This blog post will complete the Introduction to Jones Calculus posted in the first part. It is recommended to review the first post although this is a complete review of Jones Calculus.
https://blog.ozeninc.com/resources/polarization-states-for-zemax-jones-calculus-into-part-i
Jones Calculus is a mathematical framework used in Optics to describe the polarization of light and how it changes as light passes through optical elements such as polarizers, wave plates, or birefringent materials. Jones Calculus treats light as two-dimensional vector to represent its electric field polarization state. It's specifically designed for fully coherent and polarized light, such as laser beams.
Jones Calculus
Consider a Plane-Wave propagating along the z-axis (as you would design in Zemax).
The unit vector containing
J is the Jones Vector.
This is the notation Zemax uses for all its inputs to Polarization. Remember, phi is the difference of X-Phase and Y-Phase. The coefficient out front in general is a complex number. The Polarization described by the Jones Vector will not change unless the wave reflects or refracts off an optical element. Various states were reviewed in Part 1. Table 2.1 below shows the various states of polarization using Jones vectors.
Table 2.1 Various common polarization states for input into Zemax.
Zemax accepts Jones Matrix inputs for Polarization States, however does not directly output Jones Matrix data. Jones vectors are a great way to quickly review the current polarization state, and although Zemax will not output the end polarization state in this form, it can still be quickly used to evaluate the state.
For example, if we input Linear +45 degrees and pass through an ideal QWP using the Jones Matrix surface in Sequential Mode (Jones Matrix surface is similar to the Paraxial Surface, in that the Paraxial Surface creates a perfect focus, a Jones Matrix creates a perfect phase change) we should create a -pi/2 phase difference, Right Hand Circular (RHC).
Table 2.2 Example of Linear 45 degree input through an Ideal QWP, which creates RHC.
We can confirm the state of polarization after the Jones Matrix QWP by clicking the text tab at the bottom of the Polarization Pupil Map. This is useful because the polarization state may not be easily identifiable (this case is relatively obvious, but it is always good engineering to check for confirmation).
Table 2.3 Example of Linear 45 degree input through an Ideal QWP, text tab on Polarization Pupil Map.
In the text tab of the Polarization Pupil Map, each pupil position is displayed with information about the polarization state. Here, every position is the same, but that is not always the case. Px is the Pupil Coordinate along the x-axis. Py is the Pupil Coordinate along the y-axis. Ex is the magnitude of the Electric Field, and the 1st row of the Jones vector. Ey is the magnitude of the Electric Field, and the 2nd row of the Jones vector. Intensity is normalized Intensity or transmission. Phase (Deg) is the phase difference between Phase X and Phase Y. Orientation is the major axis of the polarization ellipse in degrees, typically measured from the x-axis.
We can plug in the values like so,
and show that this is indeed Right Hand Circular Polarization. This information can also be gathered from the Polarization Ray Trace, which can be extremely useful when dealing with complicated polarization components and states. This is because we can call the POLTRACE from a Macro and compile this information into a Jones Matrix. The Macro code for this is below, but the details of this Macro is out of scope for this blog post.
------------------------------------------------------------------------------------------------------------
! This macro is intended to compute Polarization as a Jones Vector from Polarization Ray Trace.
! define the ray to be traced.
hx = 0
hy = 1
px = 0
py = 0
wavelength = PWAV()
last_surface=NSUR()
FORMAT 4.3
PRINT "Trace this ray: "
PRINT "hx = ", hx
PRINT "hy = ", hy
PRINT "px = ", px
PRINT "py = ", py
GOSUB get_trans
trans_config_1 = intensity_1
electric_real_x = Ex_real
electric_real_y = Ey_real
electric_real_z = Ez_real
electric_imag_x = Ex_imag
electric_imag_y = Ey_imag
electric_imag_z = Ez_imag
A_x_1 = SQRT(Ex_real*Ex_real + Ex_imag*Ex_imag)
A_y_1 = SQRT(Ey_real*Ey_real + Ey_imag*Ey_imag)
A_z_1 = SQRT(Ez_real*Ez_real + Ez_imag*Ez_imag)
Phase_x_1 = Phase_X_1
Phase_y_1 = Phase_Y_1
Phase_z_1 = Phase_Z_1
EMajor = SQRT(0.5*(A_x_1*A_x_1 + A_y_1*A_y_1 + SQRT((A_x_1*A_x_1*A_x_1*A_x_1) + (A_y_1*A_y_1*A_y_1*A_y_1) + 2*(A_x_1*A_x_1*A_y_1*A_y_1)*COSI(2*(Phase_x_1 - Phase_y_1)))))
Eminor = SQRT(0.5*(A_x_1*A_x_1 + A_y_1*A_y_1 - SQRT((A_x_1*A_x_1*A_x_1*A_x_1) + (A_y_1*A_y_1*A_y_1*A_y_1) + 2*(A_x_1*A_x_1*A_y_1*A_y_1)*COSI(2*(Phase_x_1 - Phase_y_1)))))
Epsilon = 0.0000000000001
Ap = (180/3.1415926535)*(1/2)*ATAN((2*A_x_1*A_y_1*COSI(Phase_x_1 - Phase_y_1)) / (A_x_1*A_x_1 - A_y_1*A_y_1 + Epsilon))
