Fraunhofer diffraction of a partially blocked spiral phase plate

Don M. Cottrell; Jeffrey A. Davis; Travis J. Hernandez

doi:10.1364/OE.19.012873

1. Introduction

The Fraunhofer diffraction pattern from optical elements containing a spiral phase pattern creates an optical vortex pattern [1,2] consisting of a bright circular ring surrounding a dark region. This optical vortex represents an approximation to a Laguerre Gaussian (LG) optical beam [3]. Optical vortices can be generated using a computer-generated hologram (CGH) [4] or a spiral phase plate (SPP) [3,5] and have been used for such applications as optical trapping and manipulation [6,7] and image processing [5,8].

In recent work, Davis et al. [9] reported a strange effect when they encoded a partially blocked SPP onto the input plane. The Fraunhofer diffraction pattern for this blocked SPP surprisingly produced a partial vortex output pattern that is rotated by 90 degrees compared with the input. The energy is sent into a different quadrant of the output plane from the input plane. The rotation direction depends on whether the phase for the SPP increases in the clockwise or counterclockwise direction. This result is very convenient for determining the sign of the charge for a vortex beam. Although computer simulations agreed with experimental results, there was no theoretical explanation given for this effect.

In this work, we present an explanation of this effect based on careful examination of classical diffraction theory and show new experimental results.

2. Fraunhofer diffraction from a cylindrically symmetric input pattern

Next, we examine the Fraunhofer diffraction for a vortex-producing spiral phase pattern $V_{m} (ρ, ϕ)$ that is written as

V_{m} (ρ, ϕ) = e^{i m ϕ} .

Here the variables ρ and ϕ are polar coordinates in the input plane. The parameter m is the topological charge of the vortex and governs the total phase shift as the angle ϕ changes from 0 to 2π radians in the clockwise direction. For the case where $m = 4$ , the phase shift increases from 0 to 8π radians. The phase shift can increase in either the clockwise or counter-clockwise direction (positive or negative values of m).

The Fraunhofer diffraction pattern produced by this phase pattern in the focal plane of a lens is given by [10,11]

E (r, Φ) = C \int_{0}^{R} ρ d ρ \int_{0}^{2 π} d ϕ e^{i m ϕ} e^{- i (k r ρ / f) \cos (ϕ - Φ)} .

The variables r and Φ are polar coordinates in the output plane, $k = 2 π / λ$ and f is the focal length of the lens.

In order to obtain the integral equation for the Bessel function, we redefine $ϕ = x + Φ - π / 2$ , and the integral in (2) can be rewritten as

E (r, Φ) = e^{i m (Φ - π / 2)} \int_{0}^{R} ρ d ρ \int_{0}^{2 π} d x e^{i m x} e^{- i (k r ρ / f) \sin x} .

Next we incorporate the definition of the Bessel function [12] as

J_{m} (z) = \frac{1}{2 π} \int_{0}^{2 π} e^{i m τ} e^{- i z \sin τ} d τ .

As a result, the output electric field can be rewritten for positive charge values as

E (r, Φ) = e^{i m (Φ - π / 2)} C \int_{0}^{R} J_{m} (k r ρ / f) ρ d ρ

We interpret this result to mean that the coordinate system for the output plane is rotated by π/2 relative to the input plane. This point will be discussed in greater detail below.

For negative values of the charge, we use the relation [12] that

J_{- m} (x) = {(- 1)}^{m} J_{m} (x) = e^{i m π} J_{m} (x) .

Consequently, Eq. (5) is modified for negative charge values as

E (r, Φ) = e^{i m (Φ + π / 2)} C \int_{0}^{R} J_{m} (k r ρ / f) ρ d ρ .

For the case of negative charge values, the output plane is now rotated by –π/2 in the opposite direction relative to the input plane.

Equations (5) and (7) are the main results of this paper for positive and negative values of the charge respectively and clearly show a rotation of $\pm π / 2$ between the output coordinate system and the input coordinate system depending on the sense of the SPP.

In searches of the Fraunhofer diffraction theory literature [13,14], we find a different interpretation of this result. The usual approach is to factor the leading term and to rewrite it as $e^{i m (Φ \pm π / 2)} = \pm i^{m} e^{i m (Φ)}$ . While this is correct, it obscures the fact that the output coordinate system is rotated relative to the input coordinate system. One verification of this effect is in computer simulations that show a rotation in the phase of the output electric field for the Fraunhofer diffraction pattern from the spiral phase plate (please see Figs. 6 and 9 in Ref [14] that clearly show this π/2 rotation in the phase of the Fraunhofer diffraction pattern). However the effect is not seen in the intensity patterns.

However in our case, this circular symmetry can be removed by the aperture and this can explain our earlier observations [9]. In order to model a partially blocked SPP, we multiply the SPP with an angular rectangle function having a limited angular range $ϕ_{0}$ as $Rect {ϕ / ϕ_{0}}$ . We then form an angular Fourier series [14–16] from this blocked aperture as

V_{m} (ρ, ϕ) Rect {ϕ / ϕ_{0}} = e^{i m ϕ} Rect {ϕ / ϕ_{0}} = \sum_{j = - \infty}^{\infty} a_{j} e^{i (m + j) ϕ} .

Here, the coefficients $a_{j}$ depend on the angular width of the $Rect$ function. Now replacing Eq. (1) with Eq. (8) in Eq. (2), the output in Eqs. (5),7) can be rewritten as

E (r, Φ) = C \sum_{j = - \infty}^{\infty} a_{j} e^{i (m + j) (Φ \mp π / 2)} \int_{0}^{R} J_{m + j} (k r ρ / f) ρ d ρ .

