OPTICAL IMAGING AND SPECTROSCOPY Phần 3 potx

We have already consideredmathematical bases suitable for field analysis in terms of the Fourier transform andHermite – Gaussian functions.. The electromagnetic description of optical wa

A function ff(x) [ V (f) may be represented as

ff(x)¼Xn[Z

In contrast with the sampling theorem and with the Haar wavelet expansion, theexpansion coefficients are not samples of ff or inner products between ff and thebasis vectors For the B-splines it turns out that we can derive complementaryfunctions fn(x) for each fn(x)¼ bm

(x n) such that hfnjfn0i ¼ dnn 0 The lementary functions can be used to produce a continuous estimate for f (x) that iscompletely consistent with the discrete measurements This interpolated function is

comp-fest(x)¼X

n[Z

hfnj f ifn(x) (3:125)Given the orthogonality relationship between the sampling functions and the comp-lementary functions, fest is by design consistent with the measurements We canfurther state that fest¼ ff if the complementary functions are such that f [ V (f),

in which case there exist discrete coefficients p(k) such that



f (x)¼Xk[Z

Using the convolution theorem, the Fourier transform of f (x) is

^f(u)¼ ^f(u) X

k[Zp(k)ei2pku

k[Zp(k)ei2pku

X

n 00 [Za(n00)ei2pn00u

(3:129)

where we use the substitution of variables n00¼ n0 k

Poisson’s summation formula is helpful in analyzing the sums in Eqn (3.129).The summation formula states that for g(x) [ L1(R)

Xn[Z

k[Z

ðkþ1

k

^g(u)e2pinudu

¼ð1

1

^g(u)e2pinudu

Since a(x) is the autocorrelation of f, its Fourier transform isj^f(u)j2 Thus by thePoisson summation formula

Xn[Z

Substitution in Eqn (3.127) yields

^f(u)¼ P f(u)^k[Z

We can evaluate Eqn (3.135) to determinef(u) and ^ f(x) if P

k[Zj^f(uþ k)j2 isfinite The requirement that there exist positive constants A and B such that

The Fourier transform of the mth-order B-spline is

Q0(u)¼ 1 For higher orders we note that jsinc(u þ k)j2(mþ1) jsinc(u þ k)j2,meaning that Qm(u) Qo(u) Thus, 0 , Qm(u) , 1 and the B-spline functions ofall orders satisfy the Riesz basis condition

In contrast with the B-splines themselves, the complementary functions f(x) donot have finite support It is possible, nevertheless, to estimate f(x) over a finiteinterval for each B-spline order by numerical methods Estimation of Qm(u) fromEqn (3.138) is the first step in numerical analysis This objective is relativelyeasily achieved because Qm(u) is periodic with period 1 in u Evaluation of thesum over the first several thousand orders for closely spaced values of 0 u  1takes a few seconds on a digital computer

Given Qm(u), we may estimate f(x) by using a numerical inverse Fourier form of Eqn (3.135) or by calculating p(k) from Eqn (3.134) Since p(k) must bereal and since Qm(u) is periodic, we obtain

Given f(x)¼ bm

(x n) and f(x), we can calculate ff(x) for target functions Forexample, Fig 3.17 shows the signals of Figs 3.8 and 3.9 projected onto the V(f) sub-spaces for B-splines of orders 0 – 3 Higher-order splines smoothly represent signalswith higher-order local polynomial curvature Note that higher-order splines are notmore localized than the lower-order functions, however, and thus do not immediatelytranslate into higher signal resolution Notice also the errors at the edges of the signalwindows in Fig 3.17 These arise from the boundary conditions used to truncate theinfinite time signal f (x) In the case of these figures, f (x) was assumed to be periodic

in the window width, such that sampling and interpolation functions extendingbeyond the window could be wrapped around the window

The interpolated signals plotted in Fig 3.17 are the projections ff(x) [ V (f) of

f (x) onto the corresponding subspaces V (f) The consistency requirement designedinto the interpolation strategy means that these functions, despite their obvious discre-pancies relative to the actual signals, would yield the same sample projections.Corrections that map the interpolated signals back onto the actual signal lie in

V?(f) Strategies for sampling and interpolation to take advantage of knownconstraints on f (x) to so as to infer correction components f?(x) are discussed inChapter 7

Figure 3.16 Complementary interpolation functions  f(x) for the B-splines of orders 0 – 3 The zeroth-order B-spline is orthonormal such that  f(x) ¼ b 0 (x).

