• 검색 결과가 없습니다.

Chapter 16. Fraunhofer Diffraction

N/A
N/A
Protected

Academic year: 2022

Share "Chapter 16. Fraunhofer Diffraction"

Copied!
30
0
0

로드 중.... (전체 텍스트 보기)

전체 글

(1)

Chapter 16. Fraunhofer Diffraction

Chapter 16. Fraunhofer Diffraction

(2)

Fraunhofer Approximation Fraunhofer Approximation

( ) (

2

)

2

2

01

= z + x ξ + y η

r

( ) ( ξ η ) ( ) ξ η

λ r d d

U jkr j

y z x

U exp

,

,

2

01

∫∫

01

=

Huygens-Fresnel Principle

Fraunhofer Approximation :

( ) ( ) ( ξ η ) ξ η

λ η π

λ U ξ j z x y d d

z j e y e

x U

y z x j k jkz

⎥⎦ ⎤

⎢⎣ ⎡ − +

= ∫ ∫

+

2

exp ,

,

) 2 (

2 2

( )

2

max 2

2

η

ξ +

〉〉 k z

) 1 (

) 2 (

1

) 2 (

) 1 1 (

) 2 (

1

2 1 2

1 1

2 2

2 2

2 2

2 2

01

η ξ

η ξ

η ξ

η ξ

y z x

y z x

z

y z z x

y z x

z

z y z

z x r

+

− +

+

+ +

+

− +

+

=

⎥ ⎥

⎢ ⎢

⎡ ⎟

⎜ ⎞

⎝ + ⎛ −

⎟ ⎠

⎜ ⎞

⎝ + ⎛ −

FT

(3)

Fraunhofer Diffraction Fraunhofer Diffraction

Fraunhofer diffraction

• Specific sort of diffraction – far-field diffraction

– plane wavefront

– Simpler maths

(4)

(참고) Fresnel Approximation

⎥ ⎥

⎢ ⎢

⎡ ⎟

⎜ ⎞

⎝ + ⎛ −

⎟ ⎠

⎜ ⎞

⎝ + ⎛ −

2 2

01

2

1 2

1 1

z y z

z x

r ξ η

( ) λ ( ) ξ η [ ( x ξ ) ( y η ) ] d ξ d η

z j k z U

j y e

x U

jkz

exp 2 ,

,

2 2

⎭ ⎬

⎩ ⎨

⎧ − + −

= ∫ ∫

( ) ( ) ( ) ξ η ( ) ( ) ξ η

λ

η λ ξ

η π

ξ e d d

e U

z e j y e

x

U z j z x y

j k y

z x j k

jkz ∫ ∫

+ + −

+

⎭ ⎬

⎩ ⎨

= 2

2 2

⎧ , 2

2 2

2

,

( )

( ) ( )

z y f z x f z

j k y

z x j k jkz

Y X

e U

z e j y e

x U

λ λ

η

η ξ

λ ξ

/ ,

/ 2

2

2 2 2

2

, )

, (

=

= +

+

⎭ ⎬

⎩ ⎨

= F

( ) ( 2 ) 2

2

01 = z + x − ξ + y − η

r

(5)

Fresnel Diffraction

• This is most general form of diffraction – No restrictions on optical layout

• near-field diffraction

• curved wavefront – Analysis difficult

Fresnel Diffraction Fresnel Diffraction

Screen

Obstruction

(6)

16-1. Fraunhofer Diffraction from a Single Slit 16-1. Fraunhofer Diffraction from a Single Slit

• Consider the geometry shown below. Assume that the slit is very long in

the direction perpendicular to the page so that we can neglect diffraction

effects in the perpendicular direction.

(7)

Fraunhofer Diffraction from a Single Slit Fraunhofer Diffraction from a Single Slit

( )

( )

0

0

0 0

exp

0.

P

P

The contribution to the electric field amplitude at point P due to the wavelet emanating from the element ds in the slit is given by

dE dE i kr t

r

Let r r for the source element ds at s Then for any element

dE dE

r

⎛ ⎞ ω

=⎜⎝ ⎟⎠ ⎡⎣ − ⎤⎦

= =

= ⎜⎛⎝ + Δ

{ (

0

) }

0

exp

, .

