Show That the Fourier Integral of Can Be Written as

MAX BORN M.A., DocPhil., F.R.S. Nobel Laureate , EMIL WOLF Ph.D., D.Sc. , in Principles of Optics (6th Edition), 1980

eight.half dozen.3 Image formation in the microscope

In the elementary theory of resolving power which nosotros have just outlined, light from the two object points was assumed to be incoherent. This assumption is justified when the two objects are self-luminous, e.g. with stars viewed past a telescope. The intensity observed at any signal in the image airplane is then equal to the sum of the intensities due to each of the object points.

In a microscope the situation is, equally a rule, much more than complicated. The object is usually non-luminous and must, therefore, be illuminated with the help of an auxiliary system. Owing to diffraction on the discontinuity of the illuminating organization (condenser), each element of the source gives rise to a diffraction blueprint in the object aeroplane of the microscope. The diffraction patterns which have centres on points that are sufficiently close to each other partly overlap, and in consequence the light vibrations at neighbouring points of the object plane are in general partially correlated. Some of this light is transmitted through the object with or without a alter of phase, whilst the remainder is scattered, reflected, or absorbed. In consequence, it is in general incommunicable to obtain, past means of a unmarried observation, or fifty-fifty by the utilize of one particular system, a faithful enlarged picture showing all the minor-scale structural variations of the object. Various methods of observation have, therefore, been developed, each suitable for the report of certain types of objects, or designed to bring out particular features.

We shall briefly outline the theory of image formation in a microscope, confining our attention beginning of all to the two farthermost cases of completely incoherent and perfectly coherent illumination. Partially coherent illumination will be discussed in § 10.v.ii.

(a) Breathless illumination

We first consider a self-luminous object (e.chiliad. an incandescent filament of an electric seedling). Allow P exist the axial point of the object and Q a neighbouring signal in the object aeroplane, at a distance Y from P, and permit P' and Q' be the images of these points (Fig. viii.31). Further let θ and θ' be the angles which the marginal rays of the axial pencils make with the axis.

Fig. 8.31. Illustrating the theory of resolving ability of the microscope.

If a' is the radius of the region (assumed to exist circular) in which the axle of lite converging on P' intersects the dorsum focal plane F' and if D' is the distance betwixt the dorsum focal airplane and the image plane, then, since θ' is small,

Further, if $\mathrm{due\; west}=\sqrt{p^2+q^2}$ is the separation of Q' from P' measured in "diffraction units" [cf. § 8.three (35) and § 8.v (7)], i.e. the sine of the bending which the ii points subtend at the centre of the diffracting aperture, then nosotros take to a good approximation,

Let north and n' be the refractive indices, λ and λ' the wavelengths in the object and epitome spaces, and λ₀ the wavelength in vacuum. And so, since according to § 8.5 (16) the kickoff minimum of the diffraction pattern of P is given by w = 0·61λ'/a', nosotros take, at the limit of resolution

(31) $Y\prime =0.61\lambda \prime \frac{\text{D}\prime }{a\prime }=0.61\frac{{\lambda }^{\prime }}{{\theta }^{\prime }}=0.61\frac{{\lambda }_0}{n^{\prime }{\theta }^{\prime }}.$

A microscope must, of course, be so designed that it gives a sharp image not only of an axial signal but too of neighbouring points of the object plane. According to § four.v.1 the sine status must therefore be satisfied, i.e. ^*

Since θ' is small nosotros may supercede sin θ' past θ'. On substituting for Y' into (31), we finally obtain

This formula gives the distance between two object points which a microscope can just resolve when the illumination is incoherent and the discontinuity is circular.

The quantity n sin θ which enters into (32) is the numerical aperture, [cf. § 4.eight (13)] and must be large if a high resolving ability is to be achieved. Means for obtaining a large numerical aperture were discussed in § 6.6.

(b) Coherent illumination–-Abbe'southward theory

We now consider the other extreme case, namely when the lite emerging from the object may be treated every bit strictly coherent. This situation is approximately realized when a thin object of relatively unproblematic construction is illuminated by light from a sufficiently pocket-size source via a condenser of low discontinuity (cf. § 10.5.two).

The first satisfactory theory of resolution with coherent illumination was formulated and besides illustrated with beautiful experiments, by East. Abbe. ^* According to Abbe, the object acts equally a diffraction grating, so that non only every element of the aperture of the objective, merely also every element of the object must exist taken into account in determining the complex disturbance at whatever detail signal in the image aeroplane. Expressed mathematically, the transition from the object to the image involves two integrations, i extending over the object plane, the other extending over the discontinuity. In Abbe'south theory, diffraction by the object is first considered and the effect of the aperture is taken into account in the second stage. An alternative process, in which the gild is reversed, is as well permissible and leads naturally to the same upshot. ^†

To illustrate Abbe's theory we consider first the imaging of a grating-like object which is illuminated by a aeroplane wave incident ordinarily on to the object aeroplane (Köhler's central illumination). The wave is diffracted by the object and gives rise to a Fraunhofer diffraction pattern of the grating (cf. § 8.6.1), in the dorsum focal plane F' of the objective. In Fig. 8.32 the maxima (spectra of successive orders) of this blueprint are denoted by … S _-2, S _-1, S ₀, S ₁, S ₂, …. Every point in the focal aeroplane may be considered to be a middle of a coherent secondary disturbance, whose strength is proportional to the amplitude at that point. The light waves that go on from these secondary sources will then interfere with each other and will give rise to the image of the object in the epitome aeroplane Ii' of the objective. To obtain a faithful image it is necessary that all the spectra contribute to the germination of the image. Strictly this is never possible because of the finite aperture of the objective. We shall see later that the exclusion of some of the spectra may result in completely false particular appearing in the image. For practical purposes it is evidently sufficient that the aperture shall be large plenty to acknowledge all those spectra that deport an appreciable corporeality of free energy.

Fig. viii.32. Illustrating the Abbe theory of image formation in a microscope, with coherent illumination.

Let us express these considerations in more precise terms without restricting ourselves to a grating-like object. If x, y are the coordinates of a typical betoken in the object plane and f is the distance of the focal aeroplane F' from the lens objective, the disturbance at a signal

of the F' plane (see Fig. 8.32) is given by the Fraunhofer formula

(34) $U(\xi ,\eta )=C_1{\int {\int }_{A}F(x,y)e}^{-i\mathrm{chiliad}\left[\frac{\xi }{f}\mathrm{10}+\frac{\eta }{f}y\right]}dxdy.$

where F is the manual function of the object, C ₁ is a abiding, and the integration is taken over the area A of the object plane II covered by the object.

Next consider the transition from the back focal plane F' to the image airplane Ii'. If, as before, D' denotes the distance betwixt F' and II', and V(ten', y') is the disturbance at a typical signal

of the image airplane, nosotros have for Fraunhofer diffraction on the aperture B in F'

(36) $V({\mathrm{10}}^{\prime },y^{\prime })=C_2{\int {\int }_{B}U(\xi ,\eta )e}^{-ik\left[\frac{{\mathrm{10}}^{\prime }}{\text{D}\prime }\xi +\frac{y^{\prime }}{\text{D}\prime }\eta \right]}d\xi d\eta ,$

information technology beingness assumed that a'/D' ≪ 1 (meet figure viii.31). Substitution from (34) into (36) gives

(37) $V(x^{\prime },y^{\prime })=C_1{C}_2\int {\int }_A\int {\int }_{B}F(x,y)e^{-i\frac{g}{f}\left[\left(x+\frac{f}{\text{D}\prime }{x}^{\prime }\right)\xi +\left(y+\frac{f}{\text{D}\prime }{y}^{\prime }\right)\eta \right]}dxdyd\xi d\eta .$

Now if F(x, y) is defined as nada for all points of the object plane that prevarication outside A, the integration with respect to ten and y may formally be extended from – ∞ to + ∞. Too, if the discontinuity B is so large that |U(ξ, η)| is negligible for points of the F'-plane that lie outside B, the integrations with respect to ξ and η may too each be extended over the range from – ∞ to + ∞. Noting also that [cf. § 4.3 (x) where f' and Z' correspond to our f and – D' respectively]

where M(< 0) is the magnification between II and II', we obtain by the awarding of the Fourier integral theorem ^*

