Abstract

This tutorial describes the application of digital holography to the terahertz spectral region and demonstrates how to reconstruct images of complex dielectric targets. Using highly coherent terahertz sources, high-fidelity amplitude and phase reconstructions are achieved, but because the millimeter-scale wavelengths approach the decimeter-sized targets and optical components, undesirable aperture diffraction degrades the quality of the reconstructions. Consequently, off-axis terahertz digital holography differs significantly from its visible light counterpart. This tutorial addresses these challenges within the angular spectrum method and the Fresnel approximation for digital hologram reconstruction, from which the longitudinal and transverse resolution limits may be specified. We observed longitudinal resolution (λ/284) almost two times better than has been achieved with visible light digital holographic microscopy and demonstrate that submicrometer longitudinal resolution is possible using millimeter wavelengths for an imager limited ultimately by the phase stability of the terahertz source and/or receiver. Minimizing the number of optical components, using only large reflective optics, maximizing the angle of the off-axis reference beam, and judicious selection of spatial frequency filters all contribute to improve the quality of the reconstructed image. As in visible wavelength analog holography, the observed transverse resolution in terahertz digital holography is comparable to the wavelength but improves for features near the edge of the imaged object compared with features near the center, a behavior characterized by a modified description of the holographic transfer function introduced here. Holograms were recorded by raster scanning a sensitive superheterodyne receiver, and several visibly transparent and opaque dielectric structures were quantitatively examined to demonstrate the compelling application of terahertz digital holography for nondestructive test, evaluation, and analysis.

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2019 (1)

2017 (5)

J. T. Richard and H. O. Everitt, “Millimeter wave and terahertz synthetic aperture radar for locating metallic scatterers embedded in scattering media,” IEEE Trans. Terahertz Sci. Technol. 7, 732–740 (2017).
[Crossref]

L. Valzania, P. Zolliker, and E. Hack, “Topography of hidden objects using THz digital holography with multi-beam interferences,” Opt. Express 25, 11038–11047 (2017).
[Crossref]

H. Huang, D. Wang, W. Li, L. Rong, Z. D. Taylor, Q. Deng, B. Li, Y. Wang, W. Wu, and S. Panezai, “Continuous-wave terahertz multi-plane in-line digital holography,” Opt. Laser Eng. 94, 76–81 (2017).
[Crossref]

Q. Deng, W. Li, X. Wang, Z. Li, H. Huang, C. Shen, Z. Zhan, R. Zou, T. Jiang, and W. Wu, “High-resolution terahertz inline digital holography based on quantum cascade laser,” Opt. Eng. 56, 113102 (2017).
[Crossref]

R. Zhu, J. T. Richard, D. J. Brady, D. L. Marks, and H. O. Everitt, “Compressive sensing and adaptive sampling applied to millimeter wave inverse synthetic aperture imaging,” Opt. Express 25, 2270–2284 (2017).
[Crossref]

2016 (2)

N. V. Petrov, M. S. Kulya, A. N. Tsypkin, V. G. Bespalov, and A. Gorodetsky, “Application of terahertz pulse time-domain holography for phase imaging,” IEEE Trans. Terahertz Sci. Technol. 6, 464–472 (2016).
[Crossref]

H. Huang, L. Rong, D. Wang, W. Li, Q. Deng, B. Li, Y. Wang, Z. Zhan, X. Wang, and W. Wu, “Synthetic aperture in terahertz in-line digital holography for resolution enhancement,” Appl. Opt. 55, A43–A48 (2016).
[Crossref]

2015 (7)

M. S. Heimbeck, W. R. Ng, D. R. Golish, M. E. Gehm, and H. O. Everitt, “Terahertz digital holographic imaging of voids within visibly opaque dielectrics,” IEEE Trans. Terahertz Sci. Technol. 5, 110–116 (2015).
[Crossref]

