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## Off-axis holography [tutorial]

Off-axis holography is a popular technique to reconstruct a hologram. It allows retrieving the amplitude and the phase of a field pattern by measuring only one image with a digital camera. It relies on an intereferometric setup with a non-zero angle between the reference beam and the signal beam and requires to numerically filter the spatial frequencies.

This technique was initially presented in [1]. The schematic of the experimental setup is shown in Fig. 1. The idea is to make interfere the unknown "signal" wavefront of complex field $\dpi{300}&space;E_s$ with a plane wave of amplitude $\dpi{300}&space;E_0$ with a given angle $\dpi{300}&space;\theta$ between the direction of propagation of the two beams. We assume that $\dpi{300}&space;E_s$ and $\dpi{300}&space;E_0$ are in the same state of polarization.

Figure 1. Schematic of the setup.

The intensity measured by the camera reads:

$\dpi{300}&space;I=\underbrace{\left|E_0\right|^2+\left|E_s(x,y))\right|^2}_{\textrm{order&space;0}}+\underbrace{E_0&space;E_s^*(x,y)e^{ik&space;\sin\theta&space;x}}_{\textrm{order&space;+1}}+&space;\underbrace{E_0^*&space;E_s(x,y)e^{-ik&space;\sin\theta&space;x}}_{\textrm{order&space;-1}}$

The three terms correspond to signals centered around the spatial frequencies 0, $\dpi{300}&space;\nu&space;=&space;\frac{k&space;\sin(\theta))}{2\pi}$ and $\dpi{300}&space;-\nu$. If the spatial frequency $\dpi{300}&space;\nu$ is larger than the maximal spatial frequency in the signal to reconstruct, the Fourier transform of the measured intensity pattern shows three components corresponding orders 0, +1 and -1 that do not overlap. We note that, since $\dpi{300}&space;E_0$ is constant, the order -1 is directly proportional to the complex field $\dpi{300}&space;E_s$ we want to retrieve but simply shifted by the spatial frequency $\dpi{300}&space;\nu$. The idea is then to numerically select the order -1 in the Fourier domain and center the signal around the zero frequency. The inverse Fourier transform then directly gives access to the complex image $\dpi{300}&space;E_s$. The steps of this process are illustrated in Fig. 2 and Fig. 3.

The following images were obtained with the Matlab code attached to this article and used to simulate an off-axis holography measurement (the file is available at the bottom of this page, the code includes comments). For the sake of simplicity, in this example, we try to recover an image with a flat amplitude and with a random phase pattern, similar to the phase of speckle pattern [Fig 2.a.]. After interference with the tilted plane wave [Fig 2.b.], we calculate the intensity pattern as it would be measured on the camera [Fig 2.c.].

Figure 2. Phase wavefront to measure (a), tilted plane wave reference (b) and resulting interference pattern.

We then calculate the Fourier transform of this intensity pattern [Fig 3.a.]. We note that the angle of the tilted plane wave has to be high enough so that order 0 and -1 do not overlap. We then select only the -1 order [Fig 3.b.] and shift it around the zero frequency [Fig 3.b.].

Figure 3. Numerical spatial frequency filtering. (a) Complete Fourier transform of the intensity pattern. (b) Fourier transform filtered to keep only the -1 order. (c) Signal shifted around the zero spatial frequency.

The reconstructed complex field is obtained by calculating the inverse Fourier transform of the last step. We compare in Fig 4. the phase of the in initial image and the phase of the recovered one.

Figure 4. Comparison between the actual phase pattern (a) and the reconstructed one (b).

This technique allows to perform a one shot measurement of the complex field for one state of polarization at the cost of a loss of resolution due to the need to resolve the fringes inside one speckle grain. Such a technique can be extended to obtain a full image for both polarizations by using two reference beams with orthogonal polarizations with tilt angles $\dpi{300}&space;\theta_x$ and $\dpi{300}&space;\theta_y$ in two different directions.

[1] E. Cuche et al., Opt. Express, 39, (2000)

 simOffAxis.m [Simple Matlab code to simulate an off-axis holography measurement] 3 kB simOffAxis.py [Simple Python code to simulate an off-axis holography measurement] 3 kB

#14 SANJIV 2019-12-04 19:18
Hi Sebastin
The intensity shots were correlated over time however the phase that we reconstrcuted was loosing correlation over time. May be its better to explain my setup a bit:

we setup a Michelson interferometer . But one arm in the interferometer we had a reflecting USAF target. On the camera we get the diffraction pattern of the USAF target. The other arm in the interferometer serves as the reference beam to record the phase of the diffraction pattern. Based on the experiment, we see that the reconstructed phase is same only for first 10 minutes and drops after that. We are using a 15mW DFB laser at 1550 nm. What do you think is casing it ?
I have also tried using a uniphase microgreen laser 532 nm. using this laser also, I find the similar issue. The intensity shots of the hologram are correlated, but the phase reconstructed is decorrelating over time.

#13 Sébastien Popoff 2019-11-27 10:38
Quoting SANJIV:
I have constructed a Mach Zehender Interferomter. How can I quantify the phase stability of my laser ? Could you please help me with it ?

This tutorial is about measuring the phase of a complex wavefront, not about laser stability. You should find more appropriate pieces of information on articles about that particular subject.

That being said, if you have no phase perturbation between the arms of your interferometer, just looking at the fluctuations of the intensity resulting from the interference of your arms should give you the phase fluctuations.

Quoting SANJIV:

If I understand correctly the recorder hologram and it's fourier transform produces just +1 and -1 depending on the interference pattern. It will be just a point in fourier space right ? or am I wrong ?

No need to do off-axis holography if you just have a Gaussian beam, simply look at the fluctuations of the intensity of the interference.

