Chapter 2 Optical Resonator and Gaussian Beam optics

Chapter 2

Optical Resonator and Gaussian Beam optics

Chapter 2 Optical resonator and Gaussian beam

What is an optical resonator?

An optical resonator, the optical counterpart of an electronic resonant circuit, confines and stores light at certain resonance frequencies. It may be viewed as an optical transmission system incorporating feedback; light circulates or is repeatedly reflected within the system, without escaping.


Contents

• 2.1 Matrix optics• 2.2 Planar Mirror Resonators

– Resonator Modes– The Resonator as a Spectrum Analyzer– Two- and Three-Dimensional Resonators

• 2.3 Gaussian waves and its characteristics– The Gaussian beam– Transmission through optical components

• 2.4 Spherical-Mirror Resonators– Ray confinement– Gaussian Modes– Resonance Frequencies– Hermite-Gaussian Modes– Finite Apertures and Diffraction Loss


2.1 Brief review of Matrix optics

Light propagation in a optical system, can use a matrix M, whose elements are A, B, C, D, characterizes the optical system Completely ( known as the ray-transfer ray-transfer matrixmatrix.) to describe the rays transmission in the optical components.

One can use two parameters:

y: the high

: the angle above z axis


y1y2

2 1 1

2 1

y y d tg

For the paraxial rays tg

12

2 1

2 1

1

0 1

y yd

y1,1

y2,2

2 1

2 1 1

2

y y

yR

2

1

1y

R

Along z upward angle is positive, and downward is negative

d


Free-Space Propagation Refraction at a Planar Boundary

1

0 1

dM

1

2

1 0

0M n

n

Refraction at a Spherical Boundary

2 1 1

2 2

1 0

( )M n n n

n R n

Transmission Through a Thin Lens

Reflection from a Planar Mirror Reflection from a Spherical Mirror

1 0

11

M

f

1 0

0 1M

1 0

21

M

R


A Set of Parallel Transparent Plates.

1

0 1

i

i

d

nM

Matrices of Cascaded Optical Components

1 1....N NM M M M


Periodic Optical Systems

The reflection of light between two parallel mirrors forming an optical resonator is a periodic optical system is a cascade of identical unit system.

Difference Equation for the Ray Position

A periodic system is composed of a cascade of identical unit systems (stages), each with a ray-transfer matrix (A, B, C, D). A ray enters the system with initial position y0 and slope 0. To determine the position and slope (ym,m) of the ray at the exit of the mth stage, we apply the ABCD matrix m times,

0

0

m

m

m

y yA B

C D

1

1

m m m

m m m

y Ay B

Cy D


1

1

m m m

m m m

y Ay B

Cy D

From these equation, we have

1m mm

y Ay

B

2 11

m mm

y Ay

B

22 12m m my by F y

And then:

where

2

A Db

2 detF Ad BC M and

linear differential equations,

So that


If we assumed:

0m

my y h

So that, we have

2 22 0h bh F 2 2h b i F b

1cos bF If we defined

then

0 0 max 0sin( ) sin( )m mmy y F m y F m

cosb F 2 2 sinF b F We have

(cos sin ) ih F i Fe 0m im

my y F e

A general solution may be constructed from the two solutions with positive and negative signs by forming their linear combination. The sum of the two exponential functions can always be written as a harmonic (circular) function,


If F=1, then max 0sin( )my y m

Condition for a Harmonic Trajectory: if ym be harmonic, the cos-1b must be real, We have condition

1b 12

A Dor

The bound therefore provides a condition of stability (boundedness) of the ray trajectory

1b

If, instead, |b| > 1, is then imaginary and the solution is a hyperbolic function (cosh or sinh), which increases without bound. A harmonic solution ensures that y, is bounded for all m, with a maximum value of ymax. The bound |b|< 1 therefore provides a condition of stability (boundedness) of the ray trajectory.


Condition for a Periodic Trajectory

The harmonic function is periodic in m, if it is possible to find an integer s such that ym+s = ym, for all m. The smallest such integer is the period.

The necessary and sufficient condition for a periodic trajectory is: s = 2q, where q is an integer

Unstable b>1

Stable and periodic

Stable nonperiodic


EXERCISEA Periodic Set of Pairs of Different Lenses. Examine the trajectories of paraxial rays through a periodic system composed of a set of lenses with alternating focal lengths f1 and f2 as shown in Fig. Show that the ray trajectory is bounded (stable) if

2

1 1

2 11 2 1 2 2 1 2

1 21 0 1 01 1

1 11 10 1 0 1 1 1

(1 )(1 )

d dd

d d f fM

d d d df f

f f f f f f f


Home work

1. Ray-Transfer Matrix of a Lens System. Determine the ray-transfer matrix for an optical system made of a thin convex lens of focal length f and a thin concave lens of focal length -f separated by a distance f. Discuss the imaging properties of this composite lens.


2. 4 X 4 Ray-Transfer Matrix for Skewed Rays. Matrix methods may be generalized to describe skewed paraxial rays in circularly symmetric systems, and to astigmatic (non-circularly symmetric) systems. A ray crossing the plane z = 0 is generally characterized by four variables-the coordinates (x, y) of its position in the plane, and the angles (e,, ey) that its projections in the x-z and y-z planes make with the z axis. The emerging ray is also characterized by four variables linearly related to the initial four variables. The optical system may then be characterized completely, within the paraxial approximation, by a 4 X 4 matrix.