H. Huang, D. Wang, L. Rong, X. Zhou, Z. Li, and Y. Wang, “Application of autofocusing methods in continuous-wave terahertz in-line digital holography,” Opt. Commun. 346, 93–98 (2015).
[Crossref]

M. Locatelli, M. Ravaro, S. Bartalini, L. Consolino, M. S. Vitiello, R. Cicchi, F. Pavone, and P. De Natale, “Real-time terahertz digital holography with a quantum cascade laser,” Sci. Rep. 5, 13566 (2015).
[Crossref]

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L. Rong, T. Latychevskaia, C. Chen, D. Wang, Z. Yu, X. Zhou, Z. Li, H. Huang, Y. Wang, and Z. Zhou, “Terahertz in-line digital holography of human hepatocellular carcinoma tissue,” Sci. Rep. 5, 8445 (2015).
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P. Zolliker and E. Hack, “THz holography in reflection using a high resolution microbolometer array,” Opt. Express 23, 10957–10967 (2015).
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M. Locatelli, E. Pugliese, M. Paturzo, V. Bianco, A. Finizio, A. Pelagotti, P. Poggi, L. Miccio, P. Meucci, and R. Ferraro, “Imaging live humans through smoke and flames using far-infrared digital holography,” Opt. Express 21, 5379–5390 (2013).
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2012 (9)

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2008 (7)

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A. Khmaladze, M. K. Kim, and C. M. Lo, “Phase imaging of cells by simultaneous dual-wavelength reflection digital holography,” Opt. Express 16, 10900–10911 (2008).
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Q. Li, S. H. Ding, Y. D. Li, K. Xue, and Q. Wang, “Experimental research on resolution improvement in CW THz digital holography,” Appl. Phys. B 107, 103–110 (2012).
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Supplementary Material (4)

NameDescription
» Visualization 1       This video accompanies Figure 2.3 of the tutorial.
» Visualization 2       This video of an amplitude reconstruction accompanies Figure 4.3.
» Visualization 3       This video of a phase reconstruction accompanies Figure 4.3.
» Visualization 4       This video accompanies Figure 5.2 of the tutorial.

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Figures (39)