94 ANALYSIS

Use of Eqn (3.125) to estimate f (x) is somewhat unfortunate given that fn(x) doesnot have finite support A primary objection to the use of the original samplingtheorem [Eqn (3.92)] for signal estimation is that sinc(x) has infinite support anddecays relatively slowly in amplitude While f(x) is better behaved for low-orderB-splines, it is is still true that accurate estimation of f (x) may be computationallyexpensive if a large window is used for the support of f As the order of theB-spline tends to infinity, f(x) converges on sinc(x) [235] If we remove the require-ment that f(x) [ V (f), it is possible to generate a biorthogonal dual basis for bm

(x)with compact support [49] The compactly supported biorthogonal wavelets in thiscase introduce a complementary subspace V spanned by f(x)

The goal of the current section has been to consider how one might use a set ofdiscrete B-spline inner products to estimate a continuous signal This problem iscentral to imaging and optical signal analysis We have already encountered it inthe coded aperture and tomographic systems considered in Chapter 2, and we willencounter it again in the remaining chapters of the text We leave this problem fornow, however, to consider the use of sampling functions and multiscale represen-tations in signal and system analysis One may increase the resolution and fidelity

Figure 3.17 Projection of f (x) ¼ x 2 =10 and the signal of Fig 3.9 onto the V(f) subspace for B-splines of orders 0 – 3.

of the reconstructions in Fig 3.17 by increasing the resolution of the sampling tion in a manner similar to the wavelet approach taken in Section 3.8.

func-3.10 WAVELETS

As predicted in the Section 3.1, this chapter has developed three distinct classes ofmathematics: transformation tools, sampling tools, and analysis tools In the firstseveral sections we considered fields and field transformations We have just com-pleted three sections focusing on sampling Section 3.9 describes a method forrepresenting a function f (x) on the space V (f) spanned by the scaling functionf(x)¼ bm(x) This section extends our consideration of B-splines to wavelets,similar to our extension of Haar analysis in Section 3.8 We have already consideredmathematical bases suitable for field analysis in terms of the Fourier transform andHermite – Gaussian functions In fact, many functional families could be used toanalyze fields The choice of which family to use depends on which family arisesnaturally in the physical specification of the problem (e.g., Laguerre – Gaussian func-tions arise naturally in the specification of cylindrically symmetric fields), whichfamily arises at sampling interfaces, and which family enables the most computation-ally efficient and robust analysis of field transformations

Wavelet theory is a broad and powerful branch of mathematics, and the student iswell advised to consult standard courses and texts for deeper understanding [53,164].Wavelets often describe images and other natural signals well The intuitive matchbetween wavelets and images arises from the assumption that “features” in naturalsignals tend to cluster, meaning that higher resolution is desirable in the vicinity of

a feature than elsewhere in the signal Multiscale clustering enables wavelet tations to estimate signals with fewer samples than might be used with uniformregular sampling Under the Whittaker – Shannon sampling strategy, functionalsamples are distributed uniformly in space even in regions with no significantimage features Wavelets enable samples to be dynamically assigned to regionswith interesting features This dynamic resource allocation is the basis of naturalsignal compression

represen-B-splines may be used to generate semiorthogonal bases as in Section 3.9, gonal spaces and orthogonal wavelet bases As before, we imagine a hierarchy

biortho-of spaces

{0} ,   , V2, V1, Vo, V1, V2,   , L2(R) (3:140)Semiorthogonal bases are spanned by sets of functions that are not themselvesorthogonal but are orthogonal to a complementary set of functions Biorthogonalbases generate complementary spaces spanned by complementary sets of functions.Orthogonal bases generate a single hierarchy of spaces spanned by a single set oforthogonal functions We have already encountered an orthogonal wavelet basis inthe form of the Haar wavelets of Section 3.8 In this section we extend the Haar analy-sis to orthogonal bases based on higher-order B-splines

