Chapter 3

Foundation of Generalized Phase Contrast:

Mathematical Analysis of Common-Path Interferometers

A well-focused standard microscope forms a transparent image of a transparent object, and hence will not be suitable for visualizing the features of such an object. However, although the features remain invisible, the transparent image replicates the characteristic phase variations of the object. Thus the spatial features of the transparent object can be visualized by subjecting its transparent image to interferometric measurements. Rather than introducing an external reference beam, a common-path interferometer synthesizes the reference wave using the undiffracted light from the object. A phase contrast micro-scope, such as that of Zernike, can therefore be built around a common-path interfer-ometer. Unfortunately, the assumptions in a standard Zernike phase contrast analysis limit its domain of applicability to objects satisfying the small-scale phase approximation. Moreover, the use of a plane-wave approximation for the illumination implies a similar description of the undiffracted beam, which then mathematically focuses to a Dirac delta function and requires a matching but unphysical Dirac-delta filter.

In this chapter, we use a common-path interferometer as a model system for develop-ing the generalized phase contrast (GPC) method. The resulting generalized formula-tion encompasses input wavefronts with an arbitrarily wide phase modulation range and realistically incorporates physical effects arising from intrinsic system apertures. Gener-alized phase contrast not only allows us to prescribe optimal parameters in phase con-trast microscopy but also enables us to expand phase contrast applications into novel contexts beyond microscopy that will be treated in succeeding chapters.

3.1 Common-Path Interferometer: a Generic Phase Contrast Optical System

A commonly applied architecture for common-path interferometry is illustrated in Fig.

3.1. This architecture is based on the so-called 4f optical processing configuration and

provides an efficient platform for spatial filtering. An output interferogram of an

14 3 Foundation of Generalized Phase Contrast

unknown phase object or phase disturbance is obtained by applying a truncated on-axis filtering operation at the spatial frequency domain between two Fourier transforming lenses (L1 and L2). The first lens performs a spatial Fourier transform so that directly propagating, undiffracted light is focused to the on-axis filtering region whereas light representing the spatially varying object information is scattered to locations outside this central region.

Fig. 3.1Fig. 3.1Fig. 3.1Fig. 3.1 Generic CPI based on a 4f optical system (lenses L1 and L2). The input phase disturbance is

shown as an aperture-truncated phase function, ( ),x yφ , which generates an intensity distribution,

I(x’,y’), in the observation plane by an on-axis filtering operation in the spatial frequency plane. The values

of the filter parameters (A, B, θ) determine the type of filtering operation.

We can describe a general Fourier filter in which different phase shifts and amplitude

damping factors are applied to the “focused” and “scattered” light. In Fig. 3.1, we show a circularly symmetric Fourier filter described by the amplitude transmission factors, A

and B, as well as phase shifts, θA and θB, for the “scattered” and “focused” light, respec-tively. For simplicity, all of the succeeding discussions will simply characterize the filter

using the relative phase shift, θ = θB – θA, since, after all, it is the relative phase, θ, that affects the output and not the actual values of θA and θB. The parameters A, B and θ provides a generalized filter specification and properly choosing their values can repli-cate any one of a large number of commonly used spatial filtering types (i.e. phase contrast, dark central ground, point diffraction and field-absorption filtering). Without a Fourier filter, the second Fourier lens will simply perform an inverse transform, albeit with reversed coordinates, to form an inverted image of the input phase variations at the observation plane. By applying a Fourier filter, the second Fourier lens transforms the phase-shifted, focused on-axis light to act as a synthetic reference wave (SRW) at the

3.2 Field Distribution at the Image Plane of a CPI 15

output plane. The SRW interferes with the scattered light to generate an output inter-ference pattern that reveals features of the input phase modulation. In the following section we discuss the importance of the SRW and show how it influences, among other things, the choice of the Fourier filter parameters.

3.2 Field Distribution at the Image Plane of a CPI

Having described the generic optical system that makes up the CPI, we now turn to a detailed analytical treatment of the important elements in this system. Let us assume a circular input aperture with radius r∆ truncating the phase disturbance modulated onto a collimated, unit amplitude, monochromatic field of wavelength λ . We can

describe the complex amplitude, ( ),a x y , of the light emanating from the entrance plane

of the optical system shown in Fig. 3.1 by,

( ) ( ) ( ), circ exp ,a x y r r j x yφ = ∆ , (3.1)

using the definition that the circ-function is unity within the region 2 2r x y r= + ≤ ∆

and zero elsewhere. Similarly, we assume a circular on-axis centred spatial filter of the form:

( ) ( )( ) ( )1, 1 exp 1 circx y r rH f f A BA j f fθ− = + − ∆ (3.2)

where [ ]0; 1B∈ is the chosen filter transmittance of the focused light, [ ]0; 2θ π∈ is

the applied phase shift to the focused light and [ ]0; 1A∈ is a filter parameter describing

field transmittance for off-axis scattered light as indicated in Fig. 3.1. The spatial fre-quency coordinates are related to spatial coordinates in the filter plane such that:

( ) ( ) ( )1, ,x y f ff f f x yλ

−= and 2 2

r x yf f f= + [1].

Assuming, for simplicity, a unity-magnification imaging, the output field is obtained by performing an optical Fourier transform (denoted by { }ℑ operator) of the input

field from Eq. (3.1) followed by a multiplication of the filter parameters in Eq. (3.2) and a second optical Fourier transformation (Note: from here on, the second Fourier

operation is replaced by the inverse Fourier operation { }1−ℑ since their only difference

is a negation of coordinates, which is an image inversion of the image in the output plane). In mathematical form, the sequence of operations is shown below:

( ) ( ){ }{ } ( )( ) ( )

( )( ) ( ) ( ) ( ){ }{ }

1

1 1

, , exp ', ' circ '

exp 1 circ circ exp( ,

x y

r r

H f f a x y A j x y r r

BA j f f r r j x y

φ

θ φ

−

− −

ℑ ℑ = ∆

+ − ℑ ∆ ℑ ∆

(3.3)

16 3 Foundation of Generalized Phase Contrast

3.2.1 Assumption on the Phase Object’s Spatial Frequency Components

The task of performing the inverse Fourier transform operation, { }1−ℑ , at the right-

hand side of Eq. (3.3) is facilitated by a proper understanding of the different quantities

that factor into the operand given by ( ) ( ) ( ){ }circ circ exp( ,r rf f r r j x yφ∆ ℑ ∆ . From

the properties of the Fourier transform, this can be rewritten as

( ) ( ){ } ( ){ }( )circ circ exp( ,r rf f r r j x yφ∆ ℑ ∆ ⊗ ℑ , where ⊗ denotes the convolution

operation. The appearance of these three quantities, ( )circ r rf f∆ , ( ){ }circ r rℑ ∆ and

( ){ }exp( ,j x yφℑ , at the Fourier or spatial frequency plane are considered. The first two

terms are both circularly symmetric (or have no azimuthal dependence) and thus can both be illustrated in a 1D radial plot with spatial frequency rf as the horizontal axis, as shown

in Fig. 3.2.

Fig. 3.2Fig. 3.2Fig. 3.2Fig. 3.2 Relevant Fourier plane quantities: phase-shifting region, circ(fr/∆fr) (dashed); Airy function

ℑ{circ(r/∆r)} (dotted), and object frequency components ℑ{exp(jφ(x, y))} (arrows). frmin indicates the

smallest radial spatial frequency component of a given phase object.

However, the third term, ( ){ }exp( ,j x yφℑ , is in general not circularly symmetric but

may be characterized by diffraction orders or a sum of weighted Dirac-delta functions centered at various locations at the spatial frequency plane. To one of these delta func-

tions, specifically the zero-order or on-axis centered ( )rfδ , we assign the generally

complex-valued weight α derived from the Fourier analysis of a given input phase by Eq. (2.3). Among the remaining higher-orders that exist as off-axis delta functions, at least one (first-order) would be closest to the origin and, based on its frequency compo-

nents, its location on the frequency plane can be denoted as ( )min min,x yf f . The rf -axis

of the radial plot shown in Fig. 3.2 has been chosen to pass through ( )min min,x yf f ,

which enables us to plot the diffraction orders along the line defined by the origin and

rf+∆ rf−∆ ,minrf+∆ 0rf =

rf

3.2 Field Distribution at the Image Plane of a CPI 17

the point ( )min min,x yf f at the frequency plane. Thus, the higher-order that appears

closest to the origin intersects the rf -axis at a radial frequency 2 2min min minr x yf f f= + .

The third term appears as part of the convolution

( ){ } ( ){ }circ exp( ,r r j x yφℑ ∆ ⊗ ℑ , which generates a sum of weighted Jinc functions

( ( ){ }circ r r∝ ℑ ∆ ), each centered at the location of the corresponding delta function.

If the minimum non-zero frequency, minrf , is set to satisfy the condition

min 1rf r>> ∆ , (3.4)

then there will be a negligible overlap (refer to the arrows and the dotted line in Fig. 3.2) between the on-axis centered Jinc function and the closest neighboring Jinc function centered at minrf . Adding the cautious condition

min 2r rf f∆ < , (3.5)

we find that the term ( ) ( ){ } ( ){ }( )circ circ exp( ,r rf f r r j x yφ∆ ℑ ∆ ⊗ ℑ , which evalu-

ates the convolution within the phase shifting region of the Fourier filter, may be

approximated by ( ){ }circ r rℑ ∆ itself multiplied by the complex-valued term α .

Recalling that α is the coefficient of ( )rfδ in Fig. 3.2, we can calculate it as the average

of the phase object over the input area (i.e. Eq. (2.3) as written below:

( ) ( )2 2

12( ) exp , d d expx y r

r j x y x y j αα π φ α φ−

+ ≤∆ = ∆ = ∫∫ . (3.6)

Upon incorporating all these considerations, the inverse Fourier transform in Eq. (3.3) can then be evaluated as

( ) ( ){ }{ }

( )( ) ( ) ( )( ) ( )φ α θ

−

−

ℑ ℑ ≅

∆ + −

1

1

, ,

exp ', ' circ ' exp 1 ' ,

x yH f f a x y

A j x y r r BA j g r (3.7)

where

( ) ( ) ( ){ }{ } ( ) ( )11 0

0' circ circ 2 2 2 ' d

rf

r r r r rg r f f r r r J rf J r f fπ π π∆−= ℑ ∆ ℑ ∆ = ∆ ∆∫ (3.8)

can be considered as the generating function for the synthetic reference wave (SRW). Finally, the output intensity (or irradiance) is obtained by taking the squared modulus of the field given in Eq. (3.7). The generally complex-valued and object-dependent term, α , corresponding to the amplitude of the focused light plays a significant role in the expres-sion for the interference pattern described by Eq. (3.7). Referring to the discussion in Chapter 2, we are now able to confirm that the frequent assumption, that the amplitude of the focused light is approximately equal to the first term of the Taylor expansion in Eq. (2.3), can generally result in misleading interpretations of the interferograms generated at the CPI output when unwittingly applied beyond its limited domain of validity.

18 3 Foundation of Generalized Phase Contrast

3.2.2 The SRW Generating Function

To summarise, we performed an optical Fourier transform of the input field from Eq. (3.1), filtered the components by a multiplication with the filter transfer function in Eq. (3.2) and then subjected the product to a second optical Fourier transform (corre-sponding to an inverse Fourier transform with inverted coordinates). As a result, we

obtained an expression for the intensity ( )', 'I x y describing the interferogram at the

observation plane of the 4f CPI set-up:

( ) ( ) ( ) ( ) ( )2

2 1', ' exp ', ' circ ' exp 1 'I x y A j x y r r BA j g rφ α θ− = ∆ + − (3.9)

To proceed, we need to find an accurate working expression for the SRW to com-plement the derived output intensity expression. Earlier, we used the zero-order Hankel

transform [1] to describe the SRW generating function, g(r´). For an applied circular

input aperture with radius, r∆ , and a Fourier filter whose central phase shifting region corresponds to a spatial frequency radius, rf∆ , we obtained the following expression for

the SRW generating function by use of the zero-order Hankel transform (c.f. Eq. (3.8)):

( ) ( ) ( )1 00

' 2 2 2 ' drf

r r rg r r J rf J r f fπ π π∆

= ∆ ∆∫ (3.10)

As is evident from its origin in Eq. (3.8), the SRW generating function incorporates effects due to the finite extent of the input aperture and the phase-shifting central region of the filter. This suggests that, due to its influence on the SRW, proper matching of these apertures will significantly impact the performance of the common-path interferometer. To simplify the analysis, and yet maintain validity over different choices of the input aperture size, we will introduce a dimensionless term, η , to specify the size

of the central filtering region. This requires a “length scale” reference in the spatial Fourier domain, which we will take to be the radius of the main lobe of the Airy func-tion generated by the Fourier transform of the circular input aperture alone. Thus, denoting the Airy disc radius as 2R and the radius of the central filtering region as 1R ,

we can formally define the dimensionless filter parameter size as

( ) 11 2 0.61 rR R r fη −= = ∆ ∆ , (3.11)

where r∆ is the radius of the input aperture and rf∆ is the (spectral) radius of the

central filtering region. The 0.61 factor arises from the radial distance to the first zero crossing of the Airy function corresponding to half of the Airy mainlobe factor, of 1.22 [1].

