Binary encoded computer generated holograms for temporal phase shifting

Angela Amphawan

doi:10.1364/OE.19.023085

1. Introduction

Interferometry is of paramount importance in diverse areas in applied physics and engineering as it provides information on the phase difference between two wavefronts and insights into the nature of electromagnetic wave propagation. Recent trends towards real-time optical applications such as optical sensors [1–4], optical transceivers [5–8] and optical tomography [9–12] predicate the need for real-time interferometry [13], even in small, lightweight, portable devices. The observation of rapid changes in the phase is essential as it allows immediate measures to be taken at the input in order to improve the response of the system.

In temporal phase shifting interferometry, different phase shifts are induced relative to the wavefront analyzed by moving a mirror, rotating a phase plate or moving a diffraction grating [14–16]. Using interferograms recorded from the distinct phase shifts, it is possible to determine the phase of the wavefront at any measured plane [14–16]. Temporal phase shifting is typically used in stable environments. In dynamic environments with rapid fluctuations, a phase variation may take place within the time taken to record the set of interferograms, rendering the measurement invalid. The advent of spatial light modulators (SLMs) with rapid reconfiguration speed has improved interferogram recording time due to the exclusion of mechanical movement from a rotating, tilting or moving element for producing different path lengths or phase shifts with respect to the reference beam [17–19]. Instead, a more elegant approach for the generation of phase shifts is realized by simply programming the phase-shifted CGHs and then displaying them sequentially on the SLM. Amplitude [17], phase [18] and complex [19] SLMs have been used to achieve digital phase control in temporal phase shifting. Considering that a CGH is displayed almost instantaneously on the SLM [20], the time taken to program the CGH constitutes the main time constraint during the measurement of the interferograms. For adaptive real-time applications, rapid processing time of the CGH is crucial to avoid the intractability of frequent phase changes which may impede corrective measures at the input to improve the system response. For small, lightweight, portable devices such as optical sensors in handheld medical devices and optical transceivers in mobile phones and tablets, the size and weight of the device may place a constraint on the computational power of the processor. Under these circumstances, rapid processing of the CGH for temporal phase shifting in real-time applications may not be feasible. In legacy hardware with upgrade complexities, rapid processing of CGHs may also not be possible for real-time applications. Thus, it is necessary to simplify the required CGH processing in order to improve the time efficiency for interferogram recording to enable real-time processing.

The aim of this paper is to introduce the design of a set of binary encoded CGHs for five-frame temporal phase shifting interferometry based on the complex field generation technique in [21]. The CGH processing time using the new binary encoding scheme is smaller compared to the CGH processing time using the original binary encoding scheme for temporal phase shifting. This makes the new set of binary encoded CGH promising for real-time applications, particularly in small, lightweight portable devices or legacy hardware with low computational power. As a motivation for the use of the proposed design, the phase of the generated electric field for a real-time selective launch into a MMF is measured using the set of new binary encoded CGHs. The results obtained from digital temporal phase shifting using the new set of binary encoded CGHs are in good agreement with the results obtained by means temporal phase shifting using a piezo-driven flat mirror.

The paper is organized as follows. In Section 2, a mathematical derivation of the binary encoded CGH for five-frame digital temporal phase shifting is presented. An application of the binary encoded CGH for temporal phase shifting interferometry is demonstrated in a real-time selective launch into a multimode fiber in Section 3. The results from the experimental demonstration of the use of the new binary encoded CGHs for temporal phase shifting are then presented in Section 4. Finally, in Section 5, the results from the new digital phase shifting design are compared with results from standard mechanical phase shifting using a piezo-driven flat mirror.

