**3.2 Scheme II**

In the previous interferometric schemes, the two arms have different optical path lengths so they require the light source to have much better temporal coherence, which is possible only by using a laser beam. A quasi-monochromatic thermal light source with an extended area can be regarded as a spatially incoherent source with a short coherence time less than 0.1 nsec. Therefore a true thermal light source fails in these schemes. From an application point of view, we must look for other schemes if we wish to employ a true thermal light source. For this purpose, the two arms of the interferometer must have equal optical path lengths but different diffraction configurations. We propose the following schemes for a true thermal light source.

We first consider an interferometer in which the two arms have the same length, 2f� (Zhang et al. 2009a). As shown in Fig. 7, a lens Lo of focal length f� = 19cm is set in the centre of the object arm, while in the other arm the beam travels freely to beamsplitter BS2. Hence the diffraction through the lens in the object arm has a different configuration from that for free propagation in the other arm.

Let object T(x) and the CCD cameras be placed at the two focal planes of lens Lo, which produces the Fourier transform of the object. In the coherent regime, the interference term given by Eq. (5) exhibits the Fourier spatial spectrum of the object T(x)

$$
\langle \text{E}\_{\text{r}}^\*(\text{x})\text{E}\_0(\text{x}) \rangle \propto \widetilde{\text{T}}(\text{kx}/\text{f}\_0). \tag{21}
$$

lens. However, this Fourier spectrum will not appear in the coherent interference term due

Fig. 6. Experimentally observed 1D interference patterns in Scheme I (Fig. 5). (a) and (b) are

respectively; (c) and (d) are their difference and summation, respectively. Experimental data

In the previous interferometric schemes, the two arms have different optical path lengths so they require the light source to have much better temporal coherence, which is possible only by using a laser beam. A quasi-monochromatic thermal light source with an extended area can be regarded as a spatially incoherent source with a short coherence time less than 0.1 nsec. Therefore a true thermal light source fails in these schemes. From an application point of view, we must look for other schemes if we wish to employ a true thermal light source. For this purpose, the two arms of the interferometer must have equal optical path lengths but different diffraction configurations. We propose the following schemes for a true

We first consider an interferometer in which the two arms have the same length, 2f� (Zhang et al. 2009a). As shown in Fig. 7, a lens Lo of focal length f� = 19cm is set in the centre of the object arm, while in the other arm the beam travels freely to beamsplitter BS2. Hence the diffraction through the lens in the object arm has a different configuration from that for free

Let object T(x) and the CCD cameras be placed at the two focal planes of lens Lo, which produces the Fourier transform of the object. In the coherent regime, the interference term

∗(x)E�(x)〉 ∝ T�(kx/f�). (21)

given by Eq. (5) exhibits the Fourier spatial spectrum of the object T(x)

〈E�

interference patterns (averaged over 10000 frames) registered by CCD1 and CCD2,

and theoretical simulation are given by open circles and solid lines, respectively.

to the modulation of Delta function.

**3.2 Scheme II** 

thermal light source.

propagation in the other arm.

Fig. 7. Scheme II of incoherent interferometry, where S is a light source. A lens Lo of focal length f� is set in the centre of the object arm, and both arms have the same length 2f�.

However, in the incoherent regime, Eq. (7) gives the interference term to be

$$
\langle \mathbf{E}\_{\mathbf{r}}^{\*} (\mathbf{x}) \mathbf{E}\_{0} (\mathbf{x}) \rangle \propto \int \mathbf{T}(\mathbf{x}\_{0}) \mathbf{F}(\mathbf{x}, \mathbf{x}\_{0}; \mathbf{f}\_{0}) \mathbf{H}^{\*} (\mathbf{x}, \mathbf{x}\_{0}; 2\mathbf{f}\_{0}) d\mathbf{x}\_{0}
$$

$$
\propto \int \mathbf{T}(\mathbf{x}\_{0}) \mathbf{G}^{\*} (\mathbf{x} + \mathbf{x}\_{0}, 2\mathbf{f}\_{0}) d\mathbf{x}\_{0}
$$

$$
\approx \mathbf{G}^{\*} (\mathbf{x}, 2\mathbf{f}\_{0}) \mathbf{f}^{\*} \left[ \frac{\mathbf{k} \mathbf{x}}{2\mathbf{f}\_{0}} \right] \,, \tag{22}
$$

where the last step is valid in the far-field limit. The equation shows the Fresnel diffraction pattern of the object with a diffraction length 2f�, different from the f� in Eq. (21).