Figure 1.
Figure 1. (a) Standard Mach–Zehnder interferometer. (b) Geometry for an off-axis terahertz holographic interferometer.
Figure 2.
Figure 2. Illustration of separable Fourier spectra using off-axis holography for (a) small and (b) large reference wave signal levels [26].
Figure 3.
Figure 3. Fresnel reconstruction of real image (lower right), autocorrelation (center), and virtual image conjugate terms (upper left) for a digital hologram recorded of a Siemens star target at 0.48 THz. Visualization 1 illustrates the reconstructed wave field as it is backpropagated from the hologram plane to the object plane.
Figure 4.
Figure 4. (a) Three-dimensional drawing of a holographic imager. (b) Top view of the off-axis holographic imager, for which the detector location defines the hologram plane. [44]
Figure 5.
Figure 5. (a) Amplitude target (Siemens star), (b) hologram recorded at 0.550 THz with small insert showing representative interference fringes, and (c) real part of the spatial frequency map showing the autocorrelation, real, and virtual image terms. (d) Portion of spatial frequency map (c) labeling various spatial frequency cutoffs Wh and indicating a circular spectral filter.
Figure 6.
Figure 6. Prism wedges mounted in rigid terahertz absorbing foam, with wedge angles of 1°, 2°, 10°, and 5°, going clockwise from the upper left corner.
Figure 7.
Figure 7. (a) Phase object holograms from 0.230–0.740 THz. (b) Fourier domain maps of the associated phase object holograms.
Figure 8.
Figure 8. (a) Phase reconstruction of the wedges from Fig. 6 degraded by clutter noise. (b) Phase reconstruction with a narrower filter. (c) Phase reconstruction after tilt correction. Visualization 2 and Visualization 3 respectively provide amplitude and phase reconstructions of the backpropagation from the hologram plane to the objectplane.
Figure 9.
Figure 9. Amplitude and phase reconstructions of the four prism wedges over the frequency range 0.230 to 0.740 THz.
Figure 10.
Figure 10. Unwrapping steps of the 0.480 THz phase reconstruction, from (a) an optical photograph of the four wedges (placed on a red plate instead of in the absorbing foam so their orientations may be seen), (b) hologram phase reconstruction, (c) cropped phase reconstruction, and (d) unwrapped phase map.
Figure 11.
Figure 11. (a) Enlarged view of unwrapped phase wedges from Fig. 10(d) using the 0.48 THz holographic phase reconstruction. (b) Measurements and fitted slopes of phase wedges.
Figure 12.
Figure 12. (a) Object and hologram planes in lens-less Fourier transform holography for which the object size is small compared to d. (b) Illustration of diffraction rings in the hologram plane from a large object (or from point scatterers in the center and at the edge of the object plane).
Figure 13.
Figure 13. Angular spectrum reconstruction of the star target between 0.230 and 0.740 THz. Visualization 4 presents an amplitude reconstruction of the backpropagation from the hologram plane to the object plane using the angular spectrum method at 0.660 THz.
Figure 14.
Figure 14. Measured transfer function of the holographic imager using the angular spectrum reconstruction method, compared with the expected frequency-dependent resolution limits (dotted vertical lines) calculated using Eq. (33).
Figure 15.
Figure 15. (a) Simulated two point objects 1 mm apart at various locations in the object plane to illustrate the lateral resolution of the transfer function. (b) Calculated amplitude representation of the diffracted object waves, recorded for each location in the hologram plane at 0.6 THz.
Figure 16.
Figure 16. Transfer function at 0.6 THz for nine point scatterer locations within the object plane.