(39) $V({\mathrm{ten}}^{\prime },y^{\prime })=CF\left(\frac{x^{\prime }}{\mathrm{Yard}},\frac{y^{\prime }}{K}\right)=CF\left(x,y\right),$

where (x, y) is the object point whose image is at (10' y'), and

$C=C_1{C}_2{\lambda }^2{f}^2$

is a constant. Hence to the accurateness here implied ^† the prototype is strictly like to the object (but inverted), provided the discontinuity is large plenty.

To show that completely false detail may appear in the prototype if some of the spectra that carry observable energy are excluded, nosotros consider a one-dimensional grating-like object consisting of N equidistant congruent slits of width due south, separated by opaque regions, with period d. For simplicity the aperture will exist causeless to be rectangular with two of its sides parallel to the strips.

According to § 8.six (3)

(40) $U(\xi )={C^{\prime }}_1\left(\frac{sin\frac{k\xi s}{2f}}{\frac{\mathrm{yard}\xi \mathrm{southward}}{\mathrm{two}f}}\right)\frac{I-{\mathrm{due\; east}}^{i\mathrm{Northward}\mathrm{grand}d\xi /f}}{I-e^{iNkd\xi /f}},$

where for U ⁽⁰⁾ in that location has been substituted the expression relating to diffraction on a rectangular aperture and C'₁ is a constant (cf. § 8.five.1). If the rectangular aperture extends in the ξ direction throughout the range

$-a\le \xi \le a,$

the disturbance in the epitome aeroplane is, by (36) and (xl), given by (C' denoting a constant)

(41) $V(x^{\prime })=C^{\prime }{\int }_{-a}^a\frac{sin\frac{\mathrm{thou}\mathrm{south}\xi }{2f}}{\frac{ks\xi }{2f}}\frac{1-e^{-i\mathrm{Northward}kd\xi /f}}{1-e^{-iN\mathrm{1000}d\xi /f}}{e}^{-ik{x}^{\prime }\xi /D^{\prime }}d\xi .$

The position of master maxima of the integrand are given by the roots of the equation i − exp[- ikdξ/f] = 0, i.east. past ξ = mfλ/d, where m is an integer. Between these principal maxima in that location are weak secondary maxima. If N is big, the principal maxima are very sharp and the secondary maxima negligible in comparison. To a good approximation we may and so replace the integral by a sum of integrals, each extending from the midpoint Q _m of the interval betwixt ii successive principal maxima to the next midpoint Q _g+1. In each interval we may supplant the statement by the central value ξ = mfλ/d = 2πmf/kd, and obtain for V the following expression:

(42) $V({\mathrm{ten}}^{\prime })\sim V_0\underset{-\overline{g}<\mathrm{yard}<\overline{m}}{\Sigma }\frac{sin\frac{m\pi s}{d}}{\frac{m\pi s}{d}}{\mathrm{eastward}}^{\frac{2\pi im{\mathrm{10}}^{\prime }}{d^{\prime }}}.$

Hither

and V ₀ is the integral

(44) $V_0=C^{\prime }{\int }_{Q_{\mathrm{chiliad}}}^{Q_{m+\mathrm{i}}}\frac{\mathrm{one}-{\mathrm{east}}^{-i\mathrm{North}kd\xi /f}}{\mathrm{i}-e^{-i\mathrm{Northward}\mathrm{1000}d\xi /f}}d\xi ,$

which, apart from small-scale correction terms in the high orders, is practically independent of m. The serial (42) may be re-written in real form as

(45) $\frac{5(x^{\prime })}{{\mathrm{Five}}_0}=1+2\underset{\mathrm{one}<\mathrm{thousand}<\overline{\mathrm{thousand}}}{\Sigma }\frac{sin\frac{m\pi s}{d}}{\frac{k\pi s}{d}}{cos}^{\frac{\mathrm{two}\pi i\mathrm{chiliad}{\mathrm{10}}^{\prime }}{d^{\prime }}}.$

Suppose start that the length a of the aperture is very big. The summation may and then formally be extended over the whole infinite range $\left(\overline{m}=\infty \right)$ , and we can easily verify that the image is so strictly similar to the object. For this purpose we aggrandize the manual part F of the grating-similar object (run across Fig. 8.33),

Fig. 8.33. A grating-like object.