96 ANALYSIS

The orthonormal basis for spaces spanned by discretely shifted B-splines wasintroduced by Battle [15] and Lemarie [150] For the Battle – Lemarie basis, f(x) is

a scaling function on the space V (bm(x)) spanned by the mth-order B-spline Sincef(x) [ V (bm

(x)) there exist expansion coefficients p[n] such that

f(x)¼X

The Fourier transform of Eqn (3.141) yields

^

Our goal is to select f(x) to be an orthonormal scaling function such that

f(x n), f(x  m)

ð1

Xn[Z

a(n)ei2pnu¼X

k[Z

^

For an orthonormal scaling function, however, P

n[Za(n)ei2pnu¼ 1 and

^

a(u)¼ j ^f (u)j2, which yields the identity for orthonormal scaling functions

Xk

Referring to Eqn (142), we see that ^f (u) satisfies Eqn (145) if we select

^p(u)¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP 1

kj ^bm(uþ k)j2

where ^p(u) is finite and well defined because the B-splines form a Riesz basis, asdiscussed in Section 3.9 Since ^p(u) is periodic with period 1 in u, it generates adiscrete series p[n] for use in Eqn (3.141) Substituting ^bm(u) from Eqn (3.137)

in Eqns (3.146) and (3.142) yields

Higher orders of Sn(u) are obtained by noting that Snþ1(u)¼ S0

n(u)=n This yields

S4(u)¼p

4(2þ cos (2pu))

S6(u)¼p

6(33þ 26 cos (2 pu) þ cos (4pu))

p

^

98 ANALYSIS

^h(u)¼Xn[Z

For the Battle – Lemarie scaling functions

^h(u)¼ ffiffiffi

wave-^c(u)¼ 1ffiffiffi

 

(3:159)

The Battle – Lemarie scaling function and wavelet can be reconstructed by inverseFourier transforming Eqns (3.147) and (3.159) These functions satisfy the sameorthogonality and scaling rules as the Haar wavelets discussed earlier; specifically

hier-or system design considerations In still other cases, a particular basis may prove mhier-oreamenable to compact support of a particular signal class.

p

fn(x)¼ 2pxfn1(x) d

dxfn1(x) (3:168)Combine this relationship with Eqns (3.13) and (3.57) to show by recursionthat

F{fn(x)}¼ in

3.5 One-dimensional Numerical Analysis:

(a) Plot sin (2pux) on [0, 1] using 1024 uniformly spaced samples for

u¼ 16,32,64,128,256 At what point does aliasing become significant?Can you describe the structure of the aliased signal?

(b) Plot the discrete Fourier transform of sin (2pux) on [0, 1] using 1024 formly spaced samples for u¼ 16, 32, 64 Label the plot in frequencyunits What is the width of the Fourier features that you observe? Whatcauses this width?

uni-(c) Plot the discrete Fourier transform of b0(x) sin (2p ux) on [1:5, 2:5]using 4096 uniformly spaced samples for u¼ 16, 32, 64 Label the plot

in frequency units and explain the plot

100 ANALYSIS

3.6 Fourier Analysis of a Hermite – Gaussian:

(a) Plot the Hermite – Gaussian f5(x) over the range of the function.(b) Plot ~f5t(x) over a representative range of t

3.9 Fresnel Transformation of the Laguerre – Gaussian Functions Use the volution theorem and the fast Fourier transformation to numerically calculatethe Fresnel transformation of the Laguerre – Gaussian modes for m, n equal to0,0, 1,0, 1,1, 2,0, 2,1, and 4,3 for t¼ 0:5, t ¼ 1, and t ¼ 2 Use your com-putational result and the analytic result given by Eqn (3.83) to plot the absol-ute value and phase of the mode distribution at t¼ 0 and for the transformvalues of t in each case Submit your code, plots, and comments regardingfeatures of the modes or discrepancies between the computational methods.3.10 Haar Analysis:

con-(a) Generate and plot a function of similar complexity to f (x) in Fig 3.9.(b) Replicate Figs 3.9, 3.12, and 3.17 for your function

