Optimal Design of Minimum Energy Pulses for Bloch Equations in the case of Dominant Transverse Relaxation
aa r X i v : . [ m a t h . O C ] J a n Optimal Design of Minimum Energy Pulses for Bloch Equationsin the case of Dominant Transverse Relaxation
Dionisis Stefanatos ∗ Prefecture of Kefalonia, Argostoli, Kefalonia 28100, Greece (Dated: November 23, 2018)In this report, we apply Optimal Control Theory to design minimum energy π/ π pulses forBloch equations, in the case where transverse relaxation rate is much larger than longitudinal so thelater can be neglected. Using Pontryagin’s Maximum Principle, we derive an optimal feedback lawand subsequently use it to obtain analytical expressions for the energy and duration of the optimalpulses. PACS numbers: 33.25.+k, 02.30.Yy
Optimal Control Theory [1] has been extensively usedrecently for the design of pulses that optimize the per-formance of various Nuclear Magnetic Resonance (NMR)and Quantum Optics systems limited by the presence ofrelaxation [2–10]. In this report we use it to derive mini-mum energy π/ π pulses for Bloch equations, in thecase where transverse relaxation dominates.The Bloch equations, in a resonant rotating frame andwhen longitudinal relaxation is neglected are [11]˙ M z = ω y M x − ω x M y ˙ M x = − RM x − ω y M z ˙ M y = − RM y + ω x M z where M = ( M x , M y , M z ) is the magnetization vector, ω x , ω y are the transverse components of the magneticfield and R > a = ln[ M ( t ) /M (0)]tan θ = q M x + M y /M z tan φ = M y /M x where M = q M x + M y + M z , we obtain˙ a = − R sin θ (1)˙ θ = ω ⊥ − R sin θ cos θ (2)˙ φ = ω k cot θ (3)where ω ⊥ = ω x sin φ − ω y cos φ, ω k = ω x cos φ + ω y sin φ are the components of transverse magnetic field perpen-dicular and parallel to M ⊥ = ( M x , M y ), respectively.Note that ω k does not affect the angle θ of the pulse. Itjust rotates M around z -axis, resulting in a waste of en-ergy. Thus, optimality requires ω k = 0 ⇒ φ = constant.Equations (1) and (2) are sufficient to describe the rota-tion and from now on we use ω to denote ω ⊥ , see Fig. 1. ∗ Electronic address: [email protected]
M M ⊥ ω φ θ x y M z FIG. 1: The optimal transverse magnetic field ω is perpen-dicular to M ⊥ and its phase is constant. Without loss ofgenerality, the experimental setup can be arranged such that ω k x -axis. In this case, φ = π/ M rotates in yz -plane. Observe that when ω is unbounded, we can rotate θ in-stantaneously to the desired final value θ ( T ) = π/ π without losses in a , i.e. with a ( T ) = a (0) = 0 (equiv-alently M ( T ) = M (0)). This transfer requires an infi-nite amount of energy so it is unrealistic. A meaning-ful optimization problem is the following: For a spec-ified final value a ( T ) < a (0) = 0 (equivalently speci-fied M ( T ) < M (0)), what is the optimal control ω ( t ),0 ≤ t ≤ T , that accomplishes the transfer ( a (0) =0 , θ (0) = 0) → ( a ( T ) , θ ( T ) = π/ π ), while minimiz-ing energy R T ω ( t ) / dt ? The control Hamiltonian [1]for this problem is H = − ω / λ θ ( ω − R sin θ cos θ ) − λ a R sin θ (4)where λ θ , λ a are the Lagrange multipliers. According toPontryagin’s maximum principle [1], necessary conditionsfor optimality of ( ω ( t ) , a ( t ) , θ ( t ) , λ θ ( t ) , λ a ( t )) are ϑH/ϑω = 0 ⇒ ω = λ θ (5)˙ λ θ = − ϑH/ϑθ = λ θ R cos 2 θ + λ a R sin 2 θ (6)˙ λ a = − ϑH/ϑa = 0 ⇒ λ a = constant (7)Additionally, the optimal ( ω, a, θ, λ θ , λ a ) satisfies [1] H ( ω, a, θ, λ θ , λ a ) = 0 , ≤ t ≤ T. (8)Using (4),(5) the above condition becomes λ θ − λ θ R sin θ cos θ − λ a R sin θ = 0 (9)Note that when θ ( t π/ ) = π/ t π/ for both final values θ ( T ) that we consider here), then (9)gives λ a = λ θ ( t π/ ) / R ≥