, , .

sin

L L

i k r t

We can neglect the path difference in the amplitude term but not in the phase term

We let dE E ds where E is the electric field amplitude assumed uniform over the width of the slit The path difference s

ω

θ

⎞ ⎡ + Δ − ⎤

⎟ ⎣ ⎦

⎜ ⎟⎠

Δ

=

Δ =

( )

{ } ( ) ( )

( ) ( )

/ 2

0 0 / 2

0 0

/ 2

0

0 / 2

.

exp sin exp exp sin

exp sin

exp sin

L L b

P P b

b P L

b

Substituting we obtain

E ds E

dE i k r s t E i kr t i k s ds

r r

i k s

Integrating we obtain E E i kr t

r i k

θ ω ω θ

ω θ

θ

⎛ ⎞ ⎛ ⎞

=⎜ ⎟ ⎡⎣ + − ⎤⎦ =⎜ ⎟ ⎡⎣ − ⎤⎦

⎝ ⎠ ⎝ ⎠

⎡ ⎤

⎛ ⎞

=⎜ ⎟ ⎡⎣ − ⎤⎦ ⎢ ⎥

⎝ ⎠ ⎣ ⎦

(8)

Fraunhofer Diffraction from a Single Slit Fraunhofer Diffraction from a Single Slit

( ) ( ) ( )

( ) ( ) ( )

( )

0 0

0 0

0 0

exp exp

exp sin

1 sin 2

exp exp exp

2

exp 2

P L

P L

L

Evaluating with the integral limits we obtain

i i

E E i kr t

r i k

where

k b

Rearranging we obtain

E b

E i kr t i i

r i

E b

i kr t

r i

β β

ω θ

β θ

ω β β

β

ω β

− −

⎡ ⎤

⎛ ⎞

=⎜ ⎟ ⎡⎣ − ⎤⎦ ⎢ ⎥

⎝ ⎠ ⎣ ⎦

⎛ ⎞

=⎜ ⎟ ⎡⎣ − ⎤⎦ ⎡⎣ − − ⎤⎦

⎝ ⎠

⎛ ⎞

=⎜ ⎟ ⎡⎣ − ⎤⎦

⎝ ⎠

( ) (

0

)

0

2 2 2

* 2

0 0 2 0 2 0

0

2 sin exp sin

1 1 sin sin

2 2 sinc

L

P P L

E b

i i kr t

r

The irradiance at point P is given by

I = c E E c E b I I

r

β ω β

β

β β

ε ε β

β β

⎛ ⎞

=⎜ ⎟ ⎡⎣ − ⎤⎦

⎝ ⎠

⎛ ⎞

= ⎜ ⎟ = =

⎝ ⎠

sin β

0

sin

2

(

21

sin θ )

2

0

c I c kb

I

I = =

(9)

Fraunhofer Diffraction from a Single Slit Fraunhofer Diffraction from a Single Slit

2 2 2

* 2

0 0 2 0 2 0

0

0 0

1 1 sin sin

2 2 sinc

sinc 1 0, lim sinc lim sin 1

sin 0, 1 sin 1, 2,

2

L

P P

The irradiance at point P is given by

I = c E E c E b I I

r

The function is for

The zeroes of irradiance occur when or when k b m m

β β

β β

ε ε β

β β

β β β

β

β β θ π

⎛ ⎞

= ⎜ ⎟ = =

⎝ ⎠

= = =

= = = = ± ± K

( θ )

β sin sin

sin

2 0 2 12

0

c I c kb

I

I = =

(10)

Fraunhofer Diffraction from a Single Slit Fraunhofer Diffraction from a Single Slit

,

sin , 2 / ,

1 2 2

In terms of the length y on the observation screen y f and in terms of wavelength k we can write

y b y

b f f

Zeroes in the irradiance pattern will occur when

b y m f

m y

f b

The maximum in the irradiance pattern is at β = 0

θ λ π

π π

β λ λ

π π λ

λ

≅ =

= =

= =

2 2

sin cos sin cos sin

0

sin tan cos

. Secondary maxima are found from

d d

β β β β β β

β β β β β

β β β

β

⎛ ⎞ −

= − = =

⎜ ⎟

⎝ ⎠

⇒ = =

1.43π

2.46π

3.47π

0

θ β =

21

kb sin

β

2 0

sin c I

I =

(11)

Fraunhofer Diffraction from a Single Slit Fraunhofer Diffraction from a Single Slit

Note: x- and y-axes

switched in book, Figs. 16- 5a (here) and Fig. 16-1 do not match.