#12 SANJIV 2019-11-27 10:18
I have constructed a Mach Zehender Interferomter. How can I quantify the phase stability of my laser ? Could you please help me with it ?
If I understand correctly the recorder hologram and it's fourier transform produces just +1 and -1 depending on the interference pattern. It will be just a point in fourier space right ? or am I wrong ?

#11 Sébastien Popoff 2019-02-28 14:09
Quoting Sanjeev:
I record a speckle pattern using off axis digital holography. However if I take consecutive shots with nothing being changed, I see that the correlation of the Hologram intensity reduces with that of the first shot. Due to which my reconstruction gives different recobstructed field at different times. My laser is stable.

Hi Sanjeev,

Did you actually measure a decorrelation curve with intensity only (no reference) and compared it with decorrelation of the reconstructed field with the intereferences?

When you say your laser is stable, you mean the intensity is stable? If you have interferometric instabilities or phase fluctuations, the interference signal will fluctuate.

Try to remove the scattering medium and measure the interference signal (i.e. a simple Mach Zehnder), if the intensity signal varies too much, that means you do not have good interferometric stability.

Quoting Sanjeev:
I dont see the speckle pattern chaning visually in the camera.

That is not relevant. You need to quantify the decorrelation in time. Do not trust the visual aspect.

Sebastien

#10 Sanjeev 2019-02-27 19:45
I record a speckle pattern using off axis digital holography. However if I take consecutive shots with nothing being changed, I see that the correlation of the Hologram intensity reduces with that of the first shot. Due to which my reconstruction gives different recobstructed field at different times. My laser is stable. I dont see the speckle pattern chaning visually in the camera. However the noise seems degrading my signal. Can you please help me with it.

#9 Cheng Shen 2018-12-14 17:39
Quoting SANJIV:
Thanks Sebastien. Now I understand clearly. Unfortunately when I take the Fourier transform of the hologram I recorder experimentally, in addition to + and -1 terms I have +2 and -2, (pic attached).
https://ekbil-my.sharepoint.com/:i:/g/personal/sanjeev_a_ekb_co_il/EbytTozXzO5AmqJ-RmnjO-cB2Ram1VIBvFxlqQlbn65qSg?e=4Cnstf
https://ekbil-my.sharepoint.com/:i:/g/personal/sanjeev_a_ekb_co_il/EbytTozXzO5AmqJ-RmnjO-cB2Ram1VIBvFxlqQlbn65qSg?e=4Cnstf
Could you let me know what error I am doing in experiment ?

Quoting Sébastien Popoff:
Filtering is important to get the phase. You need to remove the zero-th order and one of the +1 or -1 order (be careful, the phase will be the opposite in the -1 compared to +1). If you do not shift it back to zero, you will just have a phase slope (equal to the angle between the reference and the signal beam) in addition to the phase pattern of the field you want to measure.

Everything is in the equation for the intensity.

Quoting Jitender:
Can you please explain is it correct if I don't filter the other components and measure the Inverse Fourier of the whole filed after first fourier transform ? The phase that I find will it be same in both cases ? Or is the shifting an important parameter?

The +2 and -2 order maybe are due to the overexpousre in your experiment. Taking overexposure as truncation, it will introduce higher order frequency

#8 Sébastien Popoff 2018-05-02 15:10
Quoting SANJIV:
Hi Sebastin
Each time I reconstruct the hologram the complex field I reconstruct is different at different time. Why is this happening ? How can I reduce this error ?

Hi Sanjiv.
Is it in experiment or in simulation.
If it is with experimental data, first try your code in simulation (with a code similar to the one I share).

If the results change over time in experiment, maybe your system is not stable (decorrelation of the medium, stability of the laser, camera, etc...). To test that, just take consecutive images (without Fourier treatment) and check their correlation to the first image.

Best,

Sebastien

#7 SANJIV 2018-05-02 12:39
Hi Sebastin
Each time I reconstruct the hologram the complex field I reconstruct is different at different time. Why is this happening ? How can I reduce this error ?

#6 SANJIV 2018-04-05 07:36
Thanks Sebastien. Now I understand clearly. Unfortunately when I take the Fourier transform of the hologram I recorder experimentally, in addition to + and -1 terms I have +2 and -2, (pic attached).
https://ekbil-my.sharepoint.com/:i:/g/personal/sanjeev_a_ekb_co_il/EbytTozXzO5AmqJ-RmnjO-cB2Ram1VIBvFxlqQlbn65qSg?e=4Cnstf

Could you let me know what error I am doing in experiment ?

Quoting Sébastien Popoff:
Filtering is important to get the phase. You need to remove the zero-th order and one of the +1 or -1 order (be careful, the phase will be the opposite in the -1 compared to +1). If you do not shift it back to zero, you will just have a phase slope (equal to the angle between the reference and the signal beam) in addition to the phase pattern of the field you want to measure.

Everything is in the equation for the intensity.

Quoting Jitender:
Can you please explain is it correct if I don't filter the other components and measure the Inverse Fourier of the whole filed after first fourier transform ? The phase that I find will it be same in both cases ? Or is the shifting an important parameter?

#5 Sébastien Popoff 2018-04-03 07:35
Filtering is important to get the phase. You need to remove the zero-th order and one of the +1 or -1 order (be careful, the phase will be the opposite in the -1 compared to +1). If you do not shift it back to zero, you will just have a phase slope (equal to the angle between the reference and the signal beam) in addition to the phase pattern of the field you want to measure.

Everything is in the equation for the intensity.

Quoting Jitender:
Can you please explain is it correct if I don't filter the other components and measure the Inverse Fourier of the whole filed after first fourier transform ? The phase that I find will it be same in both cases ? Or is the shifting an important parameter?