2D optical trapping potential for the confinement of heteronuclear molecules 3

... xd¯ y (A .3. 3) Therefore, aside from a constant phase shift term e2ikf , the output field is a Fourier Transform of the input field, hence the name of Fourier imaging in this case For the relay... behind the lens (refer to the right picture of figure A .3) Therefore, the output field is obtained by first propagating the input beam over a distance f , 80 Figure A .3: Optical setup for (Left)... multiplying the field with the lens quadratic phase factor, and then propagating it over another f distance Let us perform the explicit calculation sequentially The field at the plane just before the

Appendix A

Fourier Optics

A.1 Beam Propagation Equation

In this chapter, we summarize some important results of the Fourier Optics treatment, which treats the propagation of a diffracted beam Let us consider a monochromatic beam propagating

in a particular direction, for example along the Z axis, and incident to an aperture Σ Following Huygens-Fresnel principle, the electric field in a plane perpendicular to Z axis is a summation

of many wavelets emitted from every point of the aperture (refer to figure A.1) Adopting a paraxial approximation, the electric field at a point on a plane located at distance z away from the aperture is given by: [Goodman, 1996]

E(x, y, z) = e

ikz

iλz e

ik 2z (x2+y2)

Σ

 E(ξ, η, 0) e2zik(ξ2+η2)e2πiλz (xξ+yη)dξdη, (A.1.1)

where k = 2π/λ is the wavenumber and λ the beam wavelength Notice the linearity of the structure of equation A.1.1 above Writing the integration as an operator: E(z) = Pz(E(0)),

we can easily prove that E(z1 + z2) = Pz2· Pz1(E(0)) This property validates the fact that the equation A.1.1 can be interpreted as the propagation equation: it describes the field at a distance z away from a source by a Fourier Transform integral of the field at the plane of the source

Figure A.1: Huygens principle of diffraction Figure is taken from [Goodman, 1996]

A.2 The Effect of a Thin Spherical Lens

A lens is an optical component that is used a lot in the beam shaping schemes we have considered

in this report It is therefore important to consider how a lens alters the propagation of a beam Let us consider a lens which has a position-dependent thickness ∆(x, y) A lens consists of two spherical faces (with radius of curvature R1 and R2 respectively) separated by a certain thickness ∆0 Let us consider the thickness of the lens at position (x, y) from the center as

shown in the right side of figure A.2 The thickness due to the left part of the curved side is given by:

d1 = ∆01



R1 −

q

R12− x2− y2



2R1

where ∆01is the central thickness of this curved face Here, we have used a paraxial approxima-tion assuming that the size of the lens is small compared to its curvature (x,y R1) The total thickness of the lens will include a thickness d2 between the two curved face and the thickness

d3 ≈ ∆03− 1

2R 2(x2 + y2) of the second curved face Thus, the total thickness of the lens is given by:

d(x, y) = d1 + d2 + d3 = ∆0 −

 1 2R1

2R2



We remark that in this equation, the radius of curvatures are defined to be positive when the face is convex, and negative when it is concave Therefore, we see that a double convex lens is thickest in the middle, as opposed to a double concave lens which is thinnest in the middle

Figure A.2: The geometry of a spherical lens

As lenses are usually very thin, a beam is not appreciably distorted as it passes through a lens The most dominant effect is a phase shift due to the lens material Suppose that the lens has an index of refraction equals to n The phase shift acquired as the beam propagates through

a material is equal to exp(iklo), where lo is the optical length which is equal to the physical length multiplied by the index of refraction of the material The optical length in function of position is:

lo(x, y) = n · d(x, y)

lens

+ 1 · (∆0− d(x, y))

air

= n∆0− (n−1)( 1

2R1+

1 2R2(x

2+y2) = n∆0− 1

2f(x

2+y2)

(A.2.3) The last line of the above equation uses the lens maker equation which relates the focal length

f to the lens curvatures:

1

f = (n − 1)

 1

R1 +

1

R2



(A.2.4)

As we can see, the effect of a lens, aside from a constant phase-shift due to its thickness is a quadratic phase shift term: exp−ik

2f(x2+ y2)

A.3 Special Optical Configurations

In this section, we will apply the propagation and lens equation we have derived to explain the imaging property of the two setups used in the manuscript: the Fourier imaging by a single positive lens and the relay telescope For the first, we recall that the setup involves an input field pattern Einwhich is placed at a distance f away behind a lens, and we observe the output profile at the plane f distance away behind the lens (refer to the right picture of figure A.3) Therefore, the output field is obtained by first propagating the input beam over a distance f ,

80

Figure A.3: Optical setup for (Left) a relay telescope and (Right) a single-lens Fourier imaging.

multiplying the field with the lens quadratic phase factor, and then propagating it over another

f distance

Let us perform the explicit calculation sequentially The field at the plane just before the lens Ebl is the propagated input field Ein over a distance f :

Ebl(x, y) = e

ikf

iλf e

ik 2f (x 2 +y 2 ) Z Z 

Ein(X, Y ) e2fik(X 2 +Y 2 )

e2πiλf (xX+yY )

We notice that the quadratic phase factor e−2fik(x 2 +y 2 )

of the lens exactly cancels out the quadratic phase term in the field Ebl Hence the expression of the field at the plane just after the lens Eal is given by:

Eal(x, y) = e

ikf

iλf



Ein(X, Y ) e2fik(X 2 +Y 2 )

e2πiλf (xX+yY )

Finally, the output field profile Eout is obtained by propagating Ealover a distance f :

