chapter 6 frequency analysis of optical imaging...

Chapter 6

Frequency Analysis of Optical Imaging Systems

Prof. Hsuan-Ting Chang

January 5, 2016

Prof. Hsuan-Ting Chang Chapter 6 Frequency Analysis of Optical Imaging Systems

This chapter discusses the role of Fourier analysis in the

theory of coherent and incoherent imaging.

Outline

◮ 6.1 Generalized Treatment of Imaging Systems

◮ 6.2 Frequency Response for Diffraction-Limited CoherentImaging

◮ 6.3 Frequency Response for Diffraction-Limited IncoherentImaging

◮ 6.4 Aberrations and Their Effects on Frequency Response

◮ 6.5 Comparison of Coherent and Incoherent Imaging

◮ 6.6 Resolution Beyond the Classical Diffraction Limit


6.1 Generalized Treatment of Imaging Systems

1. First, we broaden our discussion beyond a single thinpositive lens, finding results applicable to more generalsystems of lenses.

2. Then, remove the restriction to monochromatic light,obtaining results for “quasi-monochromatic” light, bothspatially coherent and spatially incoherent.


6.1.1 A generalized model - 1

1. Suppose that an imaging system of interest is composed, notof a single lens, but perhaps of several lenses, some positive,some negative, with various distances between them.

2. We assume that the system ultimately produces a real imagein space.

3. All imaging elements may be lumped into a single “blackbox,” and that the significant properties of the system can becompletely described by specifying only the terminal

properties of the aggregate.

4. The terminals of the black box consist of the planes containingthe entrance and exit pupils.


Fig. 6.1



5. Through this chapter, we will use the symbols as follows: zorepresents the distance of the plane of the entrance pupil fromthe object plane, zi represents the distance of the plane of exitpupil from the image plane.

6. The distance zi is then the distance that will appear in thediffraction equations that represent the effect of diffraction bythe exit pupil on the point-spread function of the opticalsystem.

7. An imaging system is said to be diffraction-limited if adiverging spherical wave is converted by the system into a newwave, again perfectly spherical, that converges towards anideal point in the image plane, where the location of that idealimage point is related to the location of the original objectpoint through a simple scaling factor (magnification).



8. The terminal property of a diffraction-limited imaging systemis that a diverging spherical wave incident on the entrancepupil is converted by the system into a converging sphericalwave at the exit pupil.

9. If in the presence of a point-source object, the wavefrontleaving the exit pupil departs significantly from ideal sphericalshape, then the imaging system is said to have aberrations.


6.1.2 Effects of Diffraction on the Image – 1

1. The two points of view that regard image resolution as beinglimited by (1) the finite entrance pupil seen from the objectspace or (2) the finite exit pupil seen from the image space areentirely equivalent.

2. Abbe theory: only a certain portion of the diffractedcomponents generated by a complicated object are interceptedby this finite pupil. The components not intercepted areprecisely those generated by the high-frequency components ofthe object amplitude transmittance.



3. This viewpoint is illustrated in Fig. 6.2 for the case of anobject that is a grating with several orders and an imagingsystem composed of a single positive lens.



4. Again the image amplitude Ui is represented by asuperposition integral

Ui(u, v ) =

∫ ∫

∞

−∞

h(u, v ; ξ, η)Uo(ξ, η)dξdη,

where h(u, v ; ξ, η) is the amplitude at image coordinates (u, v )in response to a point-source object at coordinates (ξ, η), andUo is the amplitude distribution transmitted by the object.



5. From the discussion of Section 5.3, the light amplitude aboutthe ideal image point is simply the Fraunhofer diffractionpattern of the exit pupil, centered on image coordinates(u = Mξ, v = Mη.) Thus

h(u, v ; ξ, η) =A

λzi

∫ ∫

∞

−∞

P(x , y )e−j 2π

λzi[(u−Mξ)x+(v−Mη)y ]

dxdy ,

where the pupil function P is unity inside and zero outside theprojected aperture, A is a constant amplitude, zi is thedistance from the exit pupil to the image plane, and (x , y ) arecoordinates in the plane of the exit pupil.



6. By defining reduced coordinates in the object space accordingto

ξ = Mξ, η = Mη,

in which case the amplitude point-spread function becomes

h(u − ξ, v − η) =A

λzi

∫ ∫

∞

−∞

P(x , y )e−j 2π

λzi[(u−ξ)x+(v−η)y ]

dxdy .



