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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 25 — Dec. 3, 2012
  • pp: 27429–27441
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Position measurement of non-integer OAM beams with structurally invariant propagation

A. M. Nugrowati, W. G. Stam, and J. P. Woerdman  »View Author Affiliations


Optics Express, Vol. 20, Issue 25, pp. 27429-27441 (2012)
http://dx.doi.org/10.1364/OE.20.027429


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Abstract

We present a design to generate structurally propagation invariant light beams carrying non-integer orbital angular momentum (OAM) using Hermite-Laguerre-Gaussian (HLG) modes. Different from previous techniques, the symmetry axes of our beams are fixed when varying the OAM; this simplifies the calibration technique for beam positional measurement using a quadrant detector. We have also demonstrated analytically and experimentally that both the OAM value and the HLG mode orientation play an important role in the quadrant detector response. The assumption that a quadrant detector is most sensitive at the beam center does not always hold for anisotropic beam profiles, such as HLG beams.

© 2012 OSA

1. Introduction

Light carrying orbital angular momentum (OAM) is characterized by a helical wavefront shape and a doughnut-like intensity profile with a dark center (vortex). In a single round trip about the propagation axis, the phase of an OAM beam increases linearly and gains the value of 2πN, with N an integer value that is equivalent to the OAM content of such a beam. After the first investigation of the astigmatic transformation of Hermite-Gaussian (HG) modes into Laguerre-Gaussian (LG) modes [1

1. E. Abramochkin and V. Volostnikov, “Beam transformation and nontransformed beams,” Opt. Commun. 83, 123–125 (1991). [CrossRef]

], it was theoretically proven that LG laser modes carry a well defined OAM which is equivalent to the azimuthal mode index of the LG modes [2

2. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). [CrossRef] [PubMed]

]. Since then, the generation of LG modes has opened up a broad range of applications, including optical trapping with OAM beam structures [3

3. H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995). [CrossRef]

, 4

4. M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5, 343–348 (2011). [CrossRef]

], quantum communication at higher dimensional entanglement using OAM beams [5

5. S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’tHooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005). [CrossRef] [PubMed]

, 6

6. B. J. Pors, F. Miatto, G. W. ’t Hooft, E. R. Eliel, and J. P. Woerdman, “High-dimensional entanglement with orbital-angular-momentum states of light,” J. Opt. 13, 064008 (2011). [CrossRef]

], OAM beam for high sensitivity Raman spectroscopy in molecule detection [7

7. J. Sato, M. Endo, S. Yamaguchi, K. Nanri, and T. Fujioka, “Simple annular-beam generator with a laser-diode-pumped axially off-set power build-up cavity,” Opt. Commun. 277, 342–348 (2007). [CrossRef]

], stellar detection using OAM beam [8

8. G. C. G. Berkhout and M. Beijersbergen, “Measuring optical vortices in a speckle pattern using a multi-pinhole interferometer,” Opt. Express 18, 13836–13841 (2010). [CrossRef] [PubMed]

, 9

9. F. Tamburini, B. Thidé, G. Molina-Terriza, and G. Anzolin, “Twisting of light around rotating black holes,” Nat. Phys. 7, 195–197 (2011). [CrossRef]

], and nanometer precision metrology by using the effect of OAM on beam shifts [10

10. M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82, 023817 (2010). [CrossRef]

].

Recently, there is a growing interest in addressing non-integer values of OAM that potentially broadens the OAM beams applications. By applying non-integer OAM beams to existing OAM beams applications, we introduce a degree of freedom of optical manipulation. Non-integer OAM beams have found uses in high-dimensional quantum information processing [6

6. B. J. Pors, F. Miatto, G. W. ’t Hooft, E. R. Eliel, and J. P. Woerdman, “High-dimensional entanglement with orbital-angular-momentum states of light,” J. Opt. 13, 064008 (2011). [CrossRef]

] as well as edge-sensitive microscopy [11

11. S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral phase contrast imaging in microscopy,” Opt. Express 13, 689–694 (2005). [CrossRef] [PubMed]

]. In this paper, we present the technique to generate non-integer OAM beams and discuss the difference between our technique and the existing ones. Note that, the non-integer OAM value does not refer to the value of the beam vorticity but to the mean value of the OAM. Subsequently, we treat the position measurement of such a beam that is an inherent part of many applications using OAM beams.

