Lecture Outline. Scattering From a Dielectric Slab Anti Reflection Layer Bragg Gratings 8/9/2018. EE 4347 Applied Electromagnetics.

Transcription

1 Course Instructor Dr. Raymond C. Rumpf Office: A 337 Phone: (95) E Mail: rcrumpf@utep.edu EE 4347 Applied Electromagnetics Topic 3k Multiple Scattering Multiple These Scattering notes may contain copyrighted material obtained under fair use rules. Distribution of these materials is strictly prohibited Slide Lecture Outline Scattering From a Dielectric Slab Anti Reflection Layer Bragg Gratings Multiple Scattering Slide

2 Scattering From a Dielectric Slab Multiple Scattering Slide 3 Analysis of a Dielectric Slab Total Reflection E 0 n n, n,, re0 t r t E e j 3 0 t r r t E e j4 3 0 t r r t E e 3 3 j5 3 0 te0 rt Ee j 3 0 d j te0e j t3te0e j rt 3 Ee 0 j j j4 j j 4 j j5 3 0 r t r and so on. r r t r knd 0 3 j5 3 0 r t r3 t r r t E e 3 3 j t r r t E e j Total Transmission Multiple Scattering Slide 4

3 Generalizations Normal incidence, no loss: Oblique incidence, no loss: Normal incidence, lossy: Oblique incidence, lossy: knd 0 knd 0 cos knd, n n j 0 o knd cos, n n j 0 o Multiple Scattering Slide 5 Overall Reflection, r ( of 3) The overall reflection from the slab is the sum of all the individual reflected waves. rr trte trrte trrte j j4 3 3 j All of these terms arise due to multiple reflections within the slab. They can be written as a summation. n n jn trr3 te n0 j j n 3 rr3e n0 r t t e Now we put the summation back into our expression for overall reflection r. j j n rr rtte 3 3 n0 Multiple Scattering Slide 6 3

4 Overall Reflection, r ( of 3) Recall the closed form expression for a geometric series. n a ax for x x n0 The summation in our expression for overall reflection is a geometric series that we can now write in closed form. r j j n 3 n0 rr rtte r r t t e j j 3 j r3tte j 3 Multiple Scattering Slide 7 Overall Reflection, r (3 of 3) Recall how our local reflection and transmission parameters were related. r r t r t r t r This let s us express r just in terms of r, r 3, and. j r3tte r r j 3 j r3 r r e r j r r e r r r e r e r r e j j j 3 j j 3 3 r r e r j j Multiple Scattering Slide 8 4

5 Overall Transmission, t ( of ) The overall transmission through the slab is the sum of all the individual transmitted waves. ttte t rrte t rrte j j3 j This can be written as a summation. n n jn t t3rrte n0 Now we factor out some terms from the summation. j j n 3 3 n0 t t t e r r e The summation in this expression is a geometric series and can be written in closed form. j t t3te j 3 Multiple Scattering Slide 9 Overall Transmission, t ( of ) Recall how our local reflection and transmission parameters were related. r r t r t r t r The let s us express t just in terms of r, r 3, and. 3 3 t t e t j 3 j 3 3 r r e r r e j j t r r e j 3 j Multiple Scattering Slide 0 5

6 Relation Between r and t We solve our two expressions for r and t for ( + r r 3 e -j ). j r r3e r3 r e r t j j j j r r3e j r3 r e r t The expressions on the right hand side of these equations must be equal. j j r r e r r e 3 t r r t r r r e 3 j 3 The relation between r and t is therefore r j j Note: This is NOT the same relation that we saw for a single interface. Multiple Scattering Slide Plots of r and t Small Reflections (low finesse) Small Reflections (high finesse) t 0 r 0 The response resembles a cosine function and is usually approximated as such. The response resembles a comb filter. Multiple Scattering Slide 6

7 Low Finesse ( of ) To understand low finesse, assume that we have a symmetric slab (i.e. r = -r 3 ) and that these reflection coefficients are small. r r t The expressions for r and t reduce to r r e r e j j r r e j j t r e j 3 j r r e r r e j j r e re j r r e r r e j j j For small reflections, re j r r e j re j Multiple Scattering Slide 3 Low Finesse ( of ) The magnitude of r gives the sine wave response we were expecting. r r e r sin j Multiple Scattering Slide 4 7