7. It is convenient to define the ideal image for a perfect imagingsystem as

Ug (ξ, η) =1

|M |Uo

(

ξ

M,η

M

)

,

yielding a convolution equation for the image,

Ui(u, v ) =

∫ ∫

∞

−∞

h(u − ξ, v − η)Ug (ξ, η)d ξd η,

where

h(u, v ) =A

λzi

∫ ∫

∞

−∞

P(x , y )e−j 2π

λzi(ux+vy)

dxdy .



8. In this general case, for a diffraction-limited system we

can regard the image as being a convolution of the

image predicted by geometric optics with an impulse

response that is the Fraunhofer diffraction pattern of

the exit pupil.


6.1.3 Polychromatic Illumination: The Coherent and

Incoherent Cases – 1

1. The assumption of strictly monochromatic illumination isoverly restrictive because the illumination generated by realoptical sources, including lasers, is never perfectlymonochromatic.

2. In the case of monochromatic illumination it was convenientto represent the complex amplitude of the field by a complexphasor U that was a function of space coordinates. When theillumination is polychromtic but narrowband, this approachcan be generalized by representing the field by a time-varying

phasor that depends on both time and space coordinates.




3. Consider the case of the light that is transmitted by orreflected from an object illuminated by a polychromatic wave.Since the time variations of the phasor amplitude arestatistical in nature, only statistical concepts can provide asatisfactory description of the field.




4. The statistical relationships between the phasor amplitudes atthe various points on the object will influence the statisticalrelationships between the corresponding impulse responses inthe image plane. These statistical relationships will greatlyaffect the result of the time-averaging operation that yieldsthe final image intensity distribution.

5. Two types of illuminations are considered here:(1) Spatially coherent, the phasor amplitudes of the field atall object points vary in unison.(2) Spatially incoherent, the phasor amplitudes at all pointson the object are varying in totally uncorrelated fashions.




6. When the object illumination is coherent, the various impulseresponses in the image plane vary in unison, and thereforemust be added on a complex amplitude basis. Thus a

coherent imaging system is linear in complex amplitude.

7. When the object illumination is incoherent, the variousimpulse responses in the image plane vary in uncorrelated

fashions. They must therefore be added on a power orintensity basis. An incoherent imaging system is linear in

intensity, and the impulse response of such a system is

the squared magnitude of the amplitude impulse

response.




8. It is possible to express the time-varying phasor representationof the image in terms of the convolution of awavelength-independent impulse response with the timevarying phasor representation of the object

Ui(u, v ; t) =

∫ ∫

∞

−∞

h(u − ξ, v − η)Ug (ξ, η; t − τ)d ξd η,

where τ is a time delay associated with propagation from(ξ, η) to (u, v ).




9. To calculate the image intensity, we must time average theinstantaneous intensity represented by |Ui(u, v ; t)|

2. Thus theimage intensity is given by Ii(u, v ) =< |Ui(u, v ; t)|

2 >, or

Ii(u, v ) =

∫ ∫

∞

−∞

d ξ1d η1

×

∫ ∫

∞

−∞

d ξ2d η2h(u − ξ1, v − η1)h∗(u − ξ2, v − η2)

×⟨

Ug (ξ1, η1; t − τ1)U∗

g (ξ2, η2; t − τ2)⟩

.




10. For a fixed image point, the impulse response h is nonzeroover only a small region about the ideal image point.Therefore the integrand is nonzero only for points (ξ1, η1) and

(ξ2, η2) that are very close together.

11. The difference between the time delays τ1 and τ2 is negligibleunder the narrowband assumption, allowing two delays to bedropped.




Ii(u, v ) =

∫ ∫

∞

−∞

d ξ1d η1

×

∫ ∫

∞

−∞

d ξ2d η2h(u − ξ1, v − η1)h∗(u − ξ2, v − η2)

× Jg (ξ1, η1; ξ2, η2),

where

Jg(ξ1, η1; ξ2, η2) =⟨

Ug (ξ1, η1; t)U∗

g (ξ2, η2; t)⟩

is known as the mutual intensity, and is a measure of thespatial coherence of the light at the two object points.




12. When the illumination is coherent, the time-varying phasoramplitudes across the object plane differ only by complexconstants.

