prpagatin Angle (Degrees) 5 In Figure 3, as elsewhere in this paper, the -ais cincides with the ptical ais defined b the imaging sstem. The bject plane is rtated abut the -ais s that the line defined b the pints, and, is parallel t the plane f the figure. This line is assciated with the spatial perid T. The spatial perid T is shwn as the vertical ais in the figure and is als shwn as the hptenuse f a right triangle. The wavelength f light used,, is depicted in the figure as the distance between the rigin and the wavefrnt intrduced previusl. In additin, is shwn as the angle between the directin f wave prpagatin and the ptical ais;, T and have been intrduced previusl. MINIMUM SEPARATION REQUIREMENT The relatinship sin (109) T can be established b inspecting Figure 3 and appling the definitin f an angle s sine. Cnsequentl, 1 sin T (110) is the prpagatin angle assciated with the spatial perid T. Equatin (110) can be written as Separatin Distance (Light Wavelengths) Figure 4. Prpagatin angle as a functin f the separatin distance between tw pints in the bject.

26 6 1 sin (111) where equatin (91) has been invked. The prpagatin angle (in degrees) is pltted in Figure 4 as a functin f the separatin distance (in wavelengths f the light used) between the pints, and,. Upn cnsidering equatin (111) r eamining the curve in Figure 4, it can be determined readil that the minimum separatin distance required fr light t prpagate awa frm the bject plane eceeds tw wavelengths f the light used. Thus, satisfactin f the inequalit (11) is required fr light t prpagate awa frm the bject plane. As a cnsequence, prpagatin f light awa frm a single pint (as hpthesied b the Hugens-Fresnel principle) des nt ccur. Tw illuminated pints in the bject are required t frm light waves that cntribute t real image frmatin. PLANE WAVE PAIRS The equatin sin cs cs (113) which is equivalent t sin cs cs (114) fllws frm equatin (1) readil. Thus, as illustrated in Figure 5, tw equi-amplitude plane wave cmpnents are assciated with each pair f illuminated pints in the bject distributin. Fr each plane wave cmpnent that prpagates at an angle + relative t the ptical ais, an equi-amplitude plane wave cmpnent prpagates at an angle - relative t the ptical ais. PARAMETER CHANGES Sme prperties f the light that emerges frm an imaging sstem differ frm thse that enter the imaging sstem. As intrduced previusl, let unprimed parameters be used in the regin between the bject plane and the imaging sstem. Als, let primed parameters be used in the regin between the imaging sstem and the image plane. Let n be the inde f refractin in the unprimed regin and let n ' be the inde f refractin in the primed regin. Furthermre, let S be the wavelength f the light used as bserved in vacuum. Then

27 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 f the light used in the unprimed regin and wavelength f light in the primed regin. Substitutin f equatin (115) int equatin (109) leads t ' is the crrespnding

28 8 S sin nt sin ' S nt ' ' (116) readil. In turn 1 S sin nt ' 1 S sin nt ' ' (117) can be btained frm equatin (116) quite easil. The angle between the directin f wave prpagatin and the -ais is changed frm t ' as a result f the light passing thrugh the imaging sstem. The spatial perid at the image plane f a plane wave that cntributes t real image frmatin is given b T ' mt (118) where the lateral magnificatin m has been recalled. Substitutin f equatin (118) int the bttm rw f equatin (116) ields which can be cmbined with the tp rw f equatin (116) t btain a result knwn as the ptical invariant. The lateral magnificatin can be btained frm equatin (10) b simpl rearranging terms. IMAGE FORMATION Let S sin ' (119) n' mt mn'sin ' nsin (10) n sin m (11) n ' sin '

29 9, ;, ; ep i d d (1) be the cherent transfer functin f the imaging sstem;, ; cherent impulse respnse. The equatin ', ' ; l, ;, ; l is the crrespnding (13) relates the tw-dimensinal Furier transfrm f the quantum amplitude n the eit plane f the imaging sstem t the tw-dimensinal Furier transfrm f the quantum amplitude n the entrance plane f the imaging sstem. Simultaneus satisfactin f equatins (61), (6) and (13) ields readil. ', ' ; i, ;, ; ep i d di ' (14) New ntatin, intrduced in equatin (63), such that, ; u, ; ', ' ; i ui ', ' ; (15) can nw be used. Using this ntatin, equatin (14) can be written as ', ' ;, ;, ; ep ' ui u i d di (16) cnvenientl. Advanced imaging sstems are designed such that (ideall) ccurs. Thus, the change f variables d d' 0 (17) i m ' ' m ' d d i (18)