(a) Determine the 4 x 4 ray-transfer matrix of a distance d in free space.

(b) Determine the 4 X 4 ray-transfer matrix of a thin cylindrical lens with focal length f oriented in the y direction. The cylindrical lens has focal length f for rays in the y-z plane, and no focusing power for rays in the x-z plane.

Home works


2.2 Planar Mirror Resonators

Charles Fabry (1867-1945), Alfred Perot (1863-1925),


2.2 Planar Mirror Resonators

A. Resonator Modes

Resonator Modes as Standing Waves

This simple one-dimensionalresonator is known as a Fabry-Perot etalon.

A monochromatic wave of frequency v has a wavefunction as

( , ) Re ( ) exp( 2 )u r t U r i vtRepresents the transverse component of electric field.The complex amplitude U(r) satisfies the Helmholtz equation;

Where k =2v/c called wavenumber, c speed of light in the medium


( ) sinU r A kz

the modes of a resonator must be the solution of Helmholtz equation with the boundary conditions: 0

( ) 0z

U rz d

So that the general solution is standing wave:

With boundary condition, we have kd qd

q

qk

d

, 1, 2,...,2q

cq q

d

2F

c

d

q 1q

Resonance frequencies

1 2F q q

c

d

∵ 22

vk

c

q is integer.


The length of the resonator, d = q q /2, is an integer number of half wavelength

The resonance wavelength is: 2q

q

c dq

Attention: 0 /c c n Where n is the refractive index in the resonator

Resonator Modes as Traveling Waves

A mode of the resonator: is a self-reproducing wave, i.e., a wave that reproduces itself after a single round trip , The phase shift imparted by a single round trip of propagation (a distance 2d) must therefore be a multiple of 2.

0

4 42 2

nk d d d q

c

q= 1,2,3,…


Density of Modes (1D)

The density of modes M(v), which is the number of

modes per unit frequency per unit length of the resonator, is

4( )M

c

The number of modes in a resonator of length d within the frequency interval v is:

This represents the number of degrees of freedom for the optical waves existing in the resonator, i.e., the number of independent ways in which these waves may be arranged.

4d

c

For 1D resonator


Losses and Resonance Spectral Width

The magnitude ratio of two consecutive phasors is the round-trip amplitude attenuation factor r introduced by the two mirror reflections and by absorption in the medium. Thus:

42

1 0 0 0 0

ndii i kdU hU e U e U e U

So that, the sum of the sequential reflective light with field of

2 3 00 1 2 3 0... (1 ...) (1 )

UU U U U U U h h h h

2

2 0 0 02 2

2 2(1 2 cos )1 (1 ) 4 sin 2i

U I II Ue

max 0max2 2 2

,1 (2 / ) sin ( / 2) (1 )

I II I

Ffinally, we have

1/ 2

1

F Finesse of the resonator

U1

U2

U3

U0

Mirror 1 Mirror 2


The resonance spectral peak has a full width of half maximum (FWHM):

4Fc

d F

4 d

c

Due to We have max

2 21 (2 / ) sin ( / )F

II

F

, 1, 2,...,q Fq q where 2F c d

maxmin 21 (2 / )

II

F

2F

c

d

F F


max2 21 (2 / ) sin ( / 2)

II

F

maxmax 2 2

1

2 1 (2 / ) sin ( / 2)

II

F

1 0

2 2(2 / ) sin ( / 2) 1F

1 0sin( / 2) / 2F

1 0/ F

Full width half maximum is 1 02 2 / F

∵ 4 d

c

4Fc

d FSo that


Spectral response of Fabry-Perot Resonator

The intensity I is a periodic function of with period 2. The dependence of I on , which is the spectral response of the resonator, has a similar periodic behavior since = 4d/c is proportional to . This resonance profile:

max2 21 (2 / ) sin ( / )F

II

F

The maximum I = Imax, is achieved at the resonance frequencies

, 1, 2,...,q Fq q

whereas the minimum value maxmin 21 (2 / )

II

F

The FWHM of the resonance peak is4

Fc

d

F


Sources of Resonator Loss

• Absorption and scattering loss during the round trip: exp (-2sd)

• Imperfect reflectance of the mirror: R1, R2

1 21 2

1 1ln

2r s s m md R R

21 2 exp( 2 )sR R d 2 exp( 2 )rd

1 2

1 1ln

2r s d R R

we get:r is an effective overall distributed-loss coefficient, which is used generally in the system design and analysis

11

1 1ln

2m d R

11

1 1ln

2m d R

Defineding that


• If the reflectance of the mirrors is very high, approach to 1, so that

• The above formula can approximate as 1 2

1 2

1 1 1

2 2 2m m

R R R

d d d

1r s

R

d

exp( / 2)

1 exp( )r

r

d

d

F

rd

F

1 21R R R

1 21 2

1 1ln

2r s s m md R R

The finesse F can be expressed as a function of the effective loss coefficient r,

Because rd<<1, so that exp(-rd)=1-rd, we have:

