**2. What is incoherent interferometry?**

We consider a holographic interferometer as shown in Fig. 1, in which an object field Eo(x) and a reference field E�(x) are reflected from mirrors M1 and M2, respectively, to interfere at the recording plane RP. The intensity distribution of their superposed field is written as

$$\mathbf{I}(\mathbf{x}) = \langle |\mathbf{E}\_0(\mathbf{x}) + \mathbf{E}\_r(\mathbf{x})|^2 \rangle = \langle |\mathbf{E}\_0(\mathbf{x})|^2 \rangle + \langle |\mathbf{E}\_r(\mathbf{x})|^2 \rangle + \{ \langle \mathbf{E}\_r^\*(\mathbf{x})\mathbf{E}\_0(\mathbf{x}) \rangle + \mathbf{c} \cdot \mathbf{c} \},\tag{1}$$

where x is the transverse coordinate across the beam, and 〈E� ∗(x)E�(x)〉 the interference term. In the interferometric scheme the two fields must come from the same source field E�(x). Let a transmitting object O with transmittance T(x) be close to the source, so the two fields can be expressed as

$$\mathbf{E\_{o}(x,t) = \int T(x\_{1})h\_{0}(x,x\_{1})E\_{s}(x\_{1},t\_{1})dx\_{1}}\tag{2a}$$

its wave fronts, a stationary interference pattern which uniquely encodes the position and intensity of the object point. However, the method has various weaknesses that significantly limit its application. The schemes proposed for incoherent holography can only record the intensity distribution of a fluorescent object. Moreover, each elementary fringe pattern is formed by two extremely tiny portions of the light, and the summation of many weak interference patterns results in a very large bias level in the hologram, much larger than that in coherent holography. Hence incoherent holography is appropriate only for objects with a

In this chapter we introduce an incoherent interference mechanism which seems to contradict our existing knowledge of the interference requirements. Consider a holographic interferometer in which an object is placed in one arm; the object wave then interferes with the reference wave in the other arm, resulting in an interference pattern which records the object information. To obtain well-defined interference fringes in an ordinary balanced interferometer where both arms have the same path length, the optical field illuminating the interferometer must have spatial coherence, that is, its transverse coherence length must be large enough. We find that the requirement of spatial coherence is due to the particular geometry of the balanced interferometer. If the two fields to be interfered travel through different lengths or different diffraction configurations, then spatial coherence is no longer necessary for spatial interference. Thus we have proposed several unbalanced interferometers where the two beams travel different path lengths or have different diffraction configurations (Zhang et al., 2009a, 2009b). These interferometers are capable of exhibiting interference using either coherent or spatially incoherent light, but their

The chapter is organized as follows. In Section 2 we first analyze why both temporal and spatial coherence conditions are necessary in an ordinary holographic interferometer. Then we propose an unbalanced holographic interferometer with different path lengths and demonstrate that it can exhibit interference using a light source with spatial incoherence. The setup is capable of performing holography. In Section 3 we further suggest several types of unbalanced interferometers which are able to realize incoherent interferometry. In particular, some of the schemes can reduce the requirement of temporal coherence and employ a true incoherent source such as a lamp with an extended illumination area. Moreover, we show that phase reversal diffraction can occur in the incoherent

We consider a holographic interferometer as shown in Fig. 1, in which an object field Eo(x) and a reference field E�(x) are reflected from mirrors M1 and M2, respectively, to interfere at the recording plane RP. The intensity distribution of their superposed field is written as

In the interferometric scheme the two fields must come from the same source field E�(x). Let a transmitting object O with transmittance T(x) be close to the source, so the two fields can

E�(x, t) = � T(x�)h�(x, x�)E�(x�, t�)dx�, (2a)

∗(x)E�(x)〉 + c. c.�, (1)

∗(x)E�(x)〉 the interference term.

interference fringes are different, signifying that they have different origins.

interferometer. Finally, a brief summary is given in Section 4.