3.11 2D Wavelet Analysis Replicate Figs 3.13 and 3.14 for an image of yourchoosing

4.1 WAVES AND FIELDS

The optical field is an electromagnetic field The physical nature of the field is mined by the laws of electromagnetic propagation and by quantum mechanical andthermal laws describing the interaction between the field and materials In thedesign and analysis of optical systems we consider

deter-† How the field is generated Common mechanisms include

Thermal radiation generated, for example, by the Sun, a flame, or an descent lightbulb

incan-Electrical discharge by gases such as neon or mercury vapor

Fluorescence

Electrical recombination in semiconductors

While we do not consider light generation in detail in this text, differences in thecoherence properties of the source are central to our discussion Coherence theory,which relates the electromagnetic nature of the field to statistical properties ofquantum (e.g., photonic) processes, is the focus of Chapter 6

† How the field is detected The field may be detected by optically inducedchemical, physical, thermal, and electronic effects Optoelectronic detectioninterfaces for imaging and spectroscopy are the focus of Chapter 5

Optical Imaging and Spectroscopy By David J Brady

Copyright # 2009 John Wiley & Sons, Inc.

103

† How the field propagates and how propagating fields are modulated bymaterials Field propagation is described by the Maxwell equations for electro-magnetic waves, and field – matter interactions are described by materialsequations The electromagnetic description of optical waves and optical inter-actions is the focus of this chapter.

In view of the peculiarly quantum mechanical nature of optical field generation anddetection, it is important to understand that the conventional electromagnetic field ofthe Maxwell equations is not a sufficient description of optical fields The descriptionderived in this chapter provides a basis for optical analysis, but complete understand-ing of optical field propagation and field properties must incorporate the detectionand coherence processes discussed in Chapters 5 and 6 In short, the student mustunderstand the next three chapters as a group to have a vision for the peculiar andbeautiful nature of optical fields

4.2 WAVE MODEL FOR OPTICAL FIELDS

The Maxwell equations for electromagnetic propagation are

as a means of explaining electrodynamics Equation (4.3), which expresses theCoulomb attraction of electromagnetic charge, is called Gauss’ law Equation (4.4), iscalled Gauss’ law for magnetism and expresses the absence of magnetic monopoles.The fields are further related by the material equations

where P is the polarization of the material and M is the magnetization In most opticalmaterials, M ¼ 0 and P is a function of E The simplest and most common case is the

linear dielectric relationship

of anisotropic materials to tunable filters in Section 9.7

Using the material relations to eliminate B and D from the Faraday and Ampererelationships yields

r ð1EÞ ¼E  r1 þ 1r  E ¼0 (4:15)where we have assumed for the moment that 1 is a scalar We can reexpressEqn (4.15) as

4.2 WAVE MODEL FOR OPTICAL FIELDS 105

Substituting Eqn (4.16) in the wave equation yields

In isotropic homogeneous media, the wave equations reduce to

The primary attraction of plane wave analysis is that more general solutions can beexpressed as superpositions of plane waves In the remainder of this chapter werestrict our attention to isotropic materials, in which case 1 is a scalar and juj2¼

m01 n2 A general solution to Eqn (4.18) in this case is

E(r, t) ¼

ð ð ð

F(u, v, n)p(u, v, n)ei2p ntuxvy

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim1n 2 u 2 v 2 zp

E(r, n) ¼

ð

In this chapter n is treated as an implicit variable in the function E(r) ¼ E(r, n)

We according drop the harmonic time dependence e2i2pnt from Eqn (4.23) anddescribe spatial distribution of the field amplitude according to

E(r) ¼

ð ðF(u, v)p(u, v)ei2p uxþvyþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim1n 2 u 2 v 2 zp

In the remainder of this chapter we also assume that the field propagates paraxially.The paraxial approximation consists of the assumption that values of u and v forwhich jF(u, v)j is nonzero lie on a compact window on the wave normal sphere

Figure 4.1 The wave normal surface in free space.