β

2 0

sin c I

I =

(12)

16-2. Beam Spreading 16-2. Beam Spreading

( )

sin

min

1 2

The angular width of the central maximum is defined as the angular

separation Δθ between the first minima on either side of the central maximum,

y f

The first ima in the irradiance pattern will occur when

m f f

y Δθ

b b b

The

θ θ

λ λ λ

= ≅

= = ± ⇒ =

2

width W of the diffraction pattern thus increases linearly with distance from the slit, in the regions far from the slit where Fraunhofer diffraction applies

W = L Δθ L

b

= λ

L

W

(13)

16-3. Rectangular Apertures 16-3. Rectangular Apertures

When the length a and width b of the rectangular aperture are comparable, a diffraction pattern is observed in both the x - and y - dimensions, governed in each dimension by the formula we have already developed. The irradiance

pat

(

2

)(

2

)

0

sinc sinc

, 1 sin 2

tern is

I I

where k a

Zeroes in the irradiance pattern are observed when

m f m f

y or x

b a

α β

α θ

λ λ

=

=

= =

y

(14)

Circular Apertures Circular Apertures

xds dA =

∫∫

=

Area A isk

p e dA

r

E E sin θ

0

2 2

2

2 ⎟

⎜ ⎞

⎝ + ⎛

= x

s

R x = 2 R

2

s

2

ds s

R r e

E E R

R A isk p

2 2

sin 0

2 −

= ∫ −

θ

θ γ sin

,

/ R kR s

v = =

{ }

⎭ ⎬

⎩ ⎨

= ⎧

= ∫ − γ

γ

γ 2 π ( )

2 1 1

0 2 1 2

0 1

2 J

r R dv E

v r e

R

E p E A i v A

(the first order Bessel function of the first kind)

(15)

Bessel Functions Bessel Functions

832 . 3 2 sin

sin = 1 =

= θ θ

γ kR kD (first zero)

(16)

θ γ =

12

kD sin

( ) ( )

( )

1 2 1 2

1 2

2 sin

( ) 0

sin J k D

I I

k D θ θ

θ

⎡ ⎤

= ⎢ ⎥

⎢ ⎥

⎣ ⎦

⎭ ⎬

⎩ ⎨

= ⎧

γ γ π ( )

2

1

0

2

J

r R E

p

E

A

0) at

(or, 0 when

2 1 )

1

( → =

⎭ ⎬

⎩ ⎨

⎧ → γ θ

γ γ J

1 1

min

2 2

sin 3.83 2

2 First minimum in the Airy pattern is at

k D θ k D θ π D θ

λ

⎛ ⎞⎛ ⎞

≅ = = ⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

Fraunhofer Diffraction from Circular Apertures:

The Airy Pattern

Fraunhofer Diffraction from Circular Apertures:

The Airy Pattern

) 0 ( I/ I

D θ λ

θ 1 . 22

2

min = Δ

1

=

(17)

Airy patterns Airy patterns

Airy Disc

(18)

Slit and Circular Apertures Slit and Circular Apertures

Circular aperture

Single slit (sinc ftn)

Sin θ

0 λ/D 2λ/D 3λ/D

−λ/D

−2λ/D

−3λ/D

Intensity

(19)

16-4. Resolution 16-4. Resolution

• Ability to discern fine details of object – Lord Rayleigh in 1896

» resolution is function of the Airy disc.

– Rayleigh: Limit of resolution

» Two light sources must be separated by at least the diameter of first dark band.

» Called Rayleigh Criterion

(20)

Rayleigh Criterion Rayleigh Criterion

Rayleigh Limit

(21)

Rayleigh Limit Rayleigh Limit

Resolution limit of a lens:

(f = focal length) NA NA

D x f

λ λ

λ

61 . 0 22 2

. 1

22 .

min

1

⎟ =

⎜ ⎞

≈ ⎛

⎟ ⎠

⎜ ⎞

= ⎛

(22)

Fraunhofer Diffraction from a Double Slit Fraunhofer Diffraction from a Double Slit

Now for the double slit we can imagine that we place

an obstruction in the middle of the single slit. Then all that we have to do to calculate the field from the double slit is to change the limits of

( )

(( ))

( )

( )

(( ))

( )

( ) ( )

( )

( )

/ 2

0 / 2

0

/ 2

0 / 2

0

/ 2

0

0 / 2

exp exp sin

exp exp sin

exp sin exp

exp sin

L a b

P a b

L a b

a b

a b P L

a b

integration.