I(x) = 〈|E�(x) + E�(x)|�〉 = 〈|E�(x)|�〉 + 〈|E�(x)|�〉 + �〈E�

where x is the transverse coordinate across the beam, and 〈E�

**2. What is incoherent interferometry?** 

be expressed as

low number of resolution elements (Goodman, 1996).

$$\mathbf{E\_{r}(x,t)} = \int \mathbf{h\_{r}(x,x\_{2})} \mathbf{E\_{s}(x\_{2},t\_{2})} d\mathbf{x\_{2}},\tag{2b}$$

where h�(x, x�) and h�(x, x�) are the impulse response functions for the object and reference paths, t� and t� are the times taken for the source fields to propagate to the recording plane through the object and reference paths, respectively. x� (j = 1,2) is the transverse coordinate in the source field. Hence the interference term is obtained to be

$$
\langle \mathbf{E}\_{\rm r}^\*(\mathbf{x}, \mathbf{t}) \mathbf{E}\_0(\mathbf{x}, \mathbf{t}) \rangle = \int \mathbf{T}(\mathbf{x}\_1) \mathbf{h}\_0(\mathbf{x}, \mathbf{x}\_1) \mathbf{h}\_1^\*(\mathbf{x}, \mathbf{x}\_2) \langle \mathbf{E}\_{\rm s}^\*(\mathbf{x}\_2, \mathbf{t}\_2) \mathbf{E}\_{\rm s}(\mathbf{x}\_1, \mathbf{t}\_1) \rangle d\mathbf{x}\_1 d\mathbf{x}\_2. \tag{3}
$$

Fig. 1. A holographic interferometer consisting of two beamsplitters, BS1 and BS2, and two mirrors, M1 and M2; S is a source and O an object; RP is the recording plane.

The temporal coherence of the source field is characterized by the coherence time . If |t� − t�| is larger than , then 〈E� ∗(x�, t�)E�(x�, t�)〉 = 0 and interference never occurs. However, spatial coherence requires that the beam has a well-defined wave-front, such as a field emitted from a point source or a laser beam. When the source field satisfies both temporal and spatial coherence, then

$$
\langle \mathcal{E}\_{\sf s}^{\*} (\mathbf{x}\_{2}, \mathbf{t}\_{2}) \mathcal{E}\_{\sf s} (\mathbf{x}\_{1}, \mathbf{t}\_{1}) \rangle = \mathcal{E}\_{\sf s}^{\*} (\mathbf{x}\_{2}, \mathbf{t}\_{2}) \mathcal{E}\_{\sf s} (\mathbf{x}\_{1}, \mathbf{t}\_{1}). \tag{4}
$$

Thus coherent interferometry occurs in the form of

$$
\langle \mathbf{E\_r^\*(x,t)E\_o(x,t)} \rangle = \mathbf{E\_r^\*(x,t)E\_o(x,t)},\tag{5}
$$

which contains the object information. This is what has been known before.

We now consider the case when the source field satisfies temporal coherence (i.e. |t� − t�| < �) but not spatial coherence. The first-order correlation function of the source field is written as

$$
\langle \mathbf{E\_s^\*(x\_2, t\_2)} \mathbf{E\_s(x\_1, t\_1)} \rangle = \mathbf{I\_s} \mathbf{e^{l\phi}} \delta(\mathbf{x\_2} - \mathbf{x\_1}),\tag{6}
$$

where is an arbitrary phase. For simplicity, the intensity distribution I� is assumed to be homogeneous. Equation (6) describes complete spatial incoherence. The wave-front of the beam fluctuates randomly and any two positions across the beam are statistically independent. Substituting Eq. (6) into Eq. (3), we obtain

$$
\langle \mathbf{E}\_r^\*(\mathbf{x}) \mathbf{E}\_o(\mathbf{x}) \rangle = \mathbf{I}\_s \mathbf{e}^{\mathsf{l}\mathfrak{p}} \int \mathcal{T}(\mathbf{x}\_0) \mathbf{h}\_o(\mathbf{x}, \mathbf{x}\_0) \mathbf{h}\_r^\*(\mathbf{x}, \mathbf{x}\_0) d\mathbf{x}\_0. \tag{7}
$$

Incoherent Holographic Interferometry 141

The laser beam has good temporal coherence which will not be affected by the scattering of the slowly rotating ground glass. The path difference between the two arms is much smaller than the longitudinal coherence length cτ, where c is the light speed in vacuum and τ is the coherence time of the laser beam. However, the motion of the ground glass disk destroys the spatial coherence of the laser beam. This source emits what is known as pseudo-thermal light and satisfies Eq. (6) approximately. In contrast, a true thermal light source cannot