4.3 WAVE PROPAGATION 107

centered on the w axis, as illustrated in Fig 4.1 This window is centered on the z axis,such that w  u, v over the full spatial bandwidth This means that the polarizationvector p(u, v) is nearly parallel to the (x, y) plane over the entire spatial bandwidth.

In an isotropic material p(u, v) may be represented on any basis orthogonal to u

We select as an example a basis in which one of the polarization vectors is alsoorthogonal to the y axis The resulting orthonormal basis for p(u, v) is

px¼ k ui xþviyþwiz

iy

¼lwixluizffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1  l2v2p

py¼  luixþlviyþlwiz

du dv

(4:27)

Equation (4.27) is an exact vector model relating the Fourier distribution of the field

in linear polarizations to the spatial field distribution in three dimensions The inverserelationship is

Fx(u, v)pxei2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=l 2 u 2 v 2 zp

¼

ð ð

fy(x, y, z)ei2p(uxþvy)dx dy

(4:28)

From Eqn (4.28) we see that knowledge of fx(x, y, z) and fy(x, y, z) as functions

of (x, y) for any specific value of z is sufficient to calculate Fx(u, v) and Fy(u, v)

In particular, if we know f (x, y, z ¼ 0), we can then calculate

plane z ¼ 0, is called a boundary condition and the evolution of the field distributionfrom one boundary to another is called diffraction Equations (4.27) and (4.28) enable

us to computationally model diffraction in homogeneous media

4.4 DIFFRACTION

Diffraction is the process of wave propagation from one boundary to another Acanonical example of optical diffraction, propagation of a monochromatic fieldfrom the plane (x, y, z ¼ 0) to the plane (x0, y0, z ¼ d ), is illustrated in Fig 4.2.Given the electric field distribution on the input plane, we seek to estimate thefield distribution on the output plane Viewed as a transformation between a function

f (x, y) over the input plane and a function g(x0, y0) over the output plane, diffraction islinear and shift-invariant Our goal in this section is to describe the transfer functionand impulse response corresponding to diffraction from one plane to another

An arbitrary vector field f(x, y) in the plane z ¼ 0 corresponds to the Fourier spacedistribution

F(u, v) ¼

ð ðf(x, y)ei2p(uxþvy)dx dy (4:30)

where F(u, v) may be separated into x, y and z components Fx(u, v) ¼ F(u, v) 

px(u, v), Fy(u, v) ¼ F(u, v)  py(u, v), and Fz(u, v) ¼ F(u, v)  lu The Fzcomponentdoes not produce a propagating field

Let Gx(u, v) and Gy(u, v) be the Fourier distributions of the field in the x0, y0plane

at z ¼ d From Eqn (4.28) we see that

Gx(u, v) ¼ ei2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=l 2 u 2 v 2 dp

Fx(u, v)

Gy(u, v) ¼ ei2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=l 2 u 2 v 2 dp

Fy(u, v)

(4:31)

Figure 4.2 Diffraction between two planes.

4.4 DIFFRACTION 109

the factor T(u, v) ¼ ei2p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1=l 2 u 2 v 2 dp

is the transfer function for diffraction from the

z ¼ 0 plane to the z ¼ d plane

Nominally, the impulse response for diffraction is the inverse Fourier transform ofthe transfer function We continue along this line with care, however, by brieflyaccounting for the vector nature of the field Using the transfer function andEqn (4.27), we obtain

g(x0, y0) ¼

ð ðF(u, v)  pxpxþF(u, v)  pypy

ei2p ux0þvy0þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=l 2 u 2 v 2 zp

If there are no longitudinal (nonpropagating) field components on the input boundarythen F(u, v)  lu ¼ 0, and

F(u, v)  pxpxþF(u, v)  pypy¼F(u, v) (4:33)

