Photorefractive second-harmonic detection for ultrasound-modulated optical tomography
11 Photorefractive second-harmonic detection forultrasound-modulated optical tomography S TEVEN
J. B
YRNES The Charles Stark Draper Laboratory, Inc., 555 Technology Sq., Cambridge, Massachusetts, USA * Corresponding author: [email protected] April 19, 2018
Ultrasound-modulated optical tomography enables sharp 3D optical imaging of tissues and other turbidmedia, but the light modulation signals are hard to sensitively measure. A common solution, involvingphotorefractive crystals, enables the measurement of a relatively slow and low-spatial-resolution signaltracking the envelope of the ultrasound wave. We reexamine the photorefractive detection principle bothintuitively and quantitatively, and from this analysis we predict that the photodetector should addition-ally see a fast response at twice the ultrasound frequency, and correspondingly high spatial frequency.The fast and slow response usually have similar amplitudes and reveal complementary information, thusallowing ultrasound-modulated optical tomography to create dramatically sharper tomographic imagesunder the same measurement conditions, integration time, and experimental complexity.
OCIS codes: (170.1065) Acousto-optics; (070.1060) Acousto-optical signal processing; (190.5330) Photorefractive optics.
1. INTRODUCTION
Ultrasound-modulated optical tomography [1], also calledacousto-optic tomography or acousto-optic imaging, is an emerg-ing medical imaging technology promising a uniquely appeal-ing combination of features, including high spatial resolution( < ), deep penetration ( (cid:29) a r X i v : . [ phy s i c s . op ti c s ] A p r Fig. 1. (a) Schematic block diagram of the photorefractive de-tection system. A CW laser illuminates a tissue, creating aspeckle pattern of scattered light. This is overlapped with areference beam in a photorefractive crystal, and then a pho-todetector integrates the total light power received.photorefractive detection, unlike the naive approach mentionedabove, the different speckles do not cancel each other out, andtherefore a large-area detector can be used without reducing therelative strength of the modulation signal.In previous work [6, 8, 12], photorefractive detection has beentreated as having a spatial resolution equal to the envelope of theultrasound wave. We will argue that if a faster optical detectoris used (much faster than the ultrasound frequency), then thespatial resolution has much finer structure, as in Fig. 2, associ-ated with oscillations at the second harmonic of the ultrasoundfrequency. In many situations, these signals are beneficial tomeasure: they can reveal very small features and sharp edges(see Fig. 2); higher-frequency signals are often easier to detectover background; higher-frequency signals offer more data chan-nels per unit time; the fast and slow signals can be measuredsimultaneously with little extra effort, enabling measurement ofboth large-scale and small-scale structure; and for technical rea-sons discussed below, the high-frequency signal can sometimeshave a somewhat different image contrast than the low-frequncysignal, such that we can make certain features stand out (Sec. 4below).
A. Nonlinear modulation vs. nonlinear detection
Photorefractive detection, as we describe it here, involves the nonlinear detection of a linear acousto-optic modulation signal,and we will find that the second-harmonic signal discussedin this paper appears in the lowest order of perturbation the-ory. This topic should thus be distinguished from nonlinearmodulation of light in the tissue. Examples of the latter includepurely acoustic second-harmonic generation—when the acous-tic amplitude is sufficiently strong that the tissue’s mechanicalresponse is anharmonic—or acousto-optic second-harmonic gen-eration—when the acousto-optic interaction is sufficiently strongthat the light modulation is anharmonic. Example of nonlinearmodulation include Refs. [13, 14], in which second-harmonicsignals in ultrasound-modulated optical tomography were seenin CCD-based measurements.Nonlinear modulation offers intriguing potential applica-tions, particularly with focused ultrasound, but is weaker thanthe main signal at the ultrasound frequency. Photorefractivedetection offers an exciting avenue because, regardless of ul-trasound intensity, we predict a second-harmonic signal withsimilar amplitude to the slow envelope signal—and experienceshows that the latter can be measured even at high speeds withunfocused beams [6].To the extent that nonlinear modulation occurs, a photore-
Fig. 2.
