: the gaussian beam and abcd formalism steck section 6.5 ... · a gaussian beam is a paraboloidal...
TRANSCRIPT
A Gaussian beam is a paraboloidal wave whose origin is shifted to a complex location q-‐parameter
Beam width
Min. Beam width (or beam waist)
Is the Rayleigh length
Beam radius of curvature
The q-parameter encodes the beam width and radius of curvature at each position on its trajectory along z
Pre-class reading summary: The Gaussian Beam and ABCD formalism Steck section 6.5 (ABCD matrix formalism) and Saleh-Teich Chapter 3§1B and §2 (beam parameters from measurements, beam quality, beam shaping
Plan for today:
1) The ABCD law for rays and for Gaussian Beams
A Gaussian beam is a paraboloidal wave whose origin is shifted to a complex location q-‐parameter
Beam width
Min. Beam width (or beam waist)
Is the Rayleigh length
Beam radius of curvature
The q-parameter encodes the beam width and radius of curvature at each position on its trajectory along z
Pre-class reading summary: The Gaussian Beam and ABCD formalism Steck section 6.5 (ABCD matrix formalism) and Saleh-Teich Chapter 3§1B and §2 (beam parameters from measurements, beam quality, beam shaping
Optical elements change the q-parameter according to the ABCD law. The ABCD elements are the same that appear in the Ray Matrix formalism for “Ray Optics”
1a) What is the ABCD transformaIon law for a Gaussian beam? That, how is q1 related to q2 in terms of the A,B,C, and D values?
1b) What is required for this law to hold?
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
A Gaussian beam is a paraboloidal wave whose origin is shifted to a complex location q-‐parameter
Beam width
Min. Beam width (or beam waist)
Is the Rayleigh length
Beam radius of curvature
The q-parameter encodes the beam width and radius of curvature at each position on its trajectory along z
Pre-class reading summary: The Gaussian Beam and ABCD formalism Steck section 6.5 (ABCD matrix formalism) and Saleh-Teich Chapter 3§1B and §2 (beam parameters from measurements, beam quality, beam shaping
Optical elements change the q-parameter according to the ABCD law. The ABCD elements are the same that appear in the Ray Matrix formalism for “Ray Optics”
1a) What is the ABCD transformaIon law for a Gaussian beam? That, how is q1 related to q2 in terms of the A,B,C, and D values?
1b) What is required for this law to hold?
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
A Gaussian beam is a paraboloidal wave whose origin is shifted to a complex location q-‐parameter
Beam width
Min. Beam width (or beam waist)
Is the Rayleigh length
Beam radius of curvature
The q-parameter encodes the beam width and radius of curvature at each position on its trajectory along z
Pre-class reading summary: The Gaussian Beam and ABCD formalism Steck section 6.5 (ABCD matrix formalism) and Saleh-Teich Chapter 3§1B and §2 (beam parameters from measurements, beam quality, beam shaping
Optical elements change the q-parameter according to the ABCD law. The ABCD elements are the same that appear in the Ray Matrix formalism for “Ray Optics”
1a) What is the ABCD transformaIon law for a Gaussian beam? That, how is q1 related to q2 in terms of the A,B,C, and D values?
1b) What is required for this law to hold? The Gaussian beam stays Gaussian through any optical system as long as the paraxial approximation is valid. As soon as the paraxial approximation breaks down (i.e., nonlinear terms become important), the beam will become non-Gaussian.
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
2b) Using this vector representaIon for rays 1 and 2, what is the most general linear transform between 1 and 2? Again state your reasoning in your own words. What have you assumed about the rays? Why is this assumpIon required?
2c) If there is only free space (a distance d) from ray 1 to ray 2, what is the appropriate transform? How does the ray change?
z
y 1
2
2a) What is the 1x2 column vector that we can use to characterize an opIcal ray (such as ray 1 and ray 2)? Why are only 2 elements required?
2d) What is the transform if between ray 1 to ray 2 is a dielectric boundary with index of refracIon n1 and n2 respecIvely
In Ray Op'cs, we treat the propagaIon of light along a ray as pictured below
2e) What is the transform if between ray 1 to ray 2 is free space (a distance d) FOLLOWED by a dielectric boundary with index of refracIon n1 and n2 respecIvely?
