Optics for Engineers Chapter 9

Transcription

1 Optics for Engineers Chapter 9 Charles A. DiMarzio Northeastern University Mar. 204

2 Gaussian Beams Applications Many Laser Beams Minimum Uncertainty Simple Equations Good Approximation Extensible (e.g. Hermite Gaussian) Equations Solution of Helmholz Equation Solution to Laser Cavity Kogelnik and Li, 966 Spherical Gaussian Waves Gaussians Are Forever Imaginary Part of Field Gaussian Profile Spherical Wavefront Focusing and Propagation Simple Equations Relation to ABCD Matrix Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r

3 Derivation () Spherical Wave and Paraxial Approximation E sphere = Psphere 4π e jk x 2 +y 2 +z 2 x 2 + y 2 + z 2 Psphere 4πz e jkz e jkx2 +y 2 2z Radius of Curvature = z: Substitute Complex q E = Psphere 4π e jk x 2 +y 2 +q 2 x 2 + y 2 + q 2 Psphere 4π q ejkq e jk x 2 +y 2 2q Separate Real and Imaginary Parts of q in Exponents E Psphere 4π q ejkq e jk ( x 2 +y 2) R2qe k( x 2 +y 2) I2q Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 2

4 Derivation (2) Curvature and Gaussian Profile e jk( x 2 +y 2) R2q = e jkx2 +y 2 2ρ e k( x 2 +y 2) I2q = e kx2 +y 2 2b Definitions q = z + jb q = ρ j b ρ = z2 + b 2 z b = z2 + b 2 b Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 3

5 Gaussian Beam Parameters Beam Radius, w w 2 = 2b k b = πw2 λ = πd2 4λ e k( x 2 +y 2) I2q = e x2 +y 2 w 2 Curvature (Prev. Pg.) e jk( x 2 +y 2) R2q = e jkx2 +y 2 2ρ Gouy Phase q = z + jb = arctan z b z 2 + b2e j = λ b πw 2 e arctan z b = e jψ Linear Phase e jkq = e jk(z+jb) = e jkz e kb Normalized Complete Equation E 2P arctan z πw 2e be jkz e jk x 2 +y 2 2ρ e x2 +y 2 w 2 Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 4

6 Physical Meaning of Parameters E 2P e jk x 2 +y 2 πw 2ejkz 2ρ e x2 +y 2 w 2 e jψ Radius of Curvature, ρ Dashed Black Line Distance from Waist, z z = Rq Rayleigh Range, b d 0 ρ = R q Beam Diameter, d Black Diamonds b = Iq b = πw2 0 λ = πd2 0 4λ b = I q b = πw2 λ = πd2 4λ Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 5

7 Gouy Phase Phase Term ψ = arctan z b See White Circle Plot is IE Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 6

8 The Really Useful Equations Beam Diameter, d d = d 0 + z2 b 2 Radius of Curvature, ρ ρ = z + b2 z d/d 0, Scaled Beam Diameter ρ/b, Scaled Radius of Curvature z/b, Scaled Axial Distance Near Field d g d 0 Far Field d d 4 π λ d 0 z z/b, Scaled Axial Distance Near Field ρ b 2 /z Far Field ρ z Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 7

9 Six Questions Four Parameters for q z, Distance from Waist b or d 0, Waist Diameter ρ, Radius of Curvature b or d Local Diameter (at z) Two Independent Numbers for Complex q Pick Any Pair Solve for the Rest Practical Problems for Each Pair z and b known ρ and b z and b ρ and b z and ρ b and b Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 8

10 Question : Known z and b Sample Application: Starting with a collimated (plane wave) Gaussian, what are d and ρ at distance, z? Example: Green Laser Pointer: λ = 532nm. Solution b = πd2 0 4λ = 5.9m ρ = z + b2 z d = d 0 + z2 b 2 ( 4π λ d0 z ) z = 3m: ρ = 4.6m d = 2.2mm (mm) z = 30m: ρ = 3.2m d = 0.4mm (0.2mm) z = 376, 000km: ρ = 376, 000km d = 27km (27km) Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 9

11 Question 2: Known ρ and b Sample Application: What is the waist location for a focused beam? Example: Laser Radar, λ = 0.59µm, d = 30cm, b = 6.6km Solution ρ = f b = πd 2 / (4λ) q = ρ j b q = ρ j b q = ρ ρ b 2 j b ρ 2 + b 2 z = Rq = f + ( 4λf πd 2 ) 2 b = Iq = b b f 2 + d 2 0 = + d 2 ( ) πd 2 2 4λf Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 0

12 Question 2 Results z, Distance to Waist, km f, Focal Length, km d 0, Waist Diameter, cm f, Focal Length, km A. Waist Location B. Waist Diameter d, Beam Diameter, cm I 0 /P, m z, Distance, km z, Distance, km C. Beam Diameter (f = km) Axial Irradiance Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r