(46) $\begin{array}{l}F(x)=F_0\text{when}0<\left|x\right|<\mathrm{due\; south}/2\\ =0\text{when}s/2<\left|\mathrm{ten}\right|<d/2\end{array}\}$

into a Fourier serial

(47) $F(\mathrm{ten})=c_0+2\underset{m=1}{\overset{\infty }{\Sigma }}{c}_{k}cos\frac{2\pi mx}{d}.$

Then

(48) $\begin{array}{ccc}{c}_0=\frac{F_0\mathrm{due\; south}}{d},& c_m=F_0\frac{sin\frac{\pi ms}{d}}{\pi \mathrm{one\; thousand}}& (\mathrm{thou}=\mathrm{one},2,\mathrm{three\dots .})\end{array}.$

Nosotros see that apart from a constant factor this series is the aforementioned as (45).

Suppose now that the length a of the discontinuity is decreased. If a is so modest that only the zero-gild spectrum contributes to the image, i.east. if $\overline{k}=ad/\lambda f$ is merely a fraction of unity, so according to (45) V(x') = constant, so that the prototype plane is uniformly illuminated. (This upshot is, of class, not strictly true, equally we have neglected certain error terms; in reality in that location is a weak drib in intensity towards the edge.)

If in add-on to the zero-order spectrum the 2 spectra of the commencement order (Due south ₁, S _-1) are likewise admitted by the aperture, i.eastward. if $\overline{m}=ad/\lambda f$ is slightly greater than unity, so nosotros encounter from (45) that

(49) $\frac{V(x^{\prime })}{V_0}=1+2\frac{sin\frac{\pi \mathrm{southward}}{d}}{\frac{\pi s}{d}}cos\frac{2\pi {\mathrm{south}}^{\prime }}{d^{\prime }}.$

The image has now the correct periodicity 10' = d', only a considerably flattened intensity distribution. Past increasing the discontinuity more and more the image is seen to resemble the object more and more closely.

A completely false image is obtained when the lower orders are excluded. If for example all orders except the second are excluded, and then

(l) $\frac{V({\mathrm{ten}}^{\prime })}{V_0}=2\frac{sin\frac{\mathrm{ii}\pi \mathrm{southward}}{d}}{\frac{2\pi s}{d}}cos\frac{4\pi {\mathrm{10}}^{\prime }}{d^{\prime }},$

so that the prototype has the flow ten' = d'/2; the "paradigm" shows twice the number of lines that are in fact present in the object.

Finally let us estimate the resolving ability. Consider again the state of affairs illustrated in Fig. 8.31, just assume now that the light from P and Q is coherent. Then the distribution in the image airplane arises essentially from the coherent superposition of the ii Airy diffraction patterns, i centred on P', the other on Q'. The complex amplitude at a point situated between P' and Q' at altitude due west ₁ (measured in "diffraction units") from P' is given past

(51) $U(w_{\mathrm{i}})=\left[\frac{2J_1(\mathrm{thou}a{w}_1)}{\mathrm{thou}a{w}_{\mathrm{one}}}+\frac{2J_1[ka(w-w_1)]}{\mathrm{1000}a(\mathrm{westward}-{\mathrm{due\; west}}_1)}\right]U_0,$

w existence the altitude betwixt P' and Q' and the other symbols having the same pregnant as before. The intensity is, therefore, given by

(52) $I(w_1)={\left[\frac{2J_{\mathrm{ane}}(ka{w}_{\mathrm{ane}})}{ka{w}_1}+\frac{\mathrm{ii}{J}_{\mathrm{i}}[\mathrm{1000}a(w-{\mathrm{westward}}_1)]}{ka(w-w_1)}\right]}^{\mathrm{two}}{I}_0.$

