Detection of a Ferromagnetic Microwire

by Frank J. Crowne

ARL-TR-6115 September 2012

Approved for public release; distribution unlimited.
NOTICES

Disclaimers

The findings in this report are not to be construed as an official Department of the Army position 

unless so designated by other authorized documents.

Citation of manufacturer’s or trade names does not constitute an official endorsement or 

approval of the use thereof.

Destroy this report when it is no longer needed. Do not return it to the originator.
Army Research Laboratory 

Adelphi, MD 20783-1197 

ARL-TR-6115 September 2012 

Detection of a Ferromagnetic Microwire 

Frank J. Crowne 

Sensors and Electron Devices Directorate, ARL 

Approved for public release; distribution unlimited. 
ii

REPORT DOCUMENTATION PAGE Form Approved

OMB No. 0704-0188

Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the 

data needed, and completing and reviewing the collection information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing the 

burden, to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302. 

Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid 

OMB control number.

PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS.

1. REPORT DATE (DD-MM-YYYY)

September 2012

2. REPORT TYPE

Final

3. DATES COVERED (From - To)

4. TITLE AND SUBTITLE

Detection of a Ferromagnetic Microwire

5a. CONTRACT NUMBER

5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER

6. AUTHOR(S)

Frank J. Crowne

5d. PROJECT NUMBER

5e. TASK NUMBER

5f. WORK UNIT NUMBER

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

U.S. Army Research Laboratory

ATTN: RDRL-SER-E

2800 Powder Mill Road

Adelphi, MD 20783-1197

8. PERFORMING ORGANIZATION

 REPORT NUMBER

ARL-TR-6115

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR’S ACRONYM(S)

11. SPONSOR/MONITOR'S REPORT

 NUMBER(S)

12. DISTRIBUTION/AVAILABILITY STATEMENT

Approved for public release; distribution unlimited.

13. SUPPLEMENTARY NOTES

14. ABSTRACT

This report, which supplements the material presented in the IEEE Transactions in Magnetics publication entitled “TwoFrequency Excitation of a Ferromagnetic Microwire,” describes a simple configuration of antenna beams, polarizations, and 

wire placement calculations, which are used in calculating the response of a magnetic wire to ferromagnetic resonance (FMR)

excitation by an external magnetic field. 

15. SUBJECT TERMS

Ferromagnetic resonance, nonlinear response, magnetic microwires

16. SECURITY CLASSIFICATION OF: 

17. LIMITATION

 OF 

 ABSTRACT

UU

18. NUMBER

 OF 

 PAGES

28

19a. NAME OF RESPONSIBLE PERSON

Frank J. Crowne

a. REPORT

Unclassified

b. ABSTRACT

Unclassified

c. THIS PAGE

Unclassified

19b. TELEPHONE NUMBER (Include area code)

(301) 394-5759

Standard Form 298 (Rev. 8/98) 

Prescribed by ANSI Std. Z39.18 
iii

Contents 

1. Introduction 1

2. The Wire Problem 1

3. Nonlinear Response: Second Order 4

4. Nonlinear Response: Third Order 12

5. Conclusion 20

6. References 21

Distribution List 22
iv

INTENTIONALLY LEFT BLANK. 
1

1. Introduction

As described in the publication “Two-Frequency Excitation of a Ferromagnetic Microwire” (1), 

the remote sensing of ferroelectric microwires based on ferromagnetic resonance (FMR) has 

been proposed as a way to passively collect information from a novel type of “radio frequency 

identification (RFID) card” that can be attached to a target under surveillance (2). 

Implementation of this proposal requires a careful study of the external scattering problem of 

electromagnetic (EM) waves from the microwire, which is the topic of this technical report. A 

detailed description of the physics of ferromagnetic microwires can be found in reference 1 and 

ARL-TR-6114 (3). 

2. The Wire Problem

Assume that a ferroelectric microwire located at the coordinate origin and oriented vertical to the 

ground, i.e., along the “z-axis”, is illuminated by two external magnetic fields, which we refer to 

as “signal” and “pump” fields, both of which are the magnetic components of linearly polarized 

EM beams. The “illumination” geometries correspond to radar applications: if both beams 

originate from the same antenna, we speak of a “monostatic” geometry, while for signals 

originating from different antennas the geometry is referred to as “bistatic.” In this report, we 

discuss the monostatic scenario only, i.e., we assume that both beams originate from the same 

antenna and propagate horizontally and parallel to one another along, e.g., the x-axis. The 

geometry is shown in figure 1. 

Figure 1 Monostatic geometry for wire illumination—propagation vector. 
2

The two fields are characterized by both spatial and temporal phase information, i.e., if 

H H H p p p x y z , , 

and

H H H s s s x y z , , 

are real magnitudes of the (strong) pump field and the 

(weak) signal field, the amplitudes of the rectangular components of the fields are given by 

    p

h Re p p

i t x

t e x x

     

  

   

H

, 

    p

h Re p p

i t y

t e y y

     

  

   

H

, and 

    p

h Re p p

i t z

t e z z

     

  

   

H

, where spatial phase information associated with the wave 

polarization is included by letting 

p p p x y z     

. For our case, we assume that the pump field 

at the antenna is linearly polarized along the z-axis, so that 

h 0 px

, 

h 0 py

, and 

hp p z H c

, 

where 

Hpc

is the total pump-field magnitude. In the same way, the (weak) signal field at the 

antenna is linearly polarized in the yz plane, with its electric field vector making an angle  with 

the z-axis, having components 

h 0 sx 

,

h cos sy s  H c

, and 

h sin sz s  H c , where 

Hsc

is the 

total signal-field magnitude. Then, the wave components at the target are 

        0 0

h Re 0 h Re cos p s s

sin p s

i x t i x t

t e t e

                       