During the first decade after the initial realization of an OAM beam, many different integer OAM beam generation techniques have been introduced. The first demonstration used the so-called ‘π/2-mode converter’, which belongs to a family of astigmatic mode converters that applies the appropriate Gouy phase to create well defined mode indices of LG beams carrying integer OAM [12

12. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993). [CrossRef]

]. This was soon followed by the demonstration of a spiral phase plate (SPP) operating at optical wavelength [13

13. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994). [CrossRef]

] and at milimeter range [14

14. G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,” Opt. Commun. 127, 183–188 (1996). [CrossRef]

] for creating helical-wavefront to directly transform Gaussian beam to OAM beams. At the same time, computer-generated holograms with pitchfork structures were applied using a spatial light modulator (SLM) to convert Gaussian beams into LG beams [3

3. H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995). [CrossRef]

, 15

15. J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated holograms,” J. Mod. Opt. 45, 1231–1237 (1998). [CrossRef]

]. Different from the astigmatic mode converter, both SPP and SLM are not pure mode converters. They convert a fundamental Gaussian mode into a superposition of LG modes that contain the same azimuthal mode index but different radial mode index p. Although the OAM content of such a beam is well defined, the spatial field distribution evolves during propagation. This mode impurity problem holds also when employing q plates [16

16. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006). [CrossRef] [PubMed]

] that convert spin-to-orbital angular momentum in an anisotropic and inhomogeneous media to create helical waves. Mitigating the radial mode impurity to obtain a more robust beam profile during propagation when using SLM and SPP has then been the focus of several studies [17

17. G. Machavariani, N. Davidson, E. Hasman, S. Bilt, A. Ishaaya, and A. A. Friesem, “Efficient conversion of a Gaussian beam to a high purity helical beam,” Opt. Commun 209, 265–271 (2002). [CrossRef]

21

21. T. Ando, Y. Ohtake, N. Matsumoto, T. Inoue, and N. Fukuchi, “Mode purities of Laguerre–Gaussian beams generated via complex-amplitude modulation using phase-only spatial light modulators,” Opt. Lett 34, 34–36 (2009). [CrossRef]

].

In the field of non-integer OAM beam generation, only a handful of studies have been carried on. One of the initial ideas was to use off-axis illumination of an SPP; equivalently, one may use a non-integer 2π phase step SPP [5

5. S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’tHooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005). [CrossRef] [PubMed]

]. These techniques, however, yield non-integer OAM beams with neither nor p mode purity [22

22. M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. 6, 259–268 (2004). [CrossRef]

]. A more structurally stable non-integer OAM beam has been demonstrated recently, by using an SLM when applying a synthesis of a finite number of LG modes with carefully chosen Gouy phases [23

23. J. B. Götte, K. O’Holleran, D. Preece, F. Flossman, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16, 993–1006 (2008). [CrossRef] [PubMed]

]. It was, however, demonstrated only for half-integer OAM values. Another proposition is by exploiting the internal conical diffraction where a circularly polarized beam with a fundamental Gaussian mode is converted into a non-integer OAM beam with a Bessel mode, having only a limited OAM value range of || ≤ 1 [24

24. D. P. O’Dwyer, C. F. Phelan, Y. P. Rakovich, P. R. Eastham, J. G. Lunney, and J. F. Donegan, “Generation of continuously tunable fractional optical orbital angular momentum using internal conical diffraction,” Opt. Express 18, 16480–16485 (2010). [CrossRef]

].