2b) Using this vector representaIon for rays 1 and 2, what is the most general linear transform between 1 and 2? Again state your reasoning in your own words. What have you assumed about the rays? Why is this assumpIon required?
2c) If there is only free space (a distance d) from ray 1 to ray 2, what is the appropriate transform? How does the ray change?
2a) What is the 1x2 column vector that we can use to characterize an opIcal ray (such as ray 1 and ray 2)? Why are only 2 elements required?
2d) What is the transform if between ray 1 to ray 2 is a dielectric boundary with index of refracIon n1 and n2 respecIvely
2e) What is the transform if between ray 1 to ray 2 is free space (a distance d) FOLLOWED by a dielectric boundary with index of refracIon n1 and n2 respecIvely?
y: height off optical axis theta: angle makes w.r.t. axis
- Only two elements since we assume cylindrical symmetry - a ray has infinite length, so we don't quantify it.
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teichc
2b) Using this vector representaIon for rays 1 and 2, what is the most general linear transform between 1 and 2? Again state your reasoning in your own words. What have you assumed about the rays? Why is this assumpIon required?
2c) If there is only free space (a distance d) from ray 1 to ray 2, what is the appropriate transform? How does the ray change?
2d) What is the transform if between ray 1 to ray 2 is a dielectric boundary with index of refracIon n1 and n2 respecIvely
2e) What is the transform if between ray 1 to ray 2 is free space (a distance d) FOLLOWED by a dielectric boundary with index of refracIon n1 and n2 respecIvely?
Most general linear transform Is multiplication by a 2x2 matrix
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
2b) Using this vector representaIon for rays 1 and 2, what is the most general linear transform between 1 and 2? Again state your reasoning in your own words. What have you assumed about the rays? Why is this assumpIon required?
2c) If there is only free space (a distance d) from ray 1 to ray 2, what is the appropriate transform? How does the ray change?
2d) What is the transform if between ray 1 to ray 2 is a dielectric boundary with index of refracIon n1 and n2 respecIvely
2e) What is the transform if between ray 1 to ray 2 is free space (a distance d) FOLLOWED by a dielectric boundary with index of refracIon n1 and n2 respecIvely?
Most general linear transform Is multiplication by a 2x2 matrix
We have assumed that we can write
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
2b) Using this vector representaIon for rays 1 and 2, what is the most general linear transform between 1 and 2? Again state your reasoning in your own words. What have you assumed about the rays? Why is this assumpIon required?
2c) If there is only free space (a distance d) from ray 1 to ray 2, what is the appropriate transform? How does the ray change?
2d) What is the transform if between ray 1 to ray 2 is a dielectric boundary with index of refracIon n1 and n2 respecIvely
2e) What is the transform if between ray 1 to ray 2 is free space (a distance d) FOLLOWED by a dielectric boundary with index of refracIon n1 and n2 respecIvely?
Most general linear transform Is multiplication by a 2x2 matrix
We have assumed that we can write
Paraxial approximation!
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
2c) If there is only free space (a distance d) from ray 1 to ray 2, what is the appropriate transform? How does the ray change?
2d) What is the transform if between ray 1 to ray 2 is a dielectric boundary with index of refracIon n1 and n2 respecIvely
2e) What is the transform if between ray 1 to ray 2 is free space (a distance d) FOLLOWED by a dielectric boundary with index of refracIon n1 and n2 respecIvely?
Most general linear transform Is multiplication by a 2x2 matrix
We have assumed that we can write
Paraxial approximation!
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
2d) What is the transform if between ray 1 to ray 2 is a dielectric boundary with index of refracIon n1 and n2 respecIvely
2e) What is the transform if between ray 1 to ray 2 is free space (a distance d) FOLLOWED by a dielectric boundary with index of refracIon n1 and n2 respecIvely?
Most general linear transform Is multiplication by a 2x2 matrix
We have assumed that we can write
Paraxial approximation!
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
2d) What is the transform if between ray 1 to ray 2 is a dielectric boundary with index of refracIon n1 and n2 respecIvely
2e) What is the transform if between ray 1 to ray 2 is free space (a distance d) FOLLOWED by a dielectric boundary with index of refracIon n1 and n2 respecIvely?