Ug (ξ1, η1; t) = Ug (ξ1, η1)Ug (0, 0; t)

< |Ug (0, 0; t)|2 >12

Ug (ξ2, η2; t) = Ug (ξ2, η2)Ug (0, 0; t)

< |Ug (0, 0; t)|2 >12

Jg(ξ1, η1; ξ2, η2) = Ug (ξ1, η1)U∗

g (ξ2, η2)




13. The intensity result is

Ii(u, v ) =

∣

∣

∣

∣

∫ ∫

∞

−∞

h(u − ξ, v − η)Ug (ξ, η)d ξd η

∣

∣

∣

∣

2

.

14. Let Ui be a time-invariant phasor amplitude in the imagespace relative to the corresponding amplitude at the origin.

Ui(u, v ) =

∫ ∫

∞

−∞

h(u − ξ, v − η)Ug (ξ, η)d ξd η,

the same result in the monochromatic case.




15. When the object illumination is perfectly incoherent, thephasor amplitudes across the object vary in statisticallyindependent fashions.⟨

Ug (ξ1, η1; t)U∗

g (ξ2, η2; t)⟩

= κIg(ξ1, η1)δ(ξ1 − ξ2, η1 − η2),

where κ is a real constant.

16. When used in Eq. (6.9), we can obtain

Ii(u, v ) = κ

∫ ∫

∞

−∞

|h(u − ξ, v − η)|2Ig(ξ, η)d ξd η (6− 15)




17. For incoherent illumination, the image intensity is found as aconvolution of the intensity impulse response |h|2 with theideal image intensity Ig .

18. An incoherent imaging system is linear in intensity, rather thanamplitude.

19. Furthermore, the impulse response of incoherent mapping isjust the squared modulus of the amplitude impulse response.


6.2 Frequency Response for Diffraction-limited Coherent

Imaging

1. A coherent imaging system is linear in complex amplitude.Such a system provides a highly nonlinear intensity mapping.

2. If frequency analysis is to be applied in its usual form, it mustbe applied to the linear amplitude mapping.


6.2.1 The Amplitude Transfer Function – 1

1. The transfer-function concepts can be applied directly to thecoherent system, provided it is done on an amplitude basis.

2. Some frequency spectra of the input and output are defined as

Gg (fX , fY ) =

∫ ∫

∞

−∞

Ug (u, v ) exp[−j2π(fXu + fY v )]dudv

Gi(fX , fY ) =

∫ ∫

∞

−∞

Ui(u, v ) exp[−j2π(fXu + fY v )]dudv .

3. The amplitude transfer function H is defined as the Fouriertransform of the space-invariant amplitude impulse(point-spread) response h(u, v ),

H(fX , fY ) =

∫ ∫

∞

−∞

h(u, v ) exp[−j2π(fXu + fY v )]dudv .

(6.17)Prof. Hsuan-Ting Chang Chapter 6 Frequency Analysis of Optical Imaging Systems

6.2.1 The Amplitude Transfer Function – 2

4. By applying the convolution theorem, the effects of thediffraction-limited imaging system has been expressed in thefrequency domain:

Gi(fX , fY ) = H(fX , fY )Gg (fX , fY )

5. H itself is a Fraunhofer diffraction pattern and can beexpressed as a scaled Fourier transform of the pupil function:

H(fX , fY ) = F

{

A

λzi

∫ ∫

∞

−∞

P(x , y ) exp{−j2π

λzi(ux + vy )}dxdy

}

= (Aλzi)P(−λzi fX ,−λzi fY ).

6. For notational convenience, we set Aλzi = 1 and ignore thenegative signs in P. Thus

H(fX , fY ) = P(λzi fX , λzi fY )

This relation is of the utmost importance!Prof. Hsuan-Ting Chang Chapter 6 Frequency Analysis of Optical Imaging Systems

6.2.2 Examples of Amplitude Transfer Functions – 1

1. The frequency components outside the passband arecompletely eliminated.

2. If the pupil function P is indeed unity within some region andzero otherwise, then there exists a finite passband in thefrequency domain with which the diffraction-limited imagingsystem passes all frequency components without amplitude orphase distortion.


6.2.2 Examples of Amplitude Transfer Functions – 2

3. Consider the amplitude transfer functions of systems withsquare (width 2w) and circular (diameter 2w) pupils.

P1(x , y ) = rect(x

2w) rect(

y

2w)

P2(x , y ) = circ(

√

x2 + y 2

w)

The corresponding amplitude transfer functions are

H1(fX , fY ) = rect(λzi fX2w

) rect(λzi fY2w

)

H2(fX , fY ) = circ(

√

f 2X + f 2Yw/λzi

)


6.2.2 Examples of Amplitude Transfer Functions - 3

4. A cutoff frequency f0 can be defined in both cases by

f0 =w

λzi

5. Figure 6.3


6.3 Frequency Response for Diffraction-limited Incoherent

Imaging

6.3.1 The Optical Transfer Function

1. Incoherent imaging systems have been seen to obey theintensity convolution integral

Ii(u, v ) = κ

∫ ∫

∞

−∞

|h(u − ξ, v − η)|2Ig(ξ, η)d ξd η

Such systems should be frequency-analyzed as linearmappings of intensity distributions.


6.3.1 The Optical Transfer Function – 1

1. Let the normalized frequency spectra of Ig and Ii be defined by

Gg (fX , fY ) =

∫ ∫

∞

−∞Ig(u, v ) exp[−j2π(fXu + fY v )]dudv

∫ ∫

∞

−∞Ig (u, v )dudv

Gi(fX , fY ) =

∫ ∫

∞

−∞Ii(u, v ) exp[−j2π(fXu + fY v )]dudv

∫ ∫

∞

−∞Ii(u, v )dudv

This is a normalization of the spectra by their“zero-frequency” values because both Ig and Ii have aFourier transform which achieves its maximum value at theorigin.



2. The normalized transfer functions of the system is defined by

H(fX , fY ) =

∫ ∫

∞

−∞|h(u, v )|2 exp[−j2π(fXu + fY v )]dudv

∫ ∫

∞

−∞|h(u, v )|2dudv

3. With the convolution theorem, the frequency-domain relation

Gi(fX , fY ) = H(fX , fY )Gg (fX , fY )

4. The function H is known as the optical transfer function

(OTF) of the system. Its module |H| is known as themodulation transfer function (MTF).



5. The definitions of both the ATF and OTF involve theamplitude impulse response h. Their relations are

H(fX , fY ) = F{h}

and

H(fX , fY ) =F{|h|2}

∫ ∫

∞

−∞|h(u, v )|2dudv



6. With the help of Rayleigh’s theorem

H(fX , fY ) =

∫ ∫

∞

−∞H(p′, q′)H∗(p′ − fX , q

′ − fY )dp′dq′

∫ ∫

∞

−∞|H(p′, q′)|2dp′dq′

7. Using change of variables, p = p′ − fX2and q = q′ − fY

2, results

in the symmetrical expression

H(fX , fY ) =

∫ ∫

∞

−∞H(p + fX

2, q + fY

2)H∗(p − fX

2, q − fY

2)dpdq

∫ ∫

∞

−∞|H(p, q)|2dpdq

8. The OTF is the normalized autocorrelation function of

the ATF!


6.3.3 The OTF of an Aberration-Free System – 1

1. Consider a special case of a diffraction-limited incoherentsystem.

H(fX , fY ) = P(λzi fX , λzi fY )

2. it follows from Eq. (6.28) that

H(fX , fy )

=∫ ∫

∞

−∞P(x+

λzi fX2

,y+λzi fY

2)P∗(x−

λzi fX2

,y−λzi fY

2)dxdy

∫ ∫∞

−∞P2(x ,y)dxdy

where P equals either unity or zero such that P2 can bereplaced by P.


6.3.3 The OTF of an Aberration-Free System – 2

3. The numerator represents the area of overlap of two displacedpupil functions, one centered at (λzi fX/2, λzi fY /2) and theother centered at (−λzi fX/2,−λzi fY /2).

4. The denominator simply normalizes the area of overlap by thetotal area of the pupil.

5. Thus

H(fX , fy) =area of overlap

total area

6. Geometrical interpretation of the OTF implies that OTF of adiffraction-limited system is always real and nonnegative.

7. See Fig. 6.4 for an example.


Fig. 6.4


6.3.4 Examples of Diffraction-Limited OTFs – 1

1. Consider the examples of the OTFs corresponding todiffraction-limited systems with square (width 2w) and circular(diameter 2w) pupils.