With the above experimental scheme, we have used four types of sources. Figure 8 shows 1D patterns using the same sources as in the scheme of Fig. 2; the left part corresponds to a He-Ne laser beam passing through a ground glass plate while the right part, the direct laser beam. We can see that the interference patterns in the left part are very similar to those in Fig. 4. As a matter of fact, these two incoherent interferometry schemes exhibit similar Fresnel diffraction with slightly different effective diffraction lengths, Z = 39.3cm and 2f� = 38cm. When the ground glass plate is removed, coherent interference patterns are registered by CCD1 and CCD2, as shown in the right part of Figs. 8(a) and 8(b), respectively. They are the spatial Fourier spectra of the double-slit, consisting of two parts, T�(kx/f�) + c. c. and �T�(kx/f�)�� . The former and the latter can be extracted by the difference and sum of (a) and (b), as shown in the right part of Figs. 8(c) and 8(d), respectively.

As the third source we use a Na lamp emitting true thermal light of wavelength 589.3 nm, with an illumination area of 10×10 mm2, to replace the previous source in Fig. 2. In this case, the interference pattern directly appears on the CCD screen and it is not necessary to perform statistical averaging, as shown in Fig. 9(b). This is due to the fact that the response time of the CCD camera is much longer than the time scale of the thermal light fluctuations, so that averaging has already taken place in a single exposure. For comparison, Fig. 9(a)

Incoherent Holographic Interferometry 149

Fig. 9. Experimentally observed 2D interference patterns registered by CCD1 in the scheme of Fig. 7: (a) with the original pseudo-thermal light source in Fig. 2, averaged over 10000 frames; (b) with a Na lamp of extended illumination area as the light source; (c) with a Na

In this subsection we propose an interesting interferometric scheme which displays unusual behavior in incoherent interferometry. The setup is a simple modification of Fig. 2 whereby a transparent glass rod is inserted into one arm (Zhang et al. 2009b). The length of the rod is selected in such a way that the two arms of the interferometer have equal optical path lengths although their physical lengths are different. Thus the interferometer does not

The rod can also upset the balanced diffraction between the two arms. When a wave travels over a distance z in a homogeneous material of refractive index n, the impulse response

where k� is the wave number in vacuum. It should be noted that in the glass rod medium the optical path length nz is different from the diffraction length z/n. We rewrite Eq. (23) in

���(�/�) exp [ik�(nz)�exp ����(����)�

�(�/�) �, (23)

���(�/�) exp(ik�nz)G(x − x�, z/n), (24)

lamp followed by a pinhole as the light source. (Zhang et al., 2009a)

require the driving beam to have better temporal coherence.

h(x, x�) <sup>=</sup> � ��

H(x, x�; nz, z/n) <sup>=</sup> � ��

**3.3 Scheme III** 

function is given by

the form of

shows the 2D interference pattern corresponding to Fig. 8(a, left) for pseudo-thermal light. The two sets of fringes are similar, but have slightly different spacings, due to the different wavelengths of the two sources. Finally, we set a pinhole of diameter 0.36 mm after the lamp to dispel the spatial incoherence. With this point-like source, a different interference pattern, which has half the fringe spacing of that for the spatially incoherent source, appears on the CCD screen, as shown in Fig. 9(c). Consequently, the present scheme is favorable for realizing incoherent interferometry using a true incoherent source.