Our paper focuses on two issues. The first concerns with the generation of beams carrying arbitrary non-integer OAM values that is structurally stable during propagation, apart from the overall scaling due to diffraction. This can be achieved by employing the concept of generally astigmatic mode converters, as was initially introduced in Ref. [25

25. A. T. O’Neil and J. Courtial, “Mode transformations in terms of the constituent Hermite-Gaussian or LaguerreGaussian modes and the variable-phase mode converter,” Opt. Commun. 181, 35–45 (2000). [CrossRef]

]. Later on, it was theoretically demonstrated that the output of a general astigmatic transformation is the intermediate beam between HG and LG beams, known as Hermite-Laguerre-Gaussian (HLG) beams [26

26. E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A: Pure Appl. Opt. 6, S157–S161 (2004). [CrossRef]

28

28. E. G. Abramochkin, E. Razueva, and V. G. Volostnikov, “General astigmatic transform of Hermite-Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 27, 2506–2513 (2010). [CrossRef]

]. Since such a HLG mode is an analytic interpolation between a HG mode and a LG mode, it is structurally propagation invariant. Moreover, this HLG beam carries non-integer OAM.

2. Generation of structurally propagation invariant light carrying non-integer OAM

2.1. Experimental set-up

A conventional π/2 astigmatic mode converter [12

12. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993). [CrossRef]

] transforms a pure HG mode into a pure LG mode by passing an incoming HG beam through a pair of identical cylindrical lenses with focal lengths f, separated at a distance d=f2, as illustrated in Fig. 1. A mode matching lens is normally used to tailor the beam waist of the outgoing laser mode into the desired beam waist in between the cylindrical lens. A well defined integer OAM is achieved when the symmetry axes of the HG beam are oriented at an angle α = 45° with respect to the symmetry axes of the cylindrical lenses [12

12. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993). [CrossRef]

]. This can be done using an open laser cavity that is forced to operate at a high order HG mode by insertion of a thin metal wire, oriented at α = 45°.

Fig. 1 Our experimental set-up to generate HLG modes as non-integer OAM beams, equipped with a quadrant detector for measuring the beam positional shifts, discussed in Section 3.

Belonging to the family of astigmatically transformed HG beams, HLG beams can be created by tuning the beam parameter α, i.e. the angle between the symmetry axes of cylindrical lenses and the symmetry axes of the input HG beam [28

28. E. G. Abramochkin, E. Razueva, and V. G. Volostnikov, “General astigmatic transform of Hermite-Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 27, 2506–2513 (2010). [CrossRef]

]. The non-integer OAM value of HLG beam is = (nm)sin2α, with n and m the mode index of high order HG beams [26

26. E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A: Pure Appl. Opt. 6, S157–S161 (2004). [CrossRef]

,27

27. J. Visser and G. Nienhuis, “Orbital angular momentum of general astigmatic modes,” Phys. Rev. A 70, 013809 (2004). [CrossRef]

]. Another way to generate HLG beams is to tailor the required Gouy phase by simultaneously tuning the separation distance of the cylindrical lenses d and the position of the cylindrical lenses pair with respect to the mode matching lens [32

32. S. J. M. Habraken, “Cylindrical lens mode converters and OAM,” Personal Communication (2010).

]. However, for aligning purposes, the approach of varying α is more attractive when tuning the non-integer OAM value.

A general astigmatic mode converter transforms a HG mode of arbitrarily high order to a HLG mode. In essence, a pure mode transformation projects an incoming HG mode into two orthogonal axes of the astigmatic mode converter. The outgoing HLG beam is a superposition of the projected mode with the additional Gouy phase. In Fig. 2, we show the projection of an incoming HG01 mode with varying orientation angle α on a ‘π/2-mode converter’. The Gouy phase φ experienced by the projected mode after traversing the cylindrical lenses is 2α. Note that the symmetry axes of the outgoing HLG beam are always aligned to the projection axes of the cylindrical lenses.