Now in the case of breathless illumination, P' and Q' were considered as resolved when the master intensity maximum of the one pattern coincided with the first minimum of the other. The intensity at the midpoint (kaw ∼ 1·92) betwixt the ii maxima is then equal to 2[2J _i(1·92)/1·92]² ∼ 0·735 of the maximum intensity of either, i.due east. the combined intensity bend has a dip of about 26·5 per cent between the principal maxima. (This corresponds to the value 19 per cent for a slit aperture–-cf. Fig. 7.62.) If nosotros consider a dip of this amount every bit again substantially determining the limit of resolution, the critical separation w = 2westward ₁ is obtained from the relation

(53) $\begin{array}{cc}\frac{I({\mathrm{west}}_1)}{I(0)}=0.735,& (\mathrm{due\; west}=2w_{\mathrm{one}})\end{array}.$

The first root of this transcendental equation is due west ₁∼2·57/ka, so that the critical separation measured in ordinary units is

(54) $Y^{\prime }=\mathrm{ii}{w}_1\text{D}\prime \sim \frac{2.57}{\pi }\frac{\text{D}\prime {\lambda }^{\prime }}{a}=\frac{0.85{\lambda }^{\prime }}{{\theta }^{\prime }}=\frac{0.82{\lambda }_0}{n^{\prime }{\theta }^{\prime }}.$

To relate Y' to the corresponding separation Y of the object points we use the sine condition (with the approximation sin θ' ∼ θ'), and finally obtain for the limit of resolution with coherent illumination the expression

Apart from a larger numerical gene (which in any case is somewhat arbitrary as it depends on the form of the object and aperture and on the sensitivity of the receptor), we obtain the same expression every bit in the instance of incoherent illumination [eq. (32)]. Thus with light of a given wavelength the resolving ability is once again essentially adamant by the numerical aperture of the objective.

(b) Coherent illumination–-Zernike'due south phase contrast method of observation ^*

We have defined a phase object as one which alters the stage but non the amplitude of the incident wave. An object of this type is of non-uniform optical thickness, only does not absorb any of the incident calorie-free. Such objects are frequently encountered in biology, crystallography, and other fields. Information technology is evident from the preceding discussion that with ordinary methods of ascertainment footling information well-nigh phase objects can be obtained. For the complex amplitude function that specifies the disturbance in the epitome plane is then similar to the transmission function of the object ^* and, every bit the eye (or any other observing instrument) simply distinguishes changes in intensity, one can merely draw conclusions near the amplitude changes but not about the phase changes introduced by the object.

To obtain data nigh phase objects, special methods of observation must exist used, for instance, the so-called central nighttime ground method of observation where the cardinal order is excluded by a stop, or the Schlieren method, where all the spectra on one side of the central order are excluded. The about powerful method, which has the advantage that it produces an intensity distribution which is directly proportional to the phase changes introduced by the object, is due to Zernike ^† and was first described by him in 1935. It is known every bit the stage contrast method.

To explain the principle of the phase dissimilarity method, consider first a transparent object in the form of a one-dimensional phase grating. The transmission role of such an object is by definition (encounter p. 401) of the form

where ϕ(x) is a real periodic office, whose menstruum (d say) is equal to the menstruum of the grating. We assume that the magnitude of ϕ is modest compared to unity, so that we may write

If we develop F into a Fourier series

(58) $F(x)=\underset{m=-\infty }{\overset{\infty }{\Sigma }}{c}_g{e}^{\frac{2\pi i\mathrm{chiliad}x}{d}},$

then, since F is of the form (56) and ϕ is real and numerically small compared to unity,

(59) $\begin{array}{ccc}{c}_0=1,& c_{-k}=-c_m^{*}& (m\ne 0)\end{array}.$

The intensity of the thouth club spectra is proportional to |c _m|².

