Abstract

We demonstrate a simple approach, using digital holograms, to perform a complete azimuthal decomposition of an optical field. Importantly, we use a set of basis functions that are not scale dependent so that unlike other methods, no knowledge of the initial field is required for the decomposition. We illustrate the power of the method by decomposing two examples: superpositions of Bessel beams and Hermite-Gaussian beams (off-axis vortex). From the measured decomposition we show reconstruction of the amplitude, phase and orbital angular momentum density of the field with a high degree of accuracy.

© 2012 OSA

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References

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2012

R. Rop, A. Dudley, C. Lopez-Mariscal, and A. Forbes, “Measuring the rotation rates of superpositions of higher-order Bessel beams,” J. Mod. Opt.59(3), 259–267 (2012).
[CrossRef]

R. Rop, I. A. Litvin, and A. Forbes, “Generation and propagation dynamics of obstructed and unobstructed rotating orbital angular momentum-carrying Helicon beams,” J. Opt.14(3), 035702 (2012).
[CrossRef]

M. Paurisse, L. Lévèque, M. Hanna, F. Druon, and P. Georges, “Complete measurement of fiber modal content by wavefront analysis,” Opt. Express20(4), 4074–4084 (2012).
[CrossRef] [PubMed]

A. Dudley, I. A. Litvin, and A. Forbes, “Quantitative measurement of the orbital angular momentum density of light,” Appl. Opt.51(7), 823–833 (2012).
[CrossRef] [PubMed]

2011

2010

2009

2008

2004

2000

1999

1995

1989

E. Tervonen, J. Turunen, and A. Friberg, “Transverse laser mode structure determination from spatial coherence measurements: experimental results,” Appl. Phys. B49(5), 409–414 (1989).
[CrossRef]

Baek, S.

Borchardt, J.

Borghi, R.

Brüning, R.

Codemard, C.

Cutolo, A.

Druon, F.

Dudley, A.

Duparré, M.

Flamm, D.

Forbes, A.

Friberg, A.

E. Tervonen, J. Turunen, and A. Friberg, “Transverse laser mode structure determination from spatial coherence measurements: experimental results,” Appl. Phys. B49(5), 409–414 (1989).
[CrossRef]

Georges, P.

Ghalmi, S.

Gori, F.

Guattari, G.

Hanna, M.

Isernia, T.

Izzo, I.

Jeong, Y.

Kaiser, T.

Khilo, N.

Kirk, A. G.

Lévèque, L.

Li, G.

Litvin, I. A.

Liu, X.

Lopez-Mariscal, C.

R. Rop, A. Dudley, C. Lopez-Mariscal, and A. Forbes, “Measuring the rotation rates of superpositions of higher-order Bessel beams,” J. Mod. Opt.59(3), 259–267 (2012).
[CrossRef]

Nicholson, J. W.

Nilsson, J.

Padgett, M. J.

Paurisse, M.

Philippov, V.

Pierri, R.

Ramachandran, S.

Rop, R.

R. Rop, I. A. Litvin, and A. Forbes, “Generation and propagation dynamics of obstructed and unobstructed rotating orbital angular momentum-carrying Helicon beams,” J. Opt.14(3), 035702 (2012).
[CrossRef]

R. Rop, A. Dudley, C. Lopez-Mariscal, and A. Forbes, “Measuring the rotation rates of superpositions of higher-order Bessel beams,” J. Mod. Opt.59(3), 259–267 (2012).
[CrossRef]

Santarsiero, M.

Schmidt, O. A.

Schröter, S.

Schulze, C.

Soh, D. B. S.

Tervonen, E.

E. Tervonen, J. Turunen, and A. Friberg, “Transverse laser mode structure determination from spatial coherence measurements: experimental results,” Appl. Phys. B49(5), 409–414 (1989).
[CrossRef]

Turunen, J.

E. Tervonen, J. Turunen, and A. Friberg, “Transverse laser mode structure determination from spatial coherence measurements: experimental results,” Appl. Phys. B49(5), 409–414 (1989).
[CrossRef]

Vasilyeu, R.

Wei, H.

Xue, X.

Yablon, A. D.

Yao, A. M.

Zeni, L.

Adv. Opt. Photon.

Appl. Opt.