Fig. 2 An incoming HG0,1 mode at varying orientation angle α projected onto the orthogonal symmetry axes of the cylindrical lenses of a ‘π/2-mode converter’. The symmetry axes of the outgoing HLG modes are always aligned to the projection axes. The outgoing HLG modes experience Gouy phase φ = 2α.

2.2. Characterization of non-integer OAM beams

Figure 3 shows the resulting generated HLG beams as a function of varying orientation angle α. The open laser cavity is forced to operate at the first higher order HG mode, i.e HG0,1. The first two rows display the measured intensity profiles of Fig. 3(a) the incoming HG0,1 and Fig. 3(b) the outgoing HLG0,1|[α:0°,90°] beams at the far-field after the collimating lens. Our generated HLG beam profiles match with the calculation shown in Fig. 3(c). In the calculated images, we have used a color map to indicate the phase profile of the generated HLG modes. For outgoing HLG profiles being the analytic interpolation between a HG mode and a LG mode, we observe a more flat wavefront inside the high intensity areas (note the even color tone). Inside the dark intensity areas, the phase value increases non-linearly along the azimuthal direction. The phase singularity of zero-OAM beams at α = N × 90° forms a line (most left and most right images of Fig. 3(c)), whereas for integer OAM beams at α = (2N +1)×45° it forms a vortex (the center image of Fig. 3(c)), with N an integer number. Apart from the overall scaling due to diffraction and slight astigmatism due to imperfect alignment, the generated HLG beams are structurally stable during propagation, as shown in Fig. 4.

Fig. 3 (a) Measured intensity profiles of the impinging HG0,1 mode as a function of the orientation angle α with respect to the symmetry axes of the ‘π/2-mode converter’; the white lines correspond to the wire orientation in the open laser cavity of Fig. 1. (b) Measured far-field intensity profiles after the collimating lens of the outgoing Hermite-Laguerre-Gauss (HLG) modes. (c) Calculated intensity profiles to compare with the measurement results in (b). The color map in the calculated intensity profiles (c) corresponds to the HLG phase profile that gradually increases from 0 to 2π.
Fig. 4 Mode profiles at several Rayleigh distance zR, representing the near- and far-field planes for the outgoing HLG0,1|[α:60°] mode.

To characterize the OAM content of the generated HLG beams, we look at the interference patterns between the outgoing HLG beam and a reference beam that comes out of the laser cavity. A typical interference pattern of an integer OAM beam shows phase dislocation features, i.e. a pitchfork that branches out into number of lines at the dark centre of the beam, which is also the case for the centre image of Fig. 5. In the case of beams with non-integer OAM values, the branching gradually dissolves into separated shifted lines as shown by the measurement result (a) and confirmed by the calculation (b).

Fig. 5 The (a) measured and (b) calculated interference patterns showing the phase singularity of the HLG beams. The black color corresponds to zero intensity and zero phase, whereas the white color corresponds to maximal intensity and phase ϕ = 2π.

3. Quadrant detector response to HLG beam displacement

In this section we deal with the response of a quadrant detector as a beam positional detector of HLG beams. A quadrant detector is a 2×2 array of individual p-n junction photodiodes, separated by a small gap of typically less than 0.05% of the active area, as depicted in the inset of Fig. 1. The photodiodes provide the photocurrents IA, IB, IC and ID which are generated when an optical beam strikes the active area. Its position-current relation can be written as:
IxIΣ=IA+ID(IB+IC)IA+ID+IB+IC=x|U(x,y)|2dydxx|U(x,y)|2dydx|U(x,y)|2dydx=20x0|U(x,y)|2dydx200|U(x,y)|2dydx=0x0|U(x,y)|2dydx00|U(x,y)|2dydx,
(1a)
IyIΣ=IA+IB(IC+ID)IA+IB+IC+ID=0y0|U(x,y)|2dxdy00|U(x,y)|2dydx,
(1b)
with |U(x,y)|2 the intensity of the impinging beam, for shifts along the x– and y–axis of the quadrant detector, respectively. When using our generation technique, the quadrant detector axes coincide with the transverse beam axes. To obtain the nominal beam displacement, the quadrant detector signal Ix,y/IΣ has to be normalized to the slope of this relationship curve, i.e. the calibration constant K.