30 30 results as a cnsequence f an ptical disturbance s interactin with an ideal imaging sstem. Equatin (16) reduces t ui, ; u, ;, ; m m (19) after the change f variables has been invked. Man practical imaging sstems have been designed t achieve ver clse apprimatins t the change f variables given b equatin (18). Treatment f these imaging sstems is bend the scpe f this paper. The tw-dimensinal inverse Furier transfrm f equatin (19) is given b ui, ; ep i d d m m, ;, ; ep u i d d (130) which ields, ;, ;, ; m ui m m u upn evaluatin. Equatin (131) can be written as (131) 1 ui, ; u, ;, ; m m m m m (13) which relates the quantum amplitude f the ptical disturbance n the image plane t the quantum amplitude f the ptical disturbance n the bject plane. This result, the cherent real image equatin, is identical with equatin (3). Equatin (13) describes an ptical real image as an ideal (undistrted) ptical real image cnvlved with the impulse respnse linked t an imaging sstem. Ecept fr special cases, the cnvlutin peratin leads t image distrtin 14. Thus, features in the image are (rdinaril) apprimatel as wide as the sum f the widths f the tw functins being cnvlved; fine features becme indistinguishable (washed ut). The distrtin ccurs as a cnsequence f the imaging sstem's impulse respnse and related transfer functin. 14 Jack D. Gaskill, Linear Sstems, Furier Transfrms, and Optics, pp (Jhn Wile & Sns, Inc., New Yrk, 1978).

31 31 PART II. OPTICAL IMAGE RESOLUTION INTRODUCTION The cherent impulse respnse assciated with an arbitrar imaging sstem is given b, ;, ; ep i d d (133) fr light f wavelength. Equatin (133) is the tw-dimensinal inverse Furier transfrm f the imaging sstem s cherent transfer functin. The reciprcal variables and, cmmnl knwn as spatial frequencies, are needed t define the tw-dimensinal inverse Furier transfrm. The imaging sstem s cherent transfer functin is given b, ;, ; ep i d d (134) fr light f wavelength.. Equatin (134) is the tw-dimensinal Furier transfrm f the imaging sstem s cherent impulse respnse. Equatins (133) and (134) cnstitute a twdimensinal Furier transfrm pair. Cnsider a diffractin-limited imaging sstem that has a clear aperture f arbitrar shape and sie. Fr light f wavelength this imaging sstem is assciated with a passband and a cherent transfer functin. The cherent transfer functin is given b 1 inside the passband, ; (135) 0 utside the passband Quantum amplitude cmpnents with spatial frequencies that are inside the passband f the imaging sstem pass thrugh the imaging sstem and cntribute t image frmatin. Quantum amplitude cmpnents with spatial frequencies that are utside the passband f the imaging sstem d nt pass thrugh the imaging sstem and d nt cntribute t image frmatin. Image distrtin due t missing quantum amplitude cmpnents ccurs. Mst imaging sstems are circular. Althugh nn-circular imaging sstems eist, little wuld be gained b cnsidering them in the present cntet. Hencefrth, unless therwise indicated, attentin will be restricted t circular imaging sstems. Man authrs prvide treatments cncerning cherent impulse respnses and cherent transfer functins; see, fr eample, Gaskill 15 r Gdman 16. A general treatment f cherent impulse respnses and cherent transfer functins is bend the scpe f this paper. 15 Jack D. Gaskill, Linear Sstems, Furier Transfrms, and Optics, pp (Jhn Wile & Sns, Inc., New Yrk, 1978). 16 Jseph W. Gdman, Intrductin t Furier Optics, pp (McGraw-Hill, Inc., New Yrk, 1968).

32 3 PASSBAND Each plane wave cmpnent that arrives at the imaging sstem is assciated with the spatial frequencies, and. These spatial frequencies are nt independent. Thus 1 (136) fllws readil b cmbining equatins () and (4). A maimum spatial frequenc c such that (137) c is included in the passband that is linked t a circular imaging sstem. This spatial frequenc is knwn as the spatial frequenc cutff f the imaging sstem. The inequalit 1 c (138) is easil btained b substituting equatin (136) int inequalit (137). The spatial frequenc cutff f an imaging sstem is the value f the spatial frequenc bend which the assciated transfer functin is er. Inside the imaging sstem's passband 1 (139) 0 c while 1 c (140) utside the imaging sstem's passband. Accrdingl c, ; 1 0 c (141)

33 33 can be btained b substituting equatins (139) and (140) int equatin (135). Imaging sstems with nn-circular apertures are, in general, assciated with multiple spatial frequenc cutffs. General treatment f such imaging sstems is bend the scpe f this paper. The transfer functins assciated with the imaging sstems cnsidered in this paper are ver clsel apprimated b equatin (141). An imaging sstem that is assciated with the spatial frequenc cutff c is als assciated with a spatial perid cutff c. The imaging sstem's spatial perid cutff, given b 1 c (14) is equal t the minimum spatial perid f an quantum amplitude that can be transferred thrugh the imaging sstem. This minimum spatial perid is given b c T c (143) c Cnsequentl, images f peridic quantum amplitude cmpnents with spatial perids T such that T c (144) can be frmed while images f quantum amplitude cmpnents with smaller spatial perids cannt be frmed. All peridic quantum amplitude cmpnents with spatial perids that eceed pass thrugh the imaging sstem and cnsequentl cntribute t real image frmatin. c Substitutin f equatin (91) int inequalit (144) leads t c (145) readil. Accrdingl, an image f tw pints in the bject plane can be frmed (the tw pints can be reslved) prvided the distance between them equals r eceeds the minimum distance (146) c Tw pints in the bject plane are, b definitin, well-separated when inequalit (145) is satisfied. Inequalit (145) cnstitutes a fundamental tw-pint ptical reslutin criterin. Circular imaging sstems are endwed with a definite diameter. Accrdingl, a circular imaging sstem restricts the prpagatin angle f the light that can enter it t a maimum allwed value c. Similarl, a circular imaging sstem restricts the prpagatin angle f the light that can leave it t a maimum allwed value ' c. The cutff prpagatin angles cand ' c are the c