2. Mathematical derivation of binary encoded CGH for temporal phase shifting

For conventional temporal phase shifting, distinct phase shifts are induced in the reference beam with respect to the analyzed wavefront. To accomplish the same effect in digital temporal phase shifting, the phase of the reference beam is preserved while distinct phase shifts are added to the analyzed wavefront. For each distinct phase shift between the wavefront analyzed and the reference beam, an interferogram of the phase-shifted wavefront and the constant reference beam is recorded on a particular plane. For five-frame temporal phase shifting, the intensities of the interferograms are given by [15-16]:

I_{1} = I_{o} [1 + γ \cos (ϕ - 2 p)]

I_{2} = I_{o} [1 + γ \cos (ϕ - p)]

I_{3} = I_{o} [1 + γ \cos ϕ]

I_{4} = I_{o} [1 + γ \cos (ϕ + p)]

I_{5} = I_{o} [1 + γ \cos (ϕ + 2 p)]

where ϕ is the phase of the test beam, γ is the fringe visibility, in radians and p= π /2 radians is the induced phase shift.

The phase from the five interferograms is given by [15-16]:

ϕ = \tan^{- 1} {2 (I_{2} - I_{4}) / [2 I_{3} - I_{5} - I_{1}]}

The relative phase shifts between the wavefront analyzed and the constant reference beam for the five interferograms in Eqs. (1) to (5) are σ = -π, -π/2, 0, π/2 and π respectively.

In [21], a complex field was generated using a binary amplitude SLM. For temporal phase shifting of the generated complex wavefront in [21], four distinct phase shifts σ = -π, -π/2, π/2 and π may be added separately to the generated complex wavefront, followed by the binary encoding of each of the phase-shifted tilted wavefront. However, for small, lightweight, portable devices such as optical sensors in medical devices and optical transceivers in mobile phones and tablets, the size and weight of the device may place a constraint on the computational power and storage used in the processor. In legacy hardware with upgrade complexities, rapid processing of CGHs may also not be possible for real-time applications. Under such circumstances, it is necessary to simplify the required CGH processing in [21] in order to improve the time efficiency for interferogram recording. This is achieved by separate encoding schemes for different phase-shifted fields. Using the original encoding scheme, the calculation of the phase-shifted modal fields for the different phase shifts may be computationally expensive and increases the execution time for temporal phase shifting. Considering that a CGH is displayed almost instantaneously on the SLM [20], the time taken to program the CGH constitutes the main time constraint during the measurement of the interferograms. For real-time applications, the processing time for the CGH is crucial to avoid intractability of rapid phase changes and inaccurate feedback to the input. Thus, the proposed encoding scheme is a more direct approach for temporal phase shifting. A comparison of the use of the original encoding scheme and the proposed technique is shown in Fig. 1 .

Fig. 1 Comparison of new set of binary encoding scheme and original binary encoding scheme for five-frame temporal phase shifting in real-time applications

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The simplified phase mappings of the required phase shifts onto the binary CGH based on the complex wavefront generation technique in [21] will be derived as follows. Let the required phase shift relative to the fixed reference beam,

σ = - π .

After the addition of linear tilts in the x and y directions to the phase of the desired field, ξ and the required phase shift, σ, the new tilted wavefront is given by

f (x) = a (x) e x p [j (ξ (x, y) + τ_{x} x + τ_{y} y + σ)]

where a(x, y) is the amplitude of the field and ξ is the phase of the field. Let

u = Re {f (x, y)}, v = Im {f (x, y)},

then

u = a (x, y) \cos [ξ (x, y) + τ_{x} x + τ_{y} y + σ]

v = a (x, y) \sin [ξ (x, y) + τ_{x} x + τ_{y} y + σ]

| a (x, y) | = \sqrt{u^{2} + v^{2}}

The binarized output, g(x, y) is 1 when the total phase is less than α,

ξ + τ_{x} x + τ_{y} y + σ \leq α

where

α

is a constant. Taking the sine of both sides of Eq. (13),

\sin (ξ + τ_{x} x + τ_{y} y + σ) \leq \sin α

given - π / 2 \leq ξ + τ_{x} x + τ_{y} y + σ \leq π / 2

0 \leq α \leq π / 2

for the sine function to be monotonically increasing.

