1. Introduction

Charged particles, such as proton, are useful tools for transmission radiography ^[1-3]. Nuclear interaction with the nucleus, multiple Coulomb scattering (MCS) with the Coulomb field of the nucleus, and ionization with the electrons are three major types of interaction when protons pass through materials. However, MCS effect may seriously blur the final image. Zumbro lens (magnetic lens) can focus the scattered particles to obtain clear image, which has been successfully demonstrated both by simulations and experiments ^[4-17].

To eliminate the major part of chromatic blur, a matched beam of a Zumbro lens is a virtual-point source beam, but this will introduce additional blur when the sample is cylindrical. In this case, a parallel beam is more appropriate.

A method for designing a magnetic lens with parallel beam matched is developed in this paper.

2. Description of Zumbro lens

A typical Zumbro lens consists of four quadrupole magnets, as shown schematically in Fig. 1, where k and l are strength and length of quadrupoles, respectively; D₁ and D₂ are the drift space lengths, and f is the focal length.

Fig.1.Full-screen View

Sketch of Zumbro lens.

Under certain condition ^[4], the transfer matrix of a Zumbro lens becomes a minus-identity (–I) matrix, which satisfies the requirements of point-to-point imaging, being assumed for a monoenergetic incident particle beam. However, energy spread always exists, and particles off the design momentum will deviate from the central trajectory due to the chromatic aberration. This problem can be partially solved by the second order chromatic matching.

Suppose an incident particle comes from a virtual point source at a distance L upstream of the entry plane of the lens, due to both MCS in the object and the beam emittance, the particle will exit the object at an angle

where w=L⁻¹ is the beam correlation coefficient, and ϕ is the angular deviation from the ideal position-angle correlation line. The matrix elements in general are functions of the fractional momentum deviation Δ(≡δp/p). By expanding the final position of the particle in Taylor series form to the first order, it is found that w should satisfy the following equations in order to realize the second order chromatic matching design (primes indicate momentum derivatives)

where R′₁₁ and R′₁₂ are the differentials of R₁₁ and R₂₂. All positions depend on chromatic aberrations vanishing, and the final transverse position is then given by

The chromatic blur caused by the energy spread becomes R′₁₂ϕΔ, and the chromatic coefficient is R′₁₂.

At the mid-plane of a chromatically matched identity lens, the transverse position of a particle is given by

where M is the transfer matrix of the first half of Zumbro lens. Eq. (4) implies that after the first cell, the transverse position of the particle depends only on ϕ.

According to Eq. (2), the two correlation coefficients at X-plane and Y-plane can be written as follows

In the Zumbro lens, we have R′₁₁ = R′₃₃ and R′₁₂ = R′₃₄ because of the symmetry, thus one gets

This implies that the beam is convergent in one direction and divergent in the other one, as shown in Fig. 2.

Fig.2Full-screen View

Sketch of virtual-point source beam for Zumbro lens

Here the coefficient L is dependent on w_x or w_y, with the relationship |w_x|=|w_y|=1/L.

3. Demand of parallel beam radiography

Point source beam is not the best choice in some practical applications. In fact, parallel beam is much more desirable in these applications. In some radiography experiments with cylindrical samples, as shown in Fig. 3, additional blur will be introduced when a point source beam is adopted, but this can be avoided by using a parallel source beam.

Fig.3Full-screen View

Additional blur with a cylindrical sample

Unlike the chromatic blur, the additional blur described above is interrelated with the sample, and it does not exist when there is no lateral drift in the sample. As an example, for an 800 MeV proton beam, a tentative design gives R′₁₁=4.058 and R′₁₂=9.336 m. Assume that the angular deviation of the beam is 5 mrad, and the energy deviation is 0.5%, the chromatic blur is then 0.24 mm according to Eq.(3). The correlation coefficient w_x is −0.435 m⁻¹ from Eq.(5). If the cylindrical sample in Fig. 3 is sized at Φ1cm×1cm, the additional blur is 0.23 mm, which is about the same as the chromatic blur. Surely, the additional blur increases with the sample size. Parallel beam can eliminate the additional blur.

In Ref. [18], we proposed a type of magnetic lens making use of the energy-loss information of the particles. Similarly to the angle collimation in Zumbro lens, the energy can be collimated in the energy-loss lens. If both energy collimation and angle collimation are in demand, the energy-loss lens and Zumbro lens are expected to be cascaded, i.e., an energy-loss lens followed by a Zumbro lens, or vice versa. Unfortunately, the different matching conditions of the two types of lens prevent them from being cascaded, since the energy-loss lens demands a parallel matched beam.

So another magnetic lens with parallel beam matched is desired.

