OPTICAL IMAGING AND SPECTROSCOPY Phần 4 docx

4.12 Volume Holography:

(a) Plot the maximum diffraction efficiency of a volume hologram as a func￾tion of reconstruction beam angle of incidence assuming that D1=1 ¼

103 and that K ¼ ffiffiffi

2 p k0:

(b) A volume hologram is recorded with l ¼ 532 nm light. The half-angle

between the recording beams in free space is 208. The surface normal

of the holographic plate is along the bisector of the recording beams.

The index of refraction of the recording material is 1.5. What is the

period of grating recorded? Plot the maximum diffraction efficiency at

the recording Bragg angle of the hologram as a function of reconstruction

wavelength.

4.13 Computer-Generated Holograms. A computer-generated hologram (CGH) is

formed by lithographically recording a pattern that reconstructs a desired field

when illuminated using a reference wave. The CGH is constrained by details

of the lithographic process. For example CGHs formed by etching glass are

phase-only holograms. Multilevel phase CGHs are formed using multiple

step etch processes. Amplitude-only CGHs may be formed using digital prin￾ters or semiconductor lithography masks. The challenge for any CGH record￾ing technology is how best to encode the target hologram given the physical

nature of the recording process. This problem considers a particular rudimen￾tary encoding scheme as an example.

(a) Let the target signal image be the letter E function from Problem 4.5.

Model a CGH on the basis of the following transmittance function

t(x, y) ¼ 1 if arg F{E}ju¼ x

ld,v¼ y

ld

 . 0

0 otherwise (

(4:101)

where l is the intended reconstruction wavelength and d  x is the

intended observation range. F{E} is the Fourier transform of your letter

E function. Numerically calculate the Fraunhofer diffraction pattern at

range d when this transmittance function is illuminated by a plane wave.

(b) A more advanced transmittance function may be formed according to the

following algorithm:

t(x, y) ¼ 1 if arg e0:2pi½(xþyÞ=l

F{E}ju¼(x=ld),v¼( y=ld)

  . 0

0 otherwise 

(4:102)

Numerically calculate the Fraunhofer diffraction pattern at range d when

this transmittance function is illuminated by a plane wave. It is helpful

when displaying these diffraction patterns to suppress low-frequency

scattering components (which are much stronger than the holographic

scattering).

144 WAVE IMAGING

(c) A still more advanced transmittance function may be formed by multiply￾ing the letter E function by a high frequency random phase function prior

to taking its Fourier transform. Numerically calculate the Fraunhofer dif￾fraction pattern for a transmission mask formed according to

t(x, y) ¼ 1 if arg e0:2pi½(xþyÞ=lÞ

F{ef(x,y)

E}ju¼(x=ld),v¼(y=ld)

  . 0

0 otherwise 

(4:103)

where f(x, y) is a random function with a spatial coherence length much

greater than l.

(d) If all goes well, the Fraunhofer diffraction pattern under the last approach

should contain a letter E. Explain why this is so. Explain the function of

each component of the CGH encoding algorithm.

4.14 Vanderlught Correlators. A Vanderlught correlator consists of the 4F optical

system sketched in Fig. 4.25.

(a) Show that the transmittance of the intermediate focal plane acts as a shift￾invariant linear filter in the transformation between the input and output

planes.

(b) Describe how a Vanderlught correlator might be combined with a holo￾graphic transmission mask to optically correlate signals f1(x, y) and

f2(x, y). How would one create the transmission mask?

(c) What advantages or disadvantages does one encounter by filtering with a

4F system as compared to simple pupil plane filtering?

Figure 4.25 A Vanderlught correlator.

PROBLEMS 145

5

DETECTION

Despite the wide variety of applications, all digital electronic cameras have the same

basic functions:

1. Optical collection of photons (i.e., a lens)

2. Wavelength discrimination of photons (i.e., filters)

3. A detector for conversion of photons to electrons (e.g., a photodiode)

4. A method to read out the detectors [e.g., a charge-coupled device (CCD)]

5. Timing, control, and drive electronics for the sensor

6. Signal processing electronics for correlated double sampling, color processing,

and so on

7. Analog-to-digital conversion

8. Interface electronics

— E. R. Fossum [78]

5.1 THE OPTOELECTRONIC INTERFACE

This text focuses on just the first two of the digital electronic camera components

named by Professor Fossum. Given that we are starting Chapter 5 and have several

chapters yet to go, we might want to expand optical systems in more than two

levels. In an image processing text, on the other hand, the list might be (1) optics,

(2) optoelectronics, and (3–8) detailing signal conditioning and estimation steps.

Whatever one’s bias, however, it helps for optical, electronic, and signal processing

engineers to be aware of the critical issues of each major system component. This

chapter accordingly explores electronic transduction of optical signals.