The finesse is inversely proportional to the loss factor rd


Photon Lifetime of Resonator

The relationship between the resonance linewidth and the resonator loss may be viewed as a manifestation of the time-frequency uncertainty relation. Form the linewidth of the resonator, we have

/ 2

/ 2r

r

cc d

d

1p

rc

1

2 p

Because r is the loss per unit length, cr is the loss per unit time, so that we can Defining the characteristic decay time as the resonator lifetime or photon lifetime

The resonance line broadening is seen to be governed by the decay of optical energy arising fromresonator losses


The Quality Factor Q

0Q

For a resonator of loss at the rate cr (per unit time), which is equivalent to the rate cr /0 (per cycle), so that

02 pQ

0

F

Q

F

2 ( )storedenergyQ

energylosspercycle

The quality factor Q is often used to characterize electrical resonance circuits and microwave resonators, for optical resonators, the Q factor may be determined by percentage of that stored energy to the loss energy per cycle:

0

12 ( / )rQ c

∵

The quality factor is related to the resonator lifetime (photon lifetime)

∵

The quality factor is related to the finesse of the resonator by

2rc

112p

rc

Large Q factors are associated with low-loss resonators


• In summary, three parameters are convenient for characterizing the losses in an optical resonator:– the finesse F

– the loss coefficient r (cm-1),

– photon lifetime p = 1/cr, (seconds).

• In addition, the quality factor Q can also be used for this purpose


B. The Resonator as a Spectrum Analyzer

Transmission of a plane wave across a planar-mirror resonator (Fabry-Perot etalon)

U1

U2

U0

Mirror 1 Mirror 2

t1 t2r1 r2

max2 2

( )1 (2 / ) sin ( / )F

TT

F

( ) tIT I

2

max 1 2 1 22, ,

(1 )

tT t t t

1/ 2

1

F

Where:

22q

q

dqc d dd

The change of the length of the cavity will change the resonance frequency


C. Two- and Three-Dimensional Resonators

• Two-Dimensional Resonators

• Mode density

, , 1, 2,..., 1, 2,...,y zy z y z

q qk k q q

d d

2 2 2 22

( )y zk k kc

2

4( )M

c

Determine an approximate expression for the number of modes in a two-dimensional resonator with frequencies lying between 0 and , assuming that 2/c >> d, i.e. d >>/2, and allowing for two orthogonal polarizations per mode number.

the number of modes per unit frequency per unit surface of the resonator


Three-Dimensional Resonators

Physical space resonator Wave vector space

, , , , , 1, 2,...,yx zx y z x y z

qq qk k k q q q

d d d

2 2 2 2 22

( )x y zk k k kc

Mode density2

3

8( )M

c

The number of modes lying in the frequency interval between 0 and v corresponds to the number of points lying in the volume of the positive octant of a sphere of radius k in the k diagram


Optical resonators and stable condition

• A. Ray Confinement of spherical resonators

d

z

The rule of the sign: concave mirror (R < 0), convex (R > 0). The planar-mirror resonator is R1 = R2=∞

The matrix-optics methods introduced which are valid only for paraxial rays, are used to study the trajectories of rays as they travel inside the resonator


B. Stable condition of the resonator

d

z

-1

0

2

R1 R2

y0

y1

y2

1

1

m m

m m

y yAB

C D

For paraxial rays, where all angles are small, the relation between (ym+1, m+1) and(ym, m) is linear and can be written in the matrix form

reflection from a mirror of radius R1

propagation a distance d through free space

reflection from a mirror of radius R2

1 2

2 2

1 0 1 01 1

1 10 1 0 1R R

A B d d

C D


max 0sin( )my y m

2

2

1 2 1 2

1 1 1

21

2 (1 )

42 2

2 2 2( 1)( 1)

dA R

dB d R

dC R R R R

d d dD R R R

2det 1M Ad BC F

1 2

( ) / 2 2 1 1 1d d

b A DR R

It the way is harmonic, we need cos-1b must be real, that is

1b 1 2

( ) / 2 2 1 1 1 1d d

b A DR R

1 2

0 1 1 1d d

R R

for g1=1+d/R1; g2=1+d/R2

1 20 1g g


resonator is in conditionally stable, there will be:

1 2

0 1 1 1d d

R R

1 20 1g g

In summary, the confinement condition for paraxial rays in a spherical-In summary, the confinement condition for paraxial rays in a spherical-

mirror resonator, constructed of mirrors of radii Rmirror resonator, constructed of mirrors of radii R11,R,R22 seperated by a seperated by a

distance d, is 0distance d, is 0≤≤gg11gg22≤≤1, where g1, where g11=1+d/R=1+d/R11 and g and g22=1+d/R=1+d/R22

For the concave R is negative, for the convex R is positive


Stable and unstable resonators

Symmetrical

resonators

a

b

c

d

1

1

-1 0 1g

2g

e

a. Planar

(R1= R2=∞)

b. Symmetrical confocal

(R1= R2=-d)

c. Symmetrical concentric

(R1= R2=-d/2)

d. confocal/planar

(R1= -d,R2=∞)

e. concave/convex

(R1<0,R2>0)

Non stable

stable

d/(-R) = 0, 1, and 2, corresponding to planar, confocal, and concentric resonators


The stable properties of optical resonators

Crystal state resonators

1 2 1 20 1g g or g g

a. Planar

(R1= R2=∞)

b. Symmetrical confocal

(R1= R2=-d)

c. Symmetrical concentric

(R1= R2=-d/2)

Stable

unstable


Unstable resonators

1 2 1 20 1g g or g g Unstable cavity corresponds to the high loss

a. Biconvex resonator

b. plan-convex resonator

c. Some cases in plan-concave resonator

d. Some cases in concave-convex resonator

d

e. Some cases in biconcave resonator

When R2<d, unstable

dWhen R1<d and R1+R2=R1-|R2|>d

R1

1 2 1 2

1 2 1 2

(1 / )(1 / ) 0

(1 / )(1 / ) 1

g g d R d R

g g d R d R

1 2R R d


Home works

1. Resonance Frequencies of a Resonator with an Etalon. (a) Determine the spacing between adjacent resonance frequencies in a resonator constructed of two parallel planar mirrors separated by a distance d = 15 cm in air (n = 1). (b) A transparent plate of thickness d, = 2.5 cm and refractive index n = 1.5 is placed inside the resonator and is tilted slightly to prevent light reflected from the plate from reaching the mirrors. Determine the spacing between the resonance frequencies of the resonator.

2. Semiconductor lasers are often fabricated from crystals whose surfaces are cleaved along crystal planes. These surfaces act as reflectors and therefore serve as the resonator mirrors. Consider a crystal with refractive index n = 3.6 placed in air (n = 1). The light reflects between two parallel surfaces separated by the distance d = 0.2 mm. Determine the spacing between resonance frequencies vf, the overall distributed loss coefficient r, the finesse , and the spectral width ᅀ v. Assume that the loss coefficient s= 1 cm-1.

3. What time does it take for the optical energy stored in a resonator of finesse = 100, length d = 50 cm, and refractive index n = 1, to decay to one-half of its initial value?

9.1-1, 9.1-2, 9.1-4, 9.1-5, 9.2-2, 9.2-3, 9.2-5 chapter 9


2.3 Gaussian waves and its characteristics

The Gaussian beam is named after the great mathematician Karl Karl Friedrich GaussFriedrich Gauss (1777- 1855)


A. Gaussian beam

The electromagnetic wave propagation is under the way of Helmholtz equation

0 0exp{ ( k r)} exp( ) exp( )U U i t U ikz i t Normally, a plan wave (in z direction) will be

2 2 0U k U

When amplitude is not constant, the wave is

( , , ) exp( ) exp( )U A x y z ikz i t

An axis symmetric wave in the amplitude

( , ) exp( ) exp( )U A z ikz i t z

2nk

2 frequency Wave vector


Paraxial Helmholtz equation

Substitute the U into the Helmholtz equation we have:

One simple solution is spherical wave ：

2 2 0T

AA i k

z

2 2

22 2T x y

21( ) exp( )

2

AA r jk

z z

where

2 2 2x y


has the other solution, which is Gaussian wave ：

2 20

0 2( ) exp[ ]exp[ ( )]

( ) ( ) 2 ( )

WU r A ikz ik i z

W z W z R z

2 1/ 20

0

( ) [1 ( ) ]z

W z Wz

20( ) [1 ( ) ]z

R z zz

1

0

( ) tanz

zz

1/ 200 0( ) ( ) (0)z

zW W z W

z0 is Rayleigh range

where

2

1 1

( ) ( ) ( )i

q z R z W z

q parameter

2 2 0T

AA i k

z

The equation


Gaussian Beam

z

Beam

radius

E

z=0


2 2 2 20

2

( )( , , ) exp[ ] exp[ ( ) ( )]

( ) ( ) 2 ( )

A x y x yE x y z ik z i z

W z W z R z

)0(0 WW

22 20 0( ) [1 ( ) ] [1 ( ) ]

W zR z z z

z z

12

0 0

( ) arctanz z

z tgW z

Electric field of Gaussian wave propagates in z direction

Beam width at z

Waist width

Radii of wave front at z

Phase factor

2 1/ 20

0

( ) [1 ( ) ]z

W z Wz

2

00

Wz

Physical meaning of parameters


Gaussian beam at z=0

]exp[)0,,(2

0

2

0

0

W

r

W

AyxE

222 yxr

0

0

eW

A

where

at z=0, the wave front of Gaussian beam is a plan surface, but the electric field is Gaussian form

-W0 W0

E

0

0

W

A

W0 is the waist half width

220

0 0lim ( ) lim 1z z

WR z z

z

2 1/ 20

0

( ) [1 ( ) ]z

W z Wz

Beam width:

will be minimum

wave front


B. The characteristics of Gaussian beam

Gaussian beam is a axis symmetrical wave, at z=0 phase is plan and the intensity is Gaussian form, at the other z, it is Gaussian spherical wave.

z

Beam

radius


Intensity of Gaussian beam

• Intensity of Gaussian beam2

200 2

( , ) [ ] exp[ ]( ) ( )