Most general linear transform Is multiplication by a 2x2 matrix
We have assumed that we can write
Paraxial approximation!
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
2d) What is the transform if between ray 1 to ray 2 is a dielectric boundary with index of refracIon n1 and n2 respecIvely
2e) What is the transform if between ray 1 to ray 2 is free space (a distance d) FOLLOWED by a dielectric boundary with index of refracIon n1 and n2 respecIvely?
Most general linear transform Is multiplication by a 2x2 matrix
We have assumed that we can write
Paraxial approximation!
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
2d) What is the transform if between ray 1 to ray 2 is a dielectric boundary with index of refracIon n1 and n2 respecIvely
2e) What is the transform if between ray 1 to ray 2 is free space (a distance d) FOLLOWED by a dielectric boundary with index of refracIon n1 and n2 respecIvely?
Most general linear transform Is multiplication by a 2x2 matrix
We have assumed that we can write
Paraxial approximation!
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
2d) What is the transform if between ray 1 to ray 2 is a dielectric boundary with index of refracIon n1 and n2 respecIvely
2e) What is the transform if between ray 1 to ray 2 is free space (a distance d) FOLLOWED by a dielectric boundary with index of refracIon n1 and n2 respecIvely?
Most general linear transform Is multiplication by a 2x2 matrix
We have assumed that we can write
Paraxial approximation!
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
2d) What is the transform if between ray 1 to ray 2 is a dielectric boundary with index of refracIon n1 and n2 respecIvely
2e) What is the transform if between ray 1 to ray 2 is free space (a distance d) FOLLOWED by a dielectric boundary with index of refracIon n1 and n2 respecIvely?
Most general linear transform Is multiplication by a 2x2 matrix
We have assumed that we can write
Paraxial approximation!
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
2d) What is the transform if between ray 1 to ray 2 is a dielectric boundary with index of refracIon n1 and n2 respecIvely
2e) What is the transform if between ray 1 to ray 2 is free space (a distance d) FOLLOWED by a dielectric boundary with index of refracIon n1 and n2 respecIvely?
Most general linear transform Is multiplication by a 2x2 matrix
We have assumed that we can write
Paraxial approximation!
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
2d) What is the transform if between ray 1 to ray 2 is a dielectric boundary with index of refracIon n1 and n2 respecIvely
2e) What is the transform if between ray 1 to ray 2 is free space (a distance d) FOLLOWED by a dielectric boundary with index of refracIon n1 and n2 respecIvely?
Most general linear transform Is multiplication by a 2x2 matrix
We have assumed that we can write
Paraxial approximation!
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
2e) What is the transform if between ray 1 to ray 2 is free space (a distance d) FOLLOWED by a dielectric boundary with index of refracIon n1 and n2 respecIvely?
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
2e) What is the transform if between ray 1 to ray 2 is free space (a distance d) FOLLOWED by a dielectric boundary with index of refracIon n1 and n2 respecIvely?
Cascaded optical system
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
2e) What is the transform if between ray 1 to ray 2 is free space (a distance d) FOLLOWED by a dielectric boundary with index of refracIon n1 and n2 respecIvely?
Cascaded optical system
input
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
2e) What is the transform if between ray 1 to ray 2 is free space (a distance d) FOLLOWED by a dielectric boundary with index of refracIon n1 and n2 respecIvely?
Cascaded optical system
input
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
2e) What is the transform if between ray 1 to ray 2 is free space (a distance d) FOLLOWED by a dielectric boundary with index of refracIon n1 and n2 respecIvely?
Cascaded optical system
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
2e) What is the transform if between ray 1 to ray 2 is free space (a distance d) FOLLOWED by a dielectric boundary with index of refracIon n1 and n2 respecIvely?
Cascaded optical system
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
2e) What is the transform if between ray 1 to ray 2 is free space (a distance d) FOLLOWED by a dielectric boundary with index of refracIon n1 and n2 respecIvely?
Cascaded optical system
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
2e) What is the transform if between ray 1 to ray 2 is free space (a distance d) FOLLOWED by a dielectric boundary with index of refracIon n1 and n2 respecIvely?