We are, unfortunately, able to consider only components 3 and 4 of Professor

Fossum’s list before referring the interested reader to specialized literature. The

specific objectives of this chapter are to

Optical Imaging and Spectroscopy. By David J. Brady

Copyright # 2009 John Wiley & Sons, Inc.

147

† Motivate and explain the need to augment the electromagnetic field theory of

Chapter 4 with the more sophisticated coherence field theory of Chapter 6

and to clarify the nature of optical signal detection

† Introduce noise models for optical detection systems

† Describe the space–time geometry of sampling on electronic focal planes

Pursuit of these goals leads us through diverse topics ranging from the fundamental

quantum mechanics of photon–matter interaction to practical pixel readout strategies.

The first third of the chapter discusses the quantum mechanical nature of optical

signal detection. The middle third considers performance metrics and noise charac￾teristics of optoelectronic detectors. The final third overviews specific detector

arrays. Ultimately, we need the results of this chapter to develop mathematical

models for optoelectronic image detection. We delay detailed consideration of such

models until Chapter 7, however, because we also need the coherence field models

introduced in Chapter 6.

5.2 QUANTUM MECHANICS OF OPTICAL DETECTION

We introduce increasingly sophisticated models of the optical field and optical signals

over the course of this text. The geometric visibility model of Chapter 2 is sufficient

to explain simple isomorphic imaging systems and projection tomography, but is not

capable of describing the state of optical fields at arbitrary points in space. The wave

model of Chapter 4 describes the field as a distribution over all space but does not

accurately account for natural processes of information encoding in optical sources

and detectors. Detection and analysis of natural optical fields is the focus of this

chapter and Chapter 6.

Electromagnetic field theory and quantum mechanical dynamics must both be

applied to understand optical signal generation, propagation, and detection. The pos￾tulates of quantum mechanics and the Maxwell equations reflect empirical features of

optical fields and field–matter interactions that must be accounted for in optical

system design and analysis. Given the foundational significance of these theories,

it is perhaps surprising that we abstract what we need for system design from just

one section explicitly covering the Maxwell equations (Section 4.2) and one

section explicitly covering the Schro¨dinger equation (the present section). After

Section 4.2, everything that we need to know about the Maxwell fields is contained

in the fact that propagation consists of a Fresnel transformation. After the present

section, everything we need to know about quantum dynamics is contained in the

fact that charge is generated in proportion to the local irradiance.

Quantum mechanics arose as an explanation for three observations from optical

spectroscopy:

1. A hot object emits electromagnetic radiation. The energy density per unit wave￾length (e.g., the spectral density) of light emitted by a thermal source has a

148 DETECTION

temperature-dependent maximum. (A source may be red-hot or white-hot.)

The spectral density decays exponentially as wavenumber increases beyond

the emission peak.

2. The spectral density excited by electronic discharge through atomic and simple

molecular gases shows sharp discrete lines. The discrete spectra of gases are

very different from the smooth thermal spectra emitted by solids.

3. Optical absorption can result in cathode rays, which are charged particles

ejected from the surface of a metal. A minimum wavenumber is required to

create a cathode ray. Optical signals below this wavenumber, no matter how

intense, cannot generate a cathode ray.

These three puzzles of nineteenth-century spectroscopy are resolved by the postu￾late that materials radiate and absorb electromagnetic energy in discrete quanta.

A quantum of electromagnetic energy is called a photon. The energy of a photon

is proportional to the frequency n with which the photon is associated. The constant

of proportionality is Planck’s constant h, such that E ¼ hn. Quantization of electro￾magnetic energy in combination with basic statistical mechanics solves the first

observation via the Planck radiation formula for thermal radiation. The second obser￾vation is explained by quantization of the energy states of atoms and molecules,

which primarily decay in single photon emission events. The third observation is

the basis of Einstein’s “workfunction” and is explained by the existence of structured

bands of electronic energy states in solids.

The formal theory of quantum mechanics rests on the following axioms:

1. A quantum mechanical system is described by a state function jCl.

2. Every physical observable a is associated with an operator A. The operator acts

on the state C such that the expected value of a measurement is kCjAjCl.

3. Measurements are quantized such that an actual measurement of a must

produce an eigenvalue of A.

4. The quantum state evolves according to the Schro¨dinger equation

HC ¼ ih @C

@t (5:1)

where H is the Hamiltonian operator.

The first three postulates describe perspectives unique to quantum mechanics, the

fourth postulate links quantum analysis to classical mechanics through Hamil￾tonian dynamics.

There are deep associations between quantum theory and the functional spaces and

sampling theories discussed in Chapter 3: C is a point in a Hilbert space V, and V is

spanned by orthonormal state vectors {Cn}. The simplest observable operator is the

state projector Pn ¼ jCnlkCnj. The eigenvector of Pn is, of course, Cn. If PnC ¼ 0,

5.2 QUANTUM MECHANICS OF OPTICAL DETECTION 149