OPTICAL IMAGING AND SPECTROSCOPY Phần 5 ppsx

198 COHERENCE IMAGING 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 – 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 analysis 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 propagate the cross-spectral density from input to output, we turn to the more challenging 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 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 collimated 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 6.3 MEASURING COHERENCE 199 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 beamsplitter 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 may be varied by an amount d If the input field is E(t), the irradiance striking the detector is *   +  d 2 E(t) ỵ E t þ  I(d) ¼  c      G(0) 2d 2d ỵ G ỵ G À (6:34) ¼ 4 c c 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(À t) in Eqn (6.34) by Fourier filtering The Fourier transform of I(d) is     ^ ¼ G(0) d(u) þ c S n ¼ uc þ c S n ¼ À uc (6:35) I(u) 8 S(n) is the positive frequency component of ˆ(u), and G(t) is the inverse Fourier transI form of S(n) The Fourier transform pairing between the power spectrum and the mutual coherence 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=sn u 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 6.2 200 COHERENCE IMAGING TABLE 6.2 Spectral Density and Mutual Coherence Lineshape Monochromatic Gaussian Lorentzian S(n) d(n À n0 ) 2 (1=sn )eÀp [(nÀn0 ) =sn ] sn =[(n À n0 )2 þ sn ] G(t) e2pin0 t 2 e2pin0 t eÀpsn t 2pe2pin0 t eÀ2psn jtj The Gaussian and Lorentzian spectra are plotted in Fig 6.5 A common characteristic 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 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 Figure 6.5 Spectral densities and mutual coherence of Gaussian and Lorentzian spectra The mutual coherence is modulated by the phasor e2pin0 t ; the magnitude of the mutual coherence is plotted here 6.3 MEASURING COHERENCE 201 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 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 ðD E I(r) ¼ jE(P1 , n)h(r, P1 , n) ỵ E(P2 , n)h(r, P2 , n)j2 dn ẳ I(P1 ) ỵ I(P2 ) ỵ W(P1 , P2 , n)hà (r, P1 , n)h(r, P2 , n) dn ỵ W(P2 , P1 , n)h (r, P2 , n)h(r, P1 , n) dn Figure 6.6 Interference between fields from points P1 and P2 (6:36) 202 COHERENCE IMAGING Approximating h with the Fresnel kernel models the irradiance at point (x, y) on the measurement plane as I(x, y) ẳ I(P1 ) ỵ I(P2 ) ỵ W(x1 , y1 , x2 , y2 , n)     xDx ỵ yDy q exp 2pin dn exp 2pin cd cd ỵ W(x2 , y2 , x1 , y1 , n)    xDx ỵ yDy q  exp 2pin dn  exp À2pin cd cd (6:37) where d is the distance from the pinhole plane to the measurement plane and as before x y Dx ẳ x1 x2 and q ẳ "Dx ỵ "Dy 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 mathematically by rotating variables in the x, y plane such that ~ ¼ (xDx ỵ yDy)= x p p Dx2 ỵ Dy2 and ~ ẳ (xDx yDy)= Dx2 ỵ Dy2 In the rotated coordinate system y the interference term in the two-pinhole diffraction pattern becomes p!  ~ Dx2 ỵ Dy2 q  x exp 2pin dn , W(x1 , y1 , x2 , y2 , n)exp 2pin cd cd (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 interference is proportional to the mutual coherence Specically I(~) ẳ I(P1 ) ỵ I(P2 ) x p! q ~ Dx2 ỵ Dy2 x ỵ G x1 , y1 , x2 , y2 , t ¼ cd p! ~ Dx2 ỵ Dy2 q x ỵ G x2 , y2 , x1 , y1 , t ¼ 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 6.3 MEASURING COHERENCE 203 Sampling for the pinhole system is somewhat complicated by the uneven scaling of the sampling rate Ideally, one would sample t over the range (0, tc ) at resolution 1=2nmax , where Dn is the bandwidth of the field and nmax is the maximum temporal frequency This corresponds to a spatial sampling range X ¼ ctc d=D x at sampling rate cd=2nmax If the pixel pitch for sampling the interference pattern is 10l, which may be typical of current visible focal planes, one would need to ensure that d=Dx 20 In this case tc ¼ 100 fs would correspond to X ¼ 0:6 mm As with the Michelson interferometer, one isolates the cross-spectral density from I(~) by Fourier analysis The Fourier transform of Eqn (6.39) with respect to ~ yields x x the following term in the range u 0: ! ! cdu qu ^ 0) ¼ W x1 , y1 , x2 , y2 , n ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp 2pi p (6:40) I(u Dx2 ỵ Dy2 Dx2 ỵ Dy2 Thus we are able to isolate the complex coherence function by Fourier filtering In the current example we use an entire plane to characterize W(x1 , y1 , x2 , y2 , n) as a function of n with (x1 , y1 , x2 , y2 ) held constant As an example, suppose that a primary source consisting of a point radiator with a spectral radiance S(n) illuminates the pinholes Assuming that the point source is located at (x0 , y0 , z ¼ 0), Eqn (6.21) immediately yields the cross-spectral density at planes z = W(Dx, Dy, q, n) ¼ l4 S(n) l2 z2     (x0 Dx ỵ y0 Dy) q exp i2pn exp Ài2pn cz cz (6:41) and, for the two-pinhole system of Fig 6.6 I(x, y) ẳ 2I0 ỵ G[t (x, y)] þ G[À t(x, y)] (6:42) where G(t) is the mutual coherence and the inverse Fourier transform of S(n) and (x0 x)Dx ỵ ( y0 y)Dy ỵ 2q cd p x0 Dx ỵ y0 Dy 2~ Dx2 ỵ Dy2 ỵ 2q x ẳ cd t (x, y) ẳ (6:43) A plot of I(~) for q ¼ and for Dx=d ¼ 0:1 is shown in Fig 6.7 for a