PRINT "Jones Vector Data Config 1"
PRINT "A_x = ", A_x_1
PRINT "A_y = ", A_y_1
PRINT "Phase_x [deg] = ", Phase_x_1*(180/3.1415926535)
PRINT "Phase_y [deg] = ", Phase_y_1*(180/3.1415926535)
PRINT "Phase_diff [deg] = ", Phase_x_1*(180/3.1415926535) - Phase_y_1*(180/3.1415926535)
PRINT "Major Axis Length = ", EMajor
PRINT "Minor Axis Length = ", Eminor
PRINT "Angle between Major and x-axis [deg] = ", Ap
PRINT
PRINT "Program End"
END
SUB get_trans
POLDEFINE 0,1,150,60
SETCONFIG 1
intensity_1 = 0
POLTRACE hx, hy, px, py, wavelength, 1, last_surface
!GOSUB print_vector
Ex_real = VEC1(2)
Ey_real = VEC1(3)
Ez_real = VEC1(4)
Ex_imag = VEC1(5)
Ey_imag = VEC1(6)
Ez_imag = VEC1(7)
Phase_X_1 = VEC1(16)
Phase_Y_1 = VEC1(17)
Phase_Z_1 = VEC1(18)
intensity_1 = (Ex_real*Ex_real) + (Ey_real*Ey_real) + (Ez_real*Ez_real)
intensity_1 = intensity_1 + (Ex_imag*Ex_imag) + (Ey_imag*Ey_imag) + (Ez_imag*Ez_imag)
RETURN
SUB print_vector
FORMAT 1.0
PRINT "Electric Field Data from each configuration"
FORMAT 4.6
PRINT "Ex_real_1 = ", VEC1(2)
PRINT "Ey_real_1 = ", VEC1(3)
PRINT "Ez_real_1 = ", VEC1(4)
PRINT "Ex_imag_1 = ", VEC1(5)
PRINT "Ey_imag_1 = ", VEC1(6)
PRINT "Ez_imag_1 = ", VEC1(7)
PRINT
RETURN
------------------------------------------------------------------------------------------------------------
Jones Calculus Example
In Jones Calculus, optical elements are assumed to be linear. Each element is a 2x2 Matrix, M, that operates on a input Jones vector to output a new Jones vector as shown mathematically below.
Figure 2.4 Jones vector 1 mathematically operated on by optical element to output Jones Vector 2.
Multiple elements are computed as follows.
Figure 2.5 Jones Vector mathematical operation for multiple elements.
For a Mirror, the corresponding Jones matrix describing reflection is,
and the transmitted Jones matrix is
and the coefficients rs and rp are Fresnel reflection coefficients for s- and p-polarizations, respectively, while ts and tp are Fresnel transmission coefficients for s- and p-polarizations, respectively. The reflection and transmission coefficients are derived from matching boundary conditions at the interface. They are given here for context, but not derived.
The theta's are derived from Snell's Law.
A perfect electrical conductor is called the mirror matrix as shown below.
Not all interfaces will contain a perfect reflection and most real-life mirrors will induce a phase shift, which changes the polarization state. It is important to collect all coating and substrate information possible from vendors to properly simulate the optical component in Zemax. For example, the mirror matrix for an aluminum mirror looks like,
which will induce a Phase shift and change the polarization state.
For example if a RHC polarization is incident on an ALUM MATE from the coating file using the coating FP we can see just before the mirror the state is RHC, and right after is a form of elliptical.
Figure 2.6 RHC Polarization just before the Aluminum Mirror.
Figure 2.7 RHC incident on Aluminum Mirror and acquired a Phase shift. After the Mirror the polarization state is elliptical.
Use the Macro from above, and we can see what the parameters of this new polarization state is after the Aluminum Mirror.
Figure 2.8 Jones Calc Macro before (Top middle) and after Mirror (Top right).
Image and notation credit to Prof. Tom D. Milster, University of Arizona
Conclusion
Jones notation is widely used in Zemax OpticStudio as discussed in the first part. Jones Matrices also allow for correct modeling of various polarization components that would otherwise be difficult to understand. This framework makes modeling and optimizing these optical components easier.
Ozen Engineering Inc. leverages its extensive consulting expertise in CFD, FEA, optics, photonics, and electromagnetic simulations to achieve exceptional results across various engineering projects, addressing complex challenges such as antenna design, signal integrity, electromagnetic interference (EMI), and electric motor analysis using Ansys software.
We offer support, mentoring, and consulting services to enhance the performance and reliability of your electronics systems. Trust our proven track record to accelerate projects, optimize performance, and deliver high-quality, cost-effective results. For more information, please visit https://ozeninc.com.
If you want to learn more about our consulting services, please visit: https://www.ozeninc.com/consulting/
CFD: https://www.ozeninc.com/consulting/cfd-consulting/
FEA: https://www.ozeninc.com/consulting/fea-consulting/
Optics: https://www.ozeninc.com/consulting/optics-photonics/
Photonics: https://www.ozeninc.com/consulting/optics-photonics/
Electromagnetic Simulations: https://www.ozeninc.com/consulting/electromagnetic-consulting/
Thermal Analysis & Electronics Cooling: https://www.ozeninc.com/consulting/thermal-engineering-electronics-cooling/