Top: An example ultrasound waveform, showing asnapshot of a short pulse at 1MHz with v sound = fourth harmonic signal.
2. QUALITATIVE EXPLANATION
A. Index effect vs. displacement effect
We predict that the strength and presence of the second-harmonic signal depends critically on the phase relation betweenan ultrasound wave and the optical modulation it induces. Hav-ing a consistent, predictable phase relation is critical to ensuringthat the signal adds coherently, rather than randomly, over all thedifferent light paths meandering through the highly-scatteringtissue.Previous work [15, 16] has identified two mechanisms bywhich ultrasound modulates light, each with a different phaserelation. The first mechanism is the index effect , wherein thepressure of the sound wave changes the refractive index of thewater in the tissue (piezo-optic effect), which in turn changesthe optical phase of light passing through that region of tissue.The second is the displacement effect , where the light-scatteringstructures in the tissue move back and forth at the ultrasoundfrequency.As shown in Fig. 3, the two mechanisms lead to different op-tical modulation phases, and indeed, if both are present equally,they tend to create equal and opposite second-harmonic signalsthat cancel out in a photorefractive detection system.However, for typical tissue parameters and ultrasound fre-quencies, the two mechanisms are not present equally; instead,the index effect predominates over the displacement effect[15, 16]. For example, for tissue with light scattering coeffi-cient 1mm − , the index effect is predicted to account for of thetotal light modulation with 500kHz ultrasound, with 1MHzultrasound, and increasing towards 100% at higher frequency.(More generally, for water-based media, the index effect predom-inates by at least a 2:1 ratio on condition that the ultrasoundwavelength λ US < l , where l is the mean free path for photonscattering [15, 16].) Fig. 3.
Schematic depiction of speckle modulation during 1.5 ultrasound cycles, with ultrasound frequency 1MHz, showing (i) aseries of snapshots of a representative laser speckle exiting the tissue at different times, (ii) the acoustic pressure and (iii) displace-ment at the part of the tissue causing this modulation, and (iv) the photodetector signal after the photorefractive crystal. In all cases,dashed lines indicate the equilibrium with no ultrasound. (a) If the light modulation is caused by the index effect (Sec. A), the lightspeckle changes in phase with the pressure—i.e. when pressure is at its equilibrium value, the speckle is in its equilibrium position;(b) If the light modulation is instead caused by the displacement effect, the light speckles change in phase with the displacement. Ei-ther way, as explained in Sec. C, the signal (iv) shrinks when the speckles (i) are displaced from their time-averaged configuration.
Fig. 4.
Zoomed-in diagram of the photorefractive crystal ofFig. 1. The four beams exiting the crystal on the right are,from top to bottom, the diffracted scattered light, the trans-mitted reference beam, the transmitted scattered light, and thediffracted reference beam. The labels “destructive interference”and “constructive interference” are as discussed in Sec. B.