Cascaded optical system
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
2e) What is the transform if between ray 1 to ray 2 is free space (a distance d) FOLLOWED by a dielectric boundary with index of refracIon n1 and n2 respecIvely?
Cascaded system
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
2e) What is the transform if between ray 1 to ray 2 is free space (a distance d) FOLLOWED by a dielectric boundary with index of refracIon n1 and n2 respecIvely?
Cascaded system
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
2e) What is the transform if between ray 1 to ray 2 is free space (a distance d) FOLLOWED by a dielectric boundary with index of refracIon n1 and n2 respecIvely?
Cascaded system
2f) Find M for passing through a lens (focal length f) followed by free space prop. by a distance f. What does this system do to a bundle of rays hiWng the lens at some angle?
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
2e) What is the transform if between ray 1 to ray 2 is free space (a distance d) FOLLOWED by a dielectric boundary with index of refracIon n1 and n2 respecIvely?
Cascaded system
2f) Find M for passing through a lens (focal length f) followed by free space prop. by a distance f. What does this system do to a bundle of rays hiWng the lens at some angle?
2g) Find M for free space prop. thru a distance d1 followed by a lens (focal length f) and followed by another free space prop. by a distance d2. Draw this system. What happens to the transform matrix M when the distances saIsfy:
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
2e) What is the transform if between ray 1 to ray 2 is free space (a distance d) FOLLOWED by a dielectric boundary with index of refracIon n1 and n2 respecIvely?
Cascaded system
2f) Find M for passing through a lens (focal length f) followed by free space prop. by a distance f. What does this system do to a bundle of rays hiWng the lens at some angle?
2g) Find M for free space prop. thru a distance d1 followed by a lens (focal length f) and followed by another free space prop. by a distance d2. Draw this system. What happens to the transform matrix M when the distances saIsfy: 2h) In that case what does this system do to a point source d1 away from the lens?
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
2e) What is the transform if between ray 1 to ray 2 is free space (a distance d) FOLLOWED by a dielectric boundary with index of refracIon n1 and n2 respecIvely?
Cascaded system
2f) Find M for passing through a lens (focal length f) followed by free space prop. by a distance f. What does this system do to a bundle of rays hiWng the lens at some angle?
2g) Find M for free space prop. thru a distance d1 followed by a lens (focal length f) and followed by another free space prop. by a distance d2. Draw this system. What happens to the transform matrix M when the distances saIsfy: 2h) In that case what does this system do to a point source d1 away from the lens? 2i) Given (2h) what can you say this system does to transform a spherical wave? Draw it!
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
2e) What is the transform if between ray 1 to ray 2 is free space (a distance d) FOLLOWED by a dielectric boundary with index of refracIon n1 and n2 respecIvely?
Cascaded system
2f) Find M for passing through a lens (focal length f) followed by free space prop. by a distance f. What does this system do to a bundle of rays hiWng the lens at some angle?
2g) Find M for free space prop. thru a distance d1 followed by a lens (focal length f) and followed by another free space prop. by a distance d2. Draw this system. What happens to the transform matrix M when the distances saIsfy: 2h) In that case what does this system do to a point source d1 away from the lens? 2i) Given (2h) what can you say this system does to transform a spherical wave? Draw it! Bonus) What does this system do to a the q-‐parameter of a Gaussian beam?
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
2e) What is the transform if between ray 1 to ray 2 is free space (a distance d) FOLLOWED by a dielectric boundary with index of refracIon n1 and n2 respecIvely?
Cascaded system
2f) Find M for passing through a lens (focal length f) followed by free space prop. by a distance f. What does this system do to a bundle of rays hiWng the lens at some angle?
2g) Find M for free space prop. thru a distance d1 followed by a lens (focal length f) and followed by another free space prop. by a distance d2. Draw this system. What happens to the transform matrix M when the distances saIsfy: 2h) In that case what does this system do to a point source d1 away from the lens? 2i) Given (2h) what can you say this system does to transform a spherical wave? Draw it! Bonus) What does this system do to a the q-‐parameter of a Gaussian beam?
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
2e) What is the transform if between ray 1 to ray 2 is free space (a distance d) FOLLOWED by a dielectric boundary with index of refracIon n1 and n2 respecIvely?