WI z I

W z W z

-1 0 1 0

1

-1 0 1 0

1

-1 0 1 0

1

0W 0W0W

0

I

I 0

I

I0

I

I

x xx

y y y

The normalized beam intensity as a function of the radial distance at different axial distances

z=0 z=z0 z=2z0


On the beam axis ( = 0) the intensity

0

1

z

0

I

I

oz

1

0oz

0.5

The normalized beam intensity I/I0at points on the beam axis (=0) as a function of z

20 00

2

0

(0, ) [ ]( ) 1 ( )

W II z I

zW zz

Variation of axial intensity as the propagation length z


20

0

Wz


Power of the Gaussian beam

The power of Gaussian beam is calculated by the integration of the optical intensity over a transverse plane

20 0

1

2P I W

So that we can express the intensity of the beam by the power

2

2 2

2 2( , ) exp[ ]

( ) ( )

PI z

W z W z

The ratio of the power carried within a circle of radius. in the transverse plane at position z to the total power is

020

20

21( , )2 1 exp[ ]

( )I z d

P W z


Beam Radius 2 1/ 20

0

( ) [1 ( ) ]z

W z Wz

00

0

( )W

W z z zz

Beam Divergence

z

W(z)

2W0

W0

Beam waist

z0-z0

The beam radius W(z) has its minimum value W0 at the waist (z=0) reaches at z=±z0 and increases linearly with z for large z.

02W

00W

2

122

20

2

0

])[(2)(

22

zW

W

z

dz

zdW

20

0

Wz

∵

00W


The characteristics of divergence angle

• z=0 ， 2 =0

• z= 2 =

• z 2 =

0/2 W20

0W z

0

2

W

orz

zWx

)(2lim2

Define f=z0 as the confocal parameter of Gaussian beam2

00

Wf z

fz

fzf

z

zWzz

2

)1(2

lim)(2

lim22

2

The physical means of f ： the half distance between two section of width



Depth of Focus

Since the beam has its minimum width at z = 0, it achieves its best focus at the plane z = 0. In either direction, the beam gradually grows “out of focus.” The axial distance within which the beam radius lies within a factor 20.5 of its minimum value (i.e., its area lies within a factor of 2 of its minimum) is known as the depth of focus or confocal parameter

0

2z

z

The depth of focus of a Gaussian beam.

20

0

22 2

Wz f


Phase of Gaussian beam

The phase of the Gaussian beam is,

2

( , ) ( )2 ( )

kz kz z

R z

(0, ) ( )z kz z On the beam axis (p = 0) the phase

Phase of plan wave

an excess delay of the wavefront in comparison with a plane wave or a spherical waveThe excess delay is –/2 at z=-∞, and /2 at z= ∞

kz

( )z

The total accumulated excess retardation as the wave travels from z = -∞ to z =∞is. This phenomenon is known as the Guoy effect.

12

0 0

( ) arctanz z

z tgW z


Wavefront

z

fz

z

WzzR

22

20 ])(1[)(

Confocal field and its equal phase front


Parameters Required to Characterize a Gaussian Beam

How many parameters are required to describe a plane wave, a spherical wave, and a Gaussian beam?

The plane wave is completely specified by its complex amplitude and direction.

The spherical wave is specified by its amplitude and the location of its origin.

The Gaussian beam is characterized by more parameters- its peak amplitude the parameter A, its direction (the beam axis), the location of its waist, and one additional parameter: the waist radius W0 or the Rayleigh range zo,


Parameter used to describe a Gaussian beam

q-parameter is sufficient for characterizing a Gaussian

beam of known peak amplitude and beam axis

If the complex number q(z) = z + iz0, is known, the distance z to the beam waist and the Rayleigh range z0. are readily identified as the real and imaginary parts of q(z).

20

1 1 1 1

( ) ( ) ( ) ( )i

q z R z W z q z z iz

q(z) = z + iz0

the real part of q(z) z is the beam waist place

the imaginary parts of q(z) z0 is the Rayleigh range


a). Transmission Through a Thin Lens

R is positive since the wavefront of the incident beam is diverging and R’ isnegative since the wavefront of the transmitted beam is converging.

1 1 1

'R R f

C. TRANSMISSION THROUGH OPTICAL COMPONENTS

2 2 2

2 2 2 'kz k k kz k

R f R

1 1 1

'R R f

Notes:

Phase +phase induce by lens must equal to the back phase


'

1 1 1

'

W W

R R f

2'2 2 2 1

0

2 12

[1 ( ) ]'

'' '[1 ( ) ]

WW W

RR

z RW

In the thin lens transform, we have

If we know we can get0 1, ,W z f

The minus sign is due to the waist lies to the right of the lens.


0 2 2 1/ 2'

[1 ( / ') ]

WW

W R

2 2

''

1 ( '/ )

Rz

R W

0 0'W MW

2( ' ) ( )z f M z f ' 20 02 (2 )z M z

' 20 02 (2 )M

2 1/ 2(1 )rM

Mr

0zrz f

r

fM

z f

，

Waist radius

Waist location

Depth of focus

Divergence angle

magnification where

2 2 1/ 20 0 0[1 ( / ) ] [1 ( / ) ]R z z z and W W z z because

The beam waist is magnified by M, the beam depth of focus is magnified by M2, and the angular divergence is minified by the factor M.