13 Question 2 Summary Maximum Distance to Waist: ( z) max = b/2 at f = b Two Ways to Make a Waist at any ( z) < b/2 Near Field of d 0 ; Strong Focusing: ( z) ρ = f d 0 = 4 π λ d Depth of Focus: Quadratic in f Collimated Beam as f z c = 2b = 8f 2 λ πd 2 ( ) 4λf 2 πd + 2 Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 2

14 Geometric and Diffraction Diameters Diameter at Any Distance d 2 = d 2 g + d2 d Simple Calculation Intuitive Valid for Gaussians Geometric Optics z f d g = d f Diffraction Limit d d = 4 λ π d z Fraunhofer Zone d 0 d d = 4 π λ d z Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 3

15 Question 3: Known z and b Sample Application: What lens to make a waist at z for Negative z? How big is the waist? Example: Endoscope: λ =.06µm, waist at z = mm with d never > 00µm Solution: b = πd 2 /(4λ) b = b2 + z 2 b b = b ± Solve for b only if (z < b /2) b 2 4z 2 2 ρ = z + b2 z Typically 2 Solutions Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 4

16 Endoscope Focusing Fixed Diameter at Source d = 00µm, z = mm b = b ± b = πd2 4λ = 7.4mm z = mm b 2 4z 2 b = mm ± mm 2 Options b = 37µm d 0 = 3.6µm or b = 7.27mm d 0 = 99µm Probably Choose the First High Irradiance Small Depth of Field Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 5

17 Question 4: Known ρ and b Sample Application: How long must the laser be to get a given d 0 with a mirror of curvature ρ? Example: Later Solution: Solve for z if d 0 Small Enough ρ = z2 + b 2 z (2b < ρ) z = ρ ± ρ 2 4b 2 2 d = d 0 + z2 b 2 Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 6

18 Question 5: Known z and ρ Sample Application: cavity? How large is the beam from a laser Example: Later Solution: Solve for b ρ = z2 + b 2 z b = ρz z 2 b = z d 0 = 4λb π ρ z (ρ > z) or q = z + jb d = d 0 + z2 b 2 Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 7

19 Question 6: Known b and b Sample Application: How do we launch a beam of diameter, d, into a fiber with diameter, d 0? Example: Beam diameter, d = 5µm and desired diameter, d 0 = 5µm Solution: Solve for z b = z2 + b 2 b z = ± bb b 2 (b > b of course: d > d 0 ) ρ = z + b2 z Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 8

20 Fiber Launch Equations z = ± bb b 2 ρ = z + b2 z Inputs for d = 5µm and d 0 = 5µm Results z = b = πd2 0 4λ = 3µm bb b 2 = 37µm b = πd2 4λ = 8µm ρ = z + b2 z = 4.7µm Lens: f = ρ = 4.7µm, with D lens > 5µm, (<f/3) Distance: z = 37µm (A Little Shorter than f) Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 9

21 Gaussian Beam Characteristics The complex radius of curvature can be manipulated to solve problems of Gaussian beam propagation. The complex radius of curvature and its inverse contain four terms, which are related in such a way that only two of them are independent. Given two parameters it is possible to characterize the Gaussian beam completely. Six examples have been discussed. Solutions are at worst quadratic, producing zero, one, or two solutions. Other, more complicated problems can also be posed and solved using this formulation. Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 20

22 Gaussian Beam Propagation Propagation through Free Space (q = z + jb) q (z 2 ) = q (z ) + z 2 z Propagation through a Lens ( q = ρ j b ) s + s = f ρ + ρ = f ρ = ρ f q = q f or q = q P Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 2

23 Matrix Optics Again General q out = Aq in + B Cq in + D Translation Lens In Air q 2 = q + z = q + z 2 q = q + 0 P n q + n n q = n qn P n q = q P = q f Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 22

24 Refraction at a Surface General q = q + 0 n n n R q + n n q = n n n R + n qn Planar Surface Rayleigh Range in a Medium q = q n n b = πd2 0 4λ/n Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 23

25 Propagation Example with a Lens: F to F Waist at Front Focal Plane, F q 0 = jb 0 b 0 = πd2 0 4λ Three Steps q = q 0 + f q = q f q 2 = q + f Result: Rz 2 = 0 (Waist) q = jb 0 + f q = f + j f 2 b 0 q 2 = jf 2 b 0 d 2 = 4 π λ d 0 f (Fourier Transform) Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 24