Gaussian x spectrum of width 10 nm with a central wavelength of 600 nm The interference pattern produced has a period of l0 d=Dx, which is mm in this case Thus, one would hope to spatially sample at mm resolution over the 1.5 mm range to capture this interference pattern 204 COHERENCE IMAGING Figure 6.7 Irradiance pattern I(x) produced by a 10 nm spectral bandwidth source centered on 600 nm observed through a two-pinhole interferometer with Dx=d ¼ 0:1 Plot (a) details the center region of plot (b) Each configuration of the pinholes enables us to characterize W(Dx, Dy, ", ", n) as x y a function of n for a particular value of (Dx, Dy, ", ") One can imagine moving the x y pinholes around the plane to fully sample the cross-spectral density, but the two-pinhole approach is not a very efficient sampling mechanism and faces severe challenges with respect to sampling rate and range for large or small values of Dx The two-pinhole approach is nevertheless the basic strategy underlying the Michelson stellar interferometer [58] The sampling efficiency can be improved by using lens combinations to reduce the spatial pattern due to one pair of pinholes to a line, thus enabling “two slit” characterization of distinct values of Dx and " in x parallel Dual-slit sampling enables full utilization of a 2D measurement plane for independent measurements, but the mechanical complexity and limited throughput of this approach pose challenges 6.3.3 Rotational Shear Interferometry The cross-spectral density on a plane is a five-dimensional function of four spatial dimensions and temporal frequency A rotational shear interferometer (RSI) characterizes this space from nondegenerate measurements on a 2D plane The basic structure of an RSI is sketched in Fig 6.8 Figure 6.9 is a photograph of an RSI The structure is the same as for a Michelson interferometer, but the flat mirrors have been replaced by wavefront folding mirrors A wavefront folding mirror is a right angle assembly of two reflecting surfaces A light beam entering such an interferometer is inverted across the fold axis, as described below In the RSI of Fig 6.6 the fold mirrors consist of right-angle prisms The “fold axis” is the right-angle edge 6.3 Figure 6.8 MEASURING COHERENCE 205 System layout of a rotational shear interferometer Figure 6.9 Photograph of a rotational shear interferometer The fold mirrors consist of rightangle prisms, one of the prisms is mounted in a computer controlled rotation stage to adjust the longitudinal displacement and shear angle 206 COHERENCE IMAGING of the prism As illustrated in Fig 6.8, the fold axes of the mirrors are displaced from the vertical (x) axis by angle u on one arm and by Àu on the other arm The effect of angular displacement of the fold axes is to produce a field distribution from each arm rotated in the x, y plane with respect to the field from the other arm Let E(x, y) be the electromagnetic field that would be produced on the detection plane of an RSI after reflection from a flat mirror If this same field is reflected by a fold mirror with fold axis is parallel to y, the resulting reflected field is E(Àx, y) If the fold axis is parallel to x, the resulting field is E(x, Ày) If the fold axis lies at an arbitrary angle u with respect to the x axis in the xy plane, the resulting eld is E[x cos(2u) ỵ y sin(2u), x sin(2u) À y cos(2u)] With the fold axes of the mirrors on the two reflecting arms of the RSI counter rotated by u and Àu, the electromagnetic field on the detection plane is E[x cos(2u) ỵ y sin(2u), x sin(2u) y cos(2u)] ỵ E[x cos(2u) y sin(2u), Àx sin(2u) À y cos(2u)] exp(if) (6:44) where, as with a Michelson interferometer, f ¼ 4p nd=c is the phase difference between the two arms produced by a relative longitudinal displacement d between mirrors on the two arms The spectral density on the detection plane is found by taking appropriate expectation values of the square of Eqn (6.44), which yields Srsi (x, y, n) ẳ S[x cos(2u) ỵ y sin(2u), x sin(2u) y cos(2u), n] ỵ S[x cos(2u) y sin(2u), x sin(2u) y cos(2u), n] ỵ e4pi(nd=c) W[Dx ¼ 2y sin(2u), Dy ¼ 2x sin(2u), " ¼ 2x cos(2u), " ¼ À2y cos(2u), n] y x þ eÀ4pi(nd=c) W[Dx ¼ À2y sin(2u), Dy ¼ À2x sin(2u), " ¼ 2x cos(2u), " ¼ À2y cos(2u), n] y x (6:45) where S(x, y, n) and W(D x, Dy, ", ", n) are the spectral densities that would appear on x y the detection plane if the fold mirrors were replaced by flat mirrors As an example, suppose that an RSI is illuminated by a remote point source with spectral density S(n) The cross-spectral density incident on the RSI measurement plane for this case is given by Eqn (6.41) Substituting in Eqn (6.45) and ignoring constant factors, the spectral density observed on the RSI measurement plane is  n n o Srsi (x, y, n) ẳ S(n) ỵ cos 4p [ux y sin(2u) ỵ uy x sin(2u) ỵ d)] c (6:46) where ux ¼ x=z and uy ¼ y=z are the angular positions of the point source as observed at the RSI Figure 6.10 shows interference patterns detected by an RSI observing a remote point illuminated at two wavelengths In this 6.3 MEASURING COHERENCE 207 Figure 6.10 RSI raw data image for the two-color point source of Eqn (6.47): q q (a) ux2 ỵ uy2 ẳ 1:348; (b) ux2 ỵ uy2 ẳ 38 (u ẳ 28 in both cases) case S(n) ẳ I1 d(n n1 ) ỵ I2 d(n À n2 ), and the irradiance on the detector is 4pn1 sin(2u) (ux y ỵ uy x) c ! 