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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 23 — Nov. 7, 2011
  • pp: 23085–23096
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Binary encoded computer generated holograms for temporal phase shifting

Angela Amphawan  »View Author Affiliations


Optics Express, Vol. 19, Issue 23, pp. 23085-23096 (2011)
http://dx.doi.org/10.1364/OE.19.023085


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Abstract

The trend towards real-time optical applications predicates the need for real-time interferometry. For real-time interferometric applications, rapid processing of computer generated holograms is crucial as the intractability of rapid phase changes may compromise the input to the system. This paper introduces the design of a set of binary encoded computer generated holograms (CGHs) for real-time five-frame temporal phase shifting interferometry using a binary amplitude spatial light modulator. It is suitable for portable devices with constraints in computational power. The new set of binary encoded CGHs is used for measuring the phase of the generated electric field for a real-time selective launch in multimode fiber. The processing time for the new set of CGHs was reduced by up to 65% relative to the original encoding scheme. The results obtained from the new interferometric technique are in good agreement with the results obtained by phase shifting by means of a piezo-driven flat mirror.

© 2011 OSA

1. Introduction

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

14. T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley-VCH, 2005).

16

16. Z. Malacara and M. Servín, Interferogram Analysis For Optical Testing, Second Edition (Optical Science and Engineering) (CRC Press, 2005).

]. Using interferograms recorded from the distinct phase shifts, it is possible to determine the phase of the wavefront at any measured plane [14

14. T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley-VCH, 2005).

16

16. Z. Malacara and M. Servín, Interferogram Analysis For Optical Testing, Second Edition (Optical Science and Engineering) (CRC Press, 2005).

]. 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

17. Y. Bitou, “Digital phase-shifting interferometer with an electrically addressed liquid-crystal spatial light modulator,” Opt. Lett. 28(17), 1576–1578 (2003). [CrossRef] [PubMed]

19

19. C. Falldorf, M. Agour, C. V. Kopylow, and R. B. Bergmann, “Phase retrieval by means of a spatial light modulator in the Fourier domain of an imaging system,” Appl. Opt. 49(10), 1826–1830 (2010). [CrossRef] [PubMed]

]. 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

17. Y. Bitou, “Digital phase-shifting interferometer with an electrically addressed liquid-crystal spatial light modulator,” Opt. Lett. 28(17), 1576–1578 (2003). [CrossRef] [PubMed]

], phase [18

18. T. Meeser, C. v. Kopylow, and C. Falldorf, “Advanced Digital Lensless Fourier Holography by means of a Spatial Light Modulator,” in 3DTV-Conference: The True Vision - Capture, Transmission and Display of 3D Video (3DTV-CON),2010 2010), 1–4.

] and complex [19

19. C. Falldorf, M. Agour, C. V. Kopylow, and R. B. Bergmann, “Phase retrieval by means of a spatial light modulator in the Fourier domain of an imaging system,” Appl. Opt. 49(10), 1826–1830 (2010). [CrossRef] [PubMed]

] 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

20. I. W. Jung, Spatial Light Modulators and Applications Spatial Light Modulators for Applications in Coherent Communication, Adaptive Optics and Maskless Lithography (VDM Verlag,2009).

], 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

21. M. A. A. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microsc. 197(3), 219–223 (2000). [CrossRef] [PubMed]

]. 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

15. D. W. Robinson, Interferogram Analysis, Digital Fringe Pattern Measurement Techniques (Taylor & Francis, 1993).

-16

16. Z. Malacara and M. Servín, Interferogram Analysis For Optical Testing, Second Edition (Optical Science and Engineering) (CRC Press, 2005).

]:
I1=Io[1+γcos(ϕ2p)]
(1)
I2=Io[1+γcos(ϕp)]
(2)
I3=Io[1+γcosϕ]
(3)
I4=Io[1+γcos(ϕ+p)]
(4)
I5=Io[1+γcos(ϕ+2p)]
(5)
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

15. D. W. Robinson, Interferogram Analysis, Digital Fringe Pattern Measurement Techniques (Taylor & Francis, 1993).

-16

16. Z. Malacara and M. Servín, Interferogram Analysis For Optical Testing, Second Edition (Optical Science and Engineering) (CRC Press, 2005).

]:
ϕ=tan1{2(I2I4)/[2I3I5I1]}
(6)
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.

The simplified phase mappings of the required phase shifts onto the binary CGH based on the complex wavefront generation technique in [21

21. M. A. A. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microsc. 197(3), 219–223 (2000). [CrossRef] [PubMed]

] will be derived as follows. Let the required phase shift relative to the fixed reference beam,

σ=π.
(7)

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)exp[j(ξ(x,y)+τxx+τyy+σ)]
(8)
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)},
(9)
then

u=a(x,y)cos[ξ(x,y)+τxx+τyy+σ]
(10)
v=a(x,y)sin[ξ(x,y)+τxx+τyy+σ]
(11)
|a(x,y)|=u2+v2
(12)

The binarized output, g(x, y) is 1 when the total phase is less than α,
ξ+τxx+τyy+σα
(13)
where α is a constant. Taking the sine of both sides of Eq. (13),
sin(ξ+τxx+τyy+σ)sinα
(14)
givenπ/2ξ+τxx+τyy+σπ/2
(15)
0απ/2
(16)
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)=sinα
(17)

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)=sinα.