First, the orientation of the HLG beam profile influences the quadrant detector response. Due to the rectangular geometry of a quadrant detector, it is most natural to align the symmetry axes of the beam with respect to the quadrant detector displacement axes, as in the case of Fig. 3. When these axes are aligned, the quadrant detector calibration constant K for the displacement along the x–axis of HLGn,m|α mode is also valid for the displacement along the y–axis of HLGm,n|α mode.

Second, operating a quadrant detector around the HLG beam center to detect small displacement Δxw will not always give the most sensitive position measurement. This is due to the fact that for some cases, the profile cross section |U(x,y)|2 of the HLG modes along the axis of displacement, has near zero values across one displacement axis. As an example, let us observe HLG modes for α > 45° in Fig. 3(b). The low intensity values at the beam centre across the y-axis is certainly the least sensitive area to measure beam displacement along the x–axis. Interchangeably, the quadrant detector is least sensitive for beam displacement along the y–axis around the centre area of HLG modes for α < 45°. Therefore, it is important to find the region where the quadrant detector can operate with the highest sensitivity.

3.1. Analytical solutions of quadrant detector calibration

Now, we derive the analytical expression for the position-current relationship of a quadrant detector for HLG beams carrying non-integer OAM, in the case that both of the symmetry axes overlap. This expression can be easily extended for an arbitrarily high order HLGn,m|α mode. For didactic purposes, we take the example of a radial mode index p = 0 and an azimuthal mode index = 1 (i.e. HLG0,1|α), and investigate the quadrant response for the beam displacement along the x–axis. By applying the distribution function of HLG0,1|α given in Ref. [26

26. E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A: Pure Appl. Opt. 6, S157–S161 (2004). [CrossRef]

] into Eq. (1a), we can write the x-axis displacement relationship normalized to the beam radius w for HLG0,1|α to be
IxIΣ=22xπwexp[2(xw)2]cos2(α)+erf(2xw).
(2)
Note that due to the symmetry axes, the same expression is found for the beam displacement of HLG1,0|α mode along the y-axis, substituting the index x with y.

We present the 1-D cross section profile in Fig. 6(a) to help visualizing the general intensity distribution of HLG0,1|α beams. The position-current relationship curves in Fig. 6(b) reveal that there are different linear regions with a constant slope (calibration constant K) for different values of α. The linear region shifts to a higher x/w value for α > 45°, coinciding with the peak intensity cross section along the displacement axis, at around x/w = 0.7. For α < 45° the beam cross-section along the x-axis resembles that of a Gaussian profile and the range of linearity is around the beam center.

Fig. 6 (a) The cross section of the HLG0,1|α mode profile along the x-axis. (b) The corresponding response of a quadrant detector for beam displacement along the x–axis. Lines (both solid and dashed) and data points correspond to the analytical solution and experimental data, respectively.

To confirm our analytical expression, we measure the quadrant detector response for HLG0,1|α modes, where we have used a quadrant detector from NewFocus model 2921 with an active area of 10mm × 10mm. Figure 6(b) shows the the match between our data (open circles and dots) and the analytical solution (solid and dashed lines). It is important to realize that there exists two values of α for each HLGn,m|α mode that give the same values but with orthogonally oriented spatial distribution. These orthogonally oriented modes have different calibration constants K, as plotted in Fig. 7.

Fig. 7 The calibration constant K as a function of non-integer OAM is derived analytically for HLG0,1 modes with varying α. The white lines on the top row images illustrate the x-positions at which K is derived for several α values.