B. Two-wave mixing
A photorefractive crystal has the defining property that the crys-tal’s refractive index changes in response to how much lightintensity is at any given point. This gives rise to “two-wavemixing” [8, 17–19]. In this phenomenon, two light beams ofthe same wavelength shine into the crystal. The interferencepattern between these two beams produces peaks and nulls ofintensity within the crystal, which in turn creates a volumet-ric diffraction grating in the crystal. This grating diffracts eachbeam into the spatial mode of the other beam. In this case, oneof the “beams” is a complicated laser speckle pattern, but theprinciple still works: the volume grating diffracts light from thereference beam into the complicated laser speckle spatial profile,and vice-versa.Referring to Fig. 4, there is interference between the diffractedreference beam and the transmitted scattered light traveling tothe photodetector, and there is also interference between thetransmitted reference beam and diffracted scattered light travel-ing to the beam dump. For definiteness, we will assume that theformer interference is constructive, and hence (by conservationof energy) that the latter interference is destructive. This casecorresponds to positive gain for the scattered light, i.e. transfer of power from the reference to the scattered light paths. In gen-eral, the sign of gain, and correspondingly the choice of whichof the beams is amplified and which is depleted, depends onthe detailed mechanism of photorefraction, including the crys-tallographic axis, beam propagation directions, and direction ofthe electric field applied across the crystal (if any) [20]. If weinstead made the opposite assumption, i.e. transfer of power into the reference beam, the system would still work, but somesigns would be flipped in the discussion below, e.g. the presenceof ultrasound would tend to increase, rather than decrease, thephotodetector signal.What makes two-wave mixing particularly useful for ourpurposes is that the crystal’s photorefractive response time, τ PR ,is much slower than the ultrasound frequency. Therefore thediffracted reference beam is not quite a copy of the speckle pat-tern exiting the tissue, but rather a time-averaged copy of thatspeckle pattern. This is shown schematically in Fig. 3(i), where arepresentative speckle exiting the tissue (red dot) is modulated,while the corresponding diffracted speckle from the referencebeam (dashed line) stays stationary. The time-averaged copyenables “change detection” for the light: We are essentially com-paring the speckle pattern to its time-averaged copy, and if theyare very different, we infer that the light is being strongly modu-lated by the ultrasound. C. Mode overlap determines output signal
A key aspect of the photorefractive detection method is theinterference between the transmitted scattered light and thediffracted reference beam. Each beam by itself has an approxi-mately fixed total power, but to the extent that the two beamsinterfere, it sends more power to the photodetector and lesspower to the beam dump (Fig. 4).If both beams are static for a long time compared to τ PR , thena properly-configured photorefractive crystal will naturally de-velop an index profile that maximizes interference, such thatthe beams are maximally in phase and overlapped. This maxi-mizes the power to the photodetector, and minimizes the signalto the beam dump of Fig. 4. However, if the scattered light ismodulated by the ultrasound, its phase and amplitude profilewill not perfectly match the diffracted reference beam, so theinterference will be less effective, and the photodetector will see less light. This phenomenon is part of Fig. 3: the more thespeckles are altered from their time-averaged pattern (Fig. 3(i)),the less power is detected (Fig. 3(iv)).As stated above, two interfering waves convey maximumpower when they are in phase and have matching spatial profiles.This familiar fact from wave mechanics can be mathematicallyproven by writing [8]:Photodetector signal = (cid:90) d r | E S ( r ) + E D ( r ) | = C + (cid:90) d r Re [ E ∗ S ( r ) E D ( r )] (1) where Re (cid:104) E S ( r ) e − i ω t (cid:105) is the electric field of the transmit-ted scattered light in Fig. 4; Re (cid:104) E D ( r ) e − i ω t (cid:105) is the electricfield of the diffracted reference beam in Fig. 4; and C ≡ (cid:82) d r | E S ( r ) | + (cid:82) d r | E D ( r ) | is a constant approximately in-dependent of the presence or absence of ultrasound. Now weinvoke the Cauchy–Schwarz inequality: (cid:90) d r Re [ E ∗ S ( r ) E D ( r )] = (cid:114) (cid:90) d r (cid:12)(cid:12) E S ( r ) (cid:12)(cid:12) · (cid:90) d r (cid:12)(cid:12) E D ( r ) (cid:12)(cid:12) if E S & E D have the same phase & spatial mode (2a) (cid:90) d r Re [ E ∗ S ( r ) E D ( r )] < (cid:114) (cid:90) d r (cid:12)(cid:12) E S ( r ) (cid:12)(cid:12) · (cid:90) d r (cid:12)(cid:12) E D ( r ) (cid:12)(cid:12) otherwise (2b) If E S and E D have the same phase profile and spatial mode, i.e.if E S ( r ) E D ( r ) is the same positive constant everywhere, then the pho-todetector signal is maximized. If E S and E D are mis-matched atall, the photodetector signal goes down.As shown in Fig. 3(iv), the presence of ultrasound both de-creases the time-averaged photodetector signal, and causes afast modulation at twice the ultrasound frequency. The formeris the familiar slow signal whose spatial sensitivity tracks theenvelope of the ultrasound wave; the latter is the subject of thisarticle.