4pn2 sin (2u) (ux y ỵ uy x) ỵ I2 cos c ! I(x, y) ẳ I1 ỵ I2 þ I1 cos (6:47) Consistent with Eqn (6.47), the images in Fig 6.10 show beating between two harmonics, as confirmed in Fig 6.11, which shows the 2D FFT of the irradiance patterns with DC frequencies suppressed The FFT produces images of the illuminating point source at [u ¼ 2uy sin(2u)=l, v ¼ 2ux sin(2u)=l] The point image further from the origin thus corresponds to the image of the source at the bluer illuminating wavelength The figures show interference patterns for two different angular displacements of the point source from the optical axis As expected, the fringe frequency increases as the angle increases The dark vertical lines at the left edge of Fig 6.10(a) are shadows of the fold edge of the wavefront folding mirrors The total angular displacement the fold mirrors is 48, meaning u ¼ 28 Note from Eqn (6.46) that the fringe frequency is proportional to sin(2u), so u may be set to match the fringe pattern to the sampling rate on the detector plane The fringe frequency is also proportional to n and the angular position If u is fixed, nux and nuy may be determined from Fourier analysis of Eqn (6.46) n and ux , uy may be disambiguated by varying d or the orientation of the RSI relative to the scene 6.6 235 MODAL ANALYSIS where U is a unitary matrix with rows equal to the mutually orthogonal eigenvectors of W and L is a diagonal matrix with Lii equal to the eigenvalues of W Substituting Eqn (6.103) in Eqn (6.99) yields W(x1 , y1 , x2 , y2 , n) ¼ X à Ln (n)cC (x1 , y1 , n)cC (x2 , y2 , n) n n (6:104) n where the coherent modes cC are linear combinations of the original modes according to cC ¼ Uf Equation (6.104) is called a coherent-mode decomposition of the crossspectral density By this decomposition we find that the cross-spectral density can always be reduced to a superposition of densities associated with a discrete set of perfectly coherent but mutually orthogonal modes A more rigorous discussion of the coherentmode decomposition without the assumption of a finite basis is presented by Wolf [249] As discussed in Section 6.6.4, measurements of the power spectral density or irradiance may consist of projections of the cross-spectral density on particular modes In particular, the projection on the coherent modes may be measured Since such measurements must be nonnegative and since there is no significance to coherent modes with null amplitude, W is a positive definite matrix such that Ln for all n As an example, consider an object consisting of two mutually incoherent Gaussian sources in a plane The sources are separated by a distance a and are of waist size 2a One of the sources is twice as intense as the other The cross-spectral density for this source is     y  y  x1 À 0:5a x2 À 0:5a 2fo f fo W(x1 , y1 , x2 , y2 , n) ¼ S(n)fo 2^ a 2^ a 2a o 2a    ! x1 ỵ 0:5a x2 ỵ 0:5a ỵ fo fo 2^ a 2^ a ẳ S(n)[2jf ihf j ỵ jfỵ ihfỵ j] (6:105) where fo (x) is the fundamental Gaussian mode of Eqn (3.55), jfÀ i ¼ fo ( y=2a)fo [(x 0:5a)=2a)] and jfỵ i ẳ fo ( y=2a)fo [(x þ 0:5a)=2a] An orthonormal basis may be assembled from these modes using jfa i ẳ jf i ỵ jf þ i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ eÀ(p=8) 23=4 jf À i jf ỵ i p jfb i ẳ 23=4 À eÀ(p=8) (6:106) On this basis, the cross-spectral density matrix W is  W% 3:55 0:52 0:52 0:69  (6:107) 236 COHERENCE IMAGING We assemble coherent modes from the eigenfunctions of this matrix as described above; the eigenvalues are approximately 3.6 and 0.6 The coherent modes are jf1 i % À0:98jfa i À 0:17jfb i jf2 i ¼ À0:17jfa i þ 0:98jfb i (6:108) and the cross-spectral density is W(x1 , y1 , x2 , y2 , n) % S(n)½3:6jf1 ihf1 j ỵ 0:6jf2 ihf2 j (6:109) The irradiance, cross-spectral density, and coherent-mode distributions for this field are illustrated in Fig 6.24 The coherent-mode decomposition uniquely yields orthonormal decomposition modes, but this does not necessarily imply that it is more descriptive of the “true” source In the specification of the current example, the overlapping Gaussian distributions might come from distinct lasers The coherent-mode decomposition takes both sources into account to produce a decomposition into orthogonal modes It is interesting to note that the amplitude of the stronger coherent mode is greater than the amplitude of either of the individual sources used to compose the joint source We find, therefore, that it is possible to integrate power in a single coherent mode from multiple mutually incoherent sources This is a subtle point; we will return to consider its significance after showing in Section 6.6.3 that it is impossible to efficiently combine power from mutually orthogonal modes 6.6.3 Modal Transformations We turn next to the effect of propagation through an optical system and transformations of coherent modes As discussed relative to Eqn (6.97), an orthogonal basis remains orthogonal on diffraction, meaning that the coherent-mode spectrum Ln remains invariant even as the spatial structure of the coherent modes evolves Such invariance is particularly relevant in resonator and waveguide