To use a quadrant detector for displacement measurement of non-integer OAM beams having anisotropic profile distributions, such as HLG beams, one must pay attention to the linear range of the position-current relations, i.e. at the peak of the intensity cross section along the axis of displacement. Since the orientation of our generated HLG beam are aligned with the symmetry axes of a quadrant detector, we can easily obtain the linear range and the calibration constant K. This calibration procedure is particularly relevant for potential applications using HLG modes as non-integer OAM beams: in beam shifts measurements, high precision metrology, optical manipulation using tweezers or scanning near-field optical microscopy.

4. Conclusion

In this paper, we demonstrate a technique to generate HLG modes as non-integer OAM beams that are structurally propagation invariant and having a fixed symmetry axes for arbitrary non-integer OAM values. The experimentally demonstrated HLG beams agree with the calculation, both for the intensity profile distribution and the phase features measured with interferometric set-up.

Unlike an integer OAM beam, the phase of a HLG mode increases non-linearly along the azimuthal axis. Note that any integer OAM beam can be created from an arbitrarily higher order HLG mode having the appropriate orientation angle α. For example, = 1 can be constructed from HLG0,2|[α:15°], which actually produces a phase distribution that is different from that of a LG0,1 mode. In applications such as OAM beams shifts or optical manipulation using OAM beams, noticeable differences will occur when addressing an integer OAM value by using either LG modes or HLG modes.

We have derived the analytical expression and demonstrate experimentally the response of a quadrant detector towards the generated HLG beams. The obtained calibration constant K of a quadrant detector for HLG beams agrees with Ref. [31

31. N. Hermosa, A. Aiello, and J. P. Woerdman, “Quadrant detector calibration for vortex beams,” Opt. Lett. 36, 409–411 (2011). [CrossRef] [PubMed]

] only at integer , where the beam profile is isotropic or cylindrically symmetric. The assumption that a quadrant detector is most sensitive at the beam center does not always hold for general astigmatic modes, i.e. HLG modes, that has an anisotropic beam profiles.

In conclusion, we have shown that both the values and the HLG mode orientation play a role in the quadrant detector response. Furthermore, the anisotropic nature of HLG beams creates different regions having linear response of a quadrant detector when measuring beam positional shift. The beam positional measurement is most sensitive around the peak of the HLG mode profile. Our result can easily be extended to arbitrarily higher order HLG beams as solutions of light carrying higher order non-integer OAM.

Acknowledgments

This work is supported by the European Union within FET Open-FP7 ICT as part of STREP Program 255914 Phorbitech.

References and links

1.

E. Abramochkin and V. Volostnikov, “Beam transformation and nontransformed beams,” Opt. Commun. 83, 123–125 (1991). [CrossRef]

2.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). [CrossRef] [PubMed]

3.

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995). [CrossRef]

4.

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5, 343–348 (2011). [CrossRef]

5.

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’tHooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005). [CrossRef] [PubMed]

6.

B. J. Pors, F. Miatto, G. W. ’t Hooft, E. R. Eliel, and J. P. Woerdman, “High-dimensional entanglement with orbital-angular-momentum states of light,” J. Opt. 13, 064008 (2011). [CrossRef]

7.

J. Sato, M. Endo, S. Yamaguchi, K. Nanri, and T. Fujioka, “Simple annular-beam generator with a laser-diode-pumped axially off-set power build-up cavity,” Opt. Commun. 277, 342–348 (2007). [CrossRef]

8.

G. C. G. Berkhout and M. Beijersbergen, “Measuring optical vortices in a speckle pattern using a multi-pinhole interferometer,” Opt. Express 18, 13836–13841 (2010). [CrossRef] [PubMed]

9.

F. Tamburini, B. Thidé, G. Molina-Terriza, and G. Anzolin, “Twisting of light around rotating black holes,” Nat. Phys. 7, 195–197 (2011). [CrossRef]

10.

M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82, 023817 (2010). [CrossRef]

11.

S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral phase contrast imaging in microscopy,” Opt. Express 13, 689–694 (2005). [CrossRef] [PubMed]

12.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993). [CrossRef]

13.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994). [CrossRef]

14.

