OPTICAL IMAGING AND SPECTROSCOPY Phần 5 ppsx

where fa ¼ arg[W(2a, 0, n)]. Note that the relative position of the two point sources

affects the spectrum of the image field even though the points are unresolved. The

scattered spectrum observed on the optical axis as a function of a and wavelength

for a jinc distributed cross-spectral density is illustrated in Fig. 6.3. We assume

that the spectrum of the illuminating source is uniform across the observed range.

The scattered spectrum is constant if the two points are in the same position or if

the two points are widely separated. The scattered power is doubled if the two

points are at the same point as a result of constructive interference. If the two

points are separated in the transverse plane by 1–2 wavelengths, the spectrum is

weakly modulated, as illustrated Fig. 6.3(b). The spectral modulation is much

greater if the sources are displaced longitudinally or if the scattered light is observed

from an off-axis perspective. This example is considered in Problem 6.3; more

general discussion of spectral modulation by secondary scattering is presented in

Sections 6.5 and 10.3.1.

While the three examples that we have discussed have various implications for

imaging and spectroscopy, our primary goal has been to introduce the reader to analy￾sis of cross-spectral density transformations and diffraction. Equation (6.20) is quite

general and may be applied to many optical systems. Now that we know how to pro￾pagate the cross-spectral density from input to output, we turn to the more challen￾ging topic of how to measure it.

6.3 MEASURING COHERENCE

We saw in Section 6.2 that given the cross-spectral density (or equivalently

the mutual coherence) on a boundary, the cross-spectral density can be calculated

over all space. But how do we characterize the coherence function on a boundary?

We have often noted that optical detectors measure only the irradiance I(x, y, t)

over points x, y, and t in space and time. Coherence functions must be inferred

from such irradiance measurements. The goal of optical sensor design is to lay out

physical structures such that desired projections of coherence fields are revealed in

irradiance data.

Sensor performance metrics are complex and task-specific, but it is useful to

start with the assumption that one wishes simply to measure natural cross-spectral

densities or mutual coherence functions with high fidelity. We explore this

approach in simple Michelson and Young interferometers before moving on to

discuss coherence measurements of increasing sophistication based on parallel and

indirect methods.

6.3.1 Measuring Temporal Coherence

The temporal coherence of the field at a point r may be characterized using a

Michelson interferometer, as sketched in Fig. 6.4. Input light from pinhole is colli￾mated and split into two paths. Both paths are retroreflected on to a detector using

mirrors. One of the mirrors is on a translation stage such that its longitudinal position

198 COHERENCE IMAGING

may be varied by an amount d. If the input field is E(t), the irradiance striking the

detector is

I(d) ¼ 1

4

E(t) þ E t þ

2d

c



2 * +

¼ G(0)

2 þ

1

4

G 2d

c

 þ

1

4

G 2d

c

 (6:34)

where we have abbreviated the single-point mutual coherence G(r, r, t) with G(t).

G(t) is isolated from G(0) and G(t) in Eqn. (6.34) by Fourier filtering. The

Fourier transform of I(d) is

^I(u) ¼ G(0)

2

d(u) þ

c

8

S n ¼ uc

2

 þ

c

8

S n ¼ uc

2

 (6:35)

S(n) is the positive frequency component of I

ˆ(u), and G(t) is the inverse Fourier trans￾form of S(n).

The Fourier transform pairing between the power spectrum and the mutual coher￾ence corresponds to a relationship between spectral bandwidth and coherence time

through the Fourier uncertainty relationship. The bandwidth sn measures the

support of S(n), and the coherence time tc / 1=snu measures the support of G(t).

Various precise definitions for each may be given; the variance of Eqn. (3.22) may

be the best measure. For present purposes it most useful to consider the relationship

in the context of common spectral lines, as listed in Table 6.2.

Figure 6.4 Measurement of the mutual coherence using a Michelson interferometer. Light

from an input pinhole or fiber is collimated into a plane wave by lens CL and split by a beam￾splitter. Mirror M2 may be spatially shifted by an amount d along the optical axis, producing a

relative temporal delay 2d/c for light propagating along the two arms. Light reflected from M1

interferes with light from M2 on the detector.

6.3 MEASURING COHERENCE 199

The Gaussian and Lorentzian spectra are plotted in Fig. 6.5. A common character￾istic is that the spectrum is peaked at a center frequency n0 and has a characteristic

width sn. The mutual coherence function oscillates rapidly as a function of t with

period n0. The mutual coherence peaks at t ¼ 0 and has characteristic width 1=sn.