34 34 entrance angle (als knwn as the acceptance angle) and the eit angle, respectivel, f the imaging sstem. Equatin (116) becmes sin sin ' c c S nt c S nt ' ' c (147) when evaluated at cutff. After rearranging equatin (147) and substituting equatin (143) int the result S nsin c c ' c S n 'sin ' c (148) can be btained. The imaging sstem's bject side numerical aperture NA and image side numerical aperture NA i, given b NA sin n c NA n'sin ' i c (149) can be cnvenientl intrduced. Equatin (11) becmes n sinc m (150) n ' sin ' when evaluated at cutff. Substitutin f equatin (149) int equatin (150) ields easil. Substitutin f equatin (149) int equatin (148) leads t c NA m (151) NA i

35 35 S NA c ' c S NAi (15) readil. In additin NA c S ' c NA i S (153) fllws after substitutin f equatin (14) int equatin (15). As shwn b equatins (15) and (153), knwledge f a circularl smmetric imaging sstem's bject side and image side numerical apertures is sufficient t determine the imaging sstem's spatial perid and spatial frequenc cutffs, respectivel, fr an particular wavelength f light. An imaging sstem is ften characteried in terms f its numerical aperture withut distinguishing the bject side numerical aperture frm the image side numerical aperture. Rather, the term numerical aperture is used genericall t treat either numerical aperture. The fundamental limits f perfrmance fr a circular imaging sstem are ften epressed in terms f its numerical aperture withut specifing the intended numerical aperture. Cnfusin can result. RESOLUTION CRITERIA A minimum separatin distance between tw distinguishable image features eists. Tw image features that are sufficientl near each ther merge tgether t frm a single image feature. Tw image features that can be distinguished as tw image features are said t be reslved. Cmparisn f the fundamental reslutin criterin given b inequalit (145) with tw wellknwn classical reslutin criteria can be made. These reslutin criteria are the Abbė reslutin criterin (rdinaril used fr micrscpes) and the Raleigh reslutin criterin (rdinaril used fr telescpes and phtlithgraph). The Raleigh reslutin criterin is ften used t identif what is knwn as the classical diffractin limit f ptical sstem perfrmance. Neither f these criteria is based n fundamental phsics; nevertheless, the are bth rughl cnsistent with the fundamental reslutin criterin intrduced in this paper as inequalit (145). An verview f ptical reslutin has been published b A.J. den Decker and A. van den Bs 17. These authrs pint ut that ptical reslutin is nt unambiguusl defined and is interpreted in 17 A. J. den Decker and A. van den Bs, Reslutin: a Surve. J. Opt. Sc. Am. A, Vl 14, N. 3, pp (March 1997).

36 36 man was. The g n t review the cncept f ptical reslutin and man f its interpretatins. FUNDAMENTAL RESOLUTION CRITERION An image f tw pints in the bject plane can be frmed (the tw pints can be reslved) prvided the distance between them equals r eceeds the cutff distance. This fundamental twpint ptical reslutin criterin has been intrduced previusl b means f inequalit (145). Equatin (147) can be rearranged s that S T nsin c c T ' c S n 'sin ' c (154) results. Substitutin f equatin (149) int equatin (154) ields S NA Tc T ' c S NAi (155) readil. When epressed in terms f bth unprimed and primed parameters, inequalities (144) and (145) lead t S NA T T ' S NAi (156) and S NA ' S NAi (157) respectivel. Inequalit (157) cnstitutes bject side and image side fundamental tw-pint ptical reslutin criteria.