                     

     

H c

H H c c

 (1) 

Following reference 1, we introduce the following complex fields (italics) 

0 p 0 s

0 cos p s s

sin p p s s

h x h x

h h y y

h h z z

 

  

  

H c

H H c c

(2) 

and the circular field components, defined as follows: 
3

p p s

 , p p p p p p

0 , 0 , , 0 p p p p p

s s s

 , s s s s s s

0 , cos , sin , 0 , 0 s s s s s

0 p p

s

i i x y i z h ih e i e h e x y x y z z

x y z c z

i i i x z y

h ih e i e h e x y x y z z

x y c z c y z

h ih x y

h x

     

   

    

     

   

        

 





H H H

H H H H

H H H

H H H H H

 

 

cos s s

0 p p

cos s s s

ih i y c

h ih x y

h ih i x y c





  



 

  

 

    

H

H (3) 

In order to estimate the power radiated by FMR excitation of the microwire, it is necessary first 

to describe the dynamics of its magnetization in the presence of an external time-dependent 

magnetic field. These dynamics are governed by the Landau-Lifshits-Gilbert (LLG) equation (4): 

  M M M H t M eff t t Mst

  

    

 

(4) 

The material parameters of the wire include 

Mst

, the static magnetization of the material; γ, the 

gyromagnetic ratio; and α, the damping constant.

H t  eff

is an effective magnetic field, which 

includes the DC anisotropy field 

Ha

that fixes the direction of 

Mst

in the material, the 

depolarization field (5) due to the body’s shape, and the external time-dependent applied field. 

Note that in the absence of an ac magnetic field, there is no depolarization field, and so 

M H 0

st   a

under DC conditions, i.e., there is no DC torque. The first term is the torque 

exerted by the effective magnetic field 

H t  eff

, while the second term gives rise to damping via 

eddy currents (6). The form of this equation implies that 

2

0

M M M

t



 





is constant in time. 

The RF magnetic field 

h t 

generates an RF magnetization 

m

that under remote-sensing 

conditions is small compared to 

Mst

. The dynamics of this magnetization, in particular its decay 

with time, is strongly affected by the conservation of length of the total magnetization vector, 

whose endpoint is constrained to lie on a spherical surface at all times. 

Let us further specify the perturbed magnetization and effective magnetic field in the LLG

equation as follows: 

M M H H m , h m ,       st eff a

where 

M H 0

st   a

and
4

0 0

0 0

0 0

N

N

Nz

           

(5) 

is the depolarization tensor; for a long wire,

N N 1/ 2 and 0   z

. As described in reference 1, 

we can introduce a scaled dimensionless magnetization 

g

such that 

m g Mst 

. In general, this 

magnetization has three components

g ,g ,g x y z

, where 

gz

is the component of 

g

parallel to the 

wire and 

  g ,g x y

is a two-dimensional vector perpendicular to the wire, with the latter small 

compared to the former. As shown in reference 1, under weak-signal conditions the 

perpendicular components of the dimensionless magnetization can be expressed in terms of a 

single complex number 

G

at each illuminating frequency, expressions for which are 

   

   

s s s s

 1s

s s s s

p p p p p p

1p

p p p p

h ih h ih x y i t sx sy i t

e e

i i B B

h ih x y i t h ih x y i t

e e

i i B B

 

       

             

 

       

               

G

G

(6) 

In the geometry we have chosen, these expressions become 

s s cos 1,s s

1 1 s s s s

0

1,p

i i i t i t

e e c

i i B

     

   

          



G

G

H

 (7) 

while 

g 0 z1,s

g 0 z1,p





3. Nonlinear Response: Second Order 

From reference 1, we find that, to second order, the wire response functions satisfy the equations 
5

 

 

h

2 2 1

* * *

h

2 2 1

i

B z

t

i

B z

t



     





     



G G G

G G G

(8) 

where the “sources” appearing on the right side of these equations are independent of 

2

G . 

Introducing the notation 

h h s,p s,p s,p

s,p * h h s,p s,p s,p

i H

x y

H

x y i H

    

              

(9) 

and

1 1

 s,p s,p 2 2 s,p s,p

* * 1 1

 s,p s,p 2 2 s,p s,p

i i B B

i i B B

                 

                 

 (10) 

we obtain the following expressions for the source terms: 

   

2 2 s s * * *

s s s s s s

s+p * * * * * * s+p

p p p p s s s p s s s p

* * * * * *

h

1 s s p p p p 2 s p s s s p

* * * s

p p s p s s

i t i t h H e h H e

z z

i t i t

h H h H e h H h H e

z z z z

z h H h H h H h H

z z z z

i

h H h H e

z z

          

                         

    



                   

               

G

   

p * * * s p

s s p p p s

2 2 p p * * *

p p p p

2 2 s s * * *

s s s s s s

s+p * *

s

p p s s p s p p

*

h

1 2

t i t

h H h H e

z z

i t i t

h H e h H e

z z

i t i t h H e h H e

z z

i t

h H h H e h H

z z z

z

 

 

 

                 

   

         

 

           

             

  

 



    G

* * * * s+p

p s s

* * * * * *

p p s s p p s s s p s p

* * * s p * * * s p

s s p p p p p s s p s s

2 2 p p * * *

p p p p

i t

h H e

z

h H h H h H h H

z z z z

i t i t

h H h H e h H h H e

z z z z

i t i t

h H e h H e

z z







               





                

    

                        

     



    

       



























 

(11) 

Let us write the second-order response in the following spectral form: 
6

2 2,2s 2,2p 2,s+p 2,s p 2,0     

G G G G G G . (12) 

Here 

2,2s G

is the response at the 2

nd harmonic of the signal field, 

2,2s G

is the response at the 

2

nd harmonic of the pump field, 

2,s+p G

is the response at the sum of the signal and pump 

frequencies, 

2,s p

G

is the response at the difference of the signal and pump frequencies, and 

2,0 G

 is a DC response arising from rectification of the pump and signal. We introduce the 

following convention: if A is a sum of multi-frequency terms, we denote the sum of all the terms 

in A at the same frequency q by 

  A q

.