E E i kr t i k s ds

r

E i kr t i k s ds

r

Integrating we obtain

i k s i

E E i kr t

r i k

ω θ

ω θ

ω θ

θ

+

− −

− +

+

⎛ ⎞

=⎜ ⎟ ⎡⎣ − ⎤⎦ +

⎝ ⎠

⎛ ⎞

⎡ ⎤

⎜ ⎟ ⎣ ⎦

⎝ ⎠

⎛ ⎞ ⎡ ⎤

=⎜ ⎟ ⎡⎣ − ⎤⎦ ⎢ ⎥ +

⎣ ⎦

⎝ ⎠

( )

( )

( )

( ) ( ) ( )

( ) ( )

( ) ( )

/ 2

/ 2

0

0

0

0

sin sin

exp sin sin

exp exp

sin 2 2

sin sin

exp exp

2 2

exp exp ex

2

a b

a b L

P L

k s i k

i kr t i k a b i k a b

E

r i k

i k a b i k a b

b i kr t

E E i

r i

θ θ

ω θ θ

θ

θ θ

ω α

β

− −

− +

⎧ ⎡ ⎤ ⎫

⎪ ⎪

⎨ ⎢ ⎥ ⎬

⎣ ⎦

⎪ ⎪

⎩ ⎭

− ⎧

⎡ ⎤ + −

⎛ ⎞ ⎣ ⎦ ⎪ ⎡ ⎤ ⎡ ⎤

= ⎜ ⎟ ⎨ ⎢ ⎥− ⎢ ⎥

⎪ ⎣ ⎦ ⎣ ⎦

⎝ ⎠ ⎩

− − − + ⎫

⎡ ⎤ ⎡ ⎤⎪

+ ⎢ ⎥− ⎢ ⎥⎬

⎣ ⎦ ⎣ ⎦⎭⎪

⎡ − ⎤

⎛ ⎞ ⎣ ⎦

= ⎜ ⎟

⎝ ⎠

{

p

( )

exp

( )

exp

( )

exp

( )

exp

( ) }

sin sin

i i i i i

where k a and k b

β β α β β

α θ β θ

− − + − − −

⎡ ⎤ ⎡ ⎤

⎣ ⎦ ⎣ ⎦

= =

(23)

16-5. Fraunhofer Diffraction from a Double Slit 16-5. Fraunhofer Diffraction from a Double Slit

( ) ( )

( ) ( )

( ) ( )( )

( )

0 0

2 2

* 2

0 0 2 0

0

exp exp 2 cos

exp exp 2 sin

exp 2 cos 2 sin

2

1 1 4 sin

4 cos 4 c

2 2 4

L P

L P P

But we know that

i i

i i i

Substituting we obtain

b i kr t

E E i

r i

The irradiance at point P is given by

I = c E E c E b I

r

α α α

β β β

ω α β

β

ε ε α β

β

+ − =

− − =

⎡ ⎤

⎛ ⎞ ⎣ ⎦

= ⎜ ⎟

⎝ ⎠

⎛ ⎞ ⎛ ⎞

= ⎜ ⎟ ⎜ ⎟ =

⎝ ⎠

⎝ ⎠

2 2

2

2

0 0

0

os sin

1 2

E b

L

where I c

r

α β β ε

⎛ ⎞

⎜ ⎟

⎝ ⎠

⎛ ⎞

= ⎜ ⎟

⎝ ⎠

(24)

Fraunhofer Diffraction from a Double Slit Fraunhofer Diffraction from a Double Slit

The irradiance at point P from a double slit is given by the product of the

diffraction pattern from single slit and the interference pattern

from a double slit.