analysis, where one regards the modes as 3D spatial distributions In sensor systems, however, one is more likely to seek to transform the modal structure of the field to enable effective information extraction It is important to understand that the coherent-mode structure is a property of a single realization of the field, and that different objects may produce completely different coherent modes Complete characterization of a field is equivalent to discovery of both the coherent-mode spectrum and the coherent modes In many cases one has prior knowledge of the coherent modes; for example, in imaging an incoherent 2D object one may reasonably assume that the coherent modes correspond to the bandlimited Shannon basis functions In cases where the coherent modes are not known a priori, however, one has no choice but to make measurements on a nonideal basis We discuss measurement strategies in Section 6.6.4 after considering limits on modal transformations 6.6 MODAL ANALYSIS 237 Figure 6.24 Plot (a) shows the irradiance in the (x, y) plane associated with the cross-spectral density of Eqn (6.105), (b) is a cross section of the 5D function W(x1 , y1 , x2 , y2 , n) at fixed n in the plane y1 ¼ 0, y2 ¼ 0, (c) and (d) are the orthonormal coherent-mode distributions, and (e) plots the cross section of the coherent modes along the x axis 238 COHERENCE IMAGING In general, one may assume that an optical element exists to transform any individual mode into any other We are most aware of the use of lenses to transform plane waves into focusing beams, but we can easily imagine the use of holograms to couple any pair of modes Volume holograms, as discussed in Section 4.8, can approach 100% efficiency in such associations While mode-to-mode transformations are extremely useful, a more powerful mapping would map more than one input mode into a single output mode For example, one might wonder whether it would be possible to make a device to focus two distinct input plane waves onto a single output focal spot The answer to this question is “yes,” but the efficiency with which light can be combined from two distinct modes depends on the relative coherence of the modes Two mutually coherent plane waves of equal amplitude can be focused on the same spot with 100% efficiency The maximum efficiency for focusing two mutually incoherent plane waves on the same spot, in contrast, is 50% The most basic geometry for two-mode integration is illustrated in Fig 6.25, which shows two incident plane waves combined by a beamsplitter The input modes jf1 i and jf2 i interfere in output modes jfa i and jfb i The mutual intensity of the input modes is described by three numbers: J11 ¼ I1 , J22 ¼ I2 , and à J12 ¼ J21 In the bracket notation of Section 6.6.2, the mutual intensity of the input eld is  J ẳ jf1 i jf2 iị I1 J21 J12 I2  hf1 j hf2 j  Figure 6.25 Use of a beamsplitter to combine two plane waves (6:110) 6.6 MODAL ANALYSIS 239 The beamsplitter transforms the input modes into output modes jfa i and jfb i such that  jfa i jfb i   jf1 i ¼T jf2 i  (6:111) where, as discussed in Section 6.6.1 and in Problem 6.13, T is a unitary matrix The mutual intensity of the output field is described by the matrix J ¼ Ty   J12 T I2 I1 J21 (6:112) In the particularly common case of a beamsplitter that sends half of the incident power from jf1 i onto output mode jfa i and half onto output mode jfb i, a suitable form for T is T ¼ pffiffiffi  1 À1  (6:113) such that Jẳ  I1 ỵ I2 ỵ 2Re(J12 ) I1 I2 2iIm(J12 ) I1 I2 ỵ 2iIm(J12 ) I1 ỵ I2 2Re(J12 )  (6:114) We note several important points: J is Hermitian Power is conserved: Tr(J) ¼ I1 þ I2 pffiffiffiffiffiffiffi If the input modes are absolutely coherent and in phase, J12 ¼ I1 I2 If I1 ¼ I2 ¼ I in this case then the output mutual intensity is described by  J¼ 2I 0  (6:115) The student may wish as an exercise to confirm that the input field is described by a single coherent mode in this case If the input modes are relatively incoherent, then J12 ¼ If I1 ¼ I2 ¼ I in this case, then the output mutual intensity is described by  J¼ I 0 I  (6:116) 240 COHERENCE IMAGING If I1 = I2 , then Jẳ  I1 ỵ I2 I1 I2 I1 I I1 ỵ I  (6:117) meaning that output modes are partially coherent The coherent modes for this system would correspond to the individual outputs for each input mode Transformations of the form J0 ¼ Ty JT with T unitary not change the eigenvectors or eigenvalues of J and therefore maintain the coherent-mode structure unchanged There are various mechanisms, however, by which an optical system can change the coherent modes and the modal eigenvalues These include † † † Modal Filtering Various devices decrease the number of propagating modes in an optical system A pinhole is the most common example We have used pinholes several times to select a single spherical wave from a broader field distribution, most recently in our discussion of the two-pinhole interferometer Such devices are called spatial filters As discussed in Chapter 9, spatial filters are commonly used in spectroscopy to break degeneracies between spectral and spatial degrees of freedom Complex spatial filters may be implemented using diffractive devices or coded apertures In the example of