Applying the dimensionless filter size into Eq. (3.10) and then subsequently perform-ing a series expansion in 'r , we obtain the following expression for the SRW generating function:

3.2 Field Distribution at the Image Plane of a CPI 19

( ) ( ) ( ) ( )( )

( ) ( ) ( ){ }( )

22

0 2

43

3 4

' 1 1.22 0.61 1.22 '

10.61 2 1.22 0.61 1.22 ' ...

4

g r J J r r

J J r r

πη πη πη

πη πη πη πη

= − − ∆

+ − ∆ + (3.12)

In this expansion, the SRW has been expressed in radial coordinates normalized to the radius of the imaged input aperture to maintain applicability regardless of the choice of input aperture size. Moreover, this provides for convenient scaling to account for any magnification within the imaging system although, for simplicity, a direct imaging operation of unity magnification is assumed for the remainder of our analysis. It is apparent from Eq. (3.12) that the generated SRW will change as a function of the dimensionless parameter expressing the radius of the central filtering region. Addition-ally, it is clear that the SRW spatial profile will not necessarily be flat over the system output aperture. This is an important, yet often neglected, factor in determining the performance of a CPI.

Fig. Fig. Fig. Fig. 3.33.33.33.3 Plot of the spatial variation of the normalized synthetic reference wave (SRW) amplitude g(r’) as

a function of the normalized CPI observation plane radius for a range of η values from 0.2 to 0.627. This

plot shows that a large value of η produces significant curvature in the SRW across the aperture, which

will cause a distortion of the output interference pattern. In contrast, a low value of η generates a flat SRW, but at the cost of a reduction in the SRW amplitude.

Figure 3.3, shows the input-normalized amplitude of the SRW generating function

for different η values, each plot displayed as a function of the output radius coordinate, 'r , normalized to the system aperture radius, r∆ . It can be seen from the plots that as

η increases so does the strength of the SRW, as can be expected from qualitative

20 3 Foundation of Generalized Phase Contrast

arguments. However, the curvature also increases with increasing η , thus distorting the

wavefront profile of the SRW. From the point of view of optimising a CPI, it is desir-

able to properly select the filter size so that the curvature of ( )'g r is negligible over a

sufficiently large spatial region of the system aperture centred around ' 0r = . Firstly, we

can choose to limit the range of η so that ( )'g r never exceeds 1. This identifies an

upper limit determined by the first zero crossing of the Bessel function ( )0 1.22J πη

where 0.627η ≈ . Secondly, it is very important to keep η as small as possible to make

sure that the scattered object light is not propagated through the zero-order filtering region. Finding a minimum applicable η -value is less apparent, but obviously choosing a

very small value will reduce the strength of the SRW to an unacceptably low level compromising the fringe visibility in the interference with the diffracted light. The term η can thus have a significant impact on the resulting interferometric performance and is

of the same importance as the filter parameters A, B and θ when designing a CPI.

Fig. 3.4Fig. 3.4Fig. 3.4Fig. 3.4 Plot showing the evolution of the SRW generating function from 0η = to 5η = . The radial

profile of the generating function approaches ( )circ 'r r∆ as η is increased. The ideal top-hat scenario is

indicated by the reference planes drawn at g=1 and at 1r r′ ∆ = . Representative profiles are traced at

η = 0.40, 0.627, 1.0 (thick black trace), 2.0, 2.75, 4.0 (thick white line), and 5η = .

Figure 3.4 illustrates the input-normalized amplitude of the SRW generating func-

tion as a function of the output radius coordinate, 'r r∆ , but this time for a wider range

of η values. We can deduce from this plot that, assuming we can allow ( )'g r to

occasionally exceed the value 1, then we can define a second regime for choosing the value of η (or filter radius). We can denote this alternate operating regime as a so-called

large-η range, as opposed to the small-η range defined by the previously found interval

r r′ ∆ η

( ),g r η′

3.2 Field Distribution at the Image Plane of a CPI 21

range, as opposed to the small-η range defined by the previously found interval

0 0.627η≤ ≤ . Phase objects having relatively small spatial frequency content minrf need

to operate within the previously defined small-η regime while those that have a rela-

tively large minrf may do so within the large-η regime, to make the representation of

the output intensity in Eq. (3.9) as spatially accurate as possible. We saw that depending on the accuracy needed for the description of the interfero-

grams one can choose to include a number of spatial higher-order terms from the expansion in Eq. (3.12). The influence of the higher-order terms has the largest impact along the boundaries of the imaged aperture. For η -values smaller than 0.627 and when

operating within the central region of the image plane, spatial higher-order terms are of much less significance and we can safely approximate the synthetic reference wave with the first and space invariant term in Eq. (3.12):

( ) ( )0' 1 1.22g r central region J πη∈ ≈ − (3.13)

so that we can simplify Eq. (3.9) to give:

( ) ( )( ) ( )( )2

2 1', ' exp ', ' exp 1I x y A j x y K BA jφ α θ−≈ + − (3.14)

where ( )01 1.22K J πη= − . The influence of the finite on-axis filtering radius on the

focused light, incorporated in the K parameter, is thus effectively included as an extra

“filtering parameter” so that the four-parameter filter set {A, B, θ , ( )K η } together

with the complex object-dependent term, α , effectively defines the type of filtering scheme we are applying.

3.2.3 The Combined Filter Parameter

Having determined a suitable operating range for the CPI in terms of the production of

a good SRW as determined by the generating function g(r´), we must now examine the

role that the remaining filter parameters play in the optimization of a CPI.

Looking at Eq. (3.14), we see that the different filter parameters (A, B, θ ) can be

combined to form a single complex-valued term, C, the combined filter term, such that:

( ) ( )1exp exp 1CC C j BA jψ θ−= = − . (3.15)

Therefore, Eq. (3.14) can be simplified to give:

( ) ( )( )2

2', ' exp ', ' CI x y A j x y j K Cφ ψ α= − + (3.16)

where the usual filter parameters can be retrieved from the combined filter parameter using

22 3 Foundation of Generalized Phase Contrast

( )

21

11

1 2 cos

sin sin

C

C

BA C C

BA C

ψ

θ ψ

−

−−

= + +

=

(3.17)

Since it is a complex variable, the combined filter term C, which effectively describes

the complex filter space, can be considered to consist of a vector of phase Cψ and length

C as expressed by Eq. (3.15). Thus in order to obtain an overview of the operating

space covered by all the possible combinations of three independent filter parameters (A, B, θ ) we can now, instead, choose to consider a given filter in terms of the two

combined parameters Cψ and C . However, referring to Eq. (3.16), it can be seen that

the filter parameter, A, also appears independently of the combined filter term, C.

Fortunately, this issue can be resolved by considering that the term 1BA− from Eq. (3.15) must be constrained in the following way:

1

1

11

1 1, 1

1 1, 1

1 1, 1

BA A B C

BA A B

BA B A C

−

−

−−

< ⇒ = = +

= ⇒ = =

> ⇒ = = +

(3.18)

These constraints arise from the adoption of a maximum irradiance criterion mini-mising unnecessary absorption of light in the Fourier filter, which reduces both irradi-ance and the signal-to-noise ratio in the CPI output.

Any given filter can be explicitly defined by a given value of the two parameters Cψ

and C therefore we can use a single plot to display the location of a given filter graphi-

cally within the complex filter space. Such a plot is shown in Fig. 3.5 where we plot the magnitude of the combined filter parameter C against its phase Cψ .

There exist different families of operating curves in this complex filter space, each of

which can be traced out by keeping a term such as 1BA− constant while θ is varied or vice versa (these form the fine grid like structure in Fig. 3.5). Plotting the operating curves for C this way makes it relatively simple to identify particular operating regimes

for different classes of filters. For example, we are particularly interested in the operating

curve for a lossless Fourier filter, a filter in which 1 1BA− = , since this corresponds to a class of filters for which optical throughput is maximized. The lossless operating curve is shown as the bold line in Fig. 3.5. We can derive the expression for the shape of the lossless operating curve by using the following identity:

( ) ( ) ( )( )exp 1 2 sin 2 exp 2j jθ θ θ π− = + (3.19)

and combining Eq. (3.15) and Eq. (3.19) we obtain an expression for the lossless operat-ing curve, for which C is defined for two distinct regions as:

( )2sin 2 2; 3 2

0 2; 3 2

C C

C

C

C

ψ π ψ π π

ψ π π

= − ∀ ∈

= ∀ ∉ (3.20)