26 Propagation Example Continued: A Second Lens (f 3 ) Relay Telescope d 4 = 4 π Matrix Optics: First Lens M 02 = T 2 L T 0 = ( f 0 λ d 2 f 3 = f 3 f d 0 ) ( ) ( ) 0 f f 0 = ( 0 f f 0 ) q 2 = Aq 0 + B Cq 0 + q 0 = 0 + f 2 f q = f q 0 d 2 = 4 π λ d 0 f Both Lenses (f 3 = f ): Inverting : Telecentric System ( ) ( ) ( 0 f3 0 f 0 M 04 = M 24 M 02 = f 0 = 3 f 0 0 q 4 = Aq 0 + B Cq 0 + D = q = q 0 Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 25 )

27 Two : Relays Telecentric (Previous Page) M 04 = M 24 M 02 = ( ) ( ) 0 f3 0 f f 3 0 f 0 ( 0 ) 0 = Single Lens (4 f Configuration) ( 2f 0 M ac = T bc L b T ab = ) ( ) ( 0 2f f 0 ( 0 ) ) = Afocal f q 4 = q 0 = jb 0 Plane Wave No Field Curvature ρ ρ = f q c = Aq a + B Cq a + D = q a + 0 f q a = f + q a Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 26

28 Summary of Gaussian Beam Propagation Once two parameters of a Gaussian beam are known, then it s propagation can be analyzed using simple equations for translation, refraction, and focusing. Translation changes the beam size and curvature. It is represented by adding the distance to q. Refraction through a lens changes the curvature, but keeps the beam diameter constant. It is represented by subtracting the optical power of the lens from the inverse of q. Refraction through a dielectric interface changes the curvature and scale factor of q. These equations can be generalized using the ABCD matrices developed for geometric optics. Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 27

29 The Collins Chart Lines of Constant z (vertical) and Constant b (horizontal) Curves of Constant b (circles) and Constant ρ (semicircles) Now Mostly a Visualization Aid 2 Confocal Parameter z, Axial Distance Translation (Horizontal) z Changes, b Doesn t Focusing (Along Circles) ρ Changes, d & b Don t Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 28

30 Two : Relays on Collins Chart Confocal Parameter 3 2 Confocal Parameter z, Axial Distance z, Axial Distance A. Afocal Telescope B. Single Lens Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 29

31 Minimum Beam Diameter for Distance z 2 2 Confocal Parameter z, Axial Distance Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 30

32 Waist at z for diameter, d b, Confocal Parameter z, Axial Distance Confocal Parameter z, Axial Distance Confocal Parameter z, Axial Distance B. No Solution C. One Solution D. Two Solutions z > b /2 z = b /2 z < b /2 Question 3 b = b2 + z 2 b b = b ± Solve for b only if (z < b /2) b 2 4z 2 2 ρ = z + b2 z Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 3

33 Stable Laser Cavity Design A. Flat Output Coupler B. Flat Rear Mirror C. Confocal Resonator Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 32

34 Steady State in Laser Cavity The amplitude after a round trip is unchanged. This means that any loss (including power released as output) must be offset by corresponding gain. (Gain Saturation) The phase after a round trip must be unchanged. We discussed, in our study of interference, how this requirement on the axial phase change, e jkz, affects the laser frequency. The beam shape must be unchanged, so that the phase and amplitude is the same for all x and y. This is the subject to be considered in this section. Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 33

35 Design Problem Carbon Dioxide Laser: P(20), λ = 0.59µm Beam Output: Collimated, 5mm Diameter Cavity Length: m (Probably because of Gain) Solution Collimated Output: Flat Output Coupler Rear Mirror to Match Curvature at z = m b =.85m ρ = 4.44m d = 5.7mm Rear Mirror Concave, ρ = 4.44m Diameter Larger than d = 5.7mm (Typically.5X) Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 34

36 Stable Cavity Examples Output toward Bottom A. Flat Output Coupler B. Flat Rear Mirror C. Confocal Resonator D. Focusing Cavity Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 35

37 Laser Cavity on the Collins Chart Given ρ ρ 2 and z 2 3 b, Confocal Parameter z, Axial Distance Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 36

38 More Complicated Cavities Ring Laser with Focusing Lens Intracavity Lens (Potential Loss) Many Other Configurations Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 37

39 Matrix Optics for Stable Cavities Round Trip Equation q out = q in Need to Solve This: q = Aq + B Cq + D Cq 2 + (D A)q B = 0 Result (Need Imaginary Solutions) q = A D ± (A D) 2 + 4CB 2C (A D) 2 + 4CB < 0 Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 38

40 Stability Condition Argument of Square Root Must be Negative (A D) 2 + 4CB < 0 Determinant Condition (n = n on Round Trip) AD CB = Result (D A) 2 + 4DA < 4 (D + A) 2 < 4 Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 39