Cascaded system
2f) Find M for passing through a lens (focal length f) followed by free space prop. by a distance f. What does this system do to a bundle of rays hiWng the lens at some angle?
2g) Find M for free space prop. thru a distance d1 followed by a lens (focal length f) and followed by another free space prop. by a distance d2. Draw this system. What happens to the transform matrix M when the distances saIsfy: 2h) In that case what does this system do to a point source d1 away from the lens? 2i) Given (2h) what can you say this system does to transform a spherical wave? Draw it! Bonus) What does this system do to a the q-‐parameter of a Gaussian beam?
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
2f) Find M for passing through a lens (focal length f) followed by free space prop. by a distance f. What does this system do to a bundle of rays hiWng the lens at some angle?
2g) Find M for free space prop. thru a distance d1 followed by a lens (focal length f) and followed by another free space prop. by a distance d2. Draw this system. What happens to the transform matrix M when the distances saIsfy: 2h) In that case what does this system do to a point source d1 away from the lens? 2i) Given (2h) what can you say this system does to transform a spherical wave? Draw it! Bonus) What does this system do to a the q-‐parameter of a Gaussian beam?
All parallel rays get focused to a single point in the focal plane a distance from the op5cal axis
2g) Find M for free space prop. thru a distance d1 followed by a lens (focal length f) and followed by another free space prop. by a distance d2. Draw this system. What happens to the transform matrix M when the distances saIsfy: 2h) In that case what does this system do to a point source d1 away from the lens? 2i) Given (2h) what can you say this system does to transform a spherical wave? Draw it! Bonus) What does this system do to a the q-‐parameter of a Gaussian beam?
2g) Find M for free space prop. thru a distance d1 followed by a lens (focal length f) and followed by another free space prop. by a distance d2. Draw this system. What happens to the transform matrix M when the distances saIsfy: 2h) In that case what does this system do to a point source d1 away from the lens? 2i) Given (2h) what can you say this system does to transform a spherical wave? Draw it! Bonus) What does this system do to a the q-‐parameter of a Gaussian beam?
2g) Find M for free space prop. thru a distance d1 followed by a lens (focal length f) and followed by another free space prop. by a distance d2. Draw this system. What happens to the transform matrix M when the distances saIsfy: 2h) In that case what does this system do to a point source d1 away from the lens? 2i) Given (2h) what can you say this system does to transform a spherical wave? Draw it! Bonus) What does this system do to a the q-‐parameter of a Gaussian beam?
2g) Find M for free space prop. thru a distance d1 followed by a lens (focal length f) and followed by another free space prop. by a distance d2. Draw this system. What happens to the transform matrix M when the distances saIsfy: 2h) In that case what does this system do to a point source d1 away from the lens? 2i) Given (2h) what can you say this system does to transform a spherical wave? Draw it! Bonus) What does this system do to a the q-‐parameter of a Gaussian beam?
If
then
2g) Find M for free space prop. thru a distance d1 followed by a lens (focal length f) and followed by another free space prop. by a distance d2. Draw this system. What happens to the transform matrix M when the distances saIsfy: 2h) In that case what does this system do to a point source d1 away from the lens? 2i) Given (2h) what can you say this system does to transform a spherical wave? Draw it! Bonus) What does this system do to a the q-‐parameter of a Gaussian beam?
If
then
for
2g) Find M for free space prop. thru a distance d1 followed by a lens (focal length f) and followed by another free space prop. by a distance d2. Draw this system. What happens to the transform matrix M when the distances saIsfy: 2h) In that case what does this system do to a point source d1 away from the lens? 2i) Given (2h) what can you say this system does to transform a spherical wave? Draw it! Bonus) What does this system do to a the q-‐parameter of a Gaussian beam?
If
then
for
2g) Find M for free space prop. thru a distance d1 followed by a lens (focal length f) and followed by another free space prop. by a distance d2. Draw this system. What happens to the transform matrix M when the distances saIsfy: 2h) In that case what does this system do to a point source d1 away from the lens? 2i) Given (2h) what can you say this system does to transform a spherical wave? Draw it! Bonus) What does this system do to a the q-‐parameter of a Gaussian beam?