2 2

0 2

4 cos sin

I I α β

β

⎛ ⎞

= ⎜ ⎟

⎝ ⎠

(25)

Fraunhofer Diffraction from a Double Slit Fraunhofer Diffraction from a Double Slit

Single Slit

Double Slit

(26)

16-6. Fraunhofer Diffraction from Many Slits (Grating)

16-6. Fraunhofer Diffraction from Many Slits (Grating)

Now for the multiple slits we just need to again change the limits of integration. For N even slits with width b evenly spaced a distance a apart, we can place the origin of the coordinate system at t

(

0

)

1/ 2

{

((22 11)) / 2/ 2

( )

0

exp exp sin

j N j a b

L

P j a b

j

he center obstruction and label the slits with the index j

(Note that the diagram does not exactly correspond with this).

E E i kr t i k s ds

r ω = + θ

=

⎛ ⎞

=⎜ ⎟ ⎡⎣ − ⎤⎦

⎝ ⎠

∑ ∫

( )

( )

( )

}

( ) ( )

( )

( )

( )

( )

( )

2 1 / 2

2 1 / 2

2 1 / 2

/ 2 0

0 1 2 1 / 2

2 1 /

2 1 / 2

exp sin

exp sin

exp sin

exp sin sin

j a b j a b

j a b j N

P L

j j a b

j a b

j a b

i k s ds

Integrating we obtain

i k s

E E i kr t

r i k

i k s i k

θ

ω θ

θ θ

θ

+

+

=

=

+

+

⎧⎡ ⎤

⎛ ⎞ ⎪

=⎜ ⎟ ⎡⎣ − ⎤⎦ ⎨⎢ ⎥

⎝ ⎠ ⎪⎩⎣ ⎦

⎡ ⎤

+ ⎢ ⎥

⎣ ⎦

( ) ( ) ( )

( ) ( )

2

/ 2 0

0 1

exp 2 1 sin 2 1 sin

exp exp

sin 2 2

2 1 sin 2 1 sin

exp exp

2 2

j N L

j

i kr t i k j a b i k j a b

E

r i k

i k j a b i k j a b

ω θ θ

θ

θ θ

=

=

⎫⎪⎬

⎪⎭

⎧ ⎡ ⎤ ⎡ ⎤

− − + − −

⎡ ⎤ ⎡ ⎤ ⎡ ⎤

⎛ ⎞ ⎣ ⎦ ⎪ ⎣ ⎦ ⎣ ⎦

= ⎜ ⎟ ⎨ ⎢ ⎥− ⎢ ⎥

⎢ ⎥ ⎢ ⎥

⎝ ⎠ ⎪⎩ ⎣ ⎦ ⎣ ⎦

⎡− ⎡⎣ − − ⎤⎦ ⎤ ⎡− ⎡⎣ − − ⎤⎦ ⎤⎫⎪

+ ⎢ ⎥− ⎢ ⎥⎬

⎢ ⎥ ⎢ ⎥⎪

⎣ ⎦ ⎣ ⎦⎭

(27)

( ) { ( ) ( ) ( ) ( ( ) ) ( ) ( ) }

( )

/ 2 0

0 1

0

0

sin sin

exp exp 2 1 exp exp exp 2 1 exp exp

2

exp

j N L

P

j

L P

Substuting using k a and k b and rearranging we obtain

b i kr t

E E i j i i i j i i

r i

We can rewrite this as

b i kr t

E E

r

α θ β θ

ω α β β α β β

β

ω

=

=

= =

⎡ − ⎤

⎛ ⎞ ⎣ ⎦

= ⎜ ⎟ ⎡ ⎣ − ⎤ ⎡ ⎦ ⎣ − − ⎤ ⎦ + − − ⎡ ⎣ − − ⎤ ⎦

⎝ ⎠

⎛ ⎞ ⎡ ⎣

= ⎜ ⎟

⎝ ⎠

( ) { ( ) ( ) }

( ) { ( ) }

( ) { ( ) ( ) ( ) ( ) }

/ 2

1 / 2 0

0 1

/ 2 0

0 1

2 sin exp 2 1 exp 2 1

2

exp sin Re exp 2 1

exp sin Re exp exp 3 exp 5 exp 1

j N

j j N L

j j N L

j

i i j i j

i

E b i kr t i j

r

E b i kr t i i i i N

r The last te

β α α

β

ω β α

β

ω β α α α α

β

=

=

=

=

=

=

⎤⎦ ⎡ ⎣ − ⎤ ⎦ + ⎡ ⎣ − − ⎤ ⎦

⎛ ⎞ ⎛ ⎞

= ⎜ ⎝ ⎟ ⎠ ⎡ ⎣ − ⎤ ⎦ ⎜ ⎝ ⎟ ⎠ ⎡ ⎣ − ⎤ ⎦

⎛ ⎞ ⎛ ⎞

= ⎜ ⎝ ⎟ ⎠ ⎡ ⎣ − ⎤ ⎦ ⎜ ⎝ ⎟ ⎠ + + + + ⎡ ⎣ − ⎤ ⎦

L

( ) ( ) ( ) ( )