Fig. 8. Experimentally observed 1D interference patterns in the scheme of Fig. 7. Left and right parts are the patterns using spatially incoherent and coherent light, respectively. (a) and (b) are interference patterns (averaged over 10000 frames) registered by CCD1 and CCD2, while (c) and (d) are their difference and summation, respectively. Experimental data are given by open circles and theoretical simulation by solid lines.

shows the 2D interference pattern corresponding to Fig. 8(a, left) for pseudo-thermal light. The two sets of fringes are similar, but have slightly different spacings, due to the different wavelengths of the two sources. Finally, we set a pinhole of diameter 0.36 mm after the lamp to dispel the spatial incoherence. With this point-like source, a different interference pattern, which has half the fringe spacing of that for the spatially incoherent source, appears on the CCD screen, as shown in Fig. 9(c). Consequently, the present scheme is favorable for

Fig. 8. Experimentally observed 1D interference patterns in the scheme of Fig. 7. Left and right parts are the patterns using spatially incoherent and coherent light, respectively. (a) and (b) are interference patterns (averaged over 10000 frames) registered by CCD1 and CCD2, while (c) and (d) are their difference and summation, respectively. Experimental data

are given by open circles and theoretical simulation by solid lines.

realizing incoherent interferometry using a true incoherent source.

Fig. 9. Experimentally observed 2D interference patterns registered by CCD1 in the scheme of Fig. 7: (a) with the original pseudo-thermal light source in Fig. 2, averaged over 10000 frames; (b) with a Na lamp of extended illumination area as the light source; (c) with a Na lamp followed by a pinhole as the light source. (Zhang et al., 2009a)

### **3.3 Scheme III**

In this subsection we propose an interesting interferometric scheme which displays unusual behavior in incoherent interferometry. The setup is a simple modification of Fig. 2 whereby a transparent glass rod is inserted into one arm (Zhang et al. 2009b). The length of the rod is selected in such a way that the two arms of the interferometer have equal optical path lengths although their physical lengths are different. Thus the interferometer does not require the driving beam to have better temporal coherence.

The rod can also upset the balanced diffraction between the two arms. When a wave travels over a distance z in a homogeneous material of refractive index n, the impulse response function is given by

$$\mathbf{h(x,x\_0)} = \sqrt{\frac{\mathbf{k\_0}}{12\pi(\mathbf{z}/\mathbf{n})}} \exp[i\mathbf{k\_0}(\mathbf{n}\mathbf{z})] \exp\left[\frac{i\mathbf{k\_0}(\mathbf{x}-\mathbf{x\_0})^2}{2(\mathbf{z}/\mathbf{n})}\right] \tag{23}$$

where k� is the wave number in vacuum. It should be noted that in the glass rod medium the optical path length nz is different from the diffraction length z/n. We rewrite Eq. (23) in the form of

$$\mathbf{H}\{\mathbf{x}, \mathbf{x}\_0; \mathbf{n}\mathbf{z}, \mathbf{z}/\mathbf{n}\} = \sqrt{\frac{\mathbf{k}\_0}{\mathbf{i}\mathbf{z}\mathbf{n}(\mathbf{z}/\mathbf{n})}} \exp\left(\mathbf{i}\mathbf{k}\_0 \mathbf{n}\mathbf{z}\right) \mathbf{G}(\mathbf{x} - \mathbf{x}\_0, \mathbf{z}/\mathbf{n}),\tag{24}$$

$$\mathbf{H}(\mathbf{x}, \mathbf{x}\_0; \mathbf{z}\_1 + \mathbf{z}\_2, \overline{\mathbf{z}}\_1 + \overline{\mathbf{z}}\_2) = \int \mathbf{H}(\mathbf{x}, \mathbf{x}'; \mathbf{z}\_1, \overline{\mathbf{z}}\_1) \mathbf{H}(\mathbf{x}', \mathbf{x}\_0; \mathbf{z}\_2, \overline{\mathbf{z}}\_2) d\mathbf{x}'.\tag{25}$$

$$\mathbf{h}\_{\mathbf{0}} \{ \mathbf{x}, \mathbf{x}\_{\mathbf{0}} \} = \int \mathbf{H} \{ \mathbf{x}, \mathbf{x}'; \mathbf{z}\_{\mathbf{0}2}, \mathbf{z}\_{\mathbf{0}2} \} \mathbf{T} \{ \mathbf{x}' \} \mathbf{H} \{ \mathbf{x}', \mathbf{x}\_{\mathbf{0}}; \mathbf{z}\_{\mathbf{0}1}, \mathbf{z}\_{\mathbf{0}1} \} d\mathbf{x}'.\tag{26}$$

$$
\langle \mathcal{E}\_{\mathbf{r}}(\mathbf{x}) \mathcal{E}\_{\mathbf{0}}(\mathbf{x}) \rangle \propto \int \mathcal{H}^\*(\mathbf{x}, \mathbf{x}\_{0}; \mathbf{Z}, \bar{\mathbf{Z}}) \mathcal{H}(\mathbf{x}, \mathbf{x}'; \mathbf{z}\_{02}, \mathbf{z}\_{02}) \mathcal{T}(\mathbf{x}') \mathcal{H}(\mathbf{x}', \mathbf{x}\_{0}; \mathbf{z}\_{01}, \mathbf{z}\_{01}) d\mathbf{x}' d\mathbf{x}\_{0}
$$

$$
= \int \mathcal{H}(\mathbf{x}, \mathbf{x}'; \mathbf{z}\_{01} - \mathcal{Z}, \mathbf{z}\_{01} - \bar{\mathbf{Z}}) \mathcal{H}(\mathbf{x}, \mathbf{x}'; \mathbf{z}\_{02}, \mathbf{z}\_{02}) \mathcal{T}(\mathbf{x}') d\mathbf{x}'
$$

$$
\propto \exp[i\mathbf{k}\_{0}(\mathbf{z}\_{0} - \mathbf{Z})] \int \mathcal{T}(\mathbf{x}') \mathcal{G}(\mathbf{x} - \mathbf{x}', \mathbf{Z}\_{\text{eff}}) d\mathbf{x}' \,\tag{27}
$$

$$\frac{1}{\mathbf{z}\_{\rm eff}} = \frac{1}{\mathbf{z}\_{\rm oz}} + \frac{1}{\mathbf{z}\_{\rm oz} - \mathbf{Z}}.\tag{28}$$

$$\mathbf{z}\_{\rm o} = \mathbf{Z} = \mathbf{z}\_{\rm r} + (\mathbf{n} - \mathbf{1})\mathbf{l}.\tag{29}$$

$$\mathbf{Z\_{eff}} = \mathbf{z\_{o2}} \left[ \mathbf{1} - \frac{\mathbf{z\_{o2}}}{\mathbf{l(n-\frac{1}{n})}} \right]. \tag{30}$$

$$\mathbf{Z\_{eff}} = \mathbf{z\_{o2}} \left[ \mathbf{1} - \frac{\mathbf{z\_{o2}}}{\mathbf{z\_{o2}}|\_{\mathrm{mg}}} \right]. \tag{31}$$

$$\|\mathbf{z}\_{\rm o1}\|\_{\rm limg} = \mathbf{z}\_{\rm o} - \mathbf{z}\_{\rm o2}\|\_{\rm limg} = \mathbf{z}\_{\rm r} - \mathbf{l} + \frac{\mathbf{l}}{\mathbf{n}} = \overline{\mathbf{Z}}.\tag{32}$$

Incoherent Holographic Interferometry 153

A negative diffraction length in wave propagation means phase-reversal diffraction of the wave-front. Recent research on electromagnetic metamaterials shows that negative refractive index materials may be engineered. When an electromagnetic wave travels in a negatively refracting medium, its wave-front undergoes phase-reversal diffraction (Pendry, 2008), thus a slab of negative refraction material can play the same role as a curved lens to perform imaging. Interestingly, the two different physical systems, incoherent interferometry and negative refraction material, reveal some similarity in phase-reversal

Fig. 13. Illustration of lensless imaging by two schemes. (a) Via successive diffraction through two media, where one is a negative refractive index medium. (b) Using an

incoherent light interferometer; when the interferometer is opened out and the two arms are set along a line, the joint diffraction through them is comparable with that in (a) (Zhang et

Figure 13 explains the similarity of the two models. In Fig. 13(a), we consider the successive diffraction of an object through two media, a positive refraction medium of length l� and index n� > 0 followed by a negative one of length l� and index n� < 0. Let T(x�) describe a

diffraction (Zhang et al., 2009b).

al., 2009b).

amplitude-modulated object, the same pattern is obtained for both positive and negative diffraction with equal magnitude of diffraction length.