If
then
for
2g) Find M for free space prop. thru a distance d1 followed by a lens (focal length f) and followed by another free space prop. by a distance d2. Draw this system. What happens to the transform matrix M when the distances saIsfy: 2h) In that case what does this system do to a point source d1 away from the lens? 2i) Given (2h) what can you say this system does to transform a spherical wave? Draw it! Bonus) What does this system do to a the q-‐parameter of a Gaussian beam?
If
then
for
A lens is a device that maps a point emitter in the object plane at d1 to a point in the image plane at d2
Lens focal length: f
A lens is a device that maps a point emitter in the object plane at do to a point in the image plane at di
Image equation
2i) Given (2h) what can you say this system does to transform a spherical wave? Draw it! Bonus) What does this system do to a the q-‐parameter of a Gaussian beam?
Lens focal length: f
A lens is a device that maps a point emitter in the object plane at do to a point in the image plane at di
Image equation
What is the sign of R?
2i) Given (2h) what can you say this system does to transform a spherical wave? Draw it! Bonus) What does this system do to a the q-‐parameter of a Gaussian beam?
Lens focal length: f
A lens is a device that maps a point emitter in the object plane at do to a point in the image plane at di
Image equation
What is the sign of R?
2i) Given (2h) what can you say this system does to transform a spherical wave? Draw it! Bonus) What does this system do to a the q-‐parameter of a Gaussian beam?
Recall for a Gaussian Beam
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
Lens focal length: f
A lens is a device that maps a point emitter in the object plane at do to a point in the image plane at di
Image equation
What is the sign of R?
2i) Given (2h) what can you say this system does to transform a spherical wave? Draw it! Bonus) What does this system do to a the q-‐parameter of a Gaussian beam?
Recall for a Gaussian Beam
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
Lens focal length: f
A lens is a device that maps a point emitter in the object plane at do to a point in the image plane at di
Image equation
What is the sign of R?
Convention: curvature positive if source to the left
2i) Given (2h) what can you say this system does to transform a spherical wave? Draw it! Bonus) What does this system do to a the q-‐parameter of a Gaussian beam?
Recall for a Gaussian Beam
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
Lens focal length: f
A lens is a device that maps a point emitter in the object plane at do to a point in the image plane at di
Image equation
2i) Given (2h) what can you say this system does to transform a spherical wave? Draw it! Bonus) What does this system do to a the q-‐parameter of a Gaussian beam?
2i) The image equaIon implies that a spherical wave with a parIcular origin and radius of curvature hiWng the lens is transformed to a new spherical wave with a different radius of curvature and origin. Write down the transformaIon law for the radii of curvature. BEWARE THE SIGN OF R !
Lens focal length: f
A lens is a device that maps a point emitter in the object plane at do to a point in the image plane at di
Image equation
2i) Given (2h) what can you say this system does to transform a spherical wave? Draw it! Bonus) What does this system do to a the q-‐parameter of a Gaussian beam?
2i) The image equaIon implies that a spherical wave with a parIcular origin and radius of curvature hiWng the lens is transformed to a new spherical wave with a different radius of curvature and origin. Write down the transformaIon law for the radii of curvature. BEWARE THE SIGN OF R !
2j) Given q’s relaIon to R derive the transformaIon law for a Gaussian beam q-‐parameter
Lens focal length: f
A lens is a device that maps a point emitter in the object plane at do to a point in the image plane at di
Image equation
2i) The image equaIon implies that a spherical wave with a parIcular origin and radius of curvature hiWng the lens is transformed to a new spherical wave with a different radius of curvature and origin. Write down the transformaIon law for the radii of curvature. BEWARE THE SIGN OF R !
2j) Given q’s relaIon to R derive the transformaIon law for a Gaussian beam q-‐parameter
2i) Given (2h) what can you say this system does to transform a spherical wave? Draw it! Bonus) What does this system do to a the q-‐parameter of a Gaussian beam?
Lens focal length: f
A lens is a device that maps a point emitter in the object plane at do to a point in the image plane at di
Image equation
2i) Given (2h) what can you say this system does to transform a spherical wave? Draw it! Bonus) What does this system do to a the q-‐parameter of a Gaussian beam?