{ }

/ 2

1

2

*

0 0

0

Re exp exp 3 exp 5 exp 1 sin

sin

. int

1 1 sin

2 2

j N

j

L

P P P

rm is a geometric series that converges to

i i i i N N

The details of the last step are outlined in the book The irradiance at po P is given by

I c E E c E b

r

α α α α α

α

ε ε β

β

=

=

+ + + + ⎡ ⎣ − ⎤ ⎦ =

⎛ ⎞

= = ⎜ ⎟

⎝ ⎠

L

2 2 2 2

0

sin sin sin

sin

N N

α I β α

α β α

⎛ ⎞ ⎛ ⎜ ⎞ ⎟ = ⎛ ⎞ ⎛ ⎜ ⎞ ⎟

⎜ ⎟ ⎝ ⎠ ⎜ ⎟ ⎝ ⎠

⎝ ⎠ ⎝ ⎠

(28)

Fraunhofer Diffraction from Multiple Slits Fraunhofer Diffraction from Multiple Slits

2 2 2 2 2

*

0 0 0

0

1 1 sin sin sin sin

2 2 sin sin

, sin . , ' '

sin

lim

P P P L

m

The irradiance at point P is given by

E b N N

I c E E c I

r

When m the term N is a maximum For this condition from L Hospital s rule

α

β α β α

ε ε

β α β α

α π α

α

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞

= = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

⎝ ⎠ ⎝ ⎠

⎝ ⎠

=

( )

( )

sin

sin cos

lim lim

sin sin cos

1 sin sin 0, 1, 2,

2

, .

m m

d N

N d N N N

d d

The principal maxima the irradiance pattern occur for

p p

ka a m m

N N

For large N the principal maxima are bright and well separated This ana

π α π α π

α

α α α

α α α

α

π π

α θ θ π

λ

= = = ±

= = = = = = ± ± K

,

sin

lysis gives us the grating equation

a θ = mλ

sin 2

⎟⎟⎠

⎜⎜ ⎞

⎛ β

β 2

sin sin ⎟

⎜ ⎞

⎛ α

α N

N2

λ θ m a sin =

m=1

m=2 m=0

2 2

0

sin sin

sin

I I β N α

β α

⎛ ⎞ ⎛ ⎞

= ⎜ ⎝ ⎟ ⎠ ⎜ ⎝ ⎟ ⎠

(29)

Diffraction grating equation Diffraction grating equation

λ θ m

a sin =

m=1

m=2 m=0

m=1

m=0

(30)

Fraunhofer Diffraction from Multiple Slits Fraunhofer Diffraction from Multiple Slits

N = 2

N = 3

N = 4

N = 5

참조

관련 문서

The index is calculated with the latest 5-year auction data of 400 selected Classic, Modern, and Contemporary Chinese painting artists from major auction houses..

Phase profile of conventional DOE Obtained diffraction image. Optical vortices appear in the diffraction image generated by

Additive alignment, which occurs after passing orifice channel, has been visualized, showing additive alignment that is perpendicular to the flow direction at the

Refraction, diffraction, shoaling → Change of wave direction and height in nearshore area → Important for longshore sediment transport and wave height distribution →

Reciprocal Lattice.. Kittel, Introduction to Solid State Physics.. Hammond, The Basics of Crystallography and Diffraction.. Reciprocal Lattice Direction vs. Sherwood,

- flux of solute mass, that is, the mass of a solute crossing a unit area per unit time in a given direction, is proportional to the gradient of solute concentration

To focus our discussion on inverse kinematics, we wifi assume that the necessary transformations have been performed so that the goal point is a specification of the

- The most unstable one: Internal modes with very short wavelengths perpendicular to the magnetic field but long wavelengths parallel to the field.. Classification