In this case Eqn (4.32) simplifies considerably to yield

g(x0, y0) ¼

ð ðF(u, v)ei2p ux0þvy0þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=l 2 u 2 v 2 zp

du dv

¼

ð ðf(x, y)h(x0x, y0y)dx dy (4:34)where

h(x, y) ¼

ð ð

ei2p uxþvyþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=l 2 u 2 v 2 zp

l2p(d2þx2þy2)3=2

chooses to work with a simplified approximate impulse response As an example,

l  d in essentially all optical systems, meaning that the 1/ld term in Eqn (4.36)dominates the 1/d2 term In imaging system analysis, the impulse response isoften simplified by the Fresnel (near-field) approximation or the more restrictiveFraunhofer (far-field) approximation Both approximations are paraxial, meaningthat we restrict our attention to field distributions over the space close to the axis

of optical propagation (the z axis in Fig 4.2)

The Fresnel approximation is just the paraxial approximation that d  jx  x0j,

jy  y0jfor all x, y and x0, y0of interest In this case

h(x, y)  1

ildei(2pd=l)ei(p=ld) xð2þy2Þ (4:38)

Under the Fresnel approximation, diffraction in homogeneous isotropic space isdescribed by a 2D version of the Fresnel transform discussed in Section 3.5 evaluated

Figure 4.3 was generated using numerical analysis in Matlab The figure used a

2  2-mm spatial window sampled with 1024  1024 pixels The Fresnel transferfunction multiplied the DFT of the input field and an inverse DFT was used to gen-erate the diffracted field Of course, numerical analysis is not necessary for analysis ofdiffraction of these particular sources because, as discussed in Section 3.5, Hermite –Gaussian distributions are eigenfunctions of the Fresnel transform According toEqns (3.76) and (4.38), if the input field f (x, y) ¼ fn(x=w0)fm( y=w0) for real w0,then the diffracted field is

0B

1C

Afm

xw0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

w4

0þl2d2q

0B

Additional interesting Fresnel diffraction effects are observed in Figs 4.4 and 4.5.These figures were generated from the same spatial window and sampling as above,but the images are zoomed to focus on features of interest Figure 4.4 is a harmonicfield modulated by a Gaussian envelope Note that the harmonic features do not blur(features with the same frequency are present at all diffraction lengths) At d ¼ 10 mm,the diffracted field has begun to separate horizontally into multiple images of theGaussian envelope and harmonic modulation is observed only in the interferencebetween the separating spots, not within individual spots.

Figure 4.5 is a chirped harmonic field modulated by a Gaussian envelope For thisinput, the diffracting field sharpens to a focus at d ¼ 2 mm rather than blurring onpropagation After the focus, the field blurs In both Figs 4.4 and 4.5 it is interesting

to note that blur is not a fundamental process of diffraction for coherent fields In fact,

a diffracting coherent field maintains its spatial frequency bandwidth on propagation

Figure 4.3 Absolute values of the diffracted field for the Gaussian spots e p ½(x 2 þy 2 )=20 2  ,

e p f½(x100) 2 þy 2 =50 2 g

, ep f½(xþ200)2þy2=1002g, and ep f½x2þ(y200)2=2502g under the Fresnel mation for various diffraction distances All units are in microns and l ¼ 1 mm.

approxi-Blur in the normally observed sense of optical fields is a property of partially coherent

or incoherent fields, as discussed in Chapter 6

As illustrated in Fig 4.4, however, Fourier components of diffracting fields tend toseparate on propagation This effect is easily explained in the context of Fraunhoferdiffraction theory Fraunhofer diffraction is most easily derived from the integral form

of Fresnel diffraction

g(x0, y0) ¼e

i(2pd=l)ild

ð ð

ei(p=ld)[(x  x0)2þ( y  y0)2] f (x, y)dx dy

¼ei(2pd=l)ei(p=ld)(x02þy02)

ild

ð ðexp i2pxx

0þyy0ld

Assuming that x2ld and y2ld over the support of f (x, y), we may drop thesecond exponential in the integrand to obtain

Figure 4.5 Absolute value of the diffracted field for f (x, y) ¼ ep ½(x2þy 2 )=250 2  ½1 þ cos (5p10 4 x 2 ) All units are in micrometers; l ¼ 1 mm.