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,” Opt. Commun. 127, 183–188 (1996). [CrossRef]

15.

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated holograms,” J. Mod. Opt. 45, 1231–1237 (1998). [CrossRef]

16.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006). [CrossRef] [PubMed]

17.

G. Machavariani, N. Davidson, E. Hasman, S. Bilt, A. Ishaaya, and A. A. Friesem, “Efficient conversion of a Gaussian beam to a high purity helical beam,” Opt. Commun 209, 265–271 (2002). [CrossRef]

18.

S. A. Kennedy, M. J. Szabo, H. Teslow, J. Z. Porterfield, and E. R. I. Abraham, “Creation of Laguerre-Gaussian laser modes using diffractive optics,” Phys. Rev. A 66, 043801 (2002). [CrossRef]

19.

S.-C. Chu and K. Otsuka, “Doughnut-like beam generation of Laguerre-Gaussian mode with extremely high mode purity,” Opt. Commun. 281, 1647–1653 (2008). [CrossRef]

20.

N. Matsumoto, T. Ando, T. Inoue, Y. Ohtake, N. Fukuchi, and T. Hara, “Generation of high-quality higher-order Laguerre-Gaussian beams using liquid-crystal-on-silicon spatial light modulators,” J. Opt. Soc. Am. A 25, 1642–1651 (2008). [CrossRef]

21.

T. Ando, Y. Ohtake, N. Matsumoto, T. Inoue, and N. Fukuchi, “Mode purities of Laguerre–Gaussian beams generated via complex-amplitude modulation using phase-only spatial light modulators,” Opt. Lett 34, 34–36 (2009). [CrossRef]

22.

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. 6, 259–268 (2004). [CrossRef]

23.

J. B. Götte, K. O’Holleran, D. Preece, F. Flossman, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16, 993–1006 (2008). [CrossRef] [PubMed]

24.

D. P. O’Dwyer, C. F. Phelan, Y. P. Rakovich, P. R. Eastham, J. G. Lunney, and J. F. Donegan, “Generation of continuously tunable fractional optical orbital angular momentum using internal conical diffraction,” Opt. Express 18, 16480–16485 (2010). [CrossRef]

25.

A. T. O’Neil and J. Courtial, “Mode transformations in terms of the constituent Hermite-Gaussian or LaguerreGaussian modes and the variable-phase mode converter,” Opt. Commun. 181, 35–45 (2000). [CrossRef]

26.

E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A: Pure Appl. Opt. 6, S157–S161 (2004). [CrossRef]

27.

J. Visser and G. Nienhuis, “Orbital angular momentum of general astigmatic modes,” Phys. Rev. A 70, 013809 (2004). [CrossRef]

28.

E. G. Abramochkin, E. Razueva, and V. G. Volostnikov, “General astigmatic transform of Hermite-Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 27, 2506–2513 (2010). [CrossRef]

29.

E. J. Lee, Y. Park, C. S. Kim, and T. Kouh, “Detection sensitivity of the optical beam deflection method characterized with the optical spot size on the detector,” Curr. Appl. Phys. 10, 834–837 (2010). [CrossRef]

30.

Y. Panduputra, T. W. Ng, A. Neild, and M. Robinson, “Intensity influence on Gaussian beam laser based measurements using quadrant photodiodes,” Appl. Opt. 49, 3669–3675 (2010). [CrossRef] [PubMed]

31.

N. Hermosa, A. Aiello, and J. P. Woerdman, “Quadrant detector calibration for vortex beams,” Opt. Lett. 36, 409–411 (2011). [CrossRef] [PubMed]

32.