Eout(x, y) = e

ikf

iλf e

ik 2f (x 2 +y 2 ) Z Z 

Eal(X, Y ) e2fik(X 2 +Y 2 )

e2πiλf (xX+yY )

dXdY

2ikf

(λf )2 e2fik(x 2 +y 2 ) Z Z Z Z

Ein(¯x, ¯y) e2fik(¯ x 2 +¯ y 2 )

e2fik(X 2 +Y 2 )

e2πiλf (X ¯ x+Y ¯ y)

e2πiλf (xX+yY )

dXdY d¯xd¯y

2ikf

iλf

Ein(¯x, ¯y) e2πiλf (¯ xx+¯ yy)

Therefore, aside from a constant phase shift term e2ikf, the output field is a Fourier Transform

of the input field, hence the name of Fourier imaging in this case

For the relay telescope, we notice that the setup can be broken down into two Fourier imaging setups with lens f1 followed by f2 The field at the Fourier plane of the first lens (i.e a plane of distance f1 behind the first lens, f2 in front of the second lens) is given by the Fourier Transform of the input field Ein:

EF P = e

2ikf 1

iλf1

Ein(X, Y ) e2πiλf (Xx+Y y)

and the output field is given by the Fourier Transform of the field at the Fourier plane:

Eout(x, y) = e

2ikf 2

iλf2

EF P(X, Y ) e

2πi λf2(Xx+Y y)dXdY

= −e

2ik(f 1 +f 2 )

λ2f1f2

Ein(¯x, ¯y) eλf22πi(Xx+Y y)e2πiλf1(X ¯x+Y ¯y)dXdY d¯xd¯y

= −e

2ik(f 1 +f 2 )

λ2f1f2

Ein(¯x, ¯y) δ

 x

λf2 +

¯ x

λf1

 δ

 y

λf2 +

¯

λf1

 d¯xd¯y

= −f1

f2 Ein



−f1

f2x, −

f1

f2y



Therefore, the output field of a relay telescope arrangement is proportional to the input field with a magnification factor of −f2/f1 which we recognize from the classical optics

82

Appendix B

Gaussian Beam Properties

B.1 Gaussian Beam Propagation

In this section, we will give a brief introduction to the important properties of a TEM00Gaussian mode beam which is the idealized lasing mode of commercial lasers The Gaussian mode is one of the allowed solution of the Helmholtz equation, which governs the propagation of an electromagnetic wave in space The field of a Gaussian-mode beam propagating along the positive Z direction can be described as [Siegman, 1986]:

Eg(x, y, z) =

 2P πw(z)2

1/2

exp



−x

2+ y2 w(z)2

 exp



−ikx

2+ y2 2R(z)



eiψ(z)eikz (B.1.1)

[Siegman, 1986]

Notice that the field expression can be broken down into four components:

ˆ Amplitude distribution

In any plane perpendicular to the Z axis, the electric field amplitude follows a Gaussian distribution: πw(z)2P 2

1/2

exp−xw(z)2+y22, where P denotes the power of the beam The equivalent radius of the beam is traditionally set as the distance where the amplitude falls

to 1/e of the maximum amplitude (which is positioned at the center of the coordinate)

As we can see, this distance is given by w(z), which is called the spot size of the beam

The spot size varies as the beam propagates in space, with its evolution given by:

w(z) = w0

s

1 + z

zR

2

where zR = πw2

0/λ is known as the Rayleigh length of the beam, and w0 is called the waist of the beam The beam waist is in fact the smallest spot size of the beam, and it is positioned at z = 0 in this convention As we follow the beam propagation starting from

z = −∞, the spot size first shrunk until it is equal to the waist, then re-expands The Rayleigh length is the distance along z between the waist and the position where the spot size has grown to √2w0 This range gives an estimation of a range for which the beam spot size is approximately constant (i.e collimated beam in the classical optics point of view)

[CVI Melles Griot, ]

As we can observe from the expression of the Rayleigh length, a Gaussian beam with a larger waist has a larger Rayleigh length, meaning that they stay collimated over a longer distance In addition, when the beam is very far away from the waist (z  zR), the spot size grows approximately linearly:

In this condition, the beam is not collimated; it is either expanding or focusing as it prop-agates in space

ˆ Spherical phase curvature

The second part of the field is a phase factor with a spherical phase front: exp−ikx2R(z)2+y2 The radius of curvature of this phase term is:

2 R

We notice that the curvature at the plane of the beam waist (z = 0) is infinity, meaning that the phase front at this plane is flat Otherwise, the spherical phase front is always curving outwards with respect to the plane of the waist (see figure B.1)

ˆ Gouy phase and propagation phase

84

The last two terms are the extra phase shift called the Gouy phase: eiψ(z) and a customary phase shift due to the propagation eikz The Gouy phase shift is given by:

A compact way of describing the Gaussian beam is to utilize the complex beam parameter defined as:

The field (ignoring the constant phase shift and the Gouy phase) can then be described in term

of this single parameter:

Eg(x, y, z) = E0 exp



−ik(x

2+ y2) 2q(z)



(B.1.7)

B.2 Focusing through a Lens

Let us consider the setting where a Gaussian beam is incident on a lens with a focal length f If

we let the position of the waist to be d1 in front of the lens, we could calculate the output field

at any position behind the lens by the Fourier Optics formulation considered in the appendix

A The resulting beam after the lens is still a Gaussian mode, but with a change in the size of the waist and its position Denoting the position of the waist of the focused beam as d2 (with the convention of d2 = 0 at the lens), this position is given an equation only slightly distinct from a classical lens equation [CVI Melles Griot, ]:

1

d1+ z2R/(d1− f ) +

1

d2 =

1

and the waist of the focused beam w is given by:

In particular, in the Fourier imaging setup where d1 = f , the resulting output beam is located exactly at the back-Fourier plane of the lens (d2 = f ) and the focused beam waist is given by:

Therefore, a larger input beam will be focused as a smaller beam, which is what we expect from the Fourier Transform relation

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