In the phase contrast method of observation a thin plate of transparent material called the phase plate is placed in the back focal plane F' of the objective and past means of it the phase of the central order (S ₀ in Fig. eight.32) is retarded or advanced with respect to the diffraction spectra (Southward _ane, Due south _-1, South ₂, S _-2, …) by one-quarter of a catamenia. This means that the complex aamplitude distribution in the focal plane is altered from a distribution characterized by the coefficients c ₁₀₀₀ , to a distribution characterized by coefficients c' _g , where

(60) $\begin{array}{ccc}{c^{\prime }}_0=c_0\mathrm{east}{\pm }^{i\pi /2}=\pm i,& {c^{\prime }}_{\mathrm{1000}}=c_m& (m\ne 0)\end{array},$

the positive or negative sign being taken according equally the phase of the central order is retarded or avant-garde. The resulting light distribution in the image plane will now no longer correspond the phase grating (57), but rather a fictitious amplitude grating

Hence the intensity in the prototype aeroplane will at present be proportional to (neglecting ϕ² in comparison to unity)

(62) $I(x^{\prime })=|G(x){|}^{\mathrm{two}}=1\pm 2\varphi (\mathrm{ten}),$

where as before ten' = Mx, M being the magnification. This relation shows that with the phase contrast method of observation, phase changes introduced by the object are transformed into changes in intensity, the intensity at any point of the paradigm airplane existence (apart from an additive constant) direct proportional to the phase change due to the corresponding element of the object. ^* When the phase of the central order is retarded with respect to the diffraction spectra (upper sign in (61)), regions of the object which have greater optical thickness will appear brighter than the mean illumination, and one and then speaks of a bright phase contrast; when the phase of the central gild is advanced, regions of greater spectral thickness will appear darker and one then speaks of a dark phase contrast (Figs. 8.34 and viii.35).

Fig. 8.34. Microscope images of glass fragments (n = 1·52) mounted in clarite (n = 1·54), 100 X. (a), Bright field image; (b) and (c), Stage contrast images; (d), Dark field prototype. (Afterwards A. H. Bennett, H. Jupnik, H. Osterberg, and O. W. Richards, Trans. Amer. Microscop. Soc., 65 (1946), 126.)

Fig. viii.35. Microscope images of epithelium from frog nictitating membrane, 100 X. (a), Brilliant field image at total aperture, N.A. = 0·25; (b), Bright field paradigm with aperture half filled; (c), Phase-contrast image (bright contrast) at total aperture; (d), Phase-contrast image (nighttime contrast) at full aperture. (After A. H. Bennett, H. Jupnik, H. Osterberg, and O. W. Richards. Trans. Amer. Microscop. Soc., 65 (1946), 119.)

To obtain skilful resolution, the aperture of the illuminating system is often of annular rather than round form (cf. § 8.six.2). In this case the annular region of F' through which the directly (undiffracted) light passes plays the role of the key order S ₀ of Fig. 8.32, and it is this light which must and so be retarded or advanced by a quarter period.

The phase-changing plate may be produced by evaporating a sparse layer of a suitable dielectric substance on to a glass substrate. If n is the refractive index of the substance and d the thickness of the layer, then for a retardation of a quarter of a period one must have d = λ/four(n − 1). A retardation of the fundamental guild past this amount is, of course, equivalent to an advance of the diffracted spectra past three-quarters of a menstruation, and vice versa. It is possible to increase the sensitivity of the method by using slightly absorbing instead of a dielectric coating. We shall render to this bespeak later on.

It remains to testify that the phase contrast method is not restricted to stage objects of periodic construction. For this purpose we carve up the integral (34) into ii parts:

(63) $U(\xi ,\eta )=U_0(\xi ,\eta )+U_{\mathrm{ane}}(\xi ,\eta )$

where

(64) $\begin{array}{l}{U}_0=C_1\int {\int }_A{e}^{-\frac{ik}{f}[\xi \mathrm{10}+\eta y]}dxdy\\ U_1=C_1\int {\int }_A[F(\mathrm{ten},y)-\mathrm{i}]{\mathrm{due\; east}}^{-\frac{ik}{f}[\xi x+\eta y]}d\mathrm{ten}dy.\end{array}\}$