The two-dimensional (2D) intensity patterns detected by CCD1 are shown in Fig. 3. When the ground glass disk is moved step by step, the interference patterns are irregular speckles frame by frame and each frame is different, as shown in the two single-shot frames of Figs. 3(a) and 3(b). However, if we average over a number of exposures, as the number of frames increases, a well-defined interference pattern emerges gradually, as can

The experimental results tell us that, unlike the stationary pattern in coherent interferometry, the interference pattern here is irregularly fluctuating in time and the welldefined pattern can only be discerned after taking its statistical average. These striking features imply that incoherent interferometry originates from a different interference

We can explain the experiment using Eq. (7). The interference term in this incoherent

∗(x)E�(x)〉 = I�e�� � T(x�)H(x, x�; z�)H∗(x, x�; z�)dx�

= exp[ik(z� − z�)] � T(x�)G(x−x�, Z)dx�, (10)

G(x, z�)G∗(x, z�) = G(x, Z), (11)

��

√��� �i�c(qb��) c��(qd��). (14)

. (12)

� �� (13)

∝ exp[ik(z� − z�)] � T(x�)G(x−x�, z�)G∗(x−x�, z�)dx�

Equation (10) defines a kind of incoherent interferometry, which represents the Fresnel diffraction integral of an object under the paraxial condition, and is the same as for coherent diffraction but with an effective object distance Z replacing the real one, z�. Note that Z is negative for z� > z�, designating a phase reversal diffraction. We will discuss this issue in

∗(x)E�(x)〉 ∝ exp[ik(z� − z�)] G(x, Z)T� �

� � <sup>=</sup> � �� <sup>−</sup> � ��

In the far-field limit, Eq. (10) presents the Fourier transform T� of object T:

For a double-slit of slit width b and spacing d, its Fourier transform reads

T�(q) = � ��

satisfy the condition of temporal coherence in this setup.

be seen in Figs. 3(c)-3(g).

〈E�

interferometer can be calculated as

where function G defined by Eq. (8b) obeys

when the effective diffraction length Z is defined by

〈E�

mechanism.

Subsection 3.3.

For a conventional balanced interferometer where the two optical paths have the same length, that is h�(x, x�) = h�(x, x�), then h�(x, x�)h� ∗(x, x�) is x-independent. In the paraxial approximation, for example, the impulse response function for a propagation length z is given by

$$\mathbf{H}(\mathbf{x}, \mathbf{x}\_0; \mathbf{z}) = \sqrt{\frac{\mathbf{k}}{\mathbf{i} 2\pi \mathbf{z}}} \exp(\mathbf{i} \mathbf{k} \mathbf{z}) \exp\left[\frac{\|\mathbf{k}(\mathbf{x} - \mathbf{x}\_0)^2\|}{2\mathbf{z}}\right]$$

$$= \sqrt{\frac{\mathbf{k}}{\mathbf{i} 2\pi \mathbf{z}}} \exp(\mathbf{i} \mathbf{k} \mathbf{z}) \mathbf{G}(\mathbf{x} - \mathbf{x}\_0, \mathbf{z}), \tag{8a}$$

$$\mathbf{G}(\mathbf{x}, \mathbf{z}) \equiv \exp\left[\frac{\mathrm{ik}\mathbf{x}^2}{2\mathbf{z}}\right] \tag{8b}$$

where k is the wave number, so H(x, x�; z)H∗(x, x�; z) = k/(2πz) is x-independent. Therefore, the interference term (7) yields

$$
\langle \mathbf{E\_r^\*(x)E\_0(x)} \rangle \propto \int \mathbf{T(x\_0)dx\_0} \tag{9}
$$

and the object information has been washed out. We can also prove that, no matter where the object is placed within the object path, Eq. (9) still holds provided the two paths of the interferometer have the same length. Perhaps this consequence brought about the misunderstanding that spatial interference in holography needs not only temporal but also spatial coherence. The following experiment will show that the condition of spatial coherence is not necessary if the interferometric scheme is modified.