The response 

2,2s G

at the 2nd harmonic frequency 2s is computed as follows: 

 

   

 

   

h

2,2s 2,2s 1 2 s

* 2 2 2 * 2 s s s s

s s s s

s s 2,2s+ 2,2s 2 2

* * * h

2,2s 2,2s 1 2 s

* 2 2 s s

s s 2 s s s s

i

B z

t

i t i t i t i t h H e h H e S e S e

z z

i B z

t

i t i t h H e h H e

z z



           

                      

      

     



        



   

             

       

G G G

G G G

2 2 s s *

2,2s 2,2s 2

i t i t S e S e       

       

(13) 

 

2,2s 2 2 2,2s s s

2,2s 2

2s 2s 2s 2s

s s s 2 2 s s s s s

2

2s 2s 2s 2s

s s 2 s

s s

2 s s

2s 2s 2s 2s

S S i t i t

e e

i i B B

h H h H

z i t z i t

e e

i i B B

i t h H e h H

z z i i B B

  

      

    

           

  

 

          

    

         

 



                       

G

     

 

  

 

  

2 s

2

s 2 2 s

s s s s

2

2s 2s s s 2s 2s s s

2

s s 2 2 s s s s s

2 1 1 1 1

2s 2s s s 2s 2s s s

i t

e

h H h H

z z i t i t

e e

i i i i B B B B

h h ih h h ih

z z x y i t x y i

e e

B i i i i

     

 

      

       

   

                       





       

   

                    

s

t

 

  

  (14) 
7

where we introduce the dimensionless frequencies 

i

i

B



 



. Likewise, the response 

2,2p G

at 

the 2nd harmonic frequency 2p is

 

 

h

2,2p 2,2p 1 2p

* 2 2 2 2 p p p * p

p p p p

p p 2,2p+ 2,2p 2 2

* * * h

2,2p 2,2p 1 2p

2 p

p p 2 p p p p

i B z

t

i t i t i t i t

h H e h H e S e S e

z z

i B z

t

i t

h H e h H

z z



           

                                

    

         



        



   

           

    

G G G

G G G

* 2 2 2 p p * p

2,2p 2,2p 2

i t i t i t

e S e S e

                               

(15) 

2,2p 2 2 p p 2,2p

2,2p 2

2p 2p 2p 2p

p p p 2 2 p p p p p

2

2p 2p 2p 2p

p p 2 p

2 p p

2p 2p 2p 2p

S S i t i t

e e

i i B B

h H h H

z i t z i t

e e

i i B B

i t

h H e

z i i B B

  

           

           

          

          

    

         

 

 

     

  

          

G

     

 

  

 

 

2 p

p p

2 p 2 2 p p p p p

2

2p 2p p p 2p 2p p p

2

p p 2 p p p p p

2 1 1 p p 1 1 2s 2s 2p 2p p

i t

h H e

z

h H h H

z z i t i t

e e

i i i i B B B B

h h ih h h ih x y i t x y z z

e

i i B i

       



 

   

          

                             



       

   

                     

2 p

p

i t

e

i

 

   

  

(16) 

The DC response 

2,0 G

is
8

  h

2,0 2,0 1 0

* * * * * *

h

1 s

s s p p p s s s p p p 0 2

1 1 * *

s p s p 2 2 s s p p

2 * * 1 1

s s p p

2 2 s s p p

2,

i

B z

t

z

h H h H h H h H

z z z z

h H h H

z z i i B B

h H h H

z z i i B B



           

  

                          

   

               

 

 

 

             



G G G

G

G

1 1 * *

s

s s s

s s s s

0 2 2 1 1 * *

p p p p

p p p p

1 1 * * 2 s

s 1 1 s s

s s s s

2 1 1 * *

p p p p

1 1 p p p p

h H h H

z z i i B B

B h H h H

z z i i B B

h H h H

z z i i

B h H h H

z z i i

 

   

            

                                 



 

          



           

        



 



 (17) 
9

The sum-frequency response 

2, p s

G

is

 

 

h

2,s+p 2,s+p 1 s+p

s+p

p p s p s s s+p * s+p

* 2,s+p+ 2,s+p 2 2 s+p

p p s s s p

* * * h

2,s+p 2,s+p 1

i

B z

t

i t

h H h H e

z z i t i t

S e S e

i t

h H h H e

z z

i

B z

t



           

            

                     

                      



        





G G G

G G G

s+p

s+p

p p s s s p s+p * s+p

* 2,s+p 2,s+p 2 2 s+p

s p p p s s

i t

h H h H e

z z i t i t

S e S e

i t

h H h H e

z z



     

 

                          

                       

(18) 

2,s+p s+p 2,s+p s+p

2,s+p 2

s+p s+p s+p s+p

p p p p s s s p s s s+p s p s+p

2

s+p s+p s+p s+p

p

s+p

2

S S i t i t

e e

i i B B

h H h H i t h H h H i t z z z z

e e

i i B B

B

  