4. Design study of the new magnetic lens

Zumbro lenses have two main characteristics: 1) the transfer matrix is minus identity matrix, and 2) the mid-plane is an angle Fourier plane. The new lens will inherit the two characteristics, which are two constraints for the new lens. Also, the matched beam should be parallel beam, i.e., w in Eq. (2) should be zero. So the third constraint is

As shown in Fig. 1, Zumbro lens forms a reflection symmetric quadrupole beamline. A quadrupole beamline is reflection symmetric if its second half is the mirror image of the first half, which means the sequence of drifts and quads is traversed in the reverse order and with the opposite signs for the magnetic fields.

The new lens has also symmetric characteristic, like Zumbro lens. Two cases are proposed here.

4.1 Reflection symmetric case

The reflection symmetry forces that ^[4] R₄₄=R₁₁, R₃₃=R₂₂, R₃₄=R₁₂, R₄₃=R₂₁. In this case, the lens needs at least 8 quadrupole magnets to keep the symmetry characteristics. It consists of a repetition of two identical unit cells, each of which is reflection symmetric itself. The optical configuration is shown in Fig. 4, where k is the quadrupole strength, l is its length, f is its focal length, and D₁–D₃ are the drift space lengths.

Fig. 4Full-screen View

Configuration of the new lens.

Considering the symmetry, only the expressions in X-plane are analyzed in the following paragraphs. Like Zumbro lens described in Section 2, the transfer matrix of the whole lens is written as R, and the one of the first half is written as M.

The symmetry forces give that ^[4]

$R=-I+\tau M$ (9)

where τ =M₁₁+M₂₂ is the trace of matrix M. If the trace τ=0, it can be seen that R becomes the minus identity matrix (−I), so the first constraint is satisfied.

When energy spread is included, Eq. (9) gives the momentum derivatives of the matrix elements as follows,

${R^{\prime }}_{jk}={\tau }^{\prime }{M}_{jk}.$ (10)

Especially, we get R′₁₁= τ′M_jk.

Considering the third constraint of the new lens expressed in Eq. (8), since τ′ is not always zero (which can be verified by a simulation of thin-lens approximation), we have M₁₁ =0. As the symmetry forces M₁₁ = M₂₂, we have M₁₁=M₂₂=0. At the middle plane of the lens, the particle position is x_mid =M₁₁x₀+M₁₂x′₀, which becomes

$x_{\text{mid}}=M_{12}{x^{\prime }}_0.$ (11)

So the second constraint is satisfied naturally. Notice that in Eq. (4), the angle is expressed as ϕ, which is only the additive part in Eq. (1), while in Eq. (11), the angle is the total one in Eq. (1). However, when parallel beam is adopted, x₀′ becomes ϕ since w becomes zero. So on the X plane, M₁₁=M₂₂=0 is the only constraint required for the new lens. Conditions on the Y-plane can be got similarly, we have M₃₃=M₄₄=0.

Since the first half is reflection symmetric, M₁₁=M₂₂=0 and M₃₃=M₄₄=0 are satisfied simultaneously. Eq. (10) shows that R'₁₂= R'₃₄. This implies that the chromatic coefficients of X-plane and Y-plane are equal.

4.2 Symmetric case

The difference between a symmetric quadrupole beamline and a reflection symmetric one is that the magnetic fields of the second half are in opposite signs. The symmetry forces that R₂₂=R₁₁, R₄₄=R₃₃. In this case, six quadrupole magnets are enough. The configuration is shown in Fig. 5.

Fig. 5Full-screen View

Configuration of the new lens.

The deduction in Sect. 4.1 is still valid, but M₁₁=M₂₂=0 and M₃₃=M₄₄=0 are not definitely satisfied simultaneously since the symmetry is different. Also, the chromatic aberrations of X and Y plane are not equal since R'₁₂= R'₃₄ is invalid.

Notice that here the first half of the lens is itself symmetric. From Eq. (10) and the symmetry condition, we have M₁₁=M₂₂, so τ = 0 and M₁₁=M₂₂=0 are of equivalence. The advantage of the symmetric case is that the number of magnets is less, while the advantage of the reflection symmetric case is that the chromatic aberrations of X- and Y-plane are equal.

5. Analysis in thin-lens approximation

For a thin lens, one has df/dΔ=f, where f is the focal length of the thin lens.