In 2009 we proposed an incoherent interferometry setup as shown in Fig. 2 (Zhang et al., 2009a). In this unbalanced interferometer the object and reference arms have different lengths, z� = 16cm and z� =27cm, respectively. The spatially incoherent light source is formed by passing a He-Ne laser beam of wavelength 632.8nm through a rotating ground glass disk. The object in the experiment is a double slit T of slit width b = 125μm and spacing d = 310μm. The interference pattern can be recorded by either of two chargecoupled device (CCD) cameras.

Fig. 2. Experimental scheme for an unbalanced interferometer where the two arms have different lengths. A laser and a rotating ground glass disk G form a spatially incoherent source; M1 and M2 are mirrors, BS1 and BS2 are beamsplitters, CCD1 and CCD2 are detectors, and T is a double-slit close to BS1.

For a conventional balanced interferometer where the two optical paths have the same

approximation, for example, the impulse response function for a propagation length z is

G(x, z) � exp �����

where k is the wave number, so H(x, x�; z)H∗(x, x�; z) = k/(2πz) is x-independent. Therefore,

and the object information has been washed out. We can also prove that, no matter where the object is placed within the object path, Eq. (9) still holds provided the two paths of the interferometer have the same length. Perhaps this consequence brought about the misunderstanding that spatial interference in holography needs not only temporal but also spatial coherence. The following experiment will show that the condition of spatial

In 2009 we proposed an incoherent interferometry setup as shown in Fig. 2 (Zhang et al., 2009a). In this unbalanced interferometer the object and reference arms have different lengths, z� = 16cm and z� =27cm, respectively. The spatially incoherent light source is formed by passing a He-Ne laser beam of wavelength 632.8nm through a rotating ground glass disk. The object in the experiment is a double slit T of slit width b = 125μm and spacing d = 310μm. The interference pattern can be recorded by either of two charge-

Fig. 2. Experimental scheme for an unbalanced interferometer where the two arms have different lengths. A laser and a rotating ground glass disk G form a spatially incoherent source; M1 and M2 are mirrors, BS1 and BS2 are beamsplitters, CCD1 and CCD2 are detectors,

���� exp(ikz)exp ���(����)�

∗(x, x�) is x-independent. In the paraxial

�� �, (8b)

�� �

∗(x)E�(x)〉 ∝ � T(x�)dx�, (9)

���� exp(ikz)G(x − x�, z), (8a)

length, that is h�(x, x�) = h�(x, x�), then h�(x, x�)h�

H(x, x�; z) <sup>=</sup> � �

<sup>=</sup> � �

〈E�

coherence is not necessary if the interferometric scheme is modified.

given by

the interference term (7) yields

coupled device (CCD) cameras.

and T is a double-slit close to BS1.

The laser beam has good temporal coherence which will not be affected by the scattering of the slowly rotating ground glass. The path difference between the two arms is much smaller than the longitudinal coherence length cτ, where c is the light speed in vacuum and τ is the coherence time of the laser beam. However, the motion of the ground glass disk destroys the spatial coherence of the laser beam. This source emits what is known as pseudo-thermal light and satisfies Eq. (6) approximately. In contrast, a true thermal light source cannot satisfy the condition of temporal coherence in this setup.

The two-dimensional (2D) intensity patterns detected by CCD1 are shown in Fig. 3. When the ground glass disk is moved step by step, the interference patterns are irregular speckles frame by frame and each frame is different, as shown in the two single-shot frames of Figs. 3(a) and 3(b). However, if we average over a number of exposures, as the number of frames increases, a well-defined interference pattern emerges gradually, as can be seen in Figs. 3(c)-3(g).

The experimental results tell us that, unlike the stationary pattern in coherent interferometry, the interference pattern here is irregularly fluctuating in time and the welldefined pattern can only be discerned after taking its statistical average. These striking features imply that incoherent interferometry originates from a different interference mechanism.