2i) The image equaIon implies that a spherical wave with a parIcular origin and radius of curvature hiWng the lens is transformed to a new spherical wave with a different radius of curvature and origin. Write down the transformaIon law for the radii of curvature. BEWARE THE SIGN OF R !
2j) Given q’s relaIon to R derive the transformaIon law for a Gaussian beam q-‐parameter
Lens focal length: f
A lens is a device that maps a point emitter in the object plane at do to a point in the image plane at di
Image equation
2i) Given (2h) what can you say this system does to transform a spherical wave? Draw it! Bonus) What does this system do to a the q-‐parameter of a Gaussian beam?
2i) The image equaIon implies that a spherical wave with a parIcular origin and radius of curvature hiWng the lens is transformed to a new spherical wave with a different radius of curvature and origin. Write down the transformaIon law for the radii of curvature. BEWARE THE SIGN OF R !
2j) Given q’s relaIon to R derive the transformaIon law for a Gaussian beam q-‐parameter
A lens is a device that transforms a Gaussian q-parameter according to
A general optical system transforms q as
OR
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
A lens is a device that transforms a Gaussian q-parameter according to
3a) Find the ABCD values for the lens.
A general optical system transforms q as
OR
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
A lens is a device that transforms a Gaussian q-parameter according to
3a) Find the ABCD values for the lens.
A general optical system transforms q as
OR
3b) What does propagaIon in free space thru a distance d do to the q-‐parameter?
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
A lens is a device that transforms a Gaussian q-parameter according to
3a) Find the ABCD values for the lens.
A general optical system transforms q as
OR
3b) What does propagaIon in free space thru a distance d do to the q-‐parameter?
3c) Find the ABCD values for free space propagaIon thru distance d
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
A lens is a device that transforms a Gaussian q-parameter according to
3a) Find the ABCD values for the lens.
A general optical system transforms q as
OR
3b) What does propagaIon in free space thru a distance d do to the q-‐parameter?
3c) Find the ABCD values for free space propagaIon thru distance d
recall
3d) Suppose a lens is placed at the focus of a Gaussian beam. Sketch the situaIon. What is the q-‐parameter before and a_er the lens? Describe where and what is the minimum width of the beam a_er the lens in the limit of z0 >> f.
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
A lens is a device that transforms a Gaussian q-parameter according to
3a) Find the ABCD values for the lens.
A general optical system transforms q as
OR
3b) What does propagaIon in free space thru a distance d do to the q-‐parameter?
3c) Find the ABCD values for free space propagaIon thru distance d
recall
the imaginary parts are equal !
3d) Suppose a lens is placed at the focus of a Gaussian beam. Sketch the situaIon. What is the q-‐parameter before and a_er the lens? Describe where and what is the minimum width of the beam a_er the lens in the limit of z0 >> f.
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
A lens is a device that transforms a Gaussian q-parameter according to
3a) Find the ABCD values for the lens.
A general optical system transforms q as
OR
3b) What does propagaIon in free space thru a distance d do to the q-‐parameter?
3c) Find the ABCD values for free space propagaIon thru distance d
lens
recall
the imaginary parts are equal !
3d) Suppose a lens is placed at the focus of a Gaussian beam. Sketch the situaIon. What is the q-‐parameter before and a_er the lens? Describe where and what is the minimum width of the beam a_er the lens in the limit of z0 >> f.
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
A lens is a device that transforms a Gaussian q-parameter according to
3a) Find the ABCD values for the lens.
A general optical system transforms q as
OR
3b) What does propagaIon in free space thru a distance d do to the q-‐parameter?
3c) Find the ABCD values for free space propagaIon thru distance d
lens
recall
the imaginary parts are equal !
3d) Suppose a lens is placed at the focus of a Gaussian beam. Sketch the situaIon. What is the q-‐parameter before and a_er the lens? Describe where and what is the minimum width of the beam a_er the lens in the limit of z0 >> f.
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
A lens is a device that transforms a Gaussian q-parameter according to
3a) Find the ABCD values for the lens.
A general optical system transforms q as
OR
3b) What does propagaIon in free space thru a distance d do to the q-‐parameter?