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

(v±12)2+u214,u0
(18)

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

(u±12)2+v214,v0
(19)
(u±12)2+v214,v0
(20)
(v±12)2+u214,u0
(21)

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

g^(x1,y1)=g(ϕ)|(x1,y1)=ao2+n=1ancos(nϕ)
(25)

This gives the binarized CGH function:

g^(x1,y1)=ao2+4πn=1ancos{n[ξ(x,1y1)+τxx+1τyy1]}
(26)

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:
G(x2,y2)=Mo(x2,y2)+n=1[Mn(x2+nτx,y2+nτy)+Mn*(nτxx2,nτyy2)]
(27)
where x2 and y2 are spatial coordinates in the Fourier plane of L1, * is the complex conjugate and Mn (x1, y1) is the n-th diffraction order.

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)
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
. 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:
MSE=1pqt=1pu=1q[θ1(t,u)θ2(t,u)]2
(28)
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.

The relative power coupled into a fiber mode of a weakly guiding infinite parabolic MMF is given by the power coupling coefficient,
ηlm=|AcoreEinc(x,y)et*(x,y)dxdy|2/Acore|Einc(x,y)|2dxdyAcore|et(x,y)|2dxdy
(29)
where et is the transverse field of a fiber mode, Einc is the incident electric field of the offset beam and Acore 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
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.
. 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
Fig. 7 Phase difference (radians) between digital phase shifting using new binary encoded CGHs and mechanical digital phase shifting using piezo-driven mirror
. 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.

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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OCIS Codes
(060.0060) Fiber optics and optical communications : Fiber optics and optical communications
(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(120.5050) Instrumentation, measurement, and metrology : Phase measurement
(280.4788) Remote sensing and sensors : Optical sensing and sensors

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: August 11, 2011
Revised Manuscript: September 23, 2011
Manuscript Accepted: October 12, 2011
Published: October 31, 2011

Citation
Angela Amphawan, "Binary encoded computer generated holograms for temporal phase shifting," Opt. Express 19, 23085-23096 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-23-23085


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References

  1. K. Peters, “Polymer optical fiber sensors—a review,” Smart Mater. Struct. 20(1), 1–17 (2011). [CrossRef]
  2. H. Su, M. Zervas, C. Furlong, and G. S. Fischer, “A Miniature MRI-Compatible Fiber-optic Force Sensor Utilizing Fabry-Perot Interferometer,” MEMS Nanotech. 4, 131–136 (2011). [CrossRef]
  3. V. Cortez-Retamozo, F. K. Swirski, P. Waterman, H. Yuan, J. L. Figueiredo, A. P. Newton, R. Upadhyay, C. Vinegoni, R. Kohler, J. Blois, A. Smith, M. Nahrendorf, L. Josephson, R. Weissleder, and M. J. Pittet, “Real-time assessment of inflammation and treatment response in a mouse model of allergic airway inflammation,” J. Clin. Invest. 118(12), 4058–4066 (2008). [CrossRef] [PubMed]
  4. T. Sibillano, A. Ancona, V. Berardi, and P. M. Lugarà, “A Real-Time Spectroscopic Sensor for Monitoring Laser Welding Processes,” Sensors (Basel Switzerland) 9(5), 3376–3385 (2009). [CrossRef]
  5. N. Kaneda, Q. Yang, X. Liu, S. Chandrasekhar, W. Shieh, and Y.-K. Chen, “Real-Time 2.5 GS/s Coherent Optical Receiver for 53.3-Gb/s Sub-Banded OFDM,” J. Lightwave Technol. 28(4), 494–501 (2010). [CrossRef]
  6. E. M. Ip and J. M. Kahn, “Fiber Impairment Compensation Using Coherent Detection and Digital Signal Processing,” J. Lightwave Technol. 28(4), 502–519 (2010). [CrossRef]
  7. B. Spinnler, “Equalizer Design and Complexity for Digital Coherent Receivers,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1180–1192 (2010). [CrossRef]
  8. A. Leven, N. Kaneda, and S. Corteselli, “Real-Time Implementation of Digital Signal Processing for Coherent Optical Digital Communication Systems,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1227–1234 (2010). [CrossRef]
  9. R. S. Maldonado, J. A. Izatt, N. Sarin, D. K. Wallace, S. Freedman, C. M. Cotten, and C. A. Toth, “Optimizing hand-held spectral domain optical coherence tomography imaging for neonates, infants, and children,” Invest. Ophthalmol. Vis. Sci. 51(5), 2678–2685 (2010). [CrossRef] [PubMed]
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