We can explain the experiment using Eq. (7). The interference term in this incoherent interferometer can be calculated as

$$
\langle \mathbf{E}\_{\rm I}^\*(\mathbf{x}) \mathbf{E}\_0(\mathbf{x}) \rangle = \mathbf{I}\_3 \mathbf{e}^{\dagger \varphi} \int \mathbf{T}(\mathbf{x}\_0) \mathbf{H}(\mathbf{x}, \mathbf{x}\_0; \mathbf{z}\_0) \mathbf{H}^\*(\mathbf{x}, \mathbf{x}\_0; \mathbf{z}\_r) d\mathbf{x}\_0
$$

$$
\propto \exp[i\mathbf{k}(\mathbf{z}\_0 - \mathbf{z}\_r)] \int \mathbf{T}(\mathbf{x}\_0) \mathbf{G}(\mathbf{x} - \mathbf{x}\_0, \mathbf{z}\_0) \mathbf{G}^\*(\mathbf{x} - \mathbf{x}\_0, \mathbf{z}\_r) d\mathbf{x}\_0
$$

$$
= \exp[i\mathbf{k}(\mathbf{z}\_0 - \mathbf{z}\_r)] \int \mathbf{T}(\mathbf{x}\_0) \mathbf{G}(\mathbf{x} - \mathbf{x}\_0, \mathbf{Z}) d\mathbf{x}\_0. \tag{10}
$$

where function G defined by Eq. (8b) obeys

$$\mathbf{G}\mathbf{(x,z\_0)G^\*(x,z\_r) = G(x,Z)},\tag{11}$$

when the effective diffraction length Z is defined by

$$\frac{1}{\mathbf{z}} = \frac{1}{\mathbf{z}\_{\mathbf{o}}} - \frac{1}{\mathbf{z}\_{\mathbf{r}}}.\tag{12}$$

Equation (10) defines a kind of incoherent interferometry, which represents the Fresnel diffraction integral of an object under the paraxial condition, and is the same as for coherent diffraction but with an effective object distance Z replacing the real one, z�. Note that Z is negative for z� > z�, designating a phase reversal diffraction. We will discuss this issue in Subsection 3.3.

In the far-field limit, Eq. (10) presents the Fourier transform T� of object T:

$$
\mathbb{E}\left\langle \mathbf{E}^\*\_{\mathbf{r}}(\mathbf{x}) \mathbf{E}\_0(\mathbf{x}) \right\rangle \propto \exp[i\mathbf{k}\{\mathbf{z}\_0 - \mathbf{z}\_{\mathbf{r}}\}] \, \mathbb{G}\{\mathbf{x}, \mathbf{Z}\} \Upsilon\left(\frac{\mathbf{x}}{\mathbf{z}}\right). \tag{13}
$$

For a double-slit of slit width b and spacing d, its Fourier transform reads

$$\Upsilon(\mathbf{q}) = \left(\frac{2\mathbf{b}}{\sqrt{2\pi}}\right) \text{sinc(qb/2)} \cos(\mathbf{q}d/2). \tag{14}$$

Incoherent Holographic Interferometry 143

∗(x)E�(x)〉 ∝ exp[ik(z� − z�)] � T(x�)G(x−x�, z�)dx�

�� ��

� ≈ �T� �

�� �� ���

�, (15)

. (16)

≈ exp[ik(z� − z�)] G(x, z�)T� �

where the last step is valid in the far-field limit. However, the intensity distribution of the reference waves is homogeneous whereas the intensity of the object wave exhibits the

In holography this pattern should be avoided by diminishing the intensity of the object

To demonstrate the above theoretical explanation, Fig. 4 shows one-dimensional (1D) intensity patterns recorded by the two CCD cameras, where (a) and (b) are the interference fringes observed by CCD1 and CCD2, respectively. The left part shows the results when the interferometer is illuminated by spatially incoherent light. In our experimental scheme the effective diffraction length is calculated to be Z = 39.3cm. For a 50/50 beamsplitter BS2, if one output field is [E�(x) + E�(x)]/√2, the other should be [E�(x) − E�(x)]/√2. We can see that the two interference patterns with a phase shift are formed in the average of 10000 frames, which matches with the theoretical simulation of Eq. (13). Taking the difference and sum of the two output intensities gives, respectively, the net interference pattern and the intensity background, as shown in (c) and (d). As a matter of fact, the homogeneous intensity background verifies the spatial incoherence of

