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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
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
• Alicia V. Carpentier, Humberto Michinel and Jose R. Salgueiro, Making
Optical vortices with computer generated hologram.
• Enrique Galvez, Gaussian beams.
• Eugene Hecht, Optics.
References: