Lecture Notes on Wave Optics (03/19/14) 2.71/2.710 Introduction to Optics –Nick Fang Mathematical Preparation of Fourier Transform - Fourier Transform in time domain A time signal f (t) can be expressed as a series of frequency components F(): (,) = ∫ (, ) exp(−) ∞ −∞ (,) = 1 2 ∫ (, ) exp(+) ∞ −∞ The functions f (t) and F() are referred to as Fourier Transform pairs. - Fourier Transform in spatial domain A spatially varying signal (, )can be expressed as a series of spatial-frequency components ( , ): (, ) = 1 (2) 2 ∫ ∫ ( , ) exp( ) exp( ) ∞ −∞ ∞ −∞ ( , )=∫ ∫ (, ) exp(− ) exp(− ) ∞ −∞ ∞ −∞ Accordingly, the functions (, )and ( , )are referred to as spatial-Fourier Transform pairs. - A few famous functions o Rectangle function () ≡ { 1, || < 1 2 1 2 , || = 1 2 0, || > 1 2 o Sinc function () = sin() rect(x) x -.5 0 .5 1 1
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Lecture Notes on Wave Optics (03/19/14)Lecture Notes on Wave Optics (03/19/14) 2.71/2.710 Introduction to Optics – Nick Fang Mathematical Preparation of Fourier Transform - Fourier
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Lecture Notes on Wave Optics (03/19/14)
2.71/2.710 Introduction to Optics –Nick Fang
Mathematical Preparation of Fourier Transform
- Fourier Transform in time domain A time signal f (t) can be expressed as a series of frequency components F():
𝑓(𝑟, 𝑡) = ∫ 𝐹(𝑟, 𝜔) exp(−𝑖𝜔𝑡) 𝑑𝜔∞
−∞
𝐹(𝑟, 𝜔) =1
2𝜋∫ 𝑓(𝑟, 𝑡) exp(+𝑖𝜔𝑡) 𝑑𝑡
∞
−∞
The functions f (t) and F() are referred to as Fourier Transform pairs.
- Fourier Transform in spatial domain A spatially varying signal 𝑓(𝑥, 𝑦)can be expressed as a series of spatial-frequency