2. Figure 6.6: The area of overlap is evidently

A(fX , fY ) =

(2w − λzi |fX |)(2w − λzi |fY |),|fX | ≤ 2w/λzi|fY | ≤ 2w/λzi

0, otherwise


Fig. 6.6



3. When this area is normalized by the total area 4w 2, the resultbecomes

H(fX , fy ) = Λ(fX

2f0)Λ(

fY

2f0)

where Λ is the triangle function and f0 is the cutoff frequencyof the same system when used with coherent illumination,

f0 =w

λzi

4. The cutoff frequency of the incoherent system is at 2f0 alongthe fX and fY axes. Figure 6.7 illustrates the OTF.


Fig. 6.7



5. Figure 6.8 shows the circular case. The area of overlap isequal to four times of the shaded area B of the circular sectionA+ B .

Area(A+ B) =θ

2π(πw 2) =

cos−1(λzi fX/2w)

2π(πw 2)

while

Area(A) =1

2(λzi fX2

)

√

w 2 − (λzi fX2

)2

6. Finally, we have

H(fX , 0) =4[area(A+ B)− area(A)]

πw 2

or, for a general radial distance ρ in the frequency plane,

H(ρ) =

{

2π

[

cos−1( ρ

2ρ0)− ρ

2ρ0

√

1− ( ρ

2ρ0)2]

ρ ≤ 2ρ0

0 otherwise.


Fig. 6.8



7. The quantity ρ0 is the cutoff frequency of the coherent system,

ρ0 =w

λzi

Referring to Fig. 6.9, the OTF is again seen to extend to afrequency that is twice the coherent cutoff frequency.


Fig. 6.9


6.4 Aberrations and Their Effects on Frequency Response

1. Diffraction-limited system revisited: the presence of apoints-source object yielded at the exit pupil a perfect sphericalwave, converging toward the ideal geometrical image point.

2. We now consider the effects of aberrations, or departure ofthe exit-pupil wavefront from idea spherical form. It can raisefrom a defect as simple as a focusing error to inherentproperties of perfectly spherical lenses.


6.4.1 The Generalized Pupil Function – 1

1. When wavefront errors exist, we can imagine that the exitpupil is illuminated by a perfect spherical wave, but that aphase-shifting plate exists in the aperture, thus deforming thewavefront leaves the pupil.

2. The generalized pupil function is defined as

P(x , y ) = P(x , y ) exp[jkW (x , y )]

3. The amplitude point-spread function of an aberrated coherentsystem is simply the Fraunhofer diffraction pattern of anaperture with amplitude transmittance P.


6.4.1 The Generalized Pupil Function – 2

4. Figure 6.10 shows the geometry that defines the aberrationfunction W . If we trace a ray backward from the idea imagepoint to the coordinates (x , y ) in the exit pupil, the aberrationfunction W (x , y ) is the path-length error accumulated by thatray as it passes from the Gaussian reference sphere to theactual wavefront.


Fig. 6.10


6.4.2 Effects of Aberrations on the Amplitude Transfer

Function

1. When aberrations are present, the generalized pupil function Preplaces P. Thus the ATF is written

H(fx , fY ) = P(λzi fx , λzi fY )

= P(λzi fx , λzi fY ) exp[jkW (λzi fx , λzi fY )]

2. The sole effect of aberrations is seen to be the introduction ofphase distortion within the passband.


6.4.3 Effects of Aberrations on the OTF – 1

1. Let the function A(fX , fY ) be defined as the area of overlap of

P(x −λzi fX2

, y −λzi fY2

)

and

P(x +λzi fX2

, y +λzi fY2

)

2. Thus the new OTF of a diffraction-limited system is given by

H(fX , fY ) =

∫ ∫

A(fX , fY )dxdy

∫ ∫

A(0, 0)dxdy



3. When aberrations are present, substitution yields

H(fX , fY ) =

∫ ∫

A(fX , fY )e jk[W (x+

λzi fX2

,y+λzi fY

2)−W (x−

λzi fX2

,y−λzi fY

2)]dxdy

∫ ∫

A(0, 0)dxdy

(6− 36)



4. The aberration will never increase the MTF. Schwarz’sinequality can be used to prove this property.

5. The aberration in general will lower the contrast; the absolutecutoff frequency remains unchanged, but severe aberrationscan reduce the high-frequency portions of the OTF such thatthe effective cutoff frequency is much lower than thediffraction-limited cutoff.

6. In addition, aberrations can cause the OTF to have negative

values in certain bands of frequencies. Thus imagecomponents at that frequency undergo a contrast reversal.


6.4.4 Example of a simple aberration: A focusing error – 1

1. Consider a square aperture. A focus error means that thecenter of curvature of the spherical wavefront convergingtowards the image of an object point-source lies either to theleft or to the right of the image plane.

2. The phase distribution across the exit pupil is of the form

φ(x , y ) = −π

λza(x2 + y 2),

where za 6= zi .



3. The path length error W (x , y ) can be determined by

kW (x , y ) =−π

λza(x2 + y 2) +

π

λzi(x2 + y 2)

W (x , y ) =−1

2(1

za−

1

zi)(x2 + y 2)

4. For a square aperture of width 2w , the maximum path-lengtherror Wm at the edge of the aperture along the x or y axis is

Wm =−1

2(1

za−

1

zi)w 2

which is a convenient indication of the severity of the focusingerror.



5. Now the path-length error can be expressed as

W (x , y ) = Wm

x2 + y 2

w 2



6. Substituting this equation into Eq. (6-36) for the OTF,

H(fX , fY ) = Λ(fX

2f0)Λ(

fY

2f0)

× sinc

[

8Wm

λ(fX

2f0)(1−

|fX |

2f0)

]

× sinc

[

8Wm

λ(fY

2f0)(1−

|fY |

2f0)

]



7. Fig. 6.11

8. The diffraction-limited OTF is indeed obtained when Wm = 0.

9. For values of Wm greater than λ/2, sign reversals of the OTFoccurs.


Fig. 6.11



10. Fig. 6.12 shows the reversal of contrast.

11. The “local spatial frequency” of this target changes slowly,increasing as the radius from the center is decreased.

12. When the system is out of focus, a gradual attenuation ofcontrast and a number of contrast reversals are obtained forincreasing spatial frequency.


Fig. 6.12



13. Finally, consider Wm ≫ λ. The frequency response dropstowards zero for relatively small values of fX/2f0 and fY /2f0.So,

1−|fX |

2f0≈ 1, 1−

|fY |

2f0≈ 1,

and the OTF reduces to

H(fX , fY ) ≈ sinc

[

8Wm

λ(fX

2f0)

]

sinc

[

8Wm

λ(fY

2f0)

]

.



14. When aberrations of any kind are severe, the geometricaloptics predictions of the intensity point-spread function maybe Fourier-transformed to yield a good approximation to theOTF of the system.

15. When severe aberrations are present, the PSF is determinedprimarily by geometrical-optics effects, and diffraction plays anegligible role in determining its shape. (Fig. 6.13)


6.4.5 Apodization and its effects on frequency response –

1

1. The PSF of a diffraction-limited imaging system generally hassidelobes or side-rings of noticeable strength.

2. To reduce the strength, apodization methods have beendeveloped. The word “apodize” means “to remove the feet.”

3. Similar techniques are well known in the field of DSP, termedas “windowing.”



2

4. Apodization leads to the introduction of attenuation in theexit pupil of an imaging system, attenuation that may beinsignificant at the center but increases with distance awayfrom the center.

5. Fig. 6.14(a) shows the unapodized and apodized intensitytransmissions through a square pupil with and without aGaussian intensity apodization.


Fig. 6.14



3

6. Fig. 6.14(b) shows the crosssections of the intensitypoint-spread functions for the two cases.

7. The side lobes have been significantly suppressed byapodization, the width of the main lobe is increased, and themaxmium intensity is also reduced due to extra absorption inthe pupil.



4

8. Attenuation that increases with distance from the center ofthe pupil results in an amplitude transfer function that falls offmore rapidly with increasing frequency than it would in theabsence of apodization.

9. Fig. 6.15


Figs. 6.15 & 6.16


6.5 Comparison of Coherent and Incoherent Imaging

1. ATF and OTF are not directly comparable since the cutofffrequency of- ATF determines the maximum frequency component of theimage amplitude

- OTF determines the maximum frequency component of theimage intensity.