To further confirm whether the interference pattern is related to the spatial incoherence, we compare it with the result obtained in the same interferometer using coherent light. For this we simply remove the ground glass disk in Fig. 2. The experimental results and theoretical simulation are shown in the right part of Fig. 4, where (a) and (b) show the stationary intensity patterns registered by CCD1 and CCD2, respectively. In the coherent case, these patterns contain the interference term and the diffraction pattern of the object. Again, taking the difference and sum of (a) and (b), we obtain the net interference pattern in (c) and the diffraction pattern of the object in (d), respectively. These experimental results are very different from the left part for incoherent interferometry, and agree well with Eqs. (15) and

The above theoretical analysis and experimental demonstration tell us that, in an unbalanced interferometer, spatially incoherent light is capable of performing holographic interference in a similar way to that of coherent light. When an object is illuminated by incoherent light, the coherent information is lost in the object wave itself, but can be reproduced in an unbalanced interferometric scheme. Physically, the mechanisms of the two types of interferometry are quite different. In the incoherent case, the interference pattern fluctuates irregularly in time, and a well-defined pattern can only be formed in the statistical summation. Moreover, the diffraction pattern of an object is dependent on the effective diffraction length, which is associated with the travel distances of both the object and

(16) with a diffraction length of z� = 16cm for the coherent case.

In contrast, for the same interferometer driven by coherent light, Eq. (5) yields

〈|E�(x)|�〉 ∝ |� T(x�)G(x−x�, z�)dx�|

〈E�

diffraction pattern

wave.

the source.

reference waves.

Fig. 3. Experimentally observed 2D interference patterns recorded by CCD1 in the scheme of Fig. 2. Figs. (a) and (b) are individual single frames; (c), (d), (e), (f) and (g) are averaged over 10, 40, 400, 6400 and 10000 frames, respectively. The color-bar shows the relative intensity.

Besides the interference term, however, the output intensity distribution of the interferometer includes also the intensities of both the object and reference waves [see Eq. (1)]. In incoherent interferometry, these two intensities are homogeneously distributed.

Fig. 3. Experimentally observed 2D interference patterns recorded by CCD1 in the scheme of Fig. 2. Figs. (a) and (b) are individual single frames; (c), (d), (e), (f) and (g) are averaged over 10, 40, 400, 6400 and 10000 frames, respectively. The color-bar shows the relative intensity. Besides the interference term, however, the output intensity distribution of the interferometer includes also the intensities of both the object and reference waves [see Eq. (1)]. In incoherent interferometry, these two intensities are homogeneously distributed.

In contrast, for the same interferometer driven by coherent light, Eq. (5) yields

$$
\begin{split}
\langle \mathrm{E}\_{\mathrm{r}}^{\*} \mathrm{(x)} \mathrm{E}\_{\mathrm{0}} \mathrm{(x)} \rangle &\approx \exp[\mathrm{ik} \mathrm{(z\_{0} - z\_{r})}] \int \mathrm{T} \mathrm{(x\_{0})} \mathrm{G} \mathrm{(x - x\_{0}, z\_{0})} \mathrm{d}x\_{0} \\ &\approx \exp[\mathrm{ik} \mathrm{(z\_{0} - z\_{r})}] \, \mathrm{G} \, \mathrm{(x, z\_{0})} \widetilde{\mathrm{T}} \begin{pmatrix} \mathrm{kx} \\ \mathrm{z\_{0}} \end{pmatrix},
\end{split} \tag{15}
$$

where the last step is valid in the far-field limit. However, the intensity distribution of the reference waves is homogeneous whereas the intensity of the object wave exhibits the diffraction pattern

$$\langle\langle|\mathbf{E}\_{0}(\mathbf{x})|^{2}\rangle\otimes|\int \mathbf{T}(\mathbf{x}\_{0})\mathbf{G}(\mathbf{x}-\mathbf{x}\_{0},\mathbf{z}\_{0})d\mathbf{x}\_{0}|^{2}\approx\left|\Upsilon\left(\frac{\mathbf{k}\mathbf{x}}{\mathbf{z}\_{0}}\right)\right|^{2}.\tag{16}$$

In holography this pattern should be avoided by diminishing the intensity of the object wave.