8 Example: Do Windows Block Wifi? ( of ) Windows are typically made of fused silica (n =.5) and are around 3 mm thick. Solution Transmission through a slab of dielectric is calculated using t r r e j 3 j The parameters in this equation are nn.0.5 r nn n n3.5.0 r n n f.40 Hz knd 0 nd.5 8 c 3.00 m s m Multiple Scattering Slide 5 Example: Do Windows Block Wifi? ( of ) Windows are typically made of fused silica (n =.5) and are around 3 mm thick. Solution cont d Substituting our values into the transmission equation gives e j0.9 e t j0.450 Total power transmitted is j0.9 T t j % CONCLUSION Windows do nothing to block Wifi. Multiple Scattering Slide 6 8

9 Example: Oil on Water Oil on water is an example of thin film interference. r, 0 d Oil, n.5 Water, n.33 Multiple Scattering Slide 7 Antireflection Layer Multiple Scattering Slide 8 9

10 Problem Setup ( of ) Suppose we have an interface between two materials. This will produce reflections according to r How can we prevent a reflection at this interface? Multiple Scattering Slide 9 Problem Setup ( of ) We insert an intermediate layer that we will call an anti reflection layer. How can we choose n ar, ar, and L such that we get zero reflections from this interface? Multiple Scattering Slide 0 0

11 How to Get r = 0 from a Slab ( of 4) Recall the overall reflection from a dielectric slab. j r r3e r j To get r = 0, the numerator of this expression must be zero. r j r3e 0 The reflection coefficients r and r 3 arise from the materials in the problem that we do not wish to adjust. The trick must be in the e -j term. Solving this for we get j r r ln 3 e m r3 j r m any integer Multiple Scattering Slide How to Get r = 0 from a Slab ( of 4) Recall that r r ar ar ar ar Our expression for becomes ln j ar ar ar ar m Multiple Scattering Slide

12 How to Get r = 0 from a Slab (3 of 4) Recall that = k 0 n ar d. We substitute this into our design equation. kn d 0 ar kn d 0 ar ln j ln j ar ar ar ar ar ar ar ar ar ar m Using k 0 = / 0 and solving this for d gives m 0 ln 0 d m 4n j n ar ar ar ar This is the most general design equation and provides more freedom than the simpler one we are about to derive. Multiple Scattering Slide 3 How to Get r = 0 from a Slab (4 of 4) Our design equation is complicated and we would like to simplify it. To figure out a way to do this, we multiply out the expression inside of the natural logarithm function. ar ar ar ar ar ar ar ar This simplifies when. In fact, it reduces to just ar Recognizing that ln(-) = j, our simplified expression for d is now d 0 0 m for ar 4nar nar Multiple Scattering Slide 4

13 Interpretation of Design Equation When ar any integer d 0 0 4n m ar n m ar The second term tells us that we can adjust the length d by any integer multiple of a half wavelength. 0 m m n ar We interpret this first terms as a quarter wavelength slab of dielectric. 0 4n 4 ar Multiple Scattering Slide 5 Design Procedure Step Choose an antireflection material so that ar We have a bit of freedom here. However, when we wish to use only dielectric materials, only one choice is possible. or n nn ar ar Step Calculate thickness based on refractive index. d 0 0 any integer 4 m n n m ar ar m = 0 is the most common choice. Multiple Scattering Slide 6 3

14 Example It is desired to maximize the light through a lens. The lens is made of glass with n =.5 and resides in air with n =.0. Design an antireflection coating to maximize transmission at the center of the visible spectrum, 0 = 500 nm. Solution Step : At optical frequencies, materials cannot have a significant magnetic response. Therefore, we will design the anti reflection layer through the refractive index n ar. nar nn.0.5 n ar.39 Step : The thickness of the anti reflection layer is d 500 nm m 500 nm 0.4 nm m 0.8 nm Choose the m = 0 solution. d 0.4 nm Multiple Scattering Slide 7 Bragg Gratings Multiple Scattering Slide 8 4

15 What is a Bragg Grating? L L n n Design 0 L 4n 0 L 4n Quarter wave layers stop band n n 0 Multiple Scattering Slide 9 5