4.5 WAVE ANALYSIS OF OPTICAL ELEMENTS

Chapter 2 considered the use of optical elements to shape the mutual visibility ofsource points and detection points The visibility function, renamed the impulseresponse or point spread function, remains of central interest under the wave model.The goal of optical sensor design is to use optical elements to program the impulseresponse, within physical constraints, to usefully encode target object features intodetected data

This section presents wave models for the optical elements that we described usinggeometric models in Section 2.2 In addition, we consider diffractive opticalelements, which cannot be described by ray models As in Section 2.2, analysis ofrefraction and reflection at dielectric interfaces is a good starting point for opticalelement analysis In analogy with Fig 2.4, the effect of a dielectric interface on aplane wave is illustrated Fig 4.6 A plane wave is incident on the interface in amedium of index of refraction n1 The incident field is Ei(r) ¼ Eiexp(2piui r).The incident wave is refracted at the interface into the second medium of index ofrefraction n2, and a reflected wave is returned into the first medium The refractedand reflected fields are Et(r) ¼ Etexp(2piut r) and Er(r) ¼ Erexp(2pi ur r).Boundary conditions derived from the Maxwell equations determine the relativeamplitudes of these waves The boundary conditions may be stated as follows:

† Vector components of E and H that lie in the plane of the interface arecontinuous

† Vector components of D and B normal to the plane of the interface arecontinuous

In both cases we assume that there are no surface charges or currents, which is alwaysthe case at optical frequencies These boundary conditions are used in standard texts

Figure 4.6 Refraction of a plane wave at a planar interface.

4.5 WAVE ANALYSIS OF OPTICAL ELEMENTS 115

on optics and electromagnetics to relate the amplitudes of the refracted and reflectedwaves to the amplitude of the incident wave Typically, the power in the reflectedwave at a dielectric interface is a few percent of the incident power, and most ofthe power is transmitted Thin film layers are often used to encode the impedance

at the interface to suppress or enhance reflection

It is not necessary to model reflection and refraction in detail to understand thefunctional utility of optical elements in shaping the impulse response The mostimportant features from a wave perspective are obtained simply by noting that thefunctional form of the wave distribution must be maintained on both sides of theinterface for the boundary conditions to be satisfied To satisfy the boundaryconditions in the plane of the interface, we require that [Ei(r) þ Er(r)]  is¼

Et(r)  is, for all r on the interface is is the surface normal for the interface Tosatisfy this condition, one must require that

l2n

2 1

As in Section 2.2, we first apply Snell’s law to the analysis of prism refraction

As illustrated in Fig 4.7, a prism consists of a series of two tilted planar interfaces.The prism of Fig 4.7 consists of a dielectric of index n2embedded in a dielectric ofindex n1 If i1is the surface normal at the first interface of a prism and i2the surfacenormal at the second interface, recursive application Eqn (4.45) produces an estimate

of the output wavevector u3

We find, therefore, that the plane wave Ei(r) ¼ Eiexp (2piuir) incident on a prism

is refracted under the paraxial approximation into the plane wave

Et(r) ¼ Eiexp (2piuir)eifp

The basic idea of a transmittance function is illustrated in Fig 4.8 The wave field,

Ei(x00, y00), is incident on an optical element The field to the immediate right of theelement is t(x00, y00)Ei(x00, y00) One models a system involving the optical element byfirst propagating the field f (x, y) to the input of the element using the Fourier methodsdescribed in Section 4.4, then modulating the field by the transmittance, and finallypropagatating the modulated field to the output plane to determine g(x0, y0)

Figure 4.7 Refraction of a plane wave by a prism For n2 n1, the incident wavevector refracts to move u 2 closer to the surface normal i 1 than the incident wavevector u 1 Refraction

at the output interface moves u3away from the surface normal i2in comparison with u2.

4.5 WAVE ANALYSIS OF OPTICAL ELEMENTS 117