Fig 6.25, spatial filtering may consist simply of discarding the light on one of the output arms In that case, we have combined two input modes into one output mode Absorption A device that differentially absorbs some of the light in an optical system breaks time reversal symmetry and restructures the coherent modes Modal Decorrelation Modes propagate on different paths through the optical system If the structure of the optical system is time varying on these paths, the relative coherence of the modes may be destroyed The most common example is propagation through turbulence In the example of Fig 6.25, the two output beams may experience different time-varying phase modulation Such modulation can render the output beams relatively incoherent, which diagonalizes J In the case of the 50/50 beamsplitter with incoherent input beams, this produces relatively incoherent output beams with equal irradiances of (I1 ỵ I2 )=2 Other mechanisms for changing the coherent-mode structure include laser gain and nonlinear wavemixing For imaging and spectroscopy we focus on systems where optical components implement joint transformations on 106 – 1012 modes simultaneously We begin consideration of such systems with the coherent mode decomposition of the crossspectral density at the input W(x1 , y1 , x2 , y2 , n) ¼ X n ln (n)cà (x1 , y1 , n)cn (x2 , y2 , n) n (6:118) 6.6 MODAL ANALYSIS 241 As discussed in Section 6.6.2, the mode amplitudes ln (n) are real and positive and the modes are orthonormal The coherent modes are transformed on propagation through an optical system such that the cross-spectral density at the output is X ~à ~ W(x01 , y01 , x02 , y02 , n) ¼ ln (n)cn (x01 , y01 , n)cn (x02 , y02 , n) (6:119) n ~ In view of the impact of optical loss and spatial filtering, the functions cn (x, y, n) are not necessarily orthogonal [251] The cross-spectral density across the output aperture is described by the new coherent-mode decomposition X W(x01 , y01 , x02 , y02 , n) ¼ Ln (n)Cà (x01 , y01 , n)Cn (x02 , y02 , n) (6:120) n n where Cn (x, y, n) form a new set of orthonormal coherent modes and the functions Ln are new eigenvalues Using the nodal orthonormality, projection of Eqn (6.120) against the coherent modes yields ðððð W(x01 , y01 , x02 , y02 , n)Cà (x01 , y01 , n)  Cm (x02 , y02 , n) dx01 dy01 dx02 dy02 ¼ Lm (n) m (6:121) Substitution of the cross-spectral density of Eqn (6.119) in the integral of Eqn (6.121) yields X jcnm j2 ln ¼ Lm (6:122) n where cnm is the projection of the transformed input coherent modes on the output plane coherent modes: ðð ~à cnm ¼ cn (x, y, n)Cm (x, y, n) dx dy (6:123) We note again that the input coherent modes are orthonormal Barring laser gain, nonlinear wavemixing, or sources within the optical system, the spectral density of a mode cannot increase in magnitude This means that ðð à ~ jcn (x, y, n)j2 dx dy (6:124) Since the output coherent modes are orthonormal and complete over the output space, we also find that ðð ~à jcn (x, y, n)j2 dx dy ¼ X m jcnm j2 (6:125) 242 COHERENCE IMAGING and X jcnm j2 (6:126) m Summing Eqn (6.122) with respect m, this implies that X ln ! n X Lm (6:127) m meaning that the input power is greater than or equal to the output power Alternatively, we may express the output coherent modes in terms of the input coherent modes Of course, in a system satisfying time reversal, the output and input coherent modes are matched one to one and cnm ¼ dnm If we consider the more general reversible transformation between any set of orthogonal input modes and orthogonal output modes, conservation of power on time reversal produces the complimentary relationship to Eqn (6.126): X jcnm j2 (6:128) n While the structure of the coupling coefficients may change as a result of loss, filtering, and decoherence, all three of these mechanisms decrease, rather than increase, the effective value of the coupling coefficient between an input mode and an output mode We therefore find that Eqn (6.128) holds in any gain or source free optical system Equation (6.122) relates the power in the mth output mode to the power in input modes jcnm j2 is the fraction of the nth input-mode power that is coupled into the mth output coherent mode According to Eqn (6.128), the coefficients jcnm j2 are a weighted distribution such that no output mode can be brighter than the brightest input mode, for example Lmax lmax (6:129) The maximal brightness is obtained when cnm ¼ for n corresponding to the brightest input mode Equation (6.129) is an outcome of the second law of thermodynamics, which states that in isolated physical systems the total entropy can only remain constant or increase The entropy of a system is a measure of the total number of physical states that it might occupy If the exact state of the system is known, as in a coherent optical field, then the entropy is low If the energy of the system is distributed over many different coherent modes, theP entropy is higher In terms of normalized ~ coherent-mode amplitudes lm ¼ lm = n ln , the entropy of an optical field may be 6.6 MODAL ANALYSIS 243 expressed [190] H¼ X ~ ~ ln logln (6:130) n ~ If, for example, l1 ¼ and ln.1 ¼ 0, then H ¼ and the entropy is a minimum Alternatively, maximal