2. The OTF of a diffraction-limited system extends to afrequency that is twice the cutoff frequency of the ATF.

3. Is that incoherent illumination will yield “better” resolutionthan coherent illumination true? NO!


6.5.1 Frequency Spectrum of the Image Intensity – 1

1. Attribute of the image intensity for comparison in the twocases: frequency spectrum

2. In incoherent case, the image intensity is

Ii = |h|2 ⊗ Ig = |h|2 ⊗ |Ug |2

3. In coherent case,Ii = |h ⊗ Ug |

2



4. Let “⋆” represent the autocorrelation integral

X (fX , fY ) ⋆X (fX , fY ) =

∫ ∫

∞

−∞

X (p, q)X ∗(p− fX , q− fY )dpdq

5. The frequency spectra in two cases can be written as

Incoherent: F{Ii} = [H ⋆ H][Gg ⋆ Gg ]

Coherent: F{Ii} = HGg ⋆ HGg

where Gg is the spectrum of Ug and H is the ATF. Thefrequency content can be quite different in the two cases.



6. Consider two objects with the same intensity transmittancebut different phase distributions:

A : tA(ξ, η) = cos 2πf ξ

B : tB(ξ, η) = | cos 2πf ξ|

7. Fig. 6-17: Calculation of the spectrum of the image intensityfor object A.

8. The contrast of the image intensity distribution is poorer forthe incoherent case than for the coherent case. Thus object Ais imaged better in coherent light than in incoherent light.

9. For object B, the fundamental frequency is now 2f and2f > fo (fo is the cutoff frequency of the amplitude transferfunction). No variations of image intensity will be present forthe coherent case, while the incoherent system will form thesame image it did for object A(?).


Fig. 6.17



10. Which particular type of illumination is better from the pointof view of image spectral content depends very strongly on thedetailed structure of the object, and in particular on its phasedistribution.


6.5.2 Two-Point Resolution - 1

1. Rayleigh criterion of resolution: two incoherent point sourcesare “barely resolved” by a diffraction-limited system with acircular pupil when the center of the Airy intensity patterngenerated by one point source falls exactly on the first zero ofthe Airy pattern generated by the second.

2. The minimum resolvable separation of the geometrical imagesis δ = 0.61λzi/w and in nonparaxial caseδ = 0.61 λ

sin θ= 0.61 λ

NA, where θ is the half-angle subtended by

the exit pupil when viewed from the image plane, andNA = sin θ is the numerical aperture of the optical system.

3. Fig. 6.18: The central dip is found to fall about 27% belowpeak intensity.


Fig. 6.18


6.5.2 Two-Point Resolution - 2

4. Whether the two point-source objects, separated by the sameRayleigh distance δ, would be easier or harder to resolve withcoherent illumination than with incoherent illumination? Theanswer is found to depend on the phase distribution associatedwith the object.

5. Fig. 6.19 shows the distributions of image intensity for pointsource in phase (φ = 0, harder to resolve), in quadrature(φ = π/2, identical to Fig. 6.18), and in phase opposition(φ = π, easier to resolve).


Fig. 6.19


6.5.3 Other Effects

1. Figs. 6.20 and 6.21: Ringing effect for step-function objects.

2. Fig. 6.22: Speckle effect.


Fig. 6.20


Fig. 6.21


Fig. 6.22


6.6 Resolution Beyond The Classical Diffraction Limit

1. In principle, for a certain class of objects, resolution beyondthe classical diffraction limit is theoretically possible.

2. Often referred to as super-resolution or bandwidthextrapolation.


6.6.1 Underlying Mathematical Fundamentals

1. Theorem 1. The 2-D Fourier transform of a spatiallybounded function is an analytic function in the (fX , fY ) plane.

2. Theorem 2. If an analytic function in the (fX , fY ) plane isknown exactly in an arbitrarily small (but finite) region of thatplane, then the entire function can be found (uniquely) bymeans of analytic continuation.

3. If the finite portion of the object spectrum can be determinedexactly from the image, then, for a bounded object, the entireobject spectrum can be found by analytic continuation.


6.6.2 Intuitive Explanation of Bandwidth Extrapolation - 1

1. Fig. 6.23: Let the object be a cosinusoidal intensitydistribution of finite extent, with a frequency that exceeds theincoherent cutoff frequency.