Appl. Phys. B

E. Tervonen, J. Turunen, and A. Friberg, “Transverse laser mode structure determination from spatial coherence measurements: experimental results,” Appl. Phys. B49(5), 409–414 (1989).
[CrossRef]

J. Mod. Opt.

R. Rop, A. Dudley, C. Lopez-Mariscal, and A. Forbes, “Measuring the rotation rates of superpositions of higher-order Bessel beams,” J. Mod. Opt.59(3), 259–267 (2012).
[CrossRef]

J. Opt.

R. Rop, I. A. Litvin, and A. Forbes, “Generation and propagation dynamics of obstructed and unobstructed rotating orbital angular momentum-carrying Helicon beams,” J. Opt.14(3), 035702 (2012).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Express

Opt. Lett.

Other

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill Publishing Company, 1968).

Supplementary Material (2)

» Media 1: MOV (1255 KB)     
» Media 2: MOV (250 KB)     

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Figures (5)

Fig. 1
Fig. 1

(a) A schematic of the experimental setup for performing the modal decomposition. L: Lens (f1 = 15 mm; f2 = 75 mm; f3 = 200 mm and f4 = 200 mm); M: Mirror; SLM: Spatial Light Modulator; O: Objective; CCD: CCD Camera. The objective, O2, was placed at the focus (or Fourier plane) of lens, L4. Lens L3 and L4 perform a Fourier transform of SLM1 and SLM2, respectively, in a 2-f system. Objectives O1 and O2 are telescopes which image and magnify both the phase and amplitude of the fields at planes P1 and P3 to planes P2 and P4, respectively. (b) The Gaussian beam used to illuminate SLM1. (c) The digital hologram used to generate the optical field of interest (d) and the digital hologram (e) used to extract the weightings of the modes from the inner product (f). The digital holograms for generating (g) and decomposing (i) the field, to extract the intermodal phase. The intensity profile of the field at the plane of SLM2 (h) and CCD (j). Each hologram has a checkerboard pattern, shown as an inset to (c).

Fig. 2
Fig. 2

(a) If the phase is shifted in one of the modes in the initial field (top row), then the measured petal structure of the superposition field (middle row) is seen to rotate, as predicted by theory (bottom row); (b) comparison of the measured phase shift to the actual phase shift; (c) interference of the modes with a reference wave results in changing intensity at the origin of the Fourier plane, which can be used to infer the phase shift per mode; (d) comparison of the measured phase shift to the actual phase shift (see Media 1 and Media 2).

Fig. 3
Fig. 3

A comparison of the theoretical (top row) and the experimentally reconstructed (bottom row) images of the (a, d) intensity, (b, e) phase, and (c, f) OAM density of the light.

Fig. 4
Fig. 4

A comparison of the experimentally recorded intensity (a) and the reconstructed intensity (b), for the off-axis vortex case, created from a superposition of Hermite-Gaussian modes.

Fig. 5
Fig. 5

Graph of the measured on-axis intensity of the inner product as a function of a lateral displacement of the hologram on SLM2 for a displacement range of (a) −20 to 20 pixels (−160 µm to 160 µm) and (b) −200 to 200 pixels (−1600 µm to 1600 µm).

Equations (13)

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u(r,ϕ)= p,l c p,l R p,l (r) exp(ilϕ),
u(r,ϕ)= l [ p c p,l R p,l (r) ] exp(ilϕ)= l c l (r) exp(ilϕ),
c l (r)= p c p,l R p,l (r) = a l (r)exp[ iΔ θ l (r) ].
c l (r)= 1 2π 0 2π u(r,ϕ)exp(ilϕ)dϕ .
U l (0)= exp(i2kf) iλf 0 0 2π t l (r,ϕ)u(r,ϕ)rdrdϕ .
t l (r,ϕ)={ exp(ilϕ) RΔR/2<r<R+ΔR/2 0 otherwise .
U ˜ l (R,0)= exp(i2kf) iλf 0 0 2π t l (r,ϕ)u(r,ϕ)rdrdϕ kexp(i2kf) if c l (R)RΔR.
c l (R)= if RΔRkexp(i2kf) U ˜ l (R,0).
a l (R)=| c l (R) |= f RΔRk I l (R,0) .
I ˜ l (Δ θ l )= | c l (R)+g | 2 = a l 2 (R)+ | g | 2 +2 a l (R)| g |cos[Δ θ l (R)α].
S= ε 0 ω c 2 4 [ i( u u * u * u )+2k | u | 2 z ],
L z = 1 c 2 [ r×S ] z .
u(r,ϕ)= A m J m (kr)exp(imϕ)+ A n J n (kr)exp(inϕ)exp(iΔθ).

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