37 37 ABBÉ RESOLUTION CRITERION The Abbė reslutin criterin can be written as S NA p p ' S NAi (158) and is applicable t a cherentl illuminated peridic amplitude bject f perid p as measured n the bject plane and perid p ' as measured n the image plane. Inequalit (158) is mathematicall equivalent t inequalit (156) but it is fundamentall linked t spatial perids rather than tw-pint separatins. An amplitude bject where the spatial perids are phsicall identifiable is needed t appl the Abbė reslutin criterin. B wa f cntrast, the fundamental reslutin criterin is applicable t general bjects. RAYLEIGH RESOLUTION CRITERION The Raleigh reslutin criterin, which can be written as 1. S NA R R ' 1. S NAi (159) is a measure f the ptical imaging sstem s perfrmance. Accrding t this criterin, an image f tw pints in the bject plane can be frmed (the tw pints can be reslved) prvided the distance between them satisfies the Raleigh criterin. CONVENTIONAL OPTICAL RESOLUTION The cnventinal ratinale that underlies the Raleigh reslutin criterin is presented in Appendi A. Accrding t this ratinale a tpical image cnsists f man verlapping distributins f light. Each distributin f light is frmed b the diffractin f cherent light frm the imaging sstem s aperture. The image cnsists f the incherent superpsitin f the cherentl frmed individual distributins f light. In accrd with the Raleigh criterin, tw identical cmpnent distributins f light are barel reslved when the are separated b the distance R (bject side) r b the distance R ' (image side). The distance R ' is the laterall magnified versin f the distance R where the magnificatin is prduced b the imaging sstem. In accrd with cnventinal real image frmatin ther, reslutin and field depth can be understd (in disagreement with the nnlcal ptical real image frmatin ther) in cnnectin

38 with an islated three-dimensinal distributin f light. Reslutin is related t the width f the distributin f light in the fcal plane while image depth is related t the length f the distributin f light perpendicular t the fcal plane. As a cnsequence f this mdel, cnventinal reslutin and field depth are clsel linked t ne anther. Accrding t the nnlcal ptical real image frmatin ther, n such clse linkage ccurs. 38

39 39 PART III: FIELD DEPTH INTRODUCTION A cnfiguratin f light that clsel apprimates the cnfiguratin f light n the bject (image) plane eists n varius auiliar planes that are parallel t and include the bject (image) plane. The etreme range f distances ver which these planes can be identified is the unbstructed bject (image) field depth. Unbstructed field depth ccurs when the bject (image) plane is transparent. Obstructed field depth ccurs when the bject (image) plane is paque, and is rdinaril ne-half the unbstructed field depth. Separate treatment f bstructed field depth is nt included in this paper. In the fllwing analsis the bject plane and the image plane are bth treated as being transparent. The assciated unbstructed field depth etends n bth sides f the bject (image) plane. AUXILIARY PLANE T' ' s IMAGE PLANE Figure 7. Plane wave prpagatin thrugh the image plane and an auiliar plane.

40 40 Table I. Parameters identified in Figure 7. ' Wavelength f light used (image side) Angle between the directin f prpagatin and a perpendicular t the image and auiliar planes Quantum amplitude spatial perid f the sinusidal cmpnent assciated with the light wave (image side) Distance between the auiliar plane and the image plane s Lngitudinal displacement f a spatial perid n the auiliar plane Distance in the directin defined b IMAGE FIELD DEPTH Cnsider ptical real image frmatin where the image plane is transparent. Each plane wave pair that cntributes t real image frmatin prpagates thrugh the image plane. These plane wave pairs als prpagate thrugh varius planes that are parallel t the image plane. The plane wave pairs verlap n each f these planes. As a result, sinusidal cmpnents that clsel apprimate crrespnding sinusidal cmpnents n the image plane are frmed n auiliar planes that are parallel t and sufficientl near the image plane. These sinusidal cmpnents eist n bth sides f the image plane and are cnsidered t be part f the real image. Superpsitin f the tw members f a plane wave pair that verlap n a plane parallel t the image plane (an auiliar plane) and a distance frm it can nw be cnsidered. Gemetric relatinships defined b ne member f the plane wave pair that cntributes t real image frmatin are illustrated in Figure 7. Parameters identified in the figure are described in table I. Let A/ be the wave amplitude f each member f the plane wave pair and let amplitude spatial perid f each member f the plane wave pair. Then T ' be the quantum A 1 cs ( s) T ' (160) represents ne member f the plane wave pair while A cs ( s) T ' (161) represents the ther member f the plane wave pair. The superpsitin f bth members f the plane wave pair can be described b r 1 (16)

41 41 A A cs ( s) cs ( s) T' T' (163) which reduces t A cs cs T' s T' (164) with little effrt. The relatinships s tan (165) sin tan (166) cs ' sin (167) T ' and T ' ' cs (168) T ' can be btained b inspecting Figure 7. Hence, tan ' T ' ' (169) and s T ' ' ' (170) fllw readil. Substitutin f equatin (170) int equatin (164) ields

42 4 A ' cs cs ' ' T' T' T (171) easil. The first csine functin n the right hand side f equatin (171) is spatiall peridic where T' T' ' (17) ' is the functin s spatial perid. The range f values f fr which changes in the value f are acceptable can be epressed as T' T' ' (173) K ' where K is a subjectivel determined cnstant. The depth f image field can nw be defined as where D' (174) T' T' ' (175) K ' and T' T' ' (176) K ' have been intrduced. Substitutin f equatins (175) and (176) int equatin (174) ields T' T' ' D' (177) K ' easil. The depth f image field is given b equatin (177).