The formulas for lens transformation


Limit of Ray Optics

Consider the limiting case in which (z - f) >>zo, so that the lens is well outside the depth of focus of the incident beam, The beam may then be approximated by a spherical wave, thus

0 0'W MW

1 1 1

'z z f

r

fM M

z f

The magnification factor Mr is that based on ray optics. provides that M < Mr, the maximum magnification attainable is the ray-optics magnification Mr.

2W02W0’

z z’

0 0 r

zr and M M

z f

Imaging relation


b). Beam Shaping

Beam Focusing If a lens is placed at the waist of a Gaussian beam, so z=0, then

00 2 1/ 2

0

'[1 ( / ) ]

WW

z f

20

'1 ( / )

fz

f z

If the depth of focus of the incident beam 2z0, is much longer than the focal length f of the lens, then W0’= ( f/zo)Wo. Using z0 =W0

2/, we obtain

0 00

'W f fW

'z f

The transmitted beam is then focused at the lens’ focal plane as would be expected for parallel rays incident on a lens. This occurs because the incident Gaussian beam is well approximated by a plane wave at its waist. The spot size expected from ray optics is zero

2 1/ 20

1

[1 ( / ) ]M

z f

∵


Focus of Gaussian beam 22 00 2

201

'

(1 ) ( )

WW

Wz

f f

0 'W

fz 1

01 z

fz 1

z f2

0Wf

For given f, changes as 20'W

20'W•when decreases as z decreases

reaches minimum, and M<1, for f>0, it is focal effect

•when , increases as z increases

fz 1

0'W

0 'W

•when the bigger z, smaller f, better focus

•when reaches maximum, when , it will be

• focus


In laser scanning, laser printing, and laser fusion, it is desirable to generate the smallest possible spot size, this may be achieved by use of the shortest possible wavelength, the thickest incident beam, and the shortest focal length. Since the lens should intercept the incident beam, its diameter D must be at least 2W0. Assuming that D = 2Wo, the diameter of the focused spot is given by

0 #

42 'W F

#

fF

D

where F# is the F-number of the lens. A microscope objective with small F-number is often used.


Beam collimate

locations of the waists of the incident and transmitted beams, z and z’ are

2 20

' / 11

( / 1) ( / )

z z f

f z f z f

The beam is collimated by making the location of the new waist z’ as distant as possible from the lens. This is achieved by using the smallest ratio z0/f (short depth of focus and long focal length).


Beam expanding

A Gaussian beam is expanded and collimated using two lenses offocal lengths fi and f2,

Assuming that f1<< z and z - f1>> z0, determine the optimal distance d between the lenses such that the distance z’ to the waist of the final beam is as large as possible.

overall magnification M = W0’/Wo


C). Reflection from a Spherical Mirror

Reflection of a Gaussian beam of curvature R1 from a mirror of curvature R:

2 1W W2 1

1 1 2

R R R

R > 0 for convex mirrors and R < 0 for concave mirrors,

f = -R/2.


If the mirror is planar, i.e., R =∞, then R2= R1, so that the mirror reverses

the direction of the beam without altering its curvature If R1= ∞, i.e., the beam waist lies on the mirror, then R2= R/2. If the mirror is

concave (R < 0), R2 < 0, so that the reflected beam acquires a negative

curvature and the wavefronts converge. The mirror then focuses the beam to a smaller spot size.

If R1= -R, i.e., the incident beam has the same curvature as the mirror, then

R2= R. The wavefronts of both the incident and reflected waves coincide

with the mirror and the wave retraces its path. This is expected since the wavefront normals are also normal to the mirror, so that the mirror reflects the wave back onto itself. the mirror is concave (R < 0); the incident wave is

diverging (R1 > 0) and the reflected wave is converging (R2< 0).

R 1R 1R R


d). Transmission Through an Arbitrary Optical System

An optical system is completely characterized by the matrix M of elements (A, B, C, D) ray-transfer matrix relating the position and inclination of the transmitted ray to those of the incident ray

The q-parameters, q1 and q2, of the incident and transmitted Gaussian beams at the input and output planes of a paraxial optical system described by the (A, B, C, D) matrix are related by


The q-parameters, q1 and q2, of the incident and transmitted Gaussian beams at the input and output planes of a par-axial optical system described by the (A, B, C, D) matrix are related by

12

1

Aq Bq

Cq D

Because the q parameter identifies the width W and curvature R of the Gaussian beam, this simple law, called the ABCD law

ABCD law

Invariance of the ABCD Law to Cascading If the ABCD law is applicable to each of two optical systems with matrices Mi =(Ai, Bi, Ci, Di), i = 1,2,…, it must also apply to a system comprising their cascade (a system with matrix M = M1M2).


C. HERMITE - GAUSSIAN BEAMS

The self-reproducing waves exist in the resonator, and resonating inside of spherical mirrors, plan mirror or some other form paraboloidal wavefront mirror, are called the modes of the resonator

Hermite - Gaussian Beam Complex Amplitude

2 20

, ,

2 2( , , ) [ ] [ ] [ ] exp[ ( 1) ( )]

( ) ( ) ( ) 2 ( )l m l m l m

W x y x yU x y z A G G jkz jk j l m z

W z W z W z R z

where2

( ) ( ) exp( ), 0,1, 2,...,2l l

uG u H u l

is known as the Hermite-Gaussian function of order l, and Al,m is a constant

Hermite-Gaussian beam of order (I, m).