           

           

    



                       

    

         

 

 

   



 

G

p

s p p s

s+p s+p s+p s+p s+p

s s

s p s p

s+p s+p s+p s+p

h H h H

i z z

i t i i t B

e e

h H h H

i z z i B B

                  

                   

      

            

        

                     

(19) 

  

  

  

  

p p s s

2 s+p s+p p p s+p s+p p p s+p s+p

2

p s s p

s+p s+p s s s+p s+p s s

h H h H

z z

i i i i B B i t B B i t

e e h H h H z z

i i B B i i B B

         

                           

           

                 

   

                       

   

 

  

 

  

 

  

 

  

p p p p s s

2 1 1 1 1

s+p s+p p p s+p s+p p p s+p

2

s s p s s p

1 1 s+p s+p s s 1 1 s+p s+p s s

h h ih h h ih x y x y z z

i i i i

i t

e

B h h ih x y h h ih z x y z

i i i i



















                                      

      

     

     

   

                      

s+p i t

e

  

  



   





 

 

    

(20) 
10

Finally, the difference-frequency response 

2, p s

G

is

 

 

h

2,s p 2,s p 1 s p

* * * s p

p p s s s p s p * s p

* 2,s p+ 2,s p 2 2 * * * s p

p p s s s p

* * * h

2,s p 2,s p 1

i B z

t

i t

h H h H e

z z i t i t

S e S e

i t

h H h H e

z z

i B z

t



        

     

      

          

         

         

                         



      

  

G G G

G G G

s p

* * * s+p

p p s s s p s+p * s+p

* 2,s p 2,s p 2 2 * * * s+p

p p s s s p

i t

h H h H e

z z i t i t

S e S e

i t

h H h H e

z z

      

     

 

                          

                           

(21) 

2,s p s p 2,s p s p

2,s p 2 s p s p s p s p

* * * * * *

p p s s p p s s s p s p s p s p

2 s p s p s p s p

*

p

2

S S i t i t

e e

i i B B

h H h H h H h H

z z i t z z i t

e e

i i B B

  

               

                

    



            

            

    

             

 

 

 

 

G

 

p *

p p s s

s p s p s p s p s p s p

* s s

p s p s

s p s p s p s p

*

s p

2 s p s p p

2

h H h H

z z i i i t B i t B e e

h H h H

z z i

B

i B

h H

z

i i B B

            



  

                                          

  

      

          



            





                  

 

  

   

  

 

p s

p s p s p p p s p s p

*

s p s p

s p s p s s s p s p s s

*

p p s

2

1 s

2

h H

z

i i i t B B i t

e e

h H h H

z z

i i i i B B B B

h h ih x y z

B

             

                        

       

       

     

                                   



     

  

  

   

 

  

 

   

 

  

p p s

1 1 1 s p s p p p p s p p p s p s p

*

s s s s p p

1 1 1 1 s p s p s s s p s p s s

h h ih x y z

i i i i i t i t

e e

h h ih h h ih x y x y z z

i i i i

       

     

                                    

     

      

                                   

(22) 

The various numerators of these terms are evaluated according to our scenario: for the sumfrequency response 

2, p s

G ,
11

   

 

p p s

 , p p p p p p

0 , 0 , , 0 p p p p p

s s s

 , s s s s s s

0 , cos , sin , 0 , 0 s s s s s

sin 0 p p s s

s s p

i i x y i z h ih e i e h e x y x y z z

x y z c z

i i i x z y

h ih e i e h e x y x y z z

x y c z c y z

h h ih x y c z

h h ih x y z

     

   

    

     

   

        

  





H H H

H H H H

H H H

H H H H H

H

 

   

   

cos p s

sin 0 p p s s

cos s s p s p

i c c

h h ih x y c z

h h ih i x y c c z







  



 



   



  

   

H H

H

H H

(23) 

 

  

 

  

 

  

 

  

0 sin 0 sin s s

2 1 1 1 1

s+p s+p p p s+p s+p p p s+p s

2,s+p 2 cos cos p s p s

1 1 1 1

s+p s+p s s s+p s+p s s

c c

i i i i

i t i

e e

i i B c c c c

i i i i

     

   

                              

       

    

           

                   

   

G

H H

H H H H

     

+p

2

s+p s+p

cos p s 2 1 1 1 1

s+p s+p s s s+p s+p s s

t

i t i t i i

e e c c

B i i i i

 

 

 

     

            

         

    

                     

     

H H

(24) 

and for the difference-frequency response 

2, p s

G ,

   

p p s

 , p p p p p p

0 , 0 , , 0 p p p p p

s s s

 , s s s s s s

0 , cos , sin , 0 , 0 s s s s s

*

0 sin p p s

s

*

s p

i i x y i z h ih e i e h e x y x y z z

x y z c z

i i i x z y

h ih e i e h e x y x y z z

x y c z c y z

h ih h x y c z

h h ih x z

     

   

    

     

   

        

  





H H H

H H H H

H H H

H H H H H

H

   

   

   

cos s p s

0 sin p p s

s

cos s s p s p

i y c c

h ih h x y c z

h h ih i x y c c z







    



     

 

    



H H

H

H H

(25) 
12  

   

 

  

 

   

 

  

0 sin 0 sin s s

2

1 1 1 1 s p s p p p s p s p p p s p

2,s p 2 cos cos p s p s

1 1 1 1 s p s p s s s p s p s s

c c

i i i i

i t i

e e

i i B c c c c

i i i i

     

   

  





                          

  

                     

     

                         

G

H H

H H H H

     

s p

2

s p s p

cos p s 2 1 1 1 1 s p s p s s s p s p s s

t

i t i t i i

e e c c

B i i i i

 

 

 

 

                               

                                      

H H

(26) 

For the z – component terms, we find 

   

2 2

1 1 * * 1 2 2 s s g cos z2,s 1 1 1,s 1,s s

2 2 2 1 1 s s s s

2

1 1 2

1 2 2 2 2 2 2 2 2

cos 1 1 s s s s s 2

1 1 2 2 1 1 s s

1 1 1 1 s s s s s s s s

i i i t i t

e e c

i i B

c

B i t i t

e e

i i i i

     

         

            





     

                   

                      

G G G G H

H

1 *

g

z2,p 1,p 1,p 2

















  G G

(27) 

which do not contain sum or difference contributions. 