43 43 OBJECT FIELD DEPTH The analsis used t derive equatin (177) fr the depth f image field is als applicable t deriving analgus epressins fr the depth f bject field. Primed parameters are used fr the image side analsis; unprimed cunterparts are used fr the bject side analsis. Thus, the bject side wavelength f light used is designated as (unprimed) and the bject side quantum amplitude spatial perid is designated as T (unprimed) in the treatment rather than their image side (primed) cunterparts. The bject side cunterpart f equatin (177) is T T D (178) K The depth f bject field is given b equatin (178). VALUE OF K As intrduced previusl, K is a subjectivel determined cnstant. Selecting a suitable value fr K is perhaps best dne n the basis f eperience. Pssibl the value K 16 ma be suitable as a tentative useful value fr K. TWO POINT SEPARATION The quantum amplitude spatial perid assciated with tw pints in a cnfiguratin f light is equivalent t ne-half the distance between the tw pints. This distance is given b equatin (91) fr pints n the bject plane. Cnsequentl, T T ' ' (179) when the bject side and its image side (primed) cunterpart are bth cnsidered. Substitutin f equatin (179) int equatins (178) and (177) leads t 4 D K D ' ' ' 4 ' K ' (180) The depth f bject (image) field is independent f the imaging sstem.

44 44 CONVENTIONAL IMAGE FIELD DEPTH Equatins that are cmmnl used t estimate the field depth (bject and image) assciated with real image frmatin are dependent upn the numerical aperture designated as NA (withut regard fr the side f the imaging sstem invlved) f the imaging sstem that is used. Specificall D C (181) NA is the equatin cnventinall used t estimate the depth f bject field assciated with real image frmatin 18. Similarl, D' C ' (18) NA is the equatin that is cnventinall used t estimate the depth f image field assciated with real image frmatin These equatins are substantiall different than the cunterpart equatins derived in this paper. 18 Chris Mack, Fundamental Principles f Optical Lithgraph: The Science f Micrfabricatin, p. 105 (Jhn Wile & Sns, Ltd., Chichester, England, 007).

45 45 PART IV: REFERENCE DISTRIBUTION REAL IMAGE FORMATION INTRODUCTION Reference distributin real image frmatin (RIF) is achieved b using a reference distributin f light in additin t the usual subject distributin. This assures that tw well-separated illuminated pints are used t frm each light wave cmpnent that prpagates awa frm the bject plane and ultimatel cntributes t real image frmatin. One pint lies within the subject distributin and the ther pint lies within the reference distributin. The distance between these pints is designed t assure that it eceeds the minimum needed fr real image frmatin. Thus, inequalit (145) is satisfied intentinall fr ever pair f pints such that ne pint lies within the subject distributin and the ther pint lies within the reference distributin. An image f bth distributins f light is frmed. The spatial frequencies f the cmpnent light waves that cntribute t RIF lie within a welldefined finite bandwidth. The imaging sstem is designed such that it transfers the wave cmpnents with spatial frequencies that lie within this bandwidth t the image plane. These wave cmpnents are superpsed n the image plane t frm a real image. RIF REAL IMAGE EQUATION All spatial frequencies linked t the cmbined reference and subject distributins are included in u, ; when RIF is used. The imaging sstem is chsen t assure that these spatial frequencies lie within its passband. Thus u, ;, ; u, ; (183) fr reference distributin image frmatin. Cnsequentl, equatin (19) reduces t ui, ; u, ; m m (184) and equatin (13) reduces t 1 ui, ; u, ; m m m (185) fr RIF. Equatin (185) is the RIF real image equatin fr light f wavelength. N cnvlutin peratin is assciated with the RIF real image equatin. The RIF real image equatin defines a magnified (enlarged, unchanged r reduced and perhaps inverted) real image f bth the subject distributin and the reference distributin. Magnified versins f all spatiall peridic cmpnents linked t ne pint in the bject distributin and

46 46 anther pint in the reference distributin are included in the real image. N real image distrtin due t missing spatiall peridic cmpnents ccurs. Reslutin is cmplete. OBJECT PLANE CONFIGURATION A cnfiguratin f light that is suitable fr achieving RIF is illustrated in Figure 8. The cnfiguratin f light eists n the side f the bject plane nearest t the imaging sstem and is cnfined t the interirs f the circles shwn in the figure. Light thrughut the cnfiguratin is cherent r partiall cherent. All pints in bth the subject distributin and the reference distributin are in the same area f cherence. As indicated in the figure, the subject distributin eists inside a circular area f diameter while the reference distributin eists inside a circular area f diameter D R. As intrduced previusl, let s be wavelength f light used, as measured in vacuum, and let NA be the bject side numerical aperture f the imaging sstem. Ideal RIF ccurs when the criteria D S S S (186) NA AREA OF ILLUMINATION PERIMETER SUBJECT DISTRIBUTION REFERENCE DISTRIBUTION C/L D S S D R Figure 8. RIF bject plane cnfiguratin: plan and frnt elevatin views.

47 47 D S s (187) NA and D R s (188) NA are realied phsicall. Nis RIF ccurs when either satisf the freging inequalities. D S r D R r bth D S and D R d nt The fundamental reslutin criterin, given b equatin (157), is satisfied when inequalit (186) is satisfied; an image f ever bject pint and ever image pint is frmed. The fundamental reslutin criterin is nt satisfied when inequalities (187) and (188) are satisfied. The RIF cnfiguratin that has been presented is b n means the nl pssible RIF cnfiguratin that can be devised. Treatment f ther pssible RIF cnfiguratins is bend the scpe f this paper. NOISY RIF CONCEPT Cnsider the cnceptual illustratin f transmissin nis RIF prvided in Figure 9. Initiall, light is incident (frm the left) upn an paque screen with tw apertures in it. The side f the screen nearest t the imaging sstem (shwn as a lens) serves as an bject plane. Cherent r partiall cherent light passes thrugh the apertures t frm tw distributins f light n the bject plane. These distributins f light are, ideall, the same sie and shape as the apertures. The distributins f light are designated as the subject distributin and the reference distributin. Taken separatel, the prpagatin angles f the light that travels awa frm either f the tw distributins f light are larger than the acceptance angle f the imaging sstem. Such light des nt pass thrugh the imaging sstem and cnsequentl des nt cntribute t image frmatin. N tw pints in either the subject distributin r the reference distributin are sufficientl separated t cntribute t image frmatin. Ever pair f pints such that ne pint is in the subject distributin and ne pint is in the reference distributin is sufficientl separated t cntribute t image frmatin. Taken tgether, the prpagatin angles f the light that travels awa frm the cmbined distributins f light are smaller than the acceptance angle f the imaging sstem. This light defines a finite bandwidth and is transferred, withut amplitude r phase distrtin, thrugh the

48 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. Cnceptual nis RIF prcess. imaging sstem. Cnsequentl, light that prpagates awa frm the cmbined distributins f light cntributes t undistrted real image frmatin.

49 49 PART V: CONCLUSION A nnlcal ther f ptical real image frmatin has been develped frm the fundamental phsics assciated with an ptical real image frmatin apparatus. The ther shws hw tw separated illuminated bject pints are fundamentall required fr real image frmatin. This finding is in distinct cntrast t the rdinaril accepted assumptin that nl ne bject pint is invlved. In accrd with the new ther, ptical real images with a ver large field depth (bject and image) and n fundamental reslutin limit can be frmed. Each quantum amplitude cmpnent that cntributes t real image frmatin is assciated with a pair f illuminated pints in the bject. Bth illuminated pints are inside the area f cherence assciated with the quantum amplitude cmpnent. Image-frming light is either cherent r partiall cherent. Single phtns are capable f prviding image-frming light. A fundamental tw-pint reslutin criterin was derived as part f the new ther and cmpared with the Abbė and Raleigh reslutin criteria. Althugh neither f these criteria is based n fundamental phsics, the are bth rughl cnsistent with the fundamental reslutin criterin intrduced in this paper. Field depths fr bth the bject and image were shwn t be independent f the imaging sstem used and t be substantiall different than the are traditinall thught t be. Reference distributin real image frmatin (RIF) has been intrduced. RIF is achieved b using a reference distributin f light in additin t the usual subject distributin. This assures that tw well-separated illuminated pints are used t frm each light wave cmpnent that prpagates awa frm the bject plane and ultimatel cntributes t real image frmatin. One pint lies within the subject distributin and the ther pint lies within the reference distributin. The spatial frequencies f the cmpnent light waves that cntribute t RIF lie within a welldefined finite bandwidth. The imaging sstem transfers the wave cmpnents with spatial frequencies that lie within this bandwidth t the image plane. These wave cmpnents are superpsed n the image plane t frm a real image. Real images frmed b means f RIF are described b the RIF real image equatin. The RIF real image equatin defines a magnified (enlarged, unchanged r reduced and perhaps inverted) real image f bth the subject distributin and the reference distributin. N cnvlutin peratin is invlved. Magnified versins f all spatiall peridic cmpnents included in the bject are included in the real image. N real image distrtin due t missing spatiall peridic cmpnents ccurs. Reslutin is cmplete when real images are frmed b means f RIF.