3.2 Field Distribution at the Image Plane of a CPI 23

Fig. 3Fig. 3Fig. 3Fig. 3.5.5.5.5 Complex filter space plot of the modulus of the combined filter parameter, C , against the phase

Cψ over the complete 2π phase region. The use of these combined parameters (defined in Eqs. (3.15)

and (3.17)) allows us to simultaneously visualise all the available combinations of the terms A, B and θ. The bold curve is the operating curve for a phase-only (lossless) filter, whilst the fine grid represents

operating curves for differing values of the filter terms A, B and θ. We have marked operating regimes for

a number of CPI architectures including: (A) Zernike, (B) Henning, (C) dark central ground and (D) field absorption filtering and (E,F) point diffraction interferometers. Full filter details for these tech-

niques are summarised in Table 1.

Table 1Table 1Table 1Table 1 Comparison of filter parameters for the different CPI types highlighted in Fig. 3.5.

MethodMethodMethodMethod Filter parametersFilter parametersFilter parametersFilter parameters RefeRefeRefeReferrrrencesencesencesences Fig. 3.5 labelFig. 3.5 labelFig. 3.5 labelFig. 3.5 label

Zernike phase contrast A=1, B∈ [0.05;1], θ= 2π± [1, 2] (A1), (A2)

Henning phase contrast A=2–1/2, B=1, θ= 4π± [3, 4] (B1), (B2)

Dark central ground filter A=1, B=0, θ=0 [5– 7] (C)

Field absorption filter A<1, B=1, θ=0 [8] (D)

PDI*/Smartt interferometer A<1, B=1, θ=0, K≪ 1 [9–12] (E)

Phase-shifting PDI A=1, B=1, θ∈ [0; 2π ], K≪ 1 [13, 14] (F)

* PDI indicates: point diffraction interferometry.

24 3 Foundation of Generalized Phase Contrast

The development of this complex filter space plot makes it extremely simple to compare different filter parameter selections that have been proposed and independ-ently dealt with in the scientific literature. In Fig. 3.5, we have superimposed the operating points for some of the different CPI filter types that are commonly used. The filter type, parameters and the corresponding labels and references are summa-rised in Table 1.

The filter space plot not only provides a convenient method of comparing different CPI architectures, but also forms a useful basis for exploring the effectiveness of new CPI configurations. Referring to Fig. 3.5, it can be seen that many of the filter types used in CPIs lie on or close to the lossless operating curve defined by Eq. (3.20). The arrows on the plot indicate the general regime in the complex filter space, in which a given filter type can be operated. To our knowledge, this identification and labelling of known cases from the literature in a single general filter space representation has not been previously demonstrated. We believe that it provides a very useful basis for understanding the inherent differences or similarities between existing filtering methods. Later, we will use the complex filter space plot as a framework on which to optimize the visibility and the irradiance of a CPI operated as a wavefront sensing system.

3.3 Summary and Links

In this chapter we developed the fundamental GPC framework by analyzing common path interferometers (CPI). We considered systems based on spatial filtering around the optical axis in the spatial Fourier domain, where we have abandoned the simplifying assumptions in typical treatments. We have derived conditions for high interferometric accuracy taking account of the so-called synthetic reference wave. We also identified a combined filter parameter that places all spatial filters of different CPI schemes in the same phase space domain, drastically simplifying the comparative analysis of different filtering techniques. The Zernike filter is one among many schemes that are subsumed by this generalized treatment. We will continue to build upon this basic framework throughout the book as we adopt refinements that address the varying demands and design freedoms of different applications. Practical illustrations will be discussed in Chapter 5, where we characterise wavefront-sensing CPI systems using a combined filter parameter. We will show that the generalized mathematical treatment here can be successfully applied to the interpretation and possible modification for the optimization of a number of different but commonly used CPI architectures. In Chapter 6, we will adapt the basic framework to incorporate design freedoms available at the input plane when using GPC for wavefront engineering. There we will discuss how filter parameters can be fully optimized to match known inputs. The framework developed here is also used as the starting point for discussing the different alternative implementations in Chapter 9. Finally, we will again build upon the basic formulation developed here when

References 25

we consider CPIs with amplitude-modulated inputs when we discuss the reverse phase contrast method in Chapter 10. Meanwhile, we will develop a phasor chart method in the next chapter, which will come in handy in understanding and optimizing general-ized phase contrast systems.

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