Fig. 11. Experimentally observed 2D interference patterns in the scheme of Fig. 10. [(a)-(e)]: the double slit is placed at the positions of z�� = 31.0, 28.5, 24.2, 20.0, and 10.6 cm, respectively, where (b) is the image of the double slit (Zhang et al., 2009b).

We have learned in Section 2 that the effective diffraction length in the interferometric scheme of Fig. 2 can be either positive or negative depending on whether the reference arm is longer or shorter than the object arm. We see again the two kinds of diffraction occurring in the present setup. In particular, this scheme may be realized with a null effective diffraction length Z��� = 0, which implies that phase sensitive imaging may be accomplished without using a conventional lens. This effect cannot occur in the coherent regime. Figure 12 compares the output patterns of the interferometer in the same configuration using spatially incoherent and coherent light, when the double-slit is placed at the position z�� = 28.5 cm. The images of the double-slit when the interferometer is illuminated by the extended Na lamp are shown in the left column. After we insert a pinhole in front of the lamp to improve the spatial coherence, as expected, instead of obtaining an image we observe interference fringes, as shown in the right column of Fig. 12. In this case, lensless imaging cannot occur no matter where the object is placed.

Fig. 12. Experimentally observed 2D patterns when a double-slit object is placed at the position z�� = Z� = 28.5 cm. Left column: image patterns when the source is spatially incoherent; right column: fringe patterns when the source is spatially coherent (with a pinhole aperture in front of the source). Figure (a) was recorded by CCD1 and (b) by CCD2 (Zhang et al., 2009b).

amplitude-modulated object, the same pattern is obtained for both positive and negative

Fig. 11. Experimentally observed 2D interference patterns in the scheme of Fig. 10. [(a)-(e)]:

We have learned in Section 2 that the effective diffraction length in the interferometric scheme of Fig. 2 can be either positive or negative depending on whether the reference arm is longer or shorter than the object arm. We see again the two kinds of diffraction occurring in the present setup. In particular, this scheme may be realized with a null effective diffraction length Z��� = 0, which implies that phase sensitive imaging may be accomplished without using a conventional lens. This effect cannot occur in the coherent regime. Figure 12 compares the output patterns of the interferometer in the same configuration using spatially incoherent and coherent light, when the double-slit is placed at the position z�� = 28.5 cm. The images of the double-slit when the interferometer is illuminated by the extended Na lamp are shown in the left column. After we insert a pinhole in front of the lamp to improve the spatial coherence, as expected, instead of obtaining an image we observe interference fringes, as shown in the right column of Fig. 12. In this case, lensless imaging cannot occur

Fig. 12. Experimentally observed 2D patterns when a double-slit object is placed at the position z�� = Z� = 28.5 cm. Left column: image patterns when the source is spatially incoherent; right column: fringe patterns when the source is spatially coherent (with a pinhole aperture in front of the source). Figure (a) was recorded by CCD1 and (b) by CCD2

the double slit is placed at the positions of z�� = 31.0, 28.5, 24.2, 20.0, and 10.6 cm, respectively, where (b) is the image of the double slit (Zhang et al., 2009b).

diffraction with equal magnitude of diffraction length.

no matter where the object is placed.

(Zhang et al., 2009b).

A negative diffraction length in wave propagation means phase-reversal diffraction of the wave-front. Recent research on electromagnetic metamaterials shows that negative refractive index materials may be engineered. When an electromagnetic wave travels in a negatively refracting medium, its wave-front undergoes phase-reversal diffraction (Pendry, 2008), thus a slab of negative refraction material can play the same role as a curved lens to perform imaging. Interestingly, the two different physical systems, incoherent interferometry and negative refraction material, reveal some similarity in phase-reversal diffraction (Zhang et al., 2009b).