S. J. M. Habraken, “Cylindrical lens mode converters and OAM,” Personal Communication (2010).

OCIS Codes
(040.5160) Detectors : Photodetectors
(120.4640) Instrumentation, measurement, and metrology : Optical instruments
(260.6042) Physical optics : Singular optics

ToC Category:
Physical Optics

History
Original Manuscript: September 19, 2012
Revised Manuscript: November 1, 2012
Manuscript Accepted: November 1, 2012
Published: November 26, 2012

Citation
A. M. Nugrowati, W. G. Stam, and J. P. Woerdman, "Position measurement of non-integer OAM beams with structurally invariant propagation," Opt. Express 20, 27429-27441 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-25-27429


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References

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  12. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun.96, 123–132 (1993). [CrossRef]
  13. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun.112, 321–327 (1994). [CrossRef]
  14. G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,” Opt. Commun.127, 183–188 (1996). [CrossRef]
  15. J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated holograms,” J. Mod. Opt.45, 1231–1237 (1998). [CrossRef]
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  18. S. A. Kennedy, M. J. Szabo, H. Teslow, J. Z. Porterfield, and E. R. I. Abraham, “Creation of Laguerre-Gaussian laser modes using diffractive optics,” Phys. Rev. A66, 043801 (2002). [CrossRef]
  19. S.-C. Chu and K. Otsuka, “Doughnut-like beam generation of Laguerre-Gaussian mode with extremely high mode purity,” Opt. Commun.281, 1647–1653 (2008). [CrossRef]
  20. N. Matsumoto, T. Ando, T. Inoue, Y. Ohtake, N. Fukuchi, and T. Hara, “Generation of high-quality higher-order Laguerre-Gaussian beams using liquid-crystal-on-silicon spatial light modulators,” J. Opt. Soc. Am. A25, 1642–1651 (2008). [CrossRef]
  21. T. Ando, Y. Ohtake, N. Matsumoto, T. Inoue, and N. Fukuchi, “Mode purities of Laguerre–Gaussian beams generated via complex-amplitude modulation using phase-only spatial light modulators,” Opt. Lett34, 34–36 (2009). [CrossRef]
  22. M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt.6, 259–268 (2004). [CrossRef]
  23. J. B. Götte, K. O’Holleran, D. Preece, F. Flossman, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express16, 993–1006 (2008). [CrossRef] [PubMed]
  24. D. P. O’Dwyer, C. F. Phelan, Y. P. Rakovich, P. R. Eastham, J. G. Lunney, and J. F. Donegan, “Generation of continuously tunable fractional optical orbital angular momentum using internal conical diffraction,” Opt. Express18, 16480–16485 (2010). [CrossRef]
  25. A. T. O’Neil and J. Courtial, “Mode transformations in terms of the constituent Hermite-Gaussian or LaguerreGaussian modes and the variable-phase mode converter,” Opt. Commun.181, 35–45 (2000). [CrossRef]
  26. E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A: Pure Appl. Opt.6, S157–S161 (2004). [CrossRef]
  27. J. Visser and G. Nienhuis, “Orbital angular momentum of general astigmatic modes,” Phys. Rev. A70, 013809 (2004). [CrossRef]
  28. E. G. Abramochkin, E. Razueva, and V. G. Volostnikov, “General astigmatic transform of Hermite-Laguerre-Gaussian beams,” J. Opt. Soc. Am. A27, 2506–2513 (2010). [CrossRef]
  29. E. J. Lee, Y. Park, C. S. Kim, and T. Kouh, “Detection sensitivity of the optical beam deflection method characterized with the optical spot size on the detector,” Curr. Appl. Phys.10, 834–837 (2010). [CrossRef]
  30. Y. Panduputra, T. W. Ng, A. Neild, and M. Robinson, “Intensity influence on Gaussian beam laser based measurements using quadrant photodiodes,” Appl. Opt.49, 3669–3675 (2010). [CrossRef] [PubMed]
  31. N. Hermosa, A. Aiello, and J. P. Woerdman, “Quadrant detector calibration for vortex beams,” Opt. Lett.36, 409–411 (2011). [CrossRef] [PubMed]
  32. S. J. M. Habraken, “Cylindrical lens mode converters and OAM,” Personal Communication (2010).

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