From the Fourier series expansion of the binarized output, G(x, y), given in the first harmonic (n = 1) has an amplitude,

a (x) = \sin α

L1 Fourier transforms the first harmonic into the first diffraction order in the focal plane of the L1. The first diffraction order is Fourier transformed again by L2, and the image in the back focal plane of L2 is also has amplitude of $a (x) = \sin α .$

Combining Eqs. (11), (12) (14) and (17),

{(v \pm \frac{1}{2})}^{2} + u^{2} \geq \frac{1}{4}, u \leq 0

In the same manner, the phase mappings for relative phase shifts, $σ = - π / 2, π / 2 and π$ with respect to the fixed reference beam were derived. The following inequalities are the results of the derived phase mappings for $σ = - π / 2, π / 2 and π$ phase shifts respectively:

{(u \pm \frac{1}{2})}^{2} + v^{2} \geq \frac{1}{4}, v \geq 0

{(u \pm \frac{1}{2})}^{2} + v^{2} \geq \frac{1}{4}, v \leq 0

{(v \pm \frac{1}{2})}^{2} + u^{2} \geq \frac{1}{4}, u \leq 0

The complex plane mappings of the tilted modal electric field onto a binarized hologram are shown in Fig. 2 for -π, -π/2, 0, π/2 and π phase shifts relative to a fixed reference beam.

Fig. 2 Encoding of normalized complex plane onto a binary CGH for -π, -π/2, 0, π/2 and π phase shifts relative to a fixed reference beam. Amplitude of 0 and 1 indicate intensity transmission and no intensity transmission respectively through a pixel in the SLM active area

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3. Demonstration of application of binary encoded CGH for temporal phase shifting

As a motivation for the use of the proposed binary CGH, the phase of the generated electric field for a real-time selective launch in multimode fiber (MMF) is measured by temporal phase shifting using the new set of binary encoded CGHs. In Local Area Networks, MMF is the conventional choice for supporting high bandwidth links [22]. For optimum bandwidth enhancement in order to meet rapidly increasing customer demands [23], it is necessary to curtail modal dispersion in MMF [24-25]. Confinement of the channel to the least possible number of modes is desirable and may be achieved by selective launches which excite only a subset of propagating modes [26–33]. Recently, selective launches into MMF have been achieved by employing spatial light modulators (SLMs) [31, 34–37]. In selective launches, prior to coupling the generated modal field into the MMF, it is valuable to verify the phase of the generated electric field to maximize power coupling efficiency. This is to ensure optimum bandwidth enhancement in the channel. In selective launches with the SLM as the modal field generator, a digital temporal phase shifting interferometer may be easily constructed using the existing SLM.

The experimental setup for the real-time selective launch into a MMF based on the experiment in [37] is shown in Fig. 3 The additional parts required for the temporal phase shifting interferometer are outlined in blue. A 128x128 pixel transmissive binary amplitude spatial light modulator (SLM), two achromatic doublets [focal lengths f₁ = 300mm (L1) and f₂ = 100mm (L2)] and a fiber collimator with an aspheric lens of f₃ = 11mm (L3) were used. A visible 632.8nm Helium Neon laser was used to easily view modal field at various points along the system and to easily capture the interferograms. The MMF used was a 1km-long graded-index Thorlabs GIF625 [38], with a core diameter of 62.5μm

Fig. 3 Digital temporal phase shifting using binary encoded CGHs for a real-time selective launch into a MMF

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The expanded laser beam from the beam expander was split into two optical paths, as shown in Fig. 3. The first optical path was used for generating the modal electric field to be coupled into the MMF while the second optical path was used as a fixed reference beam for the interferometry.