41 Summary of Stable Cavity Design Stable cavities or stable optical resonators have mirrors and possibly other optical elements that cause a Gaussian beam to replicate itself upon one round trip. Resonator design can be accomplished using Gaussian beam propagation equations. Most cavities will support only one Gaussian beam, and the beam parameters are determined by the curvature of the mirrors and their spacing along with the behavior of any intervening optics. Matrix optics can be used to determine the round trip matrix, which can be used to find the Gaussian beam parameters. The Collins chart can help to evaluate the stability of a laser cavity. Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 40

42 Hermite Gaussian Modes Mode Definitions (Solutions of Helmholz Equation) h mn (x, y, z) = h m (x, z) h n (y, z) One Dimensional Functions with Hermite Polynomials, H h m (x, z) = ( 2 π Hermite Polynomials ) /4 ( ) x 2 m m!w H m e x2 w 2 e jkx2 2ρ e jψ m w H (x) = H 2 (x) = 2x H m+ (x) = 2xH m (x) 2 (m ) H m (x) Gouy Phase for Hermite Gaussian Modes ( ) ψ m = 2 + m arctan z b Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 4

43 Some Hermite Gaussian Modes 0:0 0: (0:) + i (:0) = Donut :0 : :3 2:0 2: 2:3 5:0 5: 5:3 Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 42

44 Expansion in Hermite Gaussian Modes Expand Field in Hermite Gaussian Modes Ortho Normal Basis Set E (x, y, z) = m=0 n=0 C mn h mn (x, y, z) May Take a Lot of Terms (but choose w wisely) Power in Each Mode (and Phase) P mn = C mn C mn φ = C mn Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 43

45 Finding the Coefficients, C for Given E(x, y, z) E (x, y, z) h m n (x, y, z) dx dy = E (x, y, z) h (x, y, z) dx dy = m=0 n=0 m=0 n=0 C mn h mn (x, y, z) h m n (x, y, z) dx dy = δ m,m δ n,n C mn h mn (x, y, z) h m n (x, y, z) dx dy h mn (x, y, z) h m n (x, y, z) dx dy Ortho Normality C m n = E (x, y, z) h m n (x, y, z) dx dy. Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 44

46 Picking the Best Waist Size H G Expansion Works for Any w But it Might Not Converge Nicely Pick the Best w to Maximize Power in T EM 00 Term C 00 = min [C 00 (w)] C 00 (w) = E (x, y, z) h 00 (x, y, z) dx dy Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 45

47 Expanding Uniform Circular Wave in H G Modes x C 00, Coefficien w, Gaussian radiu A. B. C. x Irradiance, db x, Radial Distance Irradiance, db D. E. F. x, Radial Distance Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 46

48 Coupling Equations and Mode Losses for Finite Apertures Integration over Finite Aperture K mnm n = aperture h mn (x, y, z) h m n (x, y, z) dx dy = δ m,m δ n,n C 00 C 0 C 0. = K 0000 K 000 K K 000 K 00 K K 000 K 00 K C 00 C 0 C 0. Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 47

49 Transverse Mode Beating: Mode Frequencies Round Trip (Additional Phase Shift beyond Chapter 7) ( 2kl + 2 ( + m + n) arctan z 2 b arctan z ) = 2πN b f (m, n, N) 2 l + ( c π ( + m + n) arctan z 2 b arctan z ) b f (m, n, N) = N c 2l c 2l ( + m + n) arctan z 2 b arctan z b π = N Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 48

50 Transverse Mode Beating: Mode Control Transverse Mode Beat Frequency f (, n, N) f (0, n, N) = 2larctan c z 2 b arctan z b π Carbon Dioxide Laser b =.85m, z = m f (, n, N) f (0, n, N) = c ( arctan m.85m 2m Solution: Include Appropriate Aperture Suppress Higher Order Transverse Modes K 0000 gρ ρ 2 > K mnmn gρ ρ 2 < for m 0, n 0 arctan 0m.85m π ) = 24MHz Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 49

51 Summary of H G Modes Hermite Gaussian modes (Equation ) form an orthogonal basis set of functions which can be used to describe arbitrary beams. The beam can be expanded in a series. Coefficients of the expansion can be calculated. Each mode maintains its own profile with a size w and radius of curvature, ρ given by the usual Gaussian beam equations. The choice of w in Equation is arbitrary, but a value chosen badly can lead to poor convergence of the series expansion. Higher order modes have increased Gouy phase shifts as shown in Equation. Beam propagation problems can be solved by manipulating the coefficients with appropriate Gouy phase shifts. Apertures can be treated using coupling coefficients which can be computed. Higher order transverse modes in a laser can mix with each other to produce unwanted mixing, or beat, signals. The higher order modes can be eliminated by placing an appropriate aperture inside the laser cavity. Mar. 204 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides9r 50