To demonstrate the above theoretical explanation, Fig. 4 shows one-dimensional (1D) intensity patterns recorded by the two CCD cameras, where (a) and (b) are the interference fringes observed by CCD1 and CCD2, respectively. The left part shows the results when the interferometer is illuminated by spatially incoherent light. In our experimental scheme the effective diffraction length is calculated to be Z = 39.3cm. For a 50/50 beamsplitter BS2, if one output field is [E�(x) + E�(x)]/√2, the other should be [E�(x) − E�(x)]/√2. We can see that the two interference patterns with a phase shift are formed in the average of 10000 frames, which matches with the theoretical simulation of Eq. (13). Taking the difference and sum of the two output intensities gives, respectively, the net interference pattern and the intensity background, as shown in (c) and (d). As a matter of fact, the homogeneous intensity background verifies the spatial incoherence of the source.

To further confirm whether the interference pattern is related to the spatial incoherence, we compare it with the result obtained in the same interferometer using coherent light. For this we simply remove the ground glass disk in Fig. 2. The experimental results and theoretical simulation are shown in the right part of Fig. 4, where (a) and (b) show the stationary intensity patterns registered by CCD1 and CCD2, respectively. In the coherent case, these patterns contain the interference term and the diffraction pattern of the object. Again, taking the difference and sum of (a) and (b), we obtain the net interference pattern in (c) and the diffraction pattern of the object in (d), respectively. These experimental results are very different from the left part for incoherent interferometry, and agree well with Eqs. (15) and (16) with a diffraction length of z� = 16cm for the coherent case.

The above theoretical analysis and experimental demonstration tell us that, in an unbalanced interferometer, spatially incoherent light is capable of performing holographic interference in a similar way to that of coherent light. When an object is illuminated by incoherent light, the coherent information is lost in the object wave itself, but can be reproduced in an unbalanced interferometric scheme. Physically, the mechanisms of the two types of interferometry are quite different. In the incoherent case, the interference pattern fluctuates irregularly in time, and a well-defined pattern can only be formed in the statistical summation. Moreover, the diffraction pattern of an object is dependent on the effective diffraction length, which is associated with the travel distances of both the object and reference waves.

Incoherent Holographic Interferometry 145

Fig. 5. Scheme I of incoherent interferometry, which is similar to the scheme of Fig. 2 but two lenses are inserted in the centre of the two arms. Both the input and output ports of the

of a field between its two focal planes, we position both the input and output ports of the interferometer at the two foci of the lenses. The impulse response function between the two

Let object T(x�) be close to the input port. In the incoherent regime, then Eq. (7) gives the

∗(x)E�(x)〉 ∝ � T(x�)F(x, x�; f�)F∗(x, x�; f�)dx�

∝ exp [i2k(f� − f�)] � T(x�)exp [−ikxx�/f]dx�

where f� and f� are the focal lengths of the lenses in the object arm and reference arm,

In the experiment we take the two lenses to be of focal lengths f� = 7.5cm and f� = 12cm, and the double-slit as the object. The experimental results are plotted in Fig. 6. After eliminating the background, Fig. 6(c) shows the Fourier spatial spectrum of the double-slit. The theoretical curves are plotted by using Eqs. (18) and (19) with the effective focal length

If a coherent plane wave drives the same interferometer, the interference term is obtained as

where Dirac Delta function δ(x) comes from the focusing effect of the lens in the reference wave, and the Fourier transform of the object is governed by the focal length f� of the object

� � <sup>=</sup> � �� <sup>−</sup> � ��

h(x, x�) = F(x, x�; f) ≡ �k/(i2πf)exp (i2kf)exp (−ikxx�/f), (17)

= exp[i2k(f� − f�)] T�[kx/f], (18)

∗(x)E�(x)〉 ∝ exp[i2k(f� − f�)] T�[kx/f�]δ(x), (20)

. (19)

interferometer are located at the two focal planes of the lenses.

focal planes for a lens of focal length f is given by

〈E�

respectively. The effective focal length f is defined as

〈E�

where k is the wave number of the beam.

interference term to be

f = 20cm.

Fig. 4. Experimentally observed 1D interference patterns in the scheme of Fig. 2. Left and right parts are the patterns obtained using spatially incoherent and coherent light, respectively. (a) and (b) are the patterns registered by CCD1 and CCD2, respectively; (c) and (d) are their difference and summation, respectively. The patterns for the incoherent case are averaged over 10000 frames. Experimental data and theoretical simulation are given by open circles and solid lines, respectively.
