Generation and characterization of

optical vortices and sorting of orbital

angular momentum states of light

Plane wave

Ψ 𝒓, 𝑡 = A𝑒𝑖(𝒌.𝒓 −𝜔𝑡)

𝑘 = Wave vector𝜔 = Angular frequency𝐴 = Amplitude of the wave

Image source: Eugene Hecht, OPTICS, 4th International Edition 2002 (Pearson Education)

Gaussian wave

• Ψ 𝑥, 𝑦, 𝑧, 𝑡 = 𝐴𝑤0

𝑤(𝑧)𝑒−𝑥2+𝑦2

𝑤 𝑧 2 𝑒𝑖(𝑘𝑧−𝜔𝑡)𝑒𝑖𝑘 𝑥2+𝑦2

2𝑅(𝑧) 𝑒−𝑖𝜙(𝑧)

• w(𝑧) =Beam Spot

• 𝑤0 = Beam waist

• 𝑅 𝑧 = Radius of curvature

• 𝜙 𝑧 =Gouy Phase

Beam Spot

• 𝑤 𝑧 = 𝑤0 (1 +𝑧2

𝑧𝑅2)

• 𝑤0 =𝑧𝑅𝜆

𝜋

• 𝑅𝑎𝑦𝑙𝑒𝑖𝑔ℎ 𝑅𝑎𝑛𝑔𝑒 𝑧𝑅 = 𝜋𝑤0

2

𝜆

• 𝜙 𝑧 = arctan𝑧

𝑧𝑅

• For 𝑧 ≫ 𝑧𝑅 , 𝑤 𝑧 =𝑤0𝑧

𝑧𝑅

• Divergence 𝜃 =𝜆

𝜋𝑤0

Higher order Gaussian Beams

• Hermite Gaussian beams

• 𝚿 𝑥, 𝑦, 𝑧, t = 𝐔 x, y, z e−𝑖𝜔𝑡

• 𝑼𝒎,𝒏 𝑥, 𝑦, 𝑧 =𝐴

𝑤 𝑧𝐻𝑚

2𝑥

𝑤 𝑧𝐻𝑛

2𝑦

𝑤 𝑧𝑒−𝑥2+𝑦2

𝑤 𝑧 2 𝑒𝑖𝑘 𝑥2+𝑦2

2𝑅(𝑧) 𝑒−𝑖𝜙(𝑧)

𝑯𝑮𝟏𝟎 𝑯𝑮𝟐𝟎

Laguerre Gaussian Beams

• 𝑼𝒑,𝒎 𝑟, 𝜃, 𝑧 =𝐴

𝑤 𝑧

2𝑟

𝑤 𝑧

|𝑚|

𝐿𝑝|𝑚| 2𝑟2

𝑤 𝑧 2 𝑒−

𝑟2

𝑤 𝑧 2𝑒𝑖𝑘𝑟2

2𝑅 𝑧 𝑒−𝑖𝜙(𝑧)

• 𝐴 = 𝑝!2

𝜋𝑝! 𝑚 +𝑝 !

1/2

𝑚 = 0 𝑚 = 1 𝑚 = 2 𝑚 = 3

L𝑮𝟎𝟏

Optical vortices

𝑨 Amplitude of the field

𝒎 Topological charge

𝑳𝒛 Orbital Angular Momentum of the light

Image source: Ebrahim Karimi, University of Ottawa

𝜓 = 𝑨𝑒𝕚𝒎𝜽

𝑳𝒛 = −𝑖𝑚ℏ

Generation of optical vortices

• Spiral Phase Plate

Δ𝜙 =2𝜋 𝑛−1 𝑑

𝜆

Where,

𝑛 = Refractive index

Δ𝜙 = phase shift

𝑑= Thickness of the plate

Computer Generated Holography (CGH)

Computer Generated Hologram and Intensity profile

𝒎 = 𝟏 𝒎 = −𝟏 𝒎 = 𝟐 𝒎 = −𝟐𝒎 = 𝟎

LCOS-Spatial Light Modulator

𝒎𝒅𝒊𝒇𝒇 = 𝒎𝒊𝒏𝒄𝒊𝒅𝒆𝒏𝒕 + 𝒏𝒎𝒇𝒐𝒓𝒌

Experimental setup

Experimentally generated vortices

𝑚 = 0 𝑚 = 1 𝑚 = −1 𝑚 = 2

𝑚 = −2 𝑚 = 3 𝑚 = −3 𝑚 = 4

Order confirmation of optical vortices

𝑚 = 0 𝑚 = 1 𝑚 = −1 𝑚 = 2 𝑚 = −2

Dove prism

Image Source: Altechna

Introduction to sorting

Experimental setup of sorter

Dove prism@ 45 degrees

633 nm

CCDCCD

Odd m Even m

Output of Sorter setup

𝑚 = 0 𝑚 = 1 𝑚 = 2 𝑚 = 3

• Alicia V. Carpentier, Humberto Michinel and Jose R. Salgueiro, Making

Optical vortices with computer generated hologram.

• Enrique Galvez, Gaussian beams.

• Eugene Hecht, Optics.

References:

Multi stage sorting