4. Nonlinear Response: Third Order

When the pump field polarization is linear and “vertical,” the third-order equation reads 

 

 

h

3 3 2

* * *

h

3 3 2

i

B z

t

i

B z

t



     





     



G G G

G G G

(28) 

where 

2

G

is the second-order response 
13  

  

 

  

 

  

2 2,2s 2,2p 2,s+p 2,s p 2,0

2

s s 2 2 s s s s s s

2 1 1 1 1

2s 2s s s 2s 2s s s

2

p p 2 p p p p

2 1 1 2s 2s p p

h h ih h h ih

z z x y i t x y i t

e e

B i i i i

h h ih h h ih x y i t x z z

e

B i i

    



   

          

   

                      

       

   

          

G G G G G G

 

  

 

  

 

  

 

  

 

p 2 p

1 1 2p 2p p p

p p p p s s

2 1 1 1 1

s+p s+p p p s+p s+p p p s+p

2

s s p s s p

1 1 s+p s+p s s

y i t

e

i i

h h ih h h ih x y x y z z

i i i i

i t

e

B h h ih x y h h ih z x y z

i i

  

   

         



      

                    

        

    

 

            

 

   

 

  

 

s+p

1 1 s+p s+p s s

*

p p p p s

s

2 1 1 1 s p s p p p s p

2 *

s s p

1 1 s p s p s s

i t

e

i i

h h ih h h ih x y x y z

z

i i i t

e

B h h ih x y z

i i

       

 

  

 

           

   

  

  

            

           



           

   

 

  

1 s p s p p p

s p

s s p

1 1 s p s p s s

* * * * * *

s

s s p p p s s s p p p 2

i i i t

e

h h ih x y z

i i

h H h H h H h H

z z z z

B

       

            

  

  



             

                       

(29) 

Denoting the quantities with dimensionless denominators by , we write 

2 2 2 2 2 s s p p

2s 2s 0 2p 2p

2 2

s+p s+p s p s p

s p s p s p s p

i t i t i t i t

e e e e

i t i t i t i t B

e e e e

         

       

    

                 

         

         

G

(30) 

where 
14  

  

 

     

 

  

 

  

2

2

s s sin cos sin cos s s

2s 1 1 1 1 1 1

2s 2s s s 2s 2s s s 2s 2s s s

s s sin cos sin cos s s

2s 1 1 1 1 1

2s 2s s s 2s 2s s s 2

h h ih x y i

z c c c

i i i i i i

h h ih x y i

z c c c

i i i i

     

  



                       



      

  



                 

H H iH

H H iH

  

       

       

2

1 s s s 2s

* *

s s p p s s p p

s s p p

0 1 1 1 1 s s p p s s p p

sin cos p p 0 0 sin cos s s s s

1 1 1 1 s s p p s s p p

sin cos s

i i

h h ih h h ih h h ih h h ih

x y x y x y x y z z z z

i i i i

c c i i c c c c

i i i i

c

     

 

   

   

               



         

               

  

H H H H H H

H

 

  

 

  

 

  

 

  

1 1 s s s s

p p 0

0

2p 1 1 1 1

2s 2s p p 2s 2s p p

p p 0

0

2p

1 1 1 1

2p 2p p p 2p 2p p p

i i

i i

h h ih x y z c

i i i i

h h ih x y z c

i i i i

    

         



  



               



 

  



               

H

H

(31) 

 

  

 

  

 

  

 

     

 

p p s s

s p

s p

1 1 1 1

s+p s+p p p s+p s+p s s

sin 0 cos s p s

cos p s

1 1 1 1 1 1

s+p s+p p p s+p s+p s s s+p s+p s s

p p s

s p

1

s+p

h h ih h h ih

x y x y z z

i i i i

i

c c c i

c c

i i i i i i

h h ih x y z

i

 

   

               

      

                       



 

  

  

H H H

H H

  

 

  

 

  

 

     

 

   

s s p

1 1 1

s+p p p s+p s+p s s

sin 0 cos s p s

cos p s

1 1 1 1 1 1

s+p s+p p p s+p s+p s s s+p s+p s s

* *

p p s

s p

s p 1 1 s p s p p p

h h ih x y z

i i i

i

c c c i

c c

i i i i i i

h h ih h h i

x y x z z

i i



 



            

      

                       

 

             

H H H

H H

 

  

 

   

 

     

 

   

 

 

s

1 1 s p s p s s

sin 0 cos s p s

cos p s

1 1 1 1 1 1 s p s p p p s p s p s s s p s p s s

p p s s

s p

s p

1 1 1 1 s p s p p p s p s p s

h y

i i

i

c c c i

c c

i i i i i i

h h ih h h ih

x y x y z z

i i i i

         

       

                             



  

   

                   

H H H

H H

 

 

   

 