5.2 Reflection symmetric case

Write the transfer matrix of Fig. 4 in thin-lens approximation. Using df/dΔ=f, the chromatic factor can be written as

${R^{\prime }}_{12}={R^{\prime }}_{34}=4(D_1+2D_2+2D_3)-\frac{8D_2{D}_3(D_1+D_2)}{f_2{}^2}-\frac{8D_1{D}_2{}^2{D}_3(D_1+2D_2+2D_3)}{f_1{}^2{f}_2{}^2}+\frac{16D_1{D}_2{}^3{D}_3{}^2(D_1+D_2)}{f_1{}^2{f}_2{}^4}$ (13)

Eq.(13) is rather complicated, but calculation shows that the chromatic factor is close to the length of the magnetic lens.

As M₃₃=M₄₄=0, it can be deduced that f₁ and f₂ are the functions of D₁, D₂ and D₃, so does R′₁₂. There are three free parameters in this system: D₁, D₂ and D₃. It is not easy to study generally. To find some quantitative laws, and for simplicity, the following three cases are considered. (a) D₁=nd, D₂=d, D₃=d; (b) D₁=d, D₂=nd, D₃=d; (c) D₁=d, D₂=d, D₃=nd. Here n varies from 0 to 5. The curves in Fig. 6 can be obtained, where the values on the x-and y-axes are n and R′₁₂, respectively. It can be seen that R′₁₂ increases with D₁ or D₂, but decreases with increasing D₃.

Fig. 6Full-screen View

The chromatic factor on three conditions.

5.3 Symmetric case

Write the transfer matrix of Fig. 5 in thin-lens approximation. Using df/dΔ=f, the chromatic factors can be written as

$\begin{array}{l}{R^{\prime }}_{12}=-\frac{1}{f_1{}^4{f}_2{}^2}2(D_2{f}_1+D_1(D_2+f_1))(f_1(D_2-2f_2)+D_1(D_2+f_1-2f_2))\\ (D_1(3D_2{}^2+f_1(f_1-2f_2)+4D_2(f_1-f_2))+D_2{f}_1(2D_2+f_1-2f_2))\end{array}$ (14) $\begin{array}{l}{R^{\prime }}_{34}=-\frac{1}{f_1{}^4{f}_2{}^2}2(D_2{f}_1+D_1(-D_2+f_1))(f_1(D_2+2f_2)+D_1(-D_2+f_1-2f_2))\\ (D_1(3D_2{}^2+f_1(f_1-2f_2)-4D_2(f_1-f_2))+D_2{f}_1(-2D_2+f_1-2f_2))\end{array}$ (15)

The two factors are not equal to each other in this case. From M₁₁=M₂₂=0 and M₃₃=M₄₄=0, it can be deduced that f₁ and f₂ are the functions of D₁ and D₂, so do R′₁₂ and R′₃₄.

Again, two cases are considered: (a) D₁=nd, D₂=d; (b) D₁=d, D₂=nd. The R′₁₂ and R′₃₄ obtained are shown in Fig. 7. One sees that R′₁₂ increases with D₁, but when D₂ increases R′₁₂ decreases first to a minimum and then increases; while R′₃₄ increases with D₁, but decreases with increasing D₂.

Fig. 7Full-screen View

The chromatic factors in X-plane (a & b) and Y-plane (c & d) on two conditions.

6 Theoretical design based on 11 MeV proton beam

An 11 MeV proton cyclotron was designed and constructed at Institute of Fluid Physics ^[19], with CW proton beams of 11 MeV and 50 μA. The lens uses the reflection symmetric structure, with (see Fig. 4) D₁=0.3 m, D₂=0.3 m, D₃=0.1 m, l=0.2 m, k₁=1.89037 T/m, k₂=5.11052 T/m, and a total length of 4.4 m).

As mentioned in Section 4, the chromatic coefficient R′₁₂ is close to the length of the lens, which is 4.549 m.

Proton trajectories in the lens, simulated using the MyBOC code ^[20], are shown in Fig. 8, where different angular deviations are represented by different line styles, and Z and X are the longitude and transverse position, respectively. It can be seen that the lens is a point-to-point system. When the incident beam is parallel, particles with the same angular deviation reach the mid-plane at a same position, so the mid-plane is the angle Fourier plane. Also, the envelope oscillation is small.

Fig. 8Full-screen View

Trajectories of protons.

7 Conclusion

A parallel incident beam is desired in practical applications, while Zumbro magnetic lens uses virtual-point source beam as its matched input. Through analyses using the transfer matrix method, we have proposed a new type of magnetic lens. It is advantageous in the matched parallel beam, while keeping the main characteristics of the Zumbro lens. Theoretic analysis, both constructions of reflective symmetry and symmetry, suggests the possible optical arrangements of the lens. The analysis shows that the new lens can fulfill all the functions of Zumbro lens. A theoretical design based on a cyclotron with 11 MeV proton beam is demonstrated. Experimental demonstration of the new imaging-lens system will be carried out on this accelerator.