The Hermite-Gaussian beam of order (0, 0) is the Gaussian beam.


H0(u) = 1, the Hermite-Gaussian function of order O, the Gaussian function. G1(u) = 2u exp( -u2/2) is an odd function, G2(u) = (4u2 - 2) exp( -u2/2) is even, G3(u) = (8u3 - 12u)exp( -u2/2) is odd,


Intensity Distribution

The optical intensity of the (I, m) Hermite-Gaussian beam is

2 2 2 20, ,

2 2( , , ) [ ] [ ] [ ]

( ) ( ) ( )l m l m l m

W x yI x y z A G G

W z W z W z


Home work 2

• Exercises in English 2,3,4 about the Gaussian Beam


A. Gaussian Modes• Gaussian beams are modes of the spherical-mirror resonator;

Gaussian beams provide solutions of the Helmholtz equation under the boundary conditions imposed by the spherical-mirror resonator

z

Beam

radius

a Gaussian beam is a circularly symmetric wave whose energy is confined about its axis (the z axis) and whose wavefront normals are paraxial rays

2.4 Gaussian beam in Spherical-Mirror Resonators


Gaussian beam intensity:

2 22 21

20

2 2( )2( )0

0 ( )

x y zx y i k z tgR zW zW

I I e eW z

2 1/ 20

0

( ) [1 ( ) ]z

W z Wz

20( )z

R z zz

20

0

Wz

Beam waist

Beam width

2 20 0

00 0

( ) 1z zz z

R R z z z zz z z z

where z0 is the distance called Rayleigh range, at which the beam wavefronts are most curved or we usually called confocal prrameter

minimum value W0 at the beam waist (z = 0).

The radius of curvature

The Rayleigh range z0

00

zW


B. Gaussian Mode of a Symmetrical Spherical-Mirror Resonator

dR1 R2

z0z1 z2

2 1z z d 20

1 11

zR z

z

20

2 22

zR z

z

21 2 1

2 1

( ),

2

d R dz z z d

R R d

2 1 2 2 10 2

2 1

( )( )( 2 )

( 2 )

d R d R d R R dz

R R d


the beam radii at the mirrors 2 1/ 20

0

[1 ( ) ] , 1, 2.ii

zW W i

z

An imaginary value of z0 signifies that the Gaussian beam is in fact a paraboloidal wave, which is an unconfined solution, so that z0 must be real. it is not difficult to show that the condition z0

2 > 0 is equivalent to

1 2

0 (1 )(1 ) 1d d

R R


Gaussian Mode of a Symmetrical Spherical-Mirror Resonator

Symmetrical resonators with concave mirrors that is R1 = R2= -/RI so that z1 = -d/2, z2 = d/2. Thus the beam center lies at the center

1/ 20 (2 1)

2

Rdz

d

2 1/ 20 (2 1)

2

RdW

d

2 21 2 1/ 2

/

{( / )[2 ( / )]}

dW W

d R d R

The confinement condition becomes

0 2d

R


Given a resonator of fixed mirror separation d, we now examine the effect of increasing mirror curvature (increasing d/lRI) on the beam radius at the waist W0, and at the mirrors Wl = W2.

As d/lRI increases, W0 decreases until it vanishes for the concentric resonator (d/lR| = 2); at this point W1 = W2 = ∞

The radius of the beam at the mirrors has its minimum value, WI = W2= (d/)1/2, when d/lRI = 1

0 2

dz 1/ 2

0 ( )2

dW

1 2 02W W W


C. Resonance Frequencies of a Gaussian beam

The phase of a Gaussian beam,2 2

1

0

( )( , , ) ( )

2 ( )

k x yzx y z kz tg z R z

At the locations of the mirrors z1 and z2 on the optical aixe (x2+y2=0), we have,

2 1 2 1 2 1(0, ) (0, ) ( ) [ ( ) ( )]z z k z z z z kd where1

0

( )z

z tgz

As the traveling wave completes a round trip between the two mirrors, therefore, itsphase changes by 2 2kz For the resonance, the phase must be in condition 2 2 2 , 1, 2, 3...kz q q

If we consider the plane wave resonance frequency 22F

ck andc d

We have q F Fq


Spherical-Mirror Resonator Resonance Frequencies (Gaussian Modes)

q F Fq

1. The frequency spacing of adjacent modes is VF = c/2d, which is the same

result as that obtained for the planar-mirror resonator.

2. For spherical-mirror resonators, this frequency spacing is independent of the curvatures of the mirrors.

3. The second term in the fomula, which does depend on the mirror curvatures, simply represents a displacement of all resonance frequencies.