3. QUANTITATIVE DERIVATION
As above, we say ω is the laser frequency and ω US is the ultra-sound frequency (assumed monochromatic for simplicity). Wewrite the scattered field (i.e. the speckle pattern coming out ofthe tissue) as Re (cid:16) E S ( r e , t ) e − i ω t (cid:17) , where E S is a complex ampli-tude and r e is any point in the exterior of the tissue where thelight exits as a speckle pattern. It will be most convenient toset r e to points on the photodetector surface, in which case E S incorporates the effects of passing the photorefractive crystal,i.e. some absorptive and diffractive loss, both usually small inpractice [8]. To simplify the descriptions, we neglect speckledecorrelation, i.e. we assume that E S ( r e , t ) depends on time only because of the presence of ultrasound, and not because ofblood flow or other motion. (Photorefractive detection contin-ues to work in the presence of speckle decorrelation, but thesignal strength goes down when the speckle decorrelation timeis shorter than the photorefractive response time τ PR . In practice,sub-millisecond values of τ PR are known to be feasible [21], andthis is sufficiently low for most in vivo applications.)The ultrasound wave inside the tissue creates local pres-sure change P ( r i , t ) and local displacement D ( r i , t ) , where t is time and r i is a point inside the tissue. For example, a planewave would have P ( r i , t ) = P cos ( k · r i − ω US t ) , D ( r i , t ) = P Z US ω US sin ( k · r i − ω US t ) , where Z US is the specific acousticimpedance (note that the displacement is 90 ◦ out of phase withthe pressure change). Then the effect of the ultrasound can bewritten using Green’s functions as: E S ( r e , t ) = E S ,0 ( r e , t )+ (cid:90) d r i G n ( r i , r e ) P ( r i , t ) + (cid:90) d r i G d ( r i , r e ) D ( r i , t ) (3) where E S ,0 is the field in the absence of ultrasound; G n is theindex-effect Green’s function, defined such that a unit changeof pressure at the point r i in the tissue causes an optical fieldamplitude change of G n ( r i , r e ) in the speckle pattern, at thepoint r e external to the tissue; and G d is the displacement-effectGreen’s function defined analogously. Eq. (3) is a first-orderapproximation, i.e. assuming that the ultrasound only modestlychanges the light flow patterns. This is usually reasonable, andsee Sec. A for further discussion. As mentioned above, we areignoring blood flow and other motion, so G n and G d do notdepend on time.If we did not have a photorefractive crystal and referencebeam, but instead just sent the scattered light into a single-pixellarge-area photodetector, it would register an intensity fluctua-tion of: ∆ I ( t ) ≡ (cid:90) d r e (cid:16) | E S ( r e , t ) | − | E S ,0 ( r e ) | (cid:17) = (cid:90) d r e (cid:18) (cid:20) E ∗ S ,0 ( r e ) (cid:90) d r i ( G n ( r i , r e ) P ( r i , t )) (cid:21)(cid:19) + (cid:90) d r e (cid:18) (cid:20) E ∗ S ,0 ( r e ) (cid:90) d r i ( G d ( r i , r e ) D ( r i , t )) (cid:21)(cid:19) + (cid:90) d r e (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) d r i ( G n ( r i , r e ) P ( r i , t ) + G d ( r i , r e ) D ( r i , t )) (cid:12)(cid:12)(cid:12)(cid:12) As discussed in Sec. 1, this fluctuation ∆ I is quite small: Theultrasound slightly changes photon phases and paths, but doesnot systematically change the total flux of photons exiting thetissue into the large detector area. (Indeed, if this modulationwere not so small, there would be no need for the photorefractivecrystal!)As described in [8] (see also Eq. (1)), under typical experi-mental conditions, the photodetector measures a quantity pro-portional to:Signal ( t ) = (cid:90) d r e (cid:0) (cid:2) E ∗ S ,0 ( r e ) E S ( r e , t ) (cid:3)(cid:1) (4) where E S ,0 is proportional to the diffracted reference beam(Fig. 4), which imitates a time-averaged version of the scatteredlight speckle pattern E S (Sec. B).In Eq. (4), there is a time-independent and ultrasound-independent offset of 2 (cid:82) | E S ,0 | , which can be experimentallysubtracted off by, for example, comparing the signal with andwithout ultrasound [8]. Discarding that, we are left with:Signal ( t ) = ∆ I ( t ) − (cid:90) d r e (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) d r i ( G n ( r i , r e ) P ( r i , t )+ G d ( r i , r e ) D ( r i , t )) (cid:12)(cid:12)(cid:12) To simplify this, first we ignore the small quantity ∆ I ( t ) , then wenote that the quantities G n ( r i , r e ) and G d ( r (cid:48) i , r (cid:48) e ) are statisticallyuncorrelated (have a random phase relation). This leads to: (cid:104) Signal ( t ) (cid:105) = (cid:42) − (cid:90) d r e (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) d r i G n ( r i , r e ) P ( r i , t ) (cid:12)(cid:12)(cid:12)(cid:12) − (cid:90) d r e (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) d r i G d ( r i , r e ) D ( r i , t ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:43) Next we try to simplify: (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) d r i G n ( r i , r e ) P ( r i , t ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:90) d r i (cid:90) d r (cid:48) i G n ( r i , r e ) P ( r i , t ) G ∗ n ( r (cid:48) i , r e ) P ( r (cid:48) i , t ) Under typical circumstances, the statistical correlation between G n ( r i , r e ) and G n ( r (cid:48) i , r e ) smoothly decays to zero with increasing (cid:12)(cid:12) r i − r (cid:48) i (cid:12)(cid:12) , which leads to: (cid:42)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) d r i G n ( r i , r e ) P ( r i , t ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:43) = B n (cid:90) d r i (cid:68) | G n ( r i , r e ) | (cid:69) P ( r i , t ) where B n = (cid:82) d r (cid:48) i P ( r (cid:48) i ) (cid:104) G n ( r i , r e ) G ∗ n ( r (cid:48) i , r e ) (cid:105) P ( r i ) (cid:68) | G n ( r i , r e ) | (cid:69) is roughly the ratio of ul-trasound pressure averaged over a sphere centered at a point topressure at the center of that sphere, where the sphere size corre-sponds to the autocorrelation length of G n mentioned above. Ifthe ultrasound waveform is approximately sinusoidal, B n will beapproximately the same real constant at each point r i , althoughthe constant will vary with ultrasound frequency, scattering co-efficient, and other parameters. The G d case is similar, so theend result is: (cid:104) Signal ( t ) (cid:105) ∝ − (cid:90) d r e (cid:90) d r i (cid:16) B n (cid:68) | G n ( r i , r e ) | (cid:69) P ( r i , t )+ B d (cid:68) | G d ( r i , r e ) | (cid:69) D ( r i , t ) (cid:17) Define the total modulation M ( M n for index effect and M d fordisplacement effect) by M n ( r i ) ≡ B n (cid:90) d r e (cid:68) | G n ( r i , r e ) | (cid:69) M d ( r i ) ≡ B d (cid:90) d r e (cid:68) | G d ( r i , r e ) | (cid:69) Mapping these quantities is an end-goal of ultrasound-modulated optical tomography, as they reveal information abouthow much light passes through each point r i , how much lightscattering is happening at that point, how much water contentis at that point, and so on. We have: (cid:104) Signal ( t ) (cid:105) ∝ − (cid:90) d r i (cid:16) M n ( r i ) P ( r i , t ) + M d ( r i ) D ( r i , t ) (cid:17) Finally, we take the simple example of a monochromatic ultra-sound plane wave in a uniform infinite medium: P ( r i , t ) = P cos ( k · r i − ω US t ) and D ( r i , t ) = P Z US ω US sin ( k · r i − ω US t ) .We find that our signal is related to the Fourier coefficients of M : (cid:104) Signal ( t ) (cid:105) P ∝ − (cid:32) ˜ M n ( ) + Z ω ˜ M d ( ) (cid:33) − Re (cid:34)(cid:32) ˜ M n ( k ) − Z ω ˜ M d ( k ) (cid:33) e − i ω US t (cid:35) (5) The first term on the right side is more generally the difference-frequency term, which tracks the envelope of the ultrasoundwave and has been frequently measured in the literature. Thesecond term is the sum-frequency term, which measures higherspatial frequency, and appears at higher temporal frequency (upto twice the highest ultrasound frequency). In the usual circum-stance that M n (cid:29) Z ω M d (see Sec. A), the fast term and slowterm have inherently equal amplitudes, though the actual ampli-tude of each will depend on the amplitude of the correspondingFourier component of the image. For example, in a tissue withno fine features or sharp edges (where “fine” and “sharp” arecompared to the ultrasound wavelength), the fast signal wouldbe much weaker than the slow signal; conversely, a tissue withlarge-scale uniformity but small-scale random texture wouldgenerally show a stronger fast signal than slow signal. On theother hand, when M n ≈ Z ω M d , as when the ultrasoundwavelength is much larger than the optical scattering mean freepath (see Sec. A), then the fast signal is expected to be muchweaker than the slow signal under most circumstances.
4. CONCLUSION
We have argued both qualitatively and quantitatively that pho-torefractive detection setups should generally see a fast signal atthe second harmonic (or more generally, at sum frequencies)of the ultrasound waves in the tissue, which, like the slow(compared to the ultrasound frequency) envelope-related sig-nal, arises at the lowest order of perturbation theory and addscoherently over all the collected speckles. Compared to thepreviously-discussed slow signal, the fast signal is different andcomplementary. In terms of spatial sensitivity, the fast signalmeasures high-spatial-frequency Fourier components, and thusis blind to large structures but sensitive to small features, finetexture, and sharp edges, while the slow signal is the opposite(see Fig. 2 for a comparison at 1MHz; the fast signal is sensitiveto sub-millimeter structures while the slow signal is sensitiveto several-millimeter structures and larger, along the directionof ultrasound propagation). In terms of signal frequency, thefast signal requires a faster photodetector and ADC (Nyquistrate of 4 × the highest ultrasound frequency), but should benefitfrom lower background, and more importantly the same appara-tus should be able to measure both the fast and slow frequencybands simultaneously. In terms of contrast, the slow signal ef-fectively measures the sum (cid:16) M n + Z ω M d (cid:17) while the fastsignal measures the difference (cid:16) M n − Z ω M d (cid:17) . These twowill often be effectively the same (if M n (cid:29) Z ω M d , as when λ US is less than twice the mean free path for photon scatter-ing [15, 16], as is typical in tissues). But in certain cases, thedifference term offers intriguing sensing possibilities. For ex-ample, with low-frequency ultrasound, it may be possible toarrange for M n ≈ Z ω M d throughout the translucent tissue,but M n (cid:29) Z ω M d in a particular area with unusually littlelight scattering, such as a cyst full of relatively clear fluid. Thehigh-frequency signal component should then show a brightcyst on a dark background.In future work, we plan to experimentally test and explorethe predictions herein. Acknowledgement
I thank all those who generously offeredfeedback and criticism of this work, especially (but not exclu-sively) Joseph Hollmann, Charles DiMarzio, Krish Kotru, and
Jeffrey Korn.
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