2. The finite-length cosine function:

Ig(u) =1

2

[

1 +m cos(2πf u)]

rect(u

L)

and the spectrum of this intensity distribution is

Gg (fX ) = sinc(LfX ) +m

2sinc[L(fX − f )] +

m

2sinc[L(fX + f )],

along with the OTF H(fX ) of the imaging system.


Fig. 6.23


6.6.2 Intuitive Explanation of Bandwidth Extrapolation - 2

3. The finite width of the cosinusoid has spread its spectralcomponents into sinc functions, and the frequency f liesbeyond the limits of the OTF. Nonetheless, the tails of thesinc functions centered at fX = ±f extend below the cutofffrequency into the observable part of the spectrum.

4. Within the passband, there does exist information thatoriginated from the consinusoidal components that lie outsidethe passband. To achieve super-resolution, it is necessary toretrieve these extremely weak components and utilize them insuch a way as to recover the signal that gave rise to them.


6.6.3 An Extrapolation Method Based on the Sampling

Theorem - 1

1. Approaches for the problem of bandwidth extrapolation:1.1 Based on the sampling theorem in the frequency domain.1.2 Based on prolate spheroidal wave-function expansions.1.3 An iterative approach suitable for digital implementation that

successively reinforces constraints in the space andspace-frequency domains.

2. Here we focus on the sampling-theorem approach due to itssimplicity. And we treat only the 1-D restoration problem.



Theorem - 2

3. Suppose that a 1-D incoherent object with intensitydistribution Ig(u) is bounded to the region (−L/2, L/2) on theu axis.

4. By the Whittaker-Shannon sampling theorem, the objectspectrum Gg (f ) can be written as

Gg (f ) =∞∑

n=−∞

Gg (n

L)sinc

[

L(f −n

L)]

.

5. Due to the limited passband of the optical system, values ofGg (

nL) can be found only for a few low-integer values of n.



Theorem - 3

6. Now, we want to extend our knowledge of the spectrum tolarger values, say for −N ≤ n ≤ N, so that

Gg (f ) ≈

N∑

n=−N

Gg (n

L)sinc

[

L(f −n

L)]

7. This representation of the image is not only within thepassband, but also the outside that passband over a frequencyregion of a size that depends on how large an N is chosen.

8. To determine the sample value outside the observablepassband, the values of Gg (f ) at any 2N + 1 distinctfrequencies fk within the passband are measured. The fk ingeneral will not coincide with the sampling points n/L.



Theorem - 4

9. The value of the object spectrum measured at frequency fkwithin the passband is represented by Gg (fk).

Gg (fk) =N∑

n=−N

Gg (n

L)sinc

[

L(fk −n

L)]

, k = 1, 2, . . . , 2N +1.

(6− 49)

10. This is a set of 2N + 1 linear equations in 2N + 1 unknowns,the Gg (f ).

11. Consider this problem in matrix form. Define a column vector~g consisting of the 2N + 1 unknown values of Gg (

nL) and a

column vector g consisting of the 2N + 1 measured values ofGg (fk).



Theorem - 5

12. It defines a (2N + 1)× (2N + 1) matrix D with entrysinc[L(fk −

nL)] in the kth row and nth column. Then the set

of Equations (6-49) can be represented by g = D~g .

13. Goal: find the vector ~g from which we can construct asatisfactory extension of the spectrum Gg beyond the normalcutoff frequency. ~g = D−1g

14. As long as the measurement frequencies fk are distinct, thedeterminant of D is nonzero, the inverse exists. Thus inprinciple, the sample values of the object spectrum outside thepassband can be determined.


6.6.4 An Iterative Extrapolation Method - 1

1. It was first applied to the bandwidth extrapolation problemfirst by Gerchberg [116] and by Papoulis [228].

2. Fig. 6.24 shows a block diagram of the iterative extrapolationalgorithm.

3. The algorithm iterates between the object domain and thespectral domain, making changes in each domain to reinforceprior knowledge or measured data.


Fig. 6.24


6.6.5 Practical Limitations

1. All methods are extremely sensitive to both noise in themeasured data and the accuracy of the assumed a prioriknowledge.

2. It is generally agreed that the Rayleigh limit to resolution

represents a practical limit to the resolution that can be

achieved with a conventional imaging system.


chapter 6 frequency analysis of optical imaging...

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