Figure 17.
Figure 17. (a) Reconstructions of the targets in two locations at 0.55 THz. (b) Frequency domain spatial Fourier transform of hologram. (c) Empirical comparison of the transfer functions for the two object locations, confirming the higher spatial frequency content when the target is placed near the edge of the object plane.
Figure 18.
Figure 18. (a) Spatial frequency map of 0.400 THz hologram and (b) real image spectrum with overlaid filter circles.
Figure 19.
Figure 19. (a) Filtered spectrum with fw=0.98cycles/mm and (b) reconstructed image. (c) Filtered spectrum with fw=0.88cycles/mm and (d) reconstructed image. (e) Filtered spectrum with fw=0.88cycles/mm and an applied Kaiser apodization window, and (f) reconstructed image. (g) Filtered spectrum with fw=0.59 cycles/mm and (h) reconstructed image. (i) Filtered spectrum with fw=0.51cycles/mm and (j) reconstructed image. (k) Filtered spectrum with fw=0.33cycles/mm and (l) reconstructed image.
Figure 20.
Figure 20. (a) Photograph, (b) hologram, and (c) phase reconstruction of the 3×3 void sample at 0.480 THz.
Figure 21.
Figure 21. (a) Magnified phase reconstruction of a single void (row 1, column 2 of Fig. 20), and (b) topographic plot (contours every 15 μm) illustrating that the void depth variations caused by three-dimensional printing are less than 50 μm.
Figure 22.
Figure 22. Magnified phase image of first three cells for 0.495 THz phase reconstruction, for which the depth of the void is labeled.
Figure 23.
Figure 23. Photograph of sample A and drawing of its internal structure [44].
Figure 24.
Figure 24. (a) Simulated (left) and measured (right) holograms and amplitude reconstructions of sample A at 0.495 THz. (b) Simulated (left) and measured (right) holograms and amplitude reconstructions of sample A at 0.710 THz [44].
Figure 25.
Figure 25. Magnified view of the structure corresponding to column 1, row 2 of sample A, reconstructed at 0.495 and 0.710 THz.
Figure 26.
Figure 26. (a) Photograph of lens and (b) phase reconstructions of holograms that were acquired at 0.485 and 0.500 THz, for which the (c) cross-section plots cross the center of the phase maps.
Figure 27.
Figure 27. Phase map of lens and corresponding cross section after one phase map was subtracted from the other.
Figure 28.
Figure 28. New phase map and center cross section of lens reconstruction, free from phase ambiguities.
Figure 29.
Figure 29. (a) Plano–convex PMP lens used as phase object for the dual-wavelength reconstruction process. (b) Holograms of the lens recorded at 0.680 and 0.725 THz [26].
Figure 30.
Figure 30. Amplitude reconstructions of PMP lens at (a) 0.680 and 0.725 THz. Phase reconstructions of lens at (b) 0.680 and 0.725 THz [26].
Figure 31.
Figure 31. Unwrapped phase map of PMP lens from three perspectives: (a) above, (b) cross section in radians, and (c) pseudo-three-dimensional [26].
Figure 32.
Figure 32. Diffraction ring–hologram plane intersections for ra for cases (1)–(4).
Figure 33.
Figure 33. Diffraction ring–hologram plane intersections for a<r<2a and rD for cases (5)–(7).
Figure 34.
Figure 34. Diffraction ring–hologram plane intersections for r8a for cases (8)–(11).
Figure 35.
Figure 35. Three transfer functions |H(x,y,fx,fy)| for the spatial frequency range 0<fx,fyfc.
Figure 36.
Figure 36. Three transfer functions |H(x,y,fx,fy)| for the spatial frequency range fc<fx,fy2fc.
Figure 37.
Figure 37. Three transfer functions |H(x,y,fx,fy)| for the spatial frequency range 2fc<fx,fy8fc2.
Figure 38.
Figure 38. Holography setup assuming an imaging lens at the hologram plane.
Figure 39.
Figure 39. Comparison between hologram reconstruction (a) without and (b) with aberrations.