entropy occurs for ln ¼ 1=N for all n, in which case H ¼ log N The more general statement of the second law in optical systems is Any time reversable optical system leaves the coherent-mode amplitudes unchanged Nonreversible effects such as modal filtering, absorption, and modal decorrelation transform the coherent-mode amplitudes so as to increase the system entropy Returning to the combination of power from multiple modes into a single mode or a smaller number of modes, one may say as a corollary to Eqn (6.129) that if energy is conserved, then Lmin ! lmin This means that the number of modes in the field is conserved when energy is conserved In the example discussed at the beginning of this section, two mutually coherent plane waves at the input can be combined into a single output mode with 100% efficiency because both input signals are part of the same coherent mode of the cross-spectral density If the input cross-spectral density consists of two modes, no single mode of the output cross-spectral density can exceed the stronger of the two input modes Conservation of modes may be related to the space – bandwidth product and the etendue of an optical system The space – bandwidth product is the product of the spatial support over which an optical signal is observed and the spatial frequency bandwidth of the field We saw in Section 3.6 that the number of discrete modes necessary to represent a signal is proportional to the space – bandwidth product The etendue, L / A2 V, of an optical system is the product of the solid angle V spanned by the wavevectors of radiation collected at the instrument aperture and the area A2 of the entrace pupil Of course, we know from many previous discussions that the angle of the wavevector is proportional to spatial frequency V is thus a measure of the spatial bandwidth, and the etendue is proportional to the space – bandwidth product, which is proportional to the number of modes that propagate through the system An energy conserving system must therefore conserve etendue 6.6.4 Modes and Measurement We have mentioned several times that measurements of the field take the form of irradiance or spectral density projections Back in Section 2.1 we introduced the concept of a visibility function to describe these projections We updated the visibility in terms of the coherent impulse response in Chapter and the incoherent or partially 244 COHERENCE IMAGING coherent response in this chapter With our understanding of coherence transformations, we are now ready to express the most general form of a first-order optical measurement as mi ¼ ðð hà (x1 , y1 , n)W(x1 , y1 , x2 , y2 , n)hi (x2 , y2 , n) dx1 dx2 dy1 dy2 dn i (6:131) where hi (x, y, n) is a coherent impulse response or visibility for the ith measurement Substituting the coherent-mode decomposition of W into Eqn (6.104) and representing the set of measurements as a vector m allows us to recast the measurement process as m ¼ HL (6:132) ð 2    hi (x, y, n)cC (x, y, n) dx dy dn hij ¼  j  (6:133) where and L is a vector of coherent-mode amplitudes Optical system design consists of selecting the measurement basis hi within physical and economic limits to enable estimation of the object under observation Ideally, design also consists of specifying the measurement matrix H in Eqn (6.132), but this assumes that we know the coherent modes in advance Common measurement scenarios in optical systems include the following cases: Focal systems, where the coherent modes of the system are known and one has the physical capacity to create a mode-specific filter to separate the coherent modes in the measurement system This scenario describes 2D focal imaging and conventional slit spectroscopy In the imaging case, for example, the coherent modes are bandlimited focused spots spanning the object plane A lens system maps the input spots to the output image, where each pixel is measured independently In the ideal case, H is the identity matrix for this system Criteria by which one might design H are discussed in Chapter 8, but for present purposes it is safe to say that if information is uniformly and independently distributed in the object pixels, then one can no better than to design H to be an identity If, on the other hand, object information is more subtly distributed, one may wish to consider generalized sampling strategies as dicussed in Section 7.5 Multidimensional systems, where the coherent modes are known but one lacks the physical capacity to independently measure each coherent mode Multispectral imaging, as discussed in Section 10.6, is the model system for this case In principle, one could create a filter assembly to separate each spectral channel and then implement focal imaging on the coherent 6.7 RADIOMETRY 245 modes, but physical implementation of such a system is impossible or inefficient, and one chooses to apply temporal multiplexing or generalized sampling instead One can directly design H in this case Interferometric systems, where the coherent modes are unknown and must be discovered as part of the measurement process Examples for this case include imaging through turbulent or inhomogeneous media and imaging under partially coherent illumination