0. Solving the above quadraticequation for λ θ and using (5), we find the optimal control ω . Note that only the positive solution of the quadraticequation has physical meaning (corresponds to increasing θ ) for the cases that we study here. The optimal ω isgiven by the following feedback law ω ( θ ) = R sin θ (cos θ + p cos θ + κ ) (10)where κ = 2 λ a /R = λ θ ( t π/ ) /R . Using (10), the valid-ity of (6) can be easily verified. Inserting (10) in (2) weobtain the differential equation for the optimal trajectory˙ θ = R sin θ p cos θ + κ (11)Integrating (1) from t = 0 to the final time t = T we get Z T sin θdt = − a ( T ) R = 1 R ln M (0) M ( T ) (12)If we use (11) to change the integration from time toangle, we obtain Z θ ( T ) θ (0) sin θ √ cos θ + κ dθ = − ln r (13)where r = M ( T ) /M (0) <
1. This condition determines κ and the results for the π/ π pulses are κ π/ = 2 r − r , κ π = 2 √ r − r (14)The duration of the optimal pulses is determined fromthe relation T = Z T dt = Z θ ( T ) θ (0) θ ( θ ) dθ (15)using again (11). Note that since θ = 0 , π are equilibriumpoints for (11), we actually start from a small positiveinitial value θ (0) = ǫ for both cases, and additionallyfor the π pulse we end to the value θ ( T ) = π − ǫ . Theduration of the optimal pulses as ǫ → T π/ = 1 R − r r (cid:20) ln (cid:18) r r (cid:19) − ln ǫ (cid:21) (16) T π = 1 R − r r (cid:20) ln (cid:18) (1 + r ) r (cid:19) − ǫ (cid:21) (17)Finally, the energy of the optimal pulses is calculatedfrom Z T ω ( t )2 dt = Z θ ( T ) θ (0) ω ( θ )2 ˙ θ ( θ ) dθ (18) A m p li t ud e ( R ) t (1/R) (a) Optimal π/ t (1/R) A m p li t ud e ( R ) (b) Optimal π pulse M z / M M ⊥ /M M(t)/M θ (t) (c) Optimal trajectory M z / M M ⊥ /M M(t)/M θ (t) (d) Optimal trajectory FIG. 2: Minimum energy π/ π (panel b) pulsesfor M (0) = M , M ( T ) = 0 . M , ǫ = 10 − . The correspondingtrajectories are also shown (panels c,d). M i n i m u m E n e r g y ( R ) r = M(T)/M(0) E π /2 E π FIG. 3: The energy of the optimal pulses as a function of theratio r = M ( T ) /M (0). using (10) and (11). The result is E π/ = R − r , E π = R r − r (19)Observe that for r → T π/ , T π → E π/ , E π → ∞ , as mentioned above.In Fig. 2 we plot the optimal π/ π pulses for M (0) = M , M ( T ) = 0 . M , ǫ = 10 − , as well as thecorresponding trajectories of normalized magnetizationvector. In Fig. 3 we plot the energy of the pulses as afunction of the ratio r = M ( T ) /M (0).To conclude, in this report we calculated minimum en-ergy π/ π pulses for Bloch equations in the casewhere transverse relaxation dominates, using OptimalControl Theory. We expect this analytical work to serveas a reference for numerical studies of more complicatedand realistic situations that incorporate for example lon-gitudinal relaxation and magnetic field inhomogeneity. Acknowledgements : The author would like to thankJr-Shin Li for his constant support. [1] L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze,and E.F. Mishchenko,
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