3c) Find the ABCD values for free space propagaIon thru distance d
lens
distance
recall
the imaginary parts are equal !
3d) Suppose a lens is placed at the focus of a Gaussian beam. Sketch the situaIon. What is the q-‐parameter before and a_er the lens? Describe where and what is the minimum width of the beam a_er the lens in the limit of z0 >> f.
Image from Fundamentals of Photonics 2nd EdiIon by Bahaa E. A. Saleh & Malvin Carl Teich
3d) Suppose a lens is placed at the focus of a Gaussian beam. Sketch the situaIon. What is the q-‐parameter before and a_er the lens? Describe where and what is the minimum width of the beam a_er the lens in the limit of z0 >> f.
recall
3d) Suppose a lens is placed at the focus of a Gaussian beam. Sketch the situaIon. What is the q-‐parameter before and a_er the lens? Describe where and what is the minimum width of the beam a_er the lens in the limit of z0 >> f.
recall
The input beam is AT ITS FOCUS, so q is purely imaginary
3d) Suppose a lens is placed at the focus of a Gaussian beam. Sketch the situaIon. What is the q-‐parameter before and a_er the lens? Describe where and what is the minimum width of the beam a_er the lens in the limit of z0 >> f.
recall
The input beam is AT ITS FOCUS, so q is purely imaginary
3d) Suppose a lens is placed at the focus of a Gaussian beam. Sketch the situaIon. What is the q-‐parameter before and a_er the lens? Describe where and what is the minimum width of the beam a_er the lens in the limit of z0 >> f.
recall
The input beam is AT ITS FOCUS, so q is purely imaginary
3d) Suppose a lens is placed at the focus of a Gaussian beam. Sketch the situaIon. What is the q-‐parameter before and a_er the lens? Describe where and what is the minimum width of the beam a_er the lens in the limit of z0 >> f.
recall
The input beam is AT ITS FOCUS, so q is purely imaginary
3d) Suppose a lens is placed at the focus of a Gaussian beam. Sketch the situaIon. What is the q-‐parameter before and a_er the lens? Describe where and what is the minimum width of the beam a_er the lens in the limit of z0 >> f.
recall
The input beam is AT ITS FOCUS, so q is purely imaginary
3d) Suppose a lens is placed at the focus of a Gaussian beam. Sketch the situaIon. What is the q-‐parameter before and a_er the lens? Describe where and what is the minimum width of the beam a_er the lens in the limit of z0 >> f.
recall
The input beam is AT ITS FOCUS, so q is purely imaginary
3d) Suppose a lens is placed at the focus of a Gaussian beam. Sketch the situaIon. What is the q-‐parameter before and a_er the lens? Describe where and what is the minimum width of the beam a_er the lens in the limit of z0 >> f.
recall
The input beam is AT ITS FOCUS, so q is purely imaginary
3d) Suppose a lens is placed at the focus of a Gaussian beam. Sketch the situaIon. What is the q-‐parameter before and a_er the lens? Describe where and what is the minimum width of the beam a_er the lens in the limit of z0 >> f.
recall
using Ideal minimum spot size
The input beam is AT ITS FOCUS, so q is purely imaginary
3d) Suppose a lens is placed at the focus of a Gaussian beam. Sketch the situaIon. What is the q-‐parameter before and a_er the lens? Describe where and what is the minimum width of the beam a_er the lens in the limit of z0 >> f.
recall What happens as z0 << f ?
using Ideal minimum spot size
The input beam is AT ITS FOCUS, so q is purely imaginary
3d) Suppose a lens is placed at the focus of a Gaussian beam. Sketch the situaIon. What is the q-‐parameter before and a_er the lens? Describe where and what is the minimum width of the beam a_er the lens in the limit of z0 >> f.
recall What happens as z0 << f ?
using Ideal minimum spot size
The input beam is AT ITS FOCUS, so q is purely imaginary
3d) Suppose a lens is placed at the focus of a Gaussian beam. Sketch the situaIon. What is the q-‐parameter before and a_er the lens? Describe where and what is the minimum width of the beam a_er the lens in the limit of z0 >> f.
recall What happens as z0 << f ?
The lens does nothing…
using Ideal minimum spot size