Tables (3)

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Table 1. Spatial Cutoff Frequencies

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Table 2. Filters Related to the Physical Quantities of the Experimental Setup

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Table 3. Optical Properties of Three-Dimensional Printed Materials

Equations (60)

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NF=(2a)2λd,
H(x,y,z)=R2+O(x,y,z)2+RO*(x,y,z)+R*O(x,y,z).
R(y)=R·eik0ysinθ,
H(x,y)=R2+O2+Reik0ysinθO*(x,y)+R*eik0ysinθO(x,y).
θmin=sin1(3Bk0).
θmin=sin1(Bk0),
A(fx,fy,z)=U(x,y,z)ei2π(fxx+fyy)dxdy.
2U(x,y,z)+k2U(x,y,z)=0.
F(fx,fy)=F{U(x,y,z)}=U(x,y,z)ei2π(fxx+fyy)dxdy.
F{2x2U(x,y,z)}=(ifx)2F(fx,fy),
2F(fx,fy,z)z2+k2(1fx2k2fy2k2)F(fx,fy,z)=0.
F(fx,fy,z)=F(fx,fy,0)eizf2fx2fy2.
A(αλ,βλ,z)=A(αλ,βλ,0)ei2πλz1α2β2,
A(αλ,βλ,0)=1·T0(αλ,βλ,0),
T0(αλ,βλ,0)=t(x,y,0)ei2πλ(αx+βy)dxdy.
to(x,y,0)=F1{F{U(x,y,d)}ei2πλd1α2β2}.
fx2+fy21λ2.
to(x,y,0)=F1{F(U(x,y,d))·ei2πλdei2πλd(fx2+fy2)}.
to(x,y,0)=ei2πλdiλd·U(x,y,d))eiπλd(x2+y2).
NEP=10·log10(kTb)+NF=174dBm/Hz+9dB,
fci=1λ1(f/#)2(NAi)λ,
OTFF{rect(xw)}2sinc2(w·fcd),
fcd=1w.
foff=sinθλ,
Fh(fxm,fyn)=FFT{Uh(m,n)},
R(m,n)=ei2πλ(msinθx+nsinθy),
Fo(fxm,fyn)=Fh(fxm,fyn)·Wh(fxm,fyn),
Hf(fxm,fyn)=ei2πλz1(λfxm)2(λfyn)2.
Ur(m,n)=IFFT{Fo(fxm,fyn)·Hf(fxm,fyn)},
Δd=Δφ·λ2π(n1),
P(x,y)=rect(x2a)rect(y2a),
H(fx,fy)=rect(λdfx2a)rect(λdfy2a).
fc=aλd.
f=r·cosαd·λ.
R(x,y,r)=arc length within boundaries of hologram plane/2πr.
WK(m)=Io(0.5π1(2mM11)2)Io(0.5π),
tm(x,y)=e(αm·Lm)·e(i·2πλ·(nm1)Lm),
tm+s(x,y)=e(23αm·Lm+αs·Ls)·e(i·2πλ·(23(nm1)Lm+(ns1)Ls)),
ΔOPDj+1=OPDj+1OPDj.
ΔOPDj+1=[ϕ(j+1)a2π+m]λaϕja2πλa,
ΔOPDj+1=[ϕ(j+1)b2π+p]λbϕjb2πλb,
m=(ϕ(j+1)aϕja)λa(ϕ(j+1)bϕjb)λb2π(λbλa).
ΔOPDj+1=12π[(ϕ(j+1)aϕja)(ϕ(j+1)bϕjb)]·λaλbλaλb.
λb=λ1·λ2|λ2λ1|,
f=r·cosαd·λ.
R(x,y,r)=arc length within boundaries of hologram plane/2πr.
Case(1):rA,thenR(x,y,r)=1,
Case(2):r>A,rB,θ1=cos1Ar,thenR(x,y,r)=2πr2θ1r2πr,
Case(3):r>B,rA2+B2,rC,θ1=cos1Ar,θ2=cos1Br,thenR(x,y,r)=2πr2(θ1+θ2)r2πr,
Case(4):r>A2+B2,rC,θ1=cos1Ar,θ2=cos1Br,thenR(x,y,r)=1(π2+θ1+θ2)r2πr.
Case(5):r>C,rA2+B2,rA2+C2,rD,θ1=cos1Ar,θ2=cos1Br,θ3=cos1Cr,thenR(x,y,r)=2πr2(θ1+θ2+θ3)r2πr,
Case(6):r>C,r>A2+B2,rA2+C2,rD,θ1=cos1Ar,θ2=cos1Br,θ3=cos1Cr,thenR(x,y,r)=2πr(π2+θ1+θ2+2θ3)r2πr,
Case(7):r>A2+B2,r>A2+C2,rD,θ2=cos1Br,θ3=cos1Cr,thenR(x,y,r)=2πr(π+θ2+θ3)r2πr.
Case(8):r>D,rA2+B2,rA2+C2,rB2+D2,rC2+D2,θ1=cos1Ar,θ2=cos1Br,θ3=cos1Cr,θ4=cos1Dr,thenR(x,y,r)=2πr2(θ1+θ2+θ3+θ4)r2πr,
Case(9):r>D,r>A2+B2,rA2+C2,rB2+D2,rC2+D2,θ2=cos1Br,θ3=cos1Cr,θ4=cos1Dr,thenR(x,y,r)=2πr(π2+θ1+θ2+2θ3+2θ4)r2πr,
Case(10):r>D,r>A2+B2,r>A2+C2,rB2+D2,rC2+D2,θ2=cos1Br,θ3=cos1Cr,θ4=cos1Dr,thenR(x,y,r)=2πr(π+θ2+θ3+2θ4)r2πr,
Case(11):r>D,r>A2+B2,r>A2+C2,r>B2+D2,rC2+D2,θ3=cos1Cr,θ4=cos1Dr,R(x,y,r)=2πr(3π2+θ3+θ4)r2πr.
|H(x,y,fx,fy)|=R(x,y,r(fx,fy)),
wcmax=0.667·u¯y16·(f/#)2,
wamax=u¯2y4·f/#.

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