Mechanical accuracy and stability must be precise to measure coherence using a

Michelson interferometer. The output irradiance I(d) oscillates with period l0=2,

where l0 ¼ c=n0. Nyquist sampling of I(d) therefore requires a sampling period of

less than l0=4, which corresponds to 100–200 nm at optical wavelengths. Fine

sampling rates on this scale are achievable using piezoelectric actuators to translate

TABLE 6.2 Spectral Density and Mutual Coherence

Lineshape S(n) G(t)

Monochromatic d(n n0) e2pin0t

Gaussian (1=sn)ep [(nn0)

2=s 2

n ] e2pin0t eps2

n t

2

Lorentzian sn=[(n n0)

2 þ s2

n ] 2pe2pin0t e2psnjtj

Figure 6.5 Spectral densities and mutual coherence of Gaussian and Lorentzian spectra. The

mutual coherence is modulated by the phasor e2pin0t ; the magnitude of the mutual coherence is

plotted here.

200 COHERENCE IMAGING

the mirror M2. Ideally, the range over which one samples should span the coherence

time tc. This corresponds to a sampling range D ¼ c=2tc.

The Michelson interferometer is used in this way is a Fourier transform

spectrometer (there are many other interferometer geometries that also produce FT

spectra). The Michelson is the first encounter in this text with a true spectrometer.

While we begin to mention spectral degrees of freedom more frequently, we delay

most of our discussion of Fourier instruments until Chapter 9. For the present purposes

it is useful to note that the FT instrument is particularly useful when one wants to

measure a spectrum using only one detector. FT instruments are favored for spectral

ranges where detectors are noisy and expensive, such as the infrared (IR) range covering

2–20 mm. Instruments in this range are sufficiently popular that the acronym FTIR

covers a major branch of spectroscopy.

6.3.2 Spatial Interferometry

One must create interference between light from multiple points to characterize

spatial coherence. The most direct way to measure W(x1, y1, x2, y2, n) samples the

interference of every pair of points as illustrated in Fig. 6.6. Pinholes at points P1

and P2 transmit the fields E(P1, n) and E(P2, n). Letting h(r, P, n) represent the

impulse response for propagation from point P on the pinhole plane to point r to

the detector plane, the irradiance at the detector array is

I(r) ¼

ð

jE(P1, n)h(r, P1, n) þ E(P2, n)h(r, P2, n)j

2 D E dn

¼ I(P1) þ I(P2) þ

ð

W(P1, P2, n)h

(r, P1, n)h(r, P2, n) dn

þ

ð

W(P2, P1, n)h

(r, P2, n)h(r, P1, n) dn (6:36)

Figure 6.6 Interference between fields from points P1 and P2.

6.3 MEASURING COHERENCE 201

Approximating h with the Fresnel kernel models the irradiance at point (x, y) on

the measurement plane as

I(x, y) ¼ I(P1) þ I(P2)

þ

ð

W(x1, y1, x2, y2, n)

 exp 2pin

xDx þ yDy

cd  exp 2pin q

cd dn

þ

ð

W(x2, y2, x1, y1, n)

 exp 2pin

xDx þ yDy

cd  exp 2pin q

cd dn

 (6:37)

where d is the distance from the pinhole plane to the measurement plane and as before

Dx ¼ x1 x2 and q ¼ xDx þ yDy.

With Fresnel diffraction, the interference pattern produced by a pair of pinholes

varies along the axis joining the pinholes and is constant along the perpendicular

bisector, as illustrated in Fig. 6.6. We isolate the 1D interference pattern mathe￾matically by rotating variables in the x, y plane such that ~x ¼ (xDx þ yDy)= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Dx2 þ Dy2 p and ~y ¼ (xDx yDy)= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Dx2 þ Dy2 p . In the rotated coordinate system

the interference term in the two-pinhole diffraction pattern becomes

ð

W(x1, y1, x2, y2, n)exp 2pin

~x ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Dx2 þ Dy2 p

cd !exp 2pin q

cd dn

, (6:38)

which is independent of ~y.

The interference term is the inverse Fourier transform of the cross-spectral density

with respect to n, which means by the Wiener–Khintchine theorem that the inter￾ference is proportional to the mutual coherence. Specifically

I(~x) ¼ I(P1) þ I(P2)

þ G x1, y1, x2, y2, t ¼ q ~x ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Dx2 þ Dy2 p

cd !

þ G x2, y2, x1, y1, t ¼ ~x ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Dx2 þ Dy2 q p

cd ! (6:39)

Like the Michelson interferometer, the two-pinhole interferometer measures the

mutual coherence. In this case, however, samples are distributed at a single moment

in time along a spatial sampling grid.

202 COHERENCE IMAGING