A typical strategy for this case involves selection of an a priori modal decomposition In some cases, the a priori basis may be “close” to a coherent mode decomposition In other cases, the a priori basis may simply have convenient mathematical properties The focal interferometry strategies discussed in Section 6.3.4 are an example of interferometric imaging on nearly orthogonal bases Imaging with a rotational shear interferometer, on the other hand, applies a Fourier basis to an unknown cross-spectral density 6.7 RADIOMETRY The discipline of radiometry focuses on measuring the energy content of optical radiation and on mapping the flow of energy through optical systems Radiometry describes the optical field using phenomenological functions related to the irradiance such as the spectral radiance, Bn (x, s, n), which is the power radiated through point x [ R3 in direction s [ S2 per unit solid angle per unit wavelength The SI unit of spectral radiance is watt per steradian per square meter per hertz The spectral radiance may be integrated along the wavelength axis to produce the radiance, which is the radiant energy propagating through x in the s direction per unit solid angle This section discusses the relationship between the spectral radiance and the crossspectral density and describes the constant radiance theorem, which limits the ability of optical systems to focus partially coherent light Unless otherwise indicated, the term radiance refers to the spectral radiance in subsequent discussion 6.7.1 Generalized Radiance As always in optical system analysis, propagation of the radiance from one boundary to the next is our first challenge Propagation of the radiance is most easily derived from the wave equations satisfied by the cross-spectral density "  # 2p n 2 W(x1 , x2 , n) ẳ r1 ỵ c (6:134) "  # 2p n W(x1 , x2 , n) ẳ r2 ỵ c where r2 is the 3D Laplacian with respect to the x1 These equations are derived by sub1 stituting the definition of the cross-spectral density into the wave equation [(Eqn (4.19)] 246 COHERENCE IMAGING The radiance is most often used to analyze diffuse objects with relatively low coherence, in which case one might assume that the cross-spectral density varies slowly with respect to " An object such that x @W(Dx, ", n) x @W(Dx, ", n) x ( @" x @D x (6:135) is said to be “quasihomogeneous” [165] (Homogeneous in this context is synonymous with spatially stationary.) A quasihomogeneous object is spatially uniform on the scale of the coherence cross section Given that the coherence cross section of an image field is approximately equal to the transverse resolution, a quasihomogeneous image must be slowly varying in comparison to the optical resolution This is not generally a good assumption in imaging, but radiance is a sufficiently useful concept that we neglect for the moment difficulties at the limits of resolution Reparameterizing W(x1 , x2 , n) in terms of Dx and " transforms Eqns (6.134) x into [198] " rDx Á r"W ¼ x # 2p n W(D x, ", n) ẳ x r2 ỵ r" þ Dx x c (6:136)  (6:137) Neglecting the Laplacian with respect to " under the quasihomogeneous approxix mation reduces Eqn (6.137) to " r2 Dx # 2p n W(D x, ", n) ẳ x ỵ c  (6:138) which is solved by W(D x, x, n) ¼ ð B(x, s, n)eÀ(2pin=c)sÁDx ds (6:139) V where according to Eqn (6.136) s Á rx Bn (x, s, n) ¼ (6:140) The “generalized radiance,” as proposed by Walther [244], is defined by the inverse transform corresponding to Eqn (6.139) B(x, s, n) ¼ ðð W(Dx, x, n)e(2pin=c)sÁDx dDx (6:141) 6.7 RADIOMETRY 247 where D x [ R2 Walther’s aim was to associate the phenomenological radiance with physical optics Ideally, a physically based radiance satisfies four criteria: Bn (x, s, n) is a linear transform of W(Dx, x, n) Bn ! 0, assuring that the radiance from an object is positive Bn ¼ outside the support of a radiating object ÐÐ cos u Bn (x, s, n) ¼ Jn (s, n), where the radiant intensity Jn (s, n) is the physical optical energy flux along s Unfortunately, the generalized radiance does not satisfy these four criteria [166] In fact, Friberg proved that no radiance function could satisfy these criteria [79] The generalized radiance remains useful, however, for paraxial analysis of quasihomogeneous fields Under the quasihomogenous approximation, the generalized radiance can be shown to be consistent with physical optics [165] The attraction of the radiance in system analysis is encapsulated in Eqn (6.140), which states that B(", s, n) is invariant along a ray in the s direction In particular, x Friberg demonstrated under the paraxial approximation that the generalized radiance may be propagated by the ABCD ray transfer matrice M according to  !  ! x x (6:142) Bn ¼ Bn M s0 s As briefly discussed in Section 10.4.3, paraxial ray tracing of the radiance fields is widely applied in image rendering algoritms for computer graphics [152,101,199] Propagation of the radiance by ray tracing is also useful for analysis astronomical sources and nonimaging optics [248] 6.7.2 The Constant Radiance Theorem The radiance is sometimes termed the “brightness” of the field The goal of this section is to show that no linear optical system can increase brightness, a result called the constant radiance theorem The constant radiance theorem expresses the primary difference between a diffuse source, such as a fluorescent lightbulb or a flame, and a laser While the fluorescent source may emit several watts and the laser just a few milliwatts, the laser can be focused on a much brighter spot In contrast, no linear optical system can make a fluorescent source brighter than the irradiance on the irradiating surface The generalized radiance is the Wigner distribution function of the coherent field   ) ð ð(  Dx Dx à (6:143) B(x, s, n) ¼ E x À , n E x ỵ , n e(2pin=c)sDx dDx 2 The Wigner function developed as a “phase space” representation of quantum mechanical states [259] As a space –frequency distribution, the Wigner function is closely related to windowed Fourier transformations and wavelet analysis 248 COHERENCE IMAGING The relationship of the Wigner distribution function and the generalized radiance was developed by Bastiaans [11] In particular, Bastianns [12] considers modal transformations of the generalized radiance Substituting Eqn (6.118) into Eqn (6.143), we find that the generalized radiance corresponding to the coherent mode decomposition of the cross-spectral density is X Bn (x, s, n) ¼ ln Bn,n (x, s, n) (6:144) n where Bn,n (x, s, n) ¼ ðð cnà     Dx Dx x , n cn x ỵ , n e(2pin=c)sÁDx dDx 2 (6:145) The Wigner distributions of the coherent modes satisfy the orthogonality relationship [11] ðððð ð ð 2   Bn,n (x, s, n)Bm,n (x, s, n) dx ds ¼  cnà (x, n)cm (x, n) dx   & m¼n ¼ m=m (6:146) Equations (6.129), (6.144), and (6.146) broadly constrain radiance transformations in optical systems According to Eqn (6.144), the radiance is the weighted sum of the radiances of the coherent modes According to Eqn (6.146), the brightness at any point in the output field will be associated primarily with a single coherent mode According to Eqn (6.129), the brightness of the brightest such mode cannot be increased in a linear optical system PROBLEMS 6.1 Coherence Time and Cross Section Consider a light source emitting the blackbody spectrum S(n) ¼ 2hn3 c2 e(hn=kb T) À (6:147) where h is Planck’s constant, kb is Boltzmann’s constant, and T is temperature (a) Plot S(n) for T ¼ 5000 K What is the wavelength corresponding to the maximum of S(n)? (b) Estimate the coherence time for this source PROBLEMS 249 (c) Suppose that the source is a 1-mm-diameter disk Derive an expression for the cross-spectral density at a range of 100 m Plot the cross-spectral density as a function of Dx and l for Dy ¼ (d) Estimate the coherence cross section [the range of Dx over which W(Dx, Dy, n) is nonnegligible] at 100 m 6.2 Temporal Coherence and Bandwidth (a) A Michelson interferometer scans the path delay d over range D with a sampling period D Estimate the wavelength range and resolution that this instrument achieves when used as a spectrometer (b) A two-point interferometer with pinhole separation Dx, as illustrated in Fig 6.6, is sampled on a focal plane at a range of d from the pinholes The pinholes are illuminated by spatially coherent light The spatial sampling period is D, and the sampling range is D Estimate the wavelength range and resolution that this instrument achieves when used to estimate the power spectral density of the illumination 6.3 Spectral Encoding in Scattered Partial Coherence Two unresolved point scatteres separated by 2a in the plane transverse to the optical axis generate the spectrum (l) ẳ So (l)[1 ỵ jinc(aa=l)] (a) Use Fourier arguments to quantify the accuracy with which one might estimate a from this spectrum Comment on the significance of this result for remote target superresolution (b) What is the scattered cross-spectral density if the scatters are displaced by a along the optical axis rather than in the transverse plane? Generate a plot similar to Fig 6.3 showing the spectral density observed along the optical axis as a function of a for this case 6.4 3D and 2D Transfer Functions Show that the incoherent transfer function, Hic (u, v) as described by Eqn (6.64), satisfies ð (6:148) B(Àldi u, Àldi v, q) dq Hic (u, v, l) ¼ 2(ldi )2 where B(x, y, q) is defined according to Eqn (6.54) 6.5 RSI Fringe Frequency The point source of Fig 6.10 was illuminated at l ¼ 532 nm and l ¼ 633 nm (a) What is the fringe period in Fig 6.10(a) and (b)? (b) Generate a plot of simulated data showing the irradiance at some particular detection pixel as a function of longitudinal delay d for this source (c) Assuming 20-mm detector pixels, what is the maximum source position ux could one observe with this instrument at u ¼ 28 for each of the two illumination wavelengths? ...  ! x1 ỵ 0:5a x2 ỵ 0:5a ỵ fo fo 2^ a 2^ a ẳ S(n)[2jf ihf j ỵ jfỵ ihfỵ j] (6:1 05) where fo (x) is the fundamental Gaussian mode of Eqn (3 .55 ), jfÀ i ¼ fo ( y=2a)fo [(x À 0:5a)=2a)] and jfỵ i ẳ... bandwidth product and the etendue of an optical system The space – bandwidth product is the product of the spatial support over which an optical signal is observed and the spatial frequency bandwidth... is  W% 3 :55 0 :52 0 :52 0:69  (6:107) 236 COHERENCE IMAGING We assemble coherent modes from the eigenfunctions of this matrix as described above; the eigenvalues are approximately 3.6 and 0.6 The