U ₀ represents the light distribution that would be obtained in the plane F' if no object were nowadays, whilst U ₁ represents the effect of diffraction. Now the "direct light" U ₀ (respective to the central social club Southward ₀ of Fig. eight.32), will be full-bodied in only a small region B ₀ of the F'-aeroplane, around the axial signal ξ = η = 0. On the other manus a very small fraction of the diffracted light will, in general, accomplish this region, almost of it being diffracted to other parts of this plane. ^*

Suppose that the region B ₀ through which the directly light passes is covered past a stage plate. The effect of the plate may be described by a manual function

For a plate that only retards or advances the light which is incident upon it, a = one; for a plate that as well absorbs calorie-free a < 1. The calorie-free emerging from the aperture volition be represented by

(66) $U^{\prime }(\xi ,\eta )=A{U}_0(\xi ,\eta )+U_{\mathrm{i}}(\xi ,\eta )$

then that, co-ordinate to (36), the distribution of the complex amplitude in the image is given by

(67) $5({\mathrm{10}}^{\prime },y^{\prime })=5_0(x^{\prime },y^{\prime })+V_1({\mathrm{10}}^{\prime },y^{\prime }),$

where

(68) $\begin{array}{l}{V}_0=A{C}_2\int {\int }_B{U}_0(\xi ,\eta )e^{-\frac{i\mathrm{chiliad}}{\text{D}\prime }[{\mathrm{10}}^{\prime }\xi +y^{\prime }\eta ]}d\xi d\eta ,\\ V_{\mathrm{ane}}=A{C}_2\int {\int }_B{U}_1(\xi ,\eta )e^{-\frac{i\mathrm{yard}}{\text{D}\prime }[x^{\prime }\xi +y^{\prime }\eta ]}d\xi d\eta .\end{array}\}$

Now the discontinuity B greatly exceeds in size the region B ₀, and since U ₀ was seen to be practically zero exterior B ₀, no appreciable error is introduced by extending the domain of integration in 5 ₀ over the whole F'-plane. Moreover, if B is assumed to be so large as to admit all the diffracted rays that carry any appreciable energy, the integral for V _i may likewise exist given space limits. Finally, if as earlier the transmission function F(10,y) is defined as zero at points of the object plane outside the region covered past the object, the integrals (64) may besides be taken with infinite limits. Nosotros then obtain, on substituting from (64) into (68), and using the Fourier integral theorem and the relation (38),

(69) $\begin{array}{l}{V}_0({\mathrm{ten}}^{\prime },y^{\prime })=CA,\\ 5_1({\mathrm{ten}}^{\prime },y^{\prime })=C\left[F\left(\frac{{\mathrm{ten}}^{\prime }}{\mathrm{Yard}},\frac{y^{\prime }}{\mathrm{1000}}\right)-1\right]=C[F(\mathrm{10},y)-\mathrm{ane}].\end{array}\}$

From (67) and (69) information technology follows that the intensity in the paradigm plane is given by

(70) $I(x^{\prime },y^{\prime })=|5(x^{\prime },y^{\prime }){|}^2=|C{|}^{\mathrm{two}}|A+F(x,y)-1{|}^2.$

With a phase object

and (lxx) reduces to ^†

(72) $I(x^{\prime },y^{\prime })=|C{|}^2[a^2+\mathrm{ii}\{1-acos\alpha -cos\varphi (x,y)+acos(\alpha -\varphi (x,y))\}].$

Since ϕ was assumed to be modest, (72) may be written equally

(73) $I(x^{\prime },y^{\prime })=|C{|}^2[a^2+2a\varphi (x,y)sin\alpha ],$

and, if the stage difference introduced by the plate represents a retardation or advance by a quarter of a menses, then α = ± π/two and (73) reduces to

(74) $I(x^{\prime },y^{\prime })=|C{|}^2[a^2\pm 2a\varphi (x,y)].$

When the plate does not absorb any of the incident light (a = ane) nosotros accept again the expression (62). The intensity changes are and so directly proportional to the phase variations of the object. With a plate that absorbs a fraction a ² of the direct light the ratio of the 2nd term to the start term in (73) has the value ± ϕ/a, so that the contrast of the image is enhanced. For example, by weakening the straight lite to one-ninth of its original value, the sensitivity of the method is increased three times.

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