Fig. 13. Illustration of lensless imaging by two schemes. (a) Via successive diffraction through two media, where one is a negative refractive index medium. (b) Using an incoherent light interferometer; when the interferometer is opened out and the two arms are set along a line, the joint diffraction through them is comparable with that in (a) (Zhang et al., 2009b).

Figure 13 explains the similarity of the two models. In Fig. 13(a), we consider the successive diffraction of an object through two media, a positive refraction medium of length l� and index n� > 0 followed by a negative one of length l� and index n� < 0. Let T(x�) describe a

Incoherent Holographic Interferometry 155

The authors thank L. A. Wu for helpful discussions. This work was supported by the National High Technology Research and Development Program of China, Project No. 2011AA120102 and the National Natural Science Foundation of China, Project No.

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transmittance object, illuminated by a plane wave E�, then the output field after diffraction is written as

$$\mathbf{E(x)} = \mathbf{E\_0} \int \mathbf{T(x\_0)} \mathbf{H(x, x\_0; n\_1 l\_1 + n\_2 l\_2, l\_1/n\_1 + l\_2/n\_2)} \mathbf{dx\_0} \tag{33}$$

When the net diffraction length in the successive propagation is null, i.e. l�/n� + l�/n� = 0, then H(x, x�; n�l� + n�l�, 0) = exp[ik�(n�l� + n�l�)] δ(x − x�) and we have

$$\mathbf{E(x) = E\_0 \exp[i\mathbf{k}\_0(\mathbf{n}\_1\mathbf{l}\_1 + \mathbf{n}\_2\mathbf{l}\_2)]} \mathbf{T(x)}.\tag{34}$$

Otherwise, Eq. (33) represents Fresnel diffraction with a diffraction length of l�/n� + l�/n�

$$\mathbf{E(x)} \ll E\_0 \exp[\mathrm{ik}\_0(\mathbf{n}\_1\mathrm{l}\_1 + \mathrm{n}\_2\mathrm{l}\_2)] \int \mathbf{T(x\_0)} \mathrm{G}(\mathbf{x} - \mathbf{x\_0}, \mathrm{l}\_1/\mathrm{n}\_1 + \mathrm{l}\_2/\mathrm{n}\_2) d\mathbf{x\_0} \tag{35}$$

which has the same form as Eq. (27)

To better compare the two schemes, in Fig. 13(b), the interferometer is opened out and the two arms are set along a line. We can see that the joint diffraction through the two arms is comparable with the geometry in Fig. 13(a).

Recently, various approaches for invisibility cloaking and transformation optics in complementary media with positive and negative refraction materials have been proposed, which can in theory accomplish exact optical cancellation between the object and its counterpart (Lai et al., 2009a, 2009b). In light of the similarity of time-reversal diffraction between incoherent interferometry and negatively refracting media, using the present scheme, we have conducted proof-of-principle experimental demonstrations of the theoretical proposals (Zhang et. al., 2010; Gan et al., 2011). The physical analogue between the two different systems may provide a convenient research platform. Moreover, a form of nonlocal imaging as well as interference effects that were previously regarded as the signature of two-photon entanglement or intensity correlation of thermal light can now be realized in incoherent interferometry, which is associated with the first-order field correlation (Gao et al., 2009; Gan et al. 2009).

### **4. Conclusion**

Interference effects in incoherent interferometry show different physics from that in coherent interferometry. In the latter case, two interfering fields have well-defined spatial distributions, whereas in the former case, these fields fluctuate randomly in space and the interference pattern appears only in the statistical average. Furthermore, incoherent interference relies entirely on the first-order spatial correlation of the two fields, so the object information is contained in the joint diffraction of the two fields. For certain optical geometries, phase reversal diffraction can occur through the first-order spatial correlation of incoherent light, thus providing a method of wave-front recovery without using a lens, and also a means for optical transformation.

Incoherent interferometry as a novel interference mechanism exhibits richer phenomena than the coherent type. Since it is not dependent on spatial coherence, the present method may find potential application in holography and other interference technologies, especially in those areas where a coherent source is unavailable. Incoherent holographic interferometry is fundamentally different from the previously known incoherent holography, while it can play the same role as coherent holography.