     

s

sin 0 cos s p s

cos p s

1 1 1 1 1 1 s p s p p p s p s p s s s p s p s s

i

c c c i

c c

i i i i i i

       

                             

H H H

H H

(32) 
15

We obtain the general source terms from the spectral decomposition of 

hz 2

G

: 

h Re 2 2

s

s

p 2 2 2 2

s s p p

1 p 2 2 0 2 2

2 * s s+p s+p s p

s

* p

p

i t i t a b h e h e z az bz

i t h e

z

i t i t i t i t i t h e e e e e

z s s p p

i t i t i t i t h e e e e z s p s p s p s

i t

h e

z

       

       

   

 

 

           

    

     

  

  

        

   

       

      

 

G G

s p i t

e p

 

 

  

   

(33) 

 

2 2 2 2

s s p p

2 2 0 2 2 s

s

s+p s+p s p s p

2 2 3 p s p s s s

s 2 s 2 s 0 s 2 s 2

i t i t i t i t

e e e e

i t s s p p

h e

z i t i t i t i t

e e e e

s p s p s p s p

i t i

i t i t i t h e h e h e h e h e

z s z s z z p z p

         

     

     

 

         

               

           

    

   

 

   

s

2 2 s p p p s p

s s s s

2 2 2 2

s s p p

p 2 2 0 2 2

p

s+p s+p s p s p

t

i t i t i i t

h e h e h e h e s p s p s p s p z z z z

i t i t i t i t

e e e e

i t s s p p

h e

z i t i t i t i t

e e e e s p s p s p s p

h

 

           

           

         

           

 

       

 

               



   

   

2 2 s p s p 3 p p p

p 2 p 2 p 0 p 2 p 2

2 2 p s p s s s

p p p p

i t i t i t i t i t

e h e h e h e h e

z s z s z z p z p

i t i t i t i t h e h e h e h e s p s p s p s p z z z z

             

   

   

           

           

(34) 
16

 

2 2 2 2

s s p p

* 2 2 0 2 2 s

s

s+p s+p s p s p

2

* * * * * 3 p s s s s

s s s s s 2 2 0 2 2

i t i t i t i t

e e e e i t s s p p

h e

z i t i t i t i t

e e e e s p s p s p s p

i t i t i t i t h e h e h e h e h

z z z z z s s p p

         

     

     

         

             

         

        

 

   

2 p s

2 2 * * * * p p s p s p

s s s s

2 2 2 2

s s p p

* p 2 2 0 2 2

p

s+p s+p s p s p

i t

e

i t i t i t i t

h e h e h e h e s p s p s p s p z z z z

i t i t i t i t

e e e e i t s s p p

h e

z i t i t i t i t

e e e e s p s p s p s p

   

           

           

        

         

         

            

   

   

2 2 s p s p 3 * * * * * p p p

p p p p p 2 2 0 2 2

2 2 * * * * p s p s s s

p p p p

i t i t i t i t i t

h e h e h e h e h e

z z z z z s s p p

i t i t i t i t h e h e h e h e s p s p s p s p z z z z



 



             

        

           

           

(35)

For ease of grouping, let us color each term according to its frequency, writing

 

     

   

h

2

2 s p

s

2 s p

p 2

s

2 p

s

p p

s 0

s

s s 2 0

p

3 s

2 p s s p

s

2 s p

s 2

2 p s

2

p

p

s 2

2

s

i t h e

s

i t

i t

h e

z

s p

i t

h e

s

i t

h e

z

s p

i t i t

h e h e

z

i

h e

z

s p

i t

i t

h e

h e

z s

i t

h

i t h

z p

e

z

h e

s

h

s p

z

e

z z z p

z

z

z

   

   

  

 

  

   

 

   

 



   

   





   







    



  

 











 G

 

 

   

* 3 s

s 2

s

p

s s *

3 2 p

p 2

* p

s

p s

p

2 p

2

2 p s

p

2 p s

p s

* s * s

s 2

*

s

p

*

2

s 0 s 2

i t h e

z

s p

i t i t h e h e

z z s p s

i

i t

h e

z

s p

i

i t

h

t

h e

p

i t

h

i t h e

z s

e

z p

i t

h e s p

e s p z

i t

i t

t

h e

z

h

z

z

e

p

h

z

s p

e

z p

 

 

   

  

 

   

 

 

   

 



   

 

   



   

  

 

 





 















   

   

   

* p

s

* * p p

p p 0 2

3

2 s p

2

* s p

p 2

* p

p 2

2 p

2

* s p

s

2

* s p

2

* s

p

* * s s

p p

2

*

p

p

p s

i t

h e

z

s p

i t i t

h e h e

p

i t

i t i h e h e s p

i t

e

i t

i t

h e

h

i

i t

h

t

h e

h e s p z

i t

z

s p

e

z p

z

s p

h e

s

e

s

z

z p

z s z z

z

   

   

   

     



 

 

 

   

   



   

 



  

   

 





 

  







 



t

(36)
17 s s s s

s s p p 2 0

* * * * s s s s

s s p 2 0

p p p

3 3 s *

* *

s p s 2 0

s s

p

s

2

p

2

i t i t i t i t h e h e h e h e

z z z z s s p s p

i t i t i t i t h e h

i t i t h

i t i t i t

h e

e h e h e

h e h e h e s p s p p

e

s s p

z z

z z s p

h e

z z s s

z

z z

z

       

    

       



  

 

   

  

  



  



  



    



 







       

     

