**3. Various schemes for incoherent interferometry**

In the last section, we indicated that incoherent interferometry does not exist in a balanced interferometer and we proposed an unbalanced interferometer where its two arms have different lengths. Based on this attribute, in this section, we propose other unbalanced interferometers to perform incoherent interference.

### **3.1 Scheme I**

As shown in Fig. 5, we modify the previous interferometer by inserting two lenses in the centre of the two arms (Zhang, July 2010). Since a lens can perform Fourier transformation

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,

open circles and solid lines, respectively.

**3.1 Scheme I** 

**3. Various schemes for incoherent interferometry** 

interferometers to perform incoherent interference.

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

In the last section, we indicated that incoherent interferometry does not exist in a balanced interferometer and we proposed an unbalanced interferometer where its two arms have different lengths. Based on this attribute, in this section, we propose other unbalanced

As shown in Fig. 5, we modify the previous interferometer by inserting two lenses in the centre of the two arms (Zhang, July 2010). Since a lens can perform Fourier transformation

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

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

$$\mathbf{h(x,x\_0) = F(x,x\_0; f) \equiv \sqrt{k/(i2\pi f)exp(i2kf)exp(-ikx\_0/f)}},\tag{17}$$

where k is the wave number of the beam.

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

$$\left(\mathbf{E}\_{\mathbf{r}}^{\*}\mathbf{(x)}\mathbf{E}\_{\mathbf{0}}\mathbf{(x)}\right) \propto \int \mathbf{T(x\_{0})F(x,x\_{0};f\_{0})F^{\*}(x,x\_{0};f\_{\mathbf{r}})dx\_{0}}$$

$$\propto \exp[i2\mathbf{k(f\_{0}-f\_{\mathbf{r}})] \int \mathbf{T(x\_{0})}\exp[-i\mathbf{ikx}\_{0}/\mathbf{f}]} \mathbf{dx}\_{0}$$

$$= \exp[i2\mathbf{k(f\_{0}-f\_{\mathbf{r}})] \int \mathbf{T(x\_{0})f}]} \tag{18}$$

where f� and f� are the focal lengths of the lenses in the object arm and reference arm, respectively. The effective focal length f is defined as

$$\frac{1}{\text{f}} = \frac{1}{\text{f}\_{\text{o}}} - \frac{1}{\text{f}\_{\text{r}}}.\tag{19}$$

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 f = 20cm.

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

$$
\langle \mathbf{E}\_{\rm r}^\*(\mathbf{x}) \mathbf{E}\_0(\mathbf{x}) \rangle \propto \exp[i2\mathbf{k}(\mathbf{f}\_0 - \mathbf{f}\_r)] \,\widetilde{\mathbf{T}}[\mathbf{k}\mathbf{x}/\mathbf{f}\_0] \delta(\mathbf{x}),\tag{20}
$$

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

Incoherent Holographic Interferometry 147

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�.

≈ G∗(x, 2f�)T� �

pattern of the object with a diffraction length 2f�, different from the f� in Eq. (21).

and (b), as shown in the right part of Figs. 8(c) and 8(d), respectively.

where the last step is valid in the far-field limit. The equation shows the Fresnel diffraction

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.

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)

∗(x)E�(x)〉 ∝ � T(x�)F(x, x�; f�)H∗(x, x�; 2f�)dx�

�� ���

. The former and the latter can be extracted by the difference and sum of (a)

�, (22)

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

〈E�

and �T�(kx/f�)��

∝ � T(x�)G∗(x+x�, 2f�)dx�

lens. However, this Fourier spectrum will not appear in the coherent interference term due to the modulation of Delta function.

Fig. 6. Experimentally observed 1D interference patterns in Scheme I (Fig. 5). (a) and (b) are interference patterns (averaged over 10000 frames) registered by CCD1 and CCD2, respectively; (c) and (d) are their difference and summation, respectively. Experimental data and theoretical simulation are given by open circles and solid lines, respectively.