The technique used for generating the electric field is illustrated in Fig. 4 and is summarized here. First, the desired electric field c(x₁, y₁) was Fourier transformed, with zero-padding to increase the resolution of fine details. Next, a linear tilt of τ_x = 88π in the horizontal direction and τ_y = 120π in the vertical direction was added to the resultant field [21]. These tilt values were chosen as they provided sufficient spatial separation of the diffraction orders in the Fourier plane so that the first diffraction order may be easily extracted later. This yielded the complex field:

f (x {}_{1}, y_{1}) = d (x {}_{1}, y_{1}) \exp [j (τ_{x} x {}_{1}+ τ_{y} y {}_{1})]

where x₁ and y₁ are spatial coordinates of the Fourier transformed field in the Fourier plane; and d (x₁, y₁) is the Fourier transform of the polarized modal electric field.

Fig. 4 Summary of technique used for selective launch in MMF, adapted from [37]

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The complex field d (x₁, y₁) may be binarized by first mapping the space of possible phase values ϕ on the space g of binarized ones g (ϕ): x → g (ϕ ) using:

ϕ = ξ (x_{1}, y_{1}) + τ_{x} x_{1} + τ_{y} y_{1}

where ξ(x₁, y₁) is the phase of the Fourier transform of the polarized modal electric field. Then, the real part of Eq. (22) is taken and approximated by a Fourier series given by the following expression:

g (ϕ) = \frac{a_{o}}{2} + \sum_{n = 1}^{\infty} [a_{n} c o s (n ϕ) + b_{n} s i n (n ϕ)]

where a_o is the constant term and a_n is the Fourier cosine coefficient, b_n is the Fourier sine coefficient. Only the real part of the complex field d (x₁, y₁) is required, thus b_n = 0. Substituting Eq. (23) into Eq. (24) yields

\hat{g} (x_{1}, y_{1}) = {g (ϕ) |}_{(x_{1}, y_{1})} = \frac{a_{o}}{2} + \sum_{n = 1}^{\infty} a_{n} \cos (n ϕ)

This gives the binarized CGH function:

\hat{g} (x_{1}, y_{1}) = \frac{a_{o}}{2} + \frac{4}{π} \sum_{n = 1}^{\infty} a_{n} \cos {n [ξ (x {}_{1}, y_{1}) ​ + τ_{x} x {}_{1}+ τ_{y} y_{1}]}

The binarized CGH function was displayed on the SLM. This was then Fourier transformed by L1. From Eq. (26), the Fourier transformed field in the back focal plane or Fourier plane of L1 is given as:

\overset{⌢}{G} (x_{2}, y_{2}) = M_{o} (x_{2}, y_{2}) + \sum_{n = 1}^{\infty} [M_{n} (x_{2} + n τ_{x}, y_{2} + n τ_{y}) + M_{n}^{*} (n τ_{x} - x_{2}, n τ_{y} - y_{2})]

where x₂ and y₂ are spatial coordinates in the Fourier plane of L1, * is the complex conjugate and M_n (x₁, y₁) is the n-th diffraction order.

The first diffraction order, M₁ was then spatially filtered using a pinhole located in the back focal plane of L1. The combination of L2 and L3 then scales M₁. Following this, the generated modal field was then interfered with the reference beam in the back focal plane of L2. This was the closest location possible for the interference as L3 was connected directly to the input endface of the MMF using a fiber collimator [38]. The five binary encoded CGHs described in the previous section were displayed on the SLM sequentially to generate five interferograms of the phase-shifted electric field with respect to the reference beam.

In this experiment, five frames were the minimum number of frames for producing satisfactory phase shift distribution. When three and four frames were used for phase shifting, many errors were found in peripheral regions where intensity values were low.

4. Experimental results

Five measured interferograms of the phase-shifted electric fields with respect to the reference field are shown in Fig. 5 (a-e) . The phase was then retrieved from the five interferograms using Eq. (6). The retrieved phase distribution is shown in Fig. 5(f). The experiment was repeated three times for each mode excited and the mean squared error (MSE) between the retrieved phase distributions were calculated using:

M S E = \frac{1}{p q} {\sum_{t = 1}^{p} \sum_{u = 1}^{q} [θ_{1} (t, u) - θ_{2} (t, u)]}^{2}

where θ₁ is the phase distribution from the first measurement, θ₂ is the phase distribution from the second measurement, t is the index number for pixels on the horizontal axis, u is the index number for pixels on the vertical axis, p = 128 is the total number of pixels on the horizontal axis, and q = 128 is the total number of pixels on the vertical axis. Using Eq. (28), the MSEs of the second and third measured phase distributions relative to the first measured phase distribution were 0.014 and 0.009 respectively.