50 50 APPENDIX A: CONVENTIONAL OPTICAL REAL IMAGE RESOLUTION INTRODUCTION APERTURE PLANE FOCAL PLANE A brief review f the generall accepted treatment f reslutin assciated with FOCUS ptical real image frmatin (in disagreement with the nnlcal ptical IMAGING SYSTEM real image frmatin ther) is prvided in this appendi. Thus, reslutin assciated with a spherical wave that cnverges t a gemetric fcus after passing thrugh a circular aperture, as Figure 10. Apertured cnverging spherical wave. cnventinall understd, is reviewed. Such a wave is ften created b a fcusing element, where the edge f the fcusing element cnstitutes the edge f the aperture. Fr this cnventinal treatment, the angle at the gemetric fcus subtended b the circular aperture is restricted t being ver small. Referring t Figure 10, a unifrm spherical mnchrmatic light wave f wavelength ' cnverges tward a gemetric fcus after passing thrugh a circular aperture. The phase f the wave is the same at all pints in the aperture; i.e., the light is cherent. As shwn in Figure 11, the -ais is chsen t pass thrugh the aperture s center and the gemetric fcus. The aperture radius a, the distance frm the aperture s center t the gemetric fcus f, and the angle ' c that is subtended b the aperture s radius at the FOCAL PLANE gemetric fcus are shwn in the figure. The angle is the fcusing element s eit angle. ' c RAYLEIGH RESOLUTION CRITERION OPTICAL AXIS APERTURE A three-dimensinal distributin f light that is dminated b a bright central regin surrunded b a set f alternating dark and bright threedimensinal vals envelps the fcal pint 19. A tpical image cnsists f man f these distributins f light, sme f which ma (and prbabl d) verlap. The center-t-center separatin (in the fcal plane) f tw reslved val-shaped distributins f light defines the Raleigh reslutin criterin. In accrd with the OPTICAL AXIS Figure 11. Gemetric parameters. FOCUS 19 Ma Brn and Emil Wlf, Principles f Optics (Sith Crrected Editin) pp (Pergamn Press, Ofrd, 1980).

51 51 Raleigh reslutin criterin, tw identical val-shaped distributins f light are barel reslved (distinguishable) when the center f ne f them cincides (n the fcal plane) with the first dark val f the ther ne. Let s R ' be the radius (n the fcal plane) f the first dark val assciated with a single val shaped distributin f light. As illustrated in Appendi B, Figure 1 ' s ' 1.0 c R (189) s that s R 0.61 ' ' c (190) when the Raleigh reslutin criterin is satisfied. In accrd with equatin (153) ' c (NA) s i (191) where (NA) i is the image side numerical aperture f the fcusing element and s is the wavelength (in vacuum) f the light used. Substitutin f equatin (191) int equatin (190) leads t s R 0.61s ' (NA) i (19) readil. The center-t-center separatin (n the fcal plane) f the tw identical val-shaped distributins f light when the are barel reslved is given b ' s ' R R (193) Cnsequentl, tw pints are reslved (barel r therwise) when the image plane Raleigh reslutin criterin 1. R ' NA S i (194) is satisfied. The prjectin f the image side Raleigh reslutin criterin thrugh the imaging sstem nt the bject plane is the bject side Raleigh reslutin criterin. Cnsequentl, the Raleigh reslutin criteria are given b

52 5 fr bth the bject plane (unprimed) and the image plane (primed). 1. S NA R R ' 1. S NAi (195) Raleigh reslutin criteria are cmmnl 0 applied far utside their ver restricted range 1 f validit. Such applicatin has led t heric attempts t use ever decreasing wavelengths f light and increasing numerical apertures t imprve reslutin. This apprach has met with a measure f success withut a true understanding f the phsics invlved. The apprach cnstitutes a fight against nature. 0 Chris Mack, Fundamental Principles f Optical Lithgraph: The Science f Micrfabricatin (Jhn Wile & Sns, Ltd., Chichester, England, 007). 1 Clin J.R. Sheppard, The Optics f Micrscp, J. Opt. A: Pure Appl. Opt. 9, pp. S1 S6 (007).

53 53 APPENDIX B. CIRCULARLY SYMMETRIC IMAGING SYSTEMS INTRODUCTION Mst imaging sstems are circularl smmetric. A change f variables frm rectangular crdinates t plane plar crdinates in bth the -plane and the -plane is advantageus fr treating circularl smmetric imaging sstems. Thus rcs rsin (196) and cs sin (197) can be apprpriatel intrduced. In additin and, ; g r, ; (198), ; g, ; (199) can be intrduced t epress the imaging sstem's cherent impulse respnse and cherent transfer functin, respectivel, in plar crdinates. The Jacbians linked t the transfrmatin frm rectangular t plar crdinates are and which reduce t cs r sin Jr, (00) sin r cs cs sin J, (01) sin cs J r, r (0)

54 54 and J, (03) respectivel. Cnsequentl and d d r dr d (04) d d d d (05) fllw. Substitutin f the freging equatins int equatins (133) and (134), as apprpriate, leads t and, ;, ; ep cs cs sin sin g r g i r d d (06) 0 0, ;, ; ep cs cs sin sin g g r i r r dr d (07) 0 0 respectivel. After intrducing the trignmetric identit cs cs sin sin cs (08) equatins (06) and (07) can be written as and respectivel., ;, ; ep cs g r g i r d d (09) 0 0, ;, ; ep cs g g r i r r dr d (10) 0 0 Integral representatins f the Bessel functin f the first kind f rder er, i.e., Frank Bwman, Intrductin t Bessel Functins, p. 57 (Dver Publicatins Inc., New Yrk, 1958).