D. Hermite - Gaussian ModesThe resolution for Helmholtz equation An entire family of solutions, the Hermite-Gaussian family, exists. Although a Hermite-Gaussian beam of order (I, m) has the same wavefronts as a Gaussian beam, its amplitude distribution differs . It follows that the entire family of Hermite-Gaussian beams represents modes of the spherical-mirror resonator

2 20

, ,

2 2( , , ) [ ] [ ] [ ] exp[ ( 1) ( )]

( ) ( ) ( ) 2 ( )l m l m l m

W x y x yU x y z A G G jkz jk j l m z

W z W z W z R z

(0, ) ( 1) ( )z kz l m z

Spherical mirror resonator Resonance Frequencies(Hermite -Gaussian Modes)

, , ( 1)l m q F Fq l m

2 2( 1) 2 , 0, 1, 2,...,kd l m q q


longitudinal or axial modes: different q and same indices (l, m) the intensity will be the same

transverse modes: The indices (I, m) label different means different spatial intensity dependences

Longitudinal modes corresponding to a given transverse mode (I, m) have resonance frequencies spaced by vF = c/2d, i.e., vI,m,q – vI’,m’,q = vF.

Transverse modes, for which the sum of the indices l+ m is the same, have the same resonance frequencies.

Two transverse modes (I, m), (I’, m’) corresponding mode q frequencies spaced

, , ', ', [( ) ( ' ')]l m q l m q Fl m l m

, , ( 1)l m q F Fq l m


*E. Finite Apertures and Diffraction LossSince the resonator mirrors are of finite extent, a portion of the optical power escapes from the resonator on each pass. An estimate of the power loss may be determined by calculating the fractional power of the beam that is not intercepted by the mirror. That is the finite apertures effect and this effect will cause diffraction loss.

If the Gaussian beam with radius W and the mirror is circular with radius a and a= 2W, each time there is a small fraction, exp( - 2a2/ W2) = 3.35 x10-4, of the beam power escapes on each pass.

For example:

Higher-order transverse modes suffer greater losses since they have greater spatial extent in the transverse plane.

In the resonator, the mirror transmission and any aperture limitation will induce loss The aperture induce loss is due to diffraction loss, and the loss depend mainly on the

diameters of laser beam, the aperture place and its diameter We can used Fresnel number N to represent the relation between the size of light

beam and the aperture, and use N to represent the loss of resonator.


The Fresnel number NF

Diffraction loss

Physical meaning ： the ratio of the accepting angle (a/d) (form one mirror to the other of the resonator )to diffractive angle of the beam (/a) .

The higher Fresnel number corresponds to a smaller loss

2 2 2

22F

a a aN

d z W

Attention: the W here is the beam width in the mirror, a is the dia. of mirror


Symmetric confocal resonator

For general stable concave mirror resonator, the Fresnel number for two mirrors are:

12 21 1 1 2

1 1 221 2

12 22 2 2 2

2 1 222 1

[ (1 )]

[ (1 )]

F

F

a a gN g g

W d g

a a gN g g

W d g

2 21 2

2 21 2

F

a aN

W W

N is the maximum number of trip that light will propagate in side resonator without escape.

1/N represent each round trip the ratio of diffraction loss to the total energy


Home work 3• The light from a Nd:YAG laser at wavelength 1.06 m is a Gaussian beam of 1 W optical power and

beam divergence 2= 1 mrad. Determine the beam waist radius, the depth of focus, the maximum

intensity, and the intensity on the beam axis at a distance z = 100 cm from the beam waist.

• Beam Focusing. An argon-ion laser produces a Gaussian beam of wavelength= 488 nm and waist radius w0 = 0.5 mm. Design a single-lens optical system for focusing the light to a spot of

diameter 100 pm. What is the shortest focal-length lens that may be used?

• Spot Size. A Gaussian beam of Rayleigh range z0 = 50 cm and wavelength =488nm is converted

into a Gaussian beam of waist radius W0’ using a lens of focal length f = 5 cm at a distance z from

its waist. Write a computer program to plot W0’ as a function of z. Verify that in the limit z - f >>z0 ,

the relations (as follows) hold; and in the limit z << z0 holds.

• Beam Refraction. A Gaussian beam is incident from air (n = 1) into a medium with a planar boundary and refractive index n = 1.5. The beam axis is normal to the boundary and the beam waist lies at the boundary. Sketch the transmitted beam. If the angular divergence of the beam in air is 1 mrad, what is the angular divergence in the medium?

0 0'W MW r

fM M

z f

0

0 2 1/ 20

'[1 ( / ) ]

WW

z f

Page 41, Problems : 1,4,5,6,10,11


Resonance Frequencies of a Resonator with an Etalon. (a) Determine the spacing between adjacent resonance frequencies in a resonator constructed of two parallel planar mirrors separated by a distance d = 15 cm in air (n = 1).(b) A transparent plate of thickness d1= 2.5 cm and refractive index n = 1.5 is placed inside the resonator and is tilted slightly to prevent light reflected from the plate from reaching the mirrors. Determine the spacing between the resonance frequencies of the resonator.

Mirrorless Resonators. Semiconductor lasers are often fabricated from crystals whose surfaces are cleaved along crystal planes. These surfaces act as reflectors and therefore serve as the resonator mirrors. Consider a crystal with refractive index n = 3.6 placed in air (n = 1). The light reflects between two parallel surfaces separated by the distance d = 0.2 mm. Determine the spacing between resonance frequencies vF, the overall distributed loss coefficient r, the finesse F, and the spectral width v. Assume that the loss coefficient (s= 1 cm-1).

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Chapter 2 Optical Resonator and Gaussian Beam optics