Fig. 5 (a-e) Interferograms from digital phase shifting technique using new binary encoded CGHs (f) Retrieved phase distribution using new digital phase shifting technique

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The relative power coupled into a fiber mode of a weakly guiding infinite parabolic MMF is given by the power coupling coefficient,

η_{l m} = {| \iint_{A_{c o r e}} E_{i n c} (x, y) e_{t} * (x, y) d x d y |}^{2} / \iint_{A_{c o r e}} {| E_{i n c} (x, y) |}^{2} d x d y \iint_{A_{c o r e}} {| e_{t} (x, y) |}^{2} d x d y

where e_tis the transverse field of a fiber mode, E_inc is the incident electric field of the offset beam and A_core is the cross sectional area of the fiber core. In the experimental setup in the power coupling efficiency is sensitive to the pitch, yaw, roll and distance of the SLM and lens with respect to the MMF. By measuring the phase at the end of the MMF prior to the launch, it is possible to adjust the pitch, yaw, roll and distance of the SLM and lens with respect to the MMF individually to find optimum power coupling into the desired mode.

5. Comparison to mechanical temporal phase shifting

The phase distribution obtained from the new set of binary CGHs using a binary amplitude SLM is compared to the phase distribution obtained from mechanical temporal phase shifting using a piezo-driven flat mirror. The measured phase distribution from mechanical temporal phase shifting using a flat mirror is shown in Fig. 6 . It is found that the measured phase distribution from the new binary CGH is in good agreement with the measured phase distribution from mechanical temporal phase shifting, as shown in Fig. 7 . The maximum phase difference is 0.18 rad. The discrepancy may be due to slight vibrations, air movement as well as thermal and electrical fluctuations in the sensor system. In peripheral regions, the discrepancy is higher as the slight environmental fluctuations have a more prominent effect on the low-intensity values in the interferograms, contributing to a higher error in the phase calculation, as evident in Fig. 7. The CGH processing time was reduced by 50% to 65% relative to the original encoding scheme when the new encoding scheme was used.

Fig. 6 (a-e) Interferograms using a flat mirror shifted by a piezo drive. (f) Retrieved phase distribution temporal phase shifting using a flat mirror shifted by a piezo drive.

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Fig. 7 Phase difference (radians) between digital phase shifting using new binary encoded CGHs and mechanical digital phase shifting using piezo-driven mirror

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6. Conclusion

The design of a set of binary encoded computer generated holograms (CGHs) for five-frame temporal phase shifting interferometry using a binary transmissive amplitude SLM is presented. The new set of binary encoded CGHs is suitable for real-time applications in portable devices or legacy hardware with constraints in computational power. To demonstrate a practical application for this, the phase of the generated electric field for a real-time selective launch in MMF is measured by temporal phase shifting using the new binary encoded CGHs. The CGH processing time was reduced by 50% to 65% relative to the original encoding scheme when the new encoding scheme was used. The results obtained from the new binary encoded CGHs are in good agreement with the results obtained by phase shifting by means of a piezo-driven flat mirror.

Acknowledgement

The author would like to thank Dr. Dominic O’Brien, Dr. Frank Payne and Dr. Martin Booth at the University of Oxford for their important advice and feedback.

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Binary encoded computer generated holograms for temporal phase shifting

Abstract

1. Introduction

2. Mathematical derivation of binary encoded CGH for temporal phase shifting

3. Demonstration of application of binary encoded CGH for temporal phase shifting

4. Experimental results

5. Comparison to mechanical temporal phase shifting

6. Conclusion

Acknowledgement

References and links

Cited By

Figures (7)

Equations (29)

Optics Express