55 55 1 J 0 r ep i r cs d 0 (11) and 1 J0 r ep i r cs d 0 (1) can nw be advantageusl intrduced. Equatins (11) and (1) can be rearranged t ield and respectivel. ep i r cs d J 0 0 r (13) ep i r cs d J 0 0 r (14) Substitutin f equatin (13) int equatin (09) leads t while, ;, ; 0 0 (15) g r g J r d, ;, ; g g r J r r dr (16) 0 can be btained after substituting equatin (14) int equatin (10). The Bessel functin f the first kind f rder er is an even functin. Cnsequentl 0 0 can be substituted int equatin (15) t btain triviall. 0 J r J r (17), ;, ; g r g J r d (18) CIRCULARLY SYMMETRIC COHERENT TRANSFER FUNCTION 0 0

56 56 Cherent transfer functins which are circularl smmetric are necessaril independent f ; cnsequentl g, ; h ; (19) can be apprpriatel intrduced. Substitutin f equatin (19) int equatin (18) ields, ; ; g r h J r d (0) The right hand side f equatin (0) is independent f. Accrdingl 0, ; hr; g r 0 (1) can be intrduced t reflect the fact that the left hand side f equatin (0) is necessaril als independent f. Substitutin f equatin (1) int equatin (0) ields ; ; h r h J r d () Substitutin f equatins (19) and (1) int equatin (16) ields 0 ; ; 0 0 h h r J r d (3) The circularl smmetric cherent transfer functin ; cherent impulse respnse ; (3). 0 h and the circularl smmetric hr are related t ne anther b means f equatins () and Substitutin f equatin (19) int equatin (199) and then substituting the result int equatin (135) leads t 1 inside the passband h; (4) 0 utside the passband which is the circularl smmetric cherent transfer functin. Equatin (4) can be written as 1 h ; 0 c c (5) where c is the spatial frequenc cutff f the imaging sstem.

57 57 CIRCULARLY SYMMETRIC COHERENT IMPULSE RESPONSE Equatin () relates the circularl smmetric cherent impulse respnse ; circularl smmetric cherent transfer functin ; hr t the h. Substitutin f the circularl smmetric transfer functin given b equatin (5) int equatin () Leads t readil. Equatin (6) can be rewritten as ; c ; h r J r d (6) c hr; r J 0 r ; r d r (7) 0 advantageusl. Man authrs, tgether with Hecht 3 and Brn and Wlf 4, treat circularl smmetric cherent impulse respnse functins and related tpics. Let J r ; 1 be the Bessel functin f the first kind f rder ne. The recursin relatin 1 d r J1 r ; r J r ; r d (8) is intrduced and discussed b man authrs, including Bas 5, Hecht 6, and Brn and Wlf 7. Equatin (8) can be rearranged and written as d r J r ; r J r ; r d (9) 1 readil. Substitutin f equatin (9) int equatin (7) ields 1 c hr; d r J r ; r immediatel. Upn evaluatin, equatin (30) ields 0 1 (30) 1 hr; r J r; r c 1 c (31) 3 Eugene Hecht, Optics, (4 th Ed.) pp (Addisn Wesle, San Francisc, 00). 4 Ma Brn and Emil Wlf, Principles f Optics (Sith Crrected Editin) pp (Pergamn Press, Ofrd, 1980). 5 Mar L. Bas, Mathematical Methds in the Phsical Sciences, p (Jhn Wile & Sns, Inc., New Yrk, 1966). 6 Eugene Hecht, Optics, (4 th Ed) p. 468 (Addisn Wesle, San Francisc, 00). 7 Ma Brn and Emil Wlf, Principles f Optics (Sith Crrected Editin), p. 396 (Pergamn Press, Ofrd, 1980).

58 Figure 1. Standardied Air frmula. in a straightfrward manner. Equatin (31) can be simplified t btain J1 c r; hr ; c c r (3) readil. Equatin (3) describes the cherent impulse respnse fr a circularl smmetric imaging sstem. The relative prbabilit densit assciated with the impulse respnse prvided b equatin (3) is given b ; * ; ; Substitutin f equatin (3) int equatin (33) ields I r h r h r (33) ; c I r r J1 c ; c r (34) triviall. This result is knwn as the Air frmula. AIRY DISK A graph f the standardied Air frmula, i.e.

59 59 Figure 13. Air pattern. Surce: Wikipedia, Air Disk. c I r; J1 c r; c r (35) is prvided in Figure 1. As illustrated in Figure 1, the ptical real image linked t the standardied Air frmula is an Air pattern. As shwn in Figures 1 and 13, an Air pattern cnsists f a central bright regin that is surrunded b a number f much fainter rings. The central bright regin is knwn as the Air disk assciated with the imaging sstem. An Air disk is bunded b a dark ring that eists at a lcatin that crrespnds t the first er f the Bessel functin J1 r ; c 8. As indicated in Figure 1, this ccurs when the value f the independent variable in equatin (35) has the value 1.0, i.e. when 1.0 (36) r c 0 where the radius f the Air disk r0 (37) c 8 Ma Brn and Emil Wlf, Principles f Optics (Sith Crrected Editin), p. 397 (Pergamn Press, Ofrd, 1980).