Consequently each term in the Fourier series is rotated by ± 90 degrees depending on the sign of the charge. These results show that each term in the Fourier series is rotated by ± 90 degrees compared with the input angular rectangle function forming a rotated partial vortex pattern. Both computer simulations and experiments show that these terms in the Fourier series remain in phase through the Fourier transform process.

With this modification of the classical diffraction theory from a circular aperture, we show that the Fraunhofer diffraction pattern will be rotated by π/2 compared with the input pattern.

3. Experiments

In our experiments, we encode the partially blocked SPPs onto a parallel-aligned LCD manufactured by Seiko Epson with 640x480 pixels and pixel spacing of 42 microns. The phase shift for each pixel exceeds 2π radians as a function of gray level at the argon laser wavelength of 514.5 nm. The Fourier transforms of the patterns are formed in the focal plane of a 38 cm focal length lens and recorded with a CCD camera.

Figure 1(a) shows a two-dimensional counterclockwise SPP where $m = - 30$ and the phase increases in the counterclockwise direction. Figure 1(d) shows the image at the focal plane of the lens and shows the well-known dark center and the surrounding bright circle. The diameter of the bright circle increases as m increases and as the focal length of the lens increases.

Fig. 1 (a) SPP with CCW ( $m = - 30$ ) angular phase, (b) partially blocked SPP with CCW ( $m = - 30$ ) angular phase, (c) partially blocked SPP with CW ( $m = + 30$ ) angular phase, Figs. 1(d-f) show the output diffraction patterns formed from the patterns in Figs. 1(a-c).

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Figure 1(b) shows the partially blocked counterclockwise SPP, again with $m = - 30$ , and where only the upper half is transmitted. As reported earlier [9], the output in Fig. 1(e) shows a half vortex pattern that is rotated counterclockwise by 90 degrees.

The energy in the output plane is always rotated by 90 degrees. However the direction depends on the clockwise or counterclockwise sense of the phase pattern. Figure 1(c) shows a partially blocked clockwise SPP, with $m = + 30$ , and again the lower half is blocked. The output shown in Fig. 1(f) is now rotated clockwise by 90 degrees.

These experiments show the value of these approach for determining the sign of the vortex charge.

We tried several additional experiments. Figure 2 shows results when the angular extent of the SPP is reduced from 180 deg. in Fig. 1(b) to 135 deg. in Fig. 2(a), to 90 deg. in Fig. 2(b), and to 45 deg. in Fig. 2(c) again for a counterclockwise SPP where $m = - 30$ . In all cases, the output pattern is rotated counterclockwise by 90 degrees. However the edges of the output pattern become less distinct as the angular extent decreases as shown in Figs. 2(d-f). We attribute this to the fact that the complete Fourier series in Eq. (9) cannot be entirely encoded resulting in an edge-smoothed partial vortex pattern. The partial vortex ring becomes less clear as the angular extent of the blocked SPP decreases. Computer simulations agree with experimental results and are not shown.

Fig. 2 Partially blocked SPP with CCW ( $m = - 30$ ) angular phase with transmissive regions having (a) 135 degrees, (b) 90 degrees, (c) 45 degrees. Figures 2(d-f) show the output diffraction patterns formed from the patterns in Figs. 2(a-c).

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Next, we show some interesting extensions of this effect. Figure 3(a) shows a multiplexed pattern where the upper quarter segment is encoded with the sum of a clockwise and counterclockwise SPP with $m = \pm 30$ . The output in Fig. 3(d) shows two quarter vortex patterns produced in both the left and right halves of the output plane. Figure 3(b) shows a multiplexed SPP where the upper quarter segment is encoded with the sum of a counterclockwise SPP with $m = - 50$ and a clockwise SPP with $m = + 30$ . The output in Fig. 3(e) shows the two quarter vortex patterns. However the radius of the former is larger than the latter. Finally we show a pattern with two separate quarter SPP patterns. The left quarter has a clockwise SPP with $m = + 30$ while the right quarter has a counterclockwise SPP with $m = - 30$ . The output plane now shows the interference pattern created by the superposition of the two quarter vortex patterns. All experimental results agree with computer simulations and with the theory.

Fig. 3 Multiplexed blocked SPP patterns with (a) combination of CCW and CW SPP patterns with $m = \pm 30$ , (b) combination of CCW SPP pattern with $m = - 50$ and CW SPP pattern with $m = + 30$ , (c) combination with the left portion having a CW SPP pattern with $m = + 30$ and the right portion having a CCW SPP pattern with $m = - 30$ . Figures 3(d-f) show the output diffraction patterns formed from the patterns in Figs. 3(a-c).

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4. Conclusions

In conclusion, we show a theoretical explanation for the output diffraction pattern formed by partially blocked spiral phase plates based on a careful reexamination of Fraunhofer diffraction theory when the rotational symmetry of the input plane is removed. These results show that the output diffraction pattern is rotated by ± 90 degrees depending on the sign of the spiral phase pattern. This approach allows an extremely simple experimental technique for measuring the sign of the vortex charge on a vortex beam. Experimental results agree with this theory. We expect further applications of this interesting result.

Acknowledgments

We thank Tomio Sonehara of Seiko Epson Corporation for the use of the LCD.

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Fraunhofer diffraction of a partially blocked spiral phase plate

Abstract

1. Introduction

2. Fraunhofer diffraction from a cylindrically symmetric input pattern

3. Experiments

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (3)

Equations (9)

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