2 2 2 2 * * p s p s p s p s

p s p 2 2

2 2 2 p s p s p s * *

2

p

p p p p * *

s 0 2

s p 2

p

s

s

s p

p

i t i t i t i t

h e h e h e h e s p s p

i t

i t i t i t i t

h e h e h e h e

z

s p s p p

i t i t i t

h e h e h e h

p p s p z z

z z z z p p

z

z z

z

z

              

           

 



 

 

       

   

  





  

   

    





 

       

       

2 2 2 2 s p s p s p s p * *

s p s p 2 2

2

3 3 p p *

2 2 2 s p s p s p s p * *

s 2 2

p p 2 2

p p

p s

s

i t 2

e

s p

i

i

i t i t i t i t

h e h e h e h e s p s p s

i t i t i t

h e h e h e h e

z

s p s p s s

t i t

h e h e

z z z

z z z z s

z z p p

   

               

        

     



         

    

  







 



 











(37)

* * * * s s

2 0 2 0

3 3 s s *

s 2 2

* p

p s p s p s s p

* * *

s p s p s p s 2 0 0

s

p

i t i t h h h h e h h h h e s p s p s p s p

i t

h h h h e h h h h s p s p s p p s p z

z z z z z z z z s s

i t i t h e h e

z z s s

z z z z z z z

 

       

 

                    



          

   



   

 

















   

   

 

2 2 p s p

2 2 * * p s p s

* * s

2 s p * *

2

s p p s 2 2

s p

p

2

p s p s 2 2

s p

i t i t

h h e h h e

z z z z s p s p p p

i t i t

h h e h h e

p s p s p

i t

e

z z p

i t

h h e h h

z z s p s p s

p

z

z

z

z

                   

   

   

 

                      

     

    

     

       



 

 











 

   

2 s p

2 2 * * s p s p

2 2

2

3 3 p p *

p 2 2

p

p

s s p

i i t

h h e h h e s

i t i t

h e

s p p

h e

z p p

s

i t

e

z

z z s

s

z z

                      

   



      



 

  

  











(38)
18

Then, the spectral decomposition can finally be written 

3 3 s s *

h

2 2 s s 2

* * s s * *

p s p 2 s 0 s p 0 p s 2

* * p *

s p s 2 p 0 p s 0 s

i t i t h e h e z

z z s s

i t i t h h h h e h h h h e s p s s s p s p s p z z z z z z z z

i t

h h h h e h h h h

z

s p z z p s p z z z s p zs p

   

   

       

                        

                       

G

   

   

 

* p

p 2

2 2 * * p s p s

p s 2 2 p s

2 2 * * p s p s

p s 2 2 p s

2

* * s p

s p 2 s

i t

e

z p

i t i t

h h e h h e s p p p s p z z z z

i t i t

h h e h h e s p p p s p z z z z

i t

h h e h

z

s p z s zs p

        

                            

                            

                 

 

   

2 s p

p 2

2 2 * * s p s p

s p 2 2 s p

3 3 p p *

p p 2 2

i t

h e

z s

i t i

h h e h h e

z

s p z s s z

s p z

i t i t

h e h e

z z p p

          

                            

   

   

(39) 

Because we are interested only in the components that arise from mixing of the pump and signal, 

we solve for the sideband (intermod) responses only: 

 

2 2

s p s p

3,sp2p, 3,sp2p,

3 3 sideband sideband 2

2 2

s p s p

3,sm2p, 3,sm2p,

i t i t

S e S e

i

t B

i t i t

S e S e

                          

   

                                       

G G

(40) 

where 

3,s 2p, p s p 2

* * *

3,s 2p, p s 2 p

* *

3,s 2p, p s p s p 2 s p

*

3,s 2p, s p 2 s p p s p

S h h h

z

s p z z p s p

S h h h

z

s p z p zs p

S h h h

z z z p

S h h h

z z p z

         

         

         

  

      

(41) 

Let 

3 sideband 3,sp2p, 3,sp2p, 3,sm2p, 3,sm2p,        

G G G G G

. Then for the upper sideband,

we have 
19  

  

 

  

2,s+p, s+p 2,s p, s+p

2,s+p 2

s+p s+p s+p s+p

p p p s s

2 1 1 s+p s+p p p s+p

2

s s p

1 1 s+p s+p s s

S S i t i t

e e

i i B B

h h ih h h x y x z z

i i i t

e

B h h ih x y z

i i

  

 

          

           

   

 

            

     

     



         

G

 

  

 

  

p

1 1 s+p s+p p p s+p

s s p

1 1 s+p s+p s s

2

s+p s+p

s p s p 2

3,s+2p,

3,s+2p 2

s+2p s+2

ih y

i i i t

e

h h ih x y z

i i

i t i t

e e

B

S

i B

    

    

         

  

   



           

  

     

                

 

 

     G

 

s+2p 3,s 2p, s+2p

p s+2p s+2p

2 *

p p s+2p s+2p

s p s p 2 2 s+2p s+2p s+2p s+2p

3 1 1 s+2p s+2p s+p 2

cos p s 2

i t S i t

e e

i B

h h i t i t z z

e e

i i B B B

i

i i

c c

B

           

      

           

                           

          

   

  

H H

  

   

   

   

s+2p

1

s+p s s

s+2p

1 1 1

s+p s+p s s s+2p s+2p

s+2p

3 1 1 1

s+p s+p s s s+2p s+2p 2

cos p s 2

1 1 1

s+p s+p s s s+2p s+2p

i t

e

i

i t i

e

i i i

i t i

e

i i i

c c

B i

i i i

   

 

    

 



             

 

              

   

  



           

H H

s+2p i t

e

 

 

 

 

(42) 

Likewise, for the lower sideband, we have 
20  

   

 

  

2,s p, s p 2,s p, s p

2,s p 2 s p s p s p s p

*

p p s

s

2

1 1 s p s p p p s p

2 *

s s p

1 1 s p s p s s

S S i t i t

e e

i i B B

h h h h ih x y z

z

i i i t

e

B h h ih x y z

i i

  

               

                

  

  

  

            

     

     



           

G

 

   

 

  

p p

1 1 s p s p p p

s p

s s p

1 1 s p s p s s

2

s p s p

s p s p 2

3,s 2p, s

2,s 2p 2

s 2p s 2p

ih x y

i i i t

e

h h ih x y z

i i

i t i t

e e

B

S i

e

i B

    

   

            

  

  



             

        

 

                

    

 

       

G

  

2p 3,s 2p, s 2p

s 2 p s 2 p

2 *

s p s p p p s 2p s 2p

2 2 s 2p s 2p s 2 p s 2 p

3 1 1 1

s 2p s 2p s p s p 2

cos p s 2

t S i t

e

i B

h h i t i t z z

e e

i i B B B

i

i i

c c

B

           

        

                 

    

                    



              

   

  

H H

 

   

s 2p

s s

s 2p

1 1 1 s p s p s s s 2p s 2p

i t

e

i

i t i

e

i i i

    

 

   

   



                 

(43) 

These expressions are the ones used in reference 1. 

5. Conclusion

The complexity of these results, even for the idealized geometry chosen here, makes them 

problematic for use in deciphering experimental data from, e.g., pump illumination by a moving 

airframe. Nevertheless, the effort could be worthwhile if the signal returns were strong enough. 

Unfortunately, as reference 1 shows, this requires very high levels of pump power even over 

short distances. 
21

6. References 

1. Crowne, F. J. Multifrequency Excitation of a Ferromagnetic Microwire. IEEE Trans. 

Magnetics, to be published. 

2. Douse, W.; King, W. unpublished. 

3. Crowne, F. J. Perturbation Theory for the Landau-Lifshits-Gilbert Equation; ARL-TR-6114; 

U.S. Army Research Laboratory: Adelphi, MD, September 2012. 

4. Landau, L. D.; Lifshits, E. M. Theory of the Dispersion of Magnetic Permeability in 

Ferromagnetic Bodies. Phys. Z. Sowietunion 1935, 8, 153.

5. Kittel, C. Phys. Rev. 1948, 73, 155.

6. Gilbert, T. L. A Phenomenological Theory of Damping in Ferromagnetic Materials. IEEE 

Trans. Magnetics 2004, 40, 3443.
22

NO. OF 

COPIES ORGANIZATION 

 1 ADMNSTR 

 ELEC DEFNS TECHL INFO CTR 

 ATTN DTIC OCP 

 8725 JOHN J KINGMAN RD STE 0944 

 FT BELVOIR VA 22060-6218 

 1 US ARMY RSRCH DEV AND ENGRG 

 CMND 

 ARMAMENT RSRCH DEV & ENGRG 

 CTR 

 ARMAMENT ENGRG & TECHNLGY 

 CTR 

 ATTN AMSRD AAR AEF T 

 J MATTS 

 BLDG 305 

 ABERDEEN PROVING GROUND MD 

 21005-5001

 1 US ARMY ARDEC 

 ATTN AMSRD AAR AEE P 

 C HAINES 

 BLDG 25 

 PICATINNY ARSENAL NJ 07806-5000

 1 US ARMY ARDEC 

 ATTN AMSRD AAR AEM C 

 A MARSTON 

 BLDG 61S 

 PICATINNY ARSENAL NJ 07806-5000

 2 US ARMY ARDEC 

 ATTN AMSRD AAR AEM L 

 S GILMAN 

 ATTN AMSRD AAR AEP F 

 O NGUYEN 

 BLDG 65S 

 PICATINNY ARSENAL NJ 07806-5000

NO. OF 

COPIES ORGANIZATION 

 1 US ARMY INFO SYS ENGRG CMND 

 ATTN AMSEL IE TD A RIVERA 

 FT HUACHUCA AZ 85613-5300

 1 US GOVERNMENT PRINT OFF 

 DEPOSITORY RECEIVING SECTION 

 ATTN MAIL STOP IDAD J TATE 

 732 NORTH CAPITOL ST NW

 WASHINGTON DC 20402 

 26 US ARMY RSRCH LAB 

 ATTN IMAL HRA MAIL & RECORDS 

 MGMT 

 ATTN RDRL CI J PELLEGRINO 

 ATTN RDRL CIO LL TECHL LIB 

 ATTN RDRL CIO LT TECHL PUB 

 ATTN RDRL SEE I S SVENSSON 

 ATTN RDRL SEE O W M GOLDING 

 ATTN RDRL SER E 

 B HUEBSCHMAN 

 ATTN RDRL SER E F CROWNE 

 (5 COPIES) 

 ATTN RDRL SER E E VIVEIROS 

 ATTN RDRL SER E G BIRDWELL

 ATTN RDRL SER E T O’REGAN

 ATTN RDRL SER E P SHAH 

 ATTN RDRL SER E 

 R DEL ROSARIO 

 ATTN RDRL SER E T IVANOV 

 ATTN RDRL SER J MAIT 

 ATTN RDRL SER L J PULSKAMP 

 ATTN RDRL SER L L CURRANO 

 ATTN RDRL SER L M CHIN 

 ATTN RDRL SER L M ERVIN 

 ATTN RDRL SER L R POLCAWICH 

 ATTN RDRL SER P AMIRTHARAJ 

 ATTN